Group representations in mathematics and physics;: Battelle Seattle, 1969, Rencontres

Lecture Notes in Physics Edited by J. Ehlers, MLinchen, K. Hepp, Z0rich, and H. A. Weidenm~Jller, Heidelberg Managing Editor: W. Beiglb6ck, Heidelberg

6 I

G. A. Goldin, R. Hermann, B. Kostant, L. Michel, C. C. Moore, L. O'Raifeartaigh, W. Rehl, D. H. Sharp, I. T. Todorov

Group Representations in Mathematics and Physics Battelle Seattle 1969 Rencontres Edited by V. Bargmann, Princeton University II

Springer-Verlag Berlin.Heidelberg • New York 1970

PREFACE

The Battelle Seattle Summer Rencontres in Mathematics and Physics have been established in order "to provide a channel of communication between mathematicians and physicists".

Each year a topic of common interest was chosen which--it was

hoped--would lead to fruitful discussions. The 1969 Rencontres dealt with "Group Representations Physics".

In the view of most participants,

expectation.

I believe,

in Mathematics and

these Rencontres lived up to

The discussions were lively, mathematicians

and physicists found many

questions of common interest to talk about, and they learned from each other.

On

the mathematical side the emphasis was on the problems of infinite-dimensional

re-

presentations,

a subject which has attracted and continues to attract many workers

in functional analysis.

On the side of physics the applications to quantum theory

of both finite- and infinite-dimensional

representations were considered, with par-

ticular attention to the theory of elementary particles,

including its most recent

branch, "current algebra", which leads beyond the framework of Lie groups and calls for new methods. As it should be, individual contacts and informal gatherings played a major role during these Rencontres. in Mathematics

In addition, there were four series of lectures--two

(by E. M. Stein and by C. C. Moore) and two in Physics (by L. Michel

and by L. O'Raifeartaigh)--and

a number of seminar talks by various participants.

The present volume contains three of the four lecture series below) and five seminar talks, written up by their authors.

(see the Note

The reader is expected

to be familiar with the main facts of the theory of group representations. other hand, to help the mathematician,

On the

Michel and O'Raifeartaigh have carefully

Note. Due to the pressure of other work Professor Stein was, Unfortunately, unable to prepare his lecture notes for publication. His course of lectures was entitled "Special Topics in the Representation Theory of Semi-Simple Groups" and dealt primarily with the analytic continuation of representations. Professor Stein has kindly supplied the following list of his publications on this subject. (a)

E. M. Stein, "Analytic Continuation of Group Representations", Advances in Academic Press, New York and London.

Mathematics, 4 (2), 172-207 (April, 1970). (b)

A. W. Knapp and E. M. Stein, "Singular Integrals and the Principal Series I. and II.", I. ~oc. Nat. Acad. Sci. U.S.A., 63 (2), 281-284 (June, 1969); II. Ibid., 66 (i), 13-17 (May, 1970).

(c)

A. W. Knapp and E. M. Stein, "The Existence of Complementary Series". To appear in Problems in Analys~s, Symposium in Honor of Solomon Bochner. Princeton University Press.

IV

stated the quantum theoretical postulates and described the experimental evidence on which the group theorectical treatment of quantum physics is based. The contributions to this volume range from systematic accounts of extensive fields to reports on current research on very specific questions. papers by Moore and by Kostant may be called "purely mathematical".

0nly the

All other

papers--although highly mathematical in content--are concerned with, or at least motivated by physical problems.

A few remarks on the interrelation between these

more physical papers may help some readers.

The papers by Michel and O'Raifeartaigh

give a very comprehensive account of the application of group theory to the most varied problems of quantum physics.

While the two papers are, in the main, comple-

mentary to each other they overlap in some measure, but wherever they do, the discussion of the questions they both treat is sufficiently different to be highly interesting and illuminating.

The contributions by Goldin-Sharp and by Hermann are

concerned with current algebra.

An introduction to the concepts involved here may

be found in the last chapter of Michel's paper and, in greater detail, in the last chapter of O'Raifeartaigh's paper.

Lastly, the papers by Todorov and by RHhl are

independent of each other and of the remaining papers in this volume. On behalf of all participants I take great pleasure in thanking the officers and the staff of the Battelle Memorial Institute, who did so much to provide an ideal setting for these Rencontres. Special thanks are due to the technical typing staff at the Battelle Seattle Research Center, for their painstaking efforts in preparing this volume for publication.

April, 1970

V. Bargmann

TABLE OF CONTENTS

CALVIN C . M O O R E Restrictions of Unitary Representations to Subgroups and Ergodic Theory: Group Extensions and Group Cohomology . . . . . . . . . LOUIS MICHEL - Applications of Group Theory to Quantum Physics Algebraic Aspects . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

36

L. O'RAIFEARTAIGH - Unitary Representations of Lie Groups in Quantum Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

144

BERTRAM KOSTANT - On Certain Unitary Representations Which Arise From a Quantization Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . .

237

I. T. TODOROV - Derivation and Solution of an Infinite-Component Wave Equation for the Relativistic Coulomb Problem . . . . . . . . . . . . . . .

254

W. RUHL - Tensor Operators

279

for the Group

SL(2,C)

. . . . . . . . . . . . .

G. A. GOLDIN and D. H. SHARP - Lie Algebras of L o c a l Currents and Their Representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ROBERT HERMANN

-

List of Attendees

Infinite Dimensional Lie Algebras and Current Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

300 . •

312 339

RESTRICTIONS OF UNITARY REPRESENTATIONS TO SUBGROUPS AND ERGODIC THEORY: GROUP EXTENSIONS AND GROUP COHOMOLOGY by Calvin C. Moore*

PREFACE

These notes are divided into two rather distinct parts, the first of which concerns the restriction of unitary representations of a group to one of its subgroups, and the connection of this with ergodic theory, while the second part concerns group extensions and the connection of this with unitary ray representations. Some background concerning representation theory is assumed and the reader should consult relevant portions of Mackey's notes [33], and survey article [34], Dixmier's book [9], and Chapter I of [3].

The square brackets refer to the common bibliography

for both Part I and Part II at the end.

PART I.

RESTRICTIONS OF UNITARY REPRESENTATIONS TO SUBGROUPS AND ERGODIC THEORY

i.

INTRODUCTION

This first part concerns the general question of what happens when one takes a unitary representation of a locally compact group and restricts it to a subgroup

H

of

G.

G, say an irreducible one,

One source of interest in this problem is

ergodic theory as we will indicate below, but we believe a thorough study of this type of question will shed much light not only on representation theory as such but will produce much useful information concerning the structure of locally compact groups and their subgroups.

Gelfand and Fomin [13] were perhaps the first to realize

the relevance of this kind of problem concerning unitary representations for ergodic theory.

They showed how one could study geodesic flows on surfaces of constant nega-

tive curvature by looking at unitary representations of the group

SL2(R ) .

This

approach was extended by Parasyuk [43], Mautner [35], Green [i], Auslander and Green [2], and the author [39].

Part of these notes are an exposition of some of the

results in [39] without proofs, and the reader is referred to this paper for further details.

We shall also discuss some related results which will appear shortly.

Chapter I of [3] contains an exposition of some aspects of the theory of unitary representations which we shall use as a general source both for Part I and Part II.

* Department of Mathematics, University of California, Berkeley, California

94720.

We shall suppose for the moment is a Lie group with Lie algebra of

G

and let

X

G

of

~(x(t))

is a one-parameter

operator

where

exp

denotes

such that

he a continuous

Then

~(exp(tX))

operators.

= exp(itA)

operators.

where

G

and given

X 6 G, the various

arbitrary

representation.

is limited

dimensional, by

of

A.

If

G

~(t) = exp(itA).

the second

is to determine

for the unitary

operator

the operator

defines

G

tion of

H

of

G

and restricts

and assumes

and

X, and for an

of

to

A

on the possi-

R, so that

G

is one-

a unitary representation

G.

concerning

of A,

Our results will con-

differently; of

H.

G

suppose

and suppose

We assume

that that

that this representation

G, and then ask what conclusions

can be drawn from this fact. of

for a given group

is semi-simple.

subgroup

is a representation

to a representation

A

representation

of these opera-

type of the operator G

the real line

A

case, namely w h e n

is a one-parameter

= exp(itA)

can be extended

is understood

or that one can put limitations

This same problem can be viewed slightly

z(exp(tX))

self adjoint

exp

In this case we can extract no information

the opposite

X E G

G, and so

= iA, but we shall not

and properties

and in fact the same situation holds for any vector group

H = {exp(tX)},

into

that hold for fixed

is for instance

then any self adjoint

cern exactly

G

One might hope to be able to say that the spectrum of

to a very few possibilities

ble eigenvalues

R

possibilities

The object is to get results

is a one-parameter

in this Rencontres.

The problem that concerns us specifically

A = idz(X).

G

It has an infinitesimal

d~(X)

the definition

that

unitary representation

(usually unbounded)

We write

in other lectures

our approach

= exp(tX)

the exponential map of

concerning

tors since this is discussed

x(t)

there is a unique

in the usual way for unbounded enter into more discussion

~

G.

group of unitary

or in other words, A

Let

be an element of

subgroup

generator,

G.

in order to illustrate

Clearly whether

concerning

one starts with a

H, or whether one starts with a representa-

that it extends

to

G

comes

in physics

for if

to exactly

the same thing.

It

is only a matter of emphasis. Such problems

are relevant

group of a quantum mechanical of

G

on the Hilbert

system,

space associated

rather one has a ray representation doesn't matter,

in

G

one has associated to the system.

of

In any case,

tum, angular momentum,

and so on.

conclude about the spectrum

level this

form part of the infinitesimal

H.

question

generators

algebraic

We have spoken about the restrictions subgroups

A = -ida(X)

interpretations

to be treated for various

such as energy,

momen-

to ask what one can

type of these operators,

tion of a larger group of some specified

one-parameter

a unitary representation

the operators

It is an obvious

or unitary

symmetry

(This is not quite true, but

G; at our present heuristic

have in many cases natural physical

fact that they

is some postulated

and in fact can be gotten around by well known methods

in Part II of these lectures.) X

G

based solely on the

of a unitary

representa-

structure.

of representations

of a group

One can raise the same kinds of questions

G

to

for larger

subgroups

of

G.

Some of the theorems below make sense in this generality

and we

shall state them in that form.

2.

STATEMENT

We turn now to the statement a special

following

H

Let

G

has property

E

(in

v

~(h)v = v

h £ H, we have

for every

~

of

of the representation

have only the obvious

for ergodic

theory.

If

H

of the Lie algebra,

generator

A = -ida(X)

presentation

~

of

~(g)v = v G

of

G

every

then the condition

essentially

0

we say that

such that

g 6 G

~(h)V c V

and every

v ~ V.

tion of any representation presentations

~

the condition

having any eigenvalues.

asserting

its closure

H

subspace

G

to

E

sional representation

of

those subgroups

the re-

that a subgroup may

if for every representation space of the re~(g)v = v

forbids

H

E

and

WM

is a one-parameter

H

E

subgroup

A = ida(X)

from

is motivated

by ergodic

theory

are satisfied,

or

(E) WM

to consider of a group

one may infer

or weakly mixing (in

closed subgroups. G

(WM).

G) if and only if It is

can never have proper-

~

of

G

induced by the trivial one-dimen-

Furthermore,

if

G

is abelian,

no proper closed sub-

Our goal is at least in some cases

to characterize

above, we see that we are going to have to assume

non-commutative,

subre-

generator

are ergodic

has property

for

the restric-

of a given group which have one or bohh of these properties,

from the two examples sufficiently

K. E.

WM

if satisfied,

these conditions

the representation

to an

the infinitesimal

of course

of the Hilbert

If again

does, and so it would suffice

group can have property

V

property

the infinitesimal

that certain group actions

be examining

forbids unless

to have any finite dimensional

ones.

The terminology

H

singles

corresponding

h ~ H, we must have

H

easy to see that a proper compact subgroup ty

stronger

has property

forbids

We note that a subgroup

for the

vectors.

for every

of

In other words,

This definition

{exp(tX)}

as an eigenvalue,

This condition,

and as we shall see later, whenever results

H

other than the obvious

of a Lie group,

vectors.

group

and every finite dimensional

presentation

G,

H, the condition says that we

is a one-parameter

has G-invariant

More precisely,

of

such that

and are looking for invariant vectors

We want also to single out a slightly have.

g 6 G.

the we

may or may not have which will be quite relevant

from having

G

for every

the G-invariant

H

a subgroup; z

as

as dis-

to introduce

H

space of this representation

to the subgroup

ones, namely

out a property which a subgroup

compact group and

concerns

generators

G) if for every representation

in the Hilbert

if we have a representation

the first of which

for infinitesimal

the theorem it is convenient

be a locally

and for every vector

restriction

eigenvalues

In order to formulate

definition.

shall say that

element

of our results,

case the study of possible

cussed above.

OF RESULTS

and that

H

is sufficiently

non-compact.

that

and G

is

#

Semi-simple Lie groups are certainly one of the most important classes of groups, and they are in a sense as non-commutative as possible.

The main result

below will characterize those subgroups of such a group which have properties and

WM.

E

This result is contained in [39] and we refer the reader to this paper for

a more detailed discussion. If

G

is a semi-simple Lie group, let

G/Z, where

Z

is the center of

~ni=l G*i

G.

G*

be its adjoint group, that is,

It is well known that

G*

is the product

of a finite number of simple Lie groups, each having center reduced to the

identity element. bra of

G

This is the global version of the decomposition of the Lie alge-

into a sum of simple ideals.

Let

Pi

denote the projection of

G

onto

G~. We shall say that a subgroup H of G is ~otally non-compact if Pi(H) has l non-compact closure in G~ for each i. Intuitively this says that H sticks out 1 non-compactly in the adjoint group of each simple factor of G. If G is simple with finite center, the condition is simply that the closure of

H

be non-compact.

Theorem i

For a semi-simple group

G

and a subgroup

H

of

G, the following are

equivalent (a)

H

is totally non-compact.

(b)

H

has property

E.

(c)

H

has property

WM.

Thus for a totally non-compact subgroup

H

of

G

and any representation

of

G

which has no G-invariant vectors, we can conclude that the restriction of

to

H

has no invariant vectors, or for that matter no finite dimensional invariant

subspaces.

Even if

H

only partially satisfies the non-compactness conditions we

can still extract information. of

H

into the first factor

For instance if GI

H c G I × G2

any finite dimensional subspace for

H

is left fixed pointwise by

lows by a detailed analysis of the proof in [39]. the case of a one-parameter subgroup operator

A = -ida(X)

and if the projection

is totally non-compact, then one can conclude that GI.

This fol-

If we specialize the theorem to

H = {exp(tX)}, X E G, we can conclude that the

has no eigenvalues provided

H

is totally non-compact.

This result overlaps with the O'Raifeartaigh theorem [42], and gives a stronger conclusion under much stronger hypotheses.

To check that a one-parameter

group is totally non-compact is in any given situation, a rather routine matter. The result above contains as special cases the results of Gelfand and Fomin, Parasyuk and Mautner mentioned above. For one-parameter groups we can in fact get much more information concerning the infinitesimal generator

A = -ida(X)

pact one-parameter subgroup. small number of possibilities.

when

X E G

The unitary type of

A

generates s totally non-comis in fact limited to a rather

~$e introduce the Hilbert space

H+(n)

which is to

consist of all square integrable functions on the interval

(0,~), with Lebesgue

measure, taking values in a standard n-dimensional Hilbert space an integer or

+~.

(-~,0), and on

Let

H-(n)

(-~,~), and let

and

H(n)

H n.

Here

n

is

denote the similar spaces of functions on

M!(ni)(M(n))

denote the unbounded self adjoint

+

operator on

H-(n)(H(n))

We note that

Hi(n ±)

which is multiplication by the scalar function

is a non-negative

(respectively a non-positive)

f(x) = x.

operator.

Theorem 2

Let

G

be semi-simple and let

parameter subgroup, and and let

A = -ida(X).

~

be a representation of

Then if we write

positive and negative parts of equivalent to

H = {exp(tX)}

be a totally non-compact one-

G

with no G-invariant Vectors,

A = A+ + Ani

A, there exists

where

A+

and A ±

such that

Ais

are the

~ unitarily

M±(n±).

The proof of this is contained in [39] and the reader is referred to that paper for the details. such that its powers

We also note that if w e have a single element g

n

g

of

G

form a totally non-compact subgroup, we can obtain an en-

tirely analogous result for the unitary type of the operator

~(g)

(see [39]).

The result above for one-parameter subgroups is best possible in that all + choices of n and n occur, and if one considers a subgroup for which the hypothesis fails, then one can find a representation for which the conclusion fails. fact, for

G = SL2(R)

In

and for a one-parameter subgroup generated by a nilpotent + n and n- occur. The situation for irreducible

matrix all possible choices of

representations of SL2(R) is quite interesting; for the principal and complemen+ + tary series, n = n- = i, and for one discrete series, n = i, and n- = 0, while + n = 0, and n- = i holds for the other discrete series. The one representation of the principal series which is not irreducible decomposes into two irreducible sum± n .

mands which behave like discrete series for

For higher dimensional semi-simple groups, the situation becomes a bit simpler.

~iore precisely,

isomorphic to ~.

if we exclude any group

G

which has a simple factor locally + SL2(R) , the only possible choices for n and n- are either 0 or

This fact is implicit in the argument contained in [39].

A = -ida(X)

alence, one of three types, M+(~), M-(~) replacing

Thus in this case

for a totally non-compact one-parameter subgroup is up to unitary equiv-

~

or

M(~).

Since changing

by its contragradient representation will interchange

X M+

into and

-X

or

M-, we

really have only two distinct cases, which we can classify as one sided spectrum or two sided spectrum. W e can raise the question of when every totally non-compact one-parameter group has two sided spectrum for every representation. in progress has shown that this is true for

SLn(R)

for

S. Scull in a dissertation n ~ 3.

B. Kostant has

proved that this is also true whenever the Weyl group of the maximal compact subgroup

6

K

of

G

contains

the element

not every element of

SPn(R)

-i.

In this connection we should also remark that

has one sided spectrum as w e shall see in a later sec-

tion of these notes. Finally we should like to indicate one application tum physics.

Let

P

denote the Poincar~

group a larger symmetry semi-simple, physical

group

and that

information

~

G

GI

X

~

GI

invariant vectors

A = -ida(X)

tent the representations

of

of mass

P

G

containing

P

i and 2 and the remarks

is a subG

To extract

can look like.

P

is

it, and Theorems

Indeed

cannot be of interest.

corresponding

The spectrum of

1

and let us assume that

A

Now

to translation controls

that can occur since in an irreducible

m, the energy operator has spectrum Theorems

on a Hilbert space. and decompose

since such vectors

is the energy. of

G P

to qaun-

P

We shall assume that

about what this decomposition

be the element of the Lie algebra of

time so that

of

to

of these results

and let us assume that

origin.

is a representation

be the largest normal subgroup

there are no let

of u n ~ w n

one would restrict

and 2 above supply information let

group,

in

to some exrepresentation

[m,~).

following

them give us the following

result.

Theorem 3

Under the above hypotheses, M-(~)

or

M(~),

Proof. subgroup

as defined

spectrum of

X

isomorphic

multiplicities

is unitarily

in Theorem

is totally non-compact to

SL2(R)

it should be noted,

to verify

either

to

M+(~),

in

that the one-parameter

GI, and that

so the result follows. is obvious

G1

has no fac-

(The result concerning

on other grounds

once one has the

A.)

This result says that in the decomposition sentations

equivalent

2.

It is a simple algebraic matter

generated by

tors locally

A

corresponding

to arbitrarily

For a survey of the representations

of

of

~

on

P, we must find repre-

small mass or zero mass or imaginary mass. P, see the article of 0'Raifeartaigh

in this

volume. As we have said before, we shall not enter into the details Theorems

1 and 2.

There is, however,

one important

of the proofs

fact, Lemma 4.2 of

the proof of Theorem i, for which we now have an alternate argument. says that a one-parameter property

E.

T. Sherman.

The argument The alternate

of the universal

argument

group of methods

[39] used in This lemma SL2(R)

SL2(k), where

covering

groups of this group defined of the covering

k

is a p-adic field,

in [40]. groups of

and

the

and also for the

To carry this out however, SL2(R)

has

due to

is global in nature and has in addition

that it works

analysis

covering

in [39] is based on infinitesimal

property

the algebraic

for

subgroup

of

SL2(k)

one needs

Contained

in

[40].

We will carry out the proof for

redefines

the meaning

over word for word

of the symbols

x(t)

=(~

w(t)

= x(t)y(-t-l)x(t)

We let

~(x(t))v =

of

= v

y(t)be and

SL2(R)

for all

(z(y(-t-l))v,v)

Itl ÷ ~.

h(t) t

and v

as ÷ v

eter group

h(t),

t -I.

s # 0

Since

g E SL2(R ) .

3.

where

that

h(t)

is a

is a unitary re-

APPLICATIONS

= i.

We let

÷ (v,v) = 1

as

is orthogonal

= a(t) It follows

(this is Mautner's

This completes

to

v,

we see that immediately

that

Since the one-paramlemma

[35]), it follows

the proof of the lemma in question.

TO ERGODIC THEORY

to the applications

of representations

little direct connection with ergodic

s(t)

~(h(s))v = v.

theory and defer for a later section a treatment the restriction

and we define

lim(~(w(t))v,v)

(~(h(t))u,v)

~(h(t))u ÷ v.

E

We shall now turn our attention

cerning

~

is

(~(w(t))v,v)

(~(h(t))u,v)

and then that

t > 0, has property for

x(t),

that

shows that

= a(t)v + s(t)

Is(t) l2 = i.

Itl ÷ ~, or equivalently,

~(g)v = v

godic

Let us suppose

Itl ÷ ~, we see that

~(h(t))u

under consideration

is a vector of unit length such that

(u,u) = i, and note that

for any

subgroup

It may be verified

An easy calculation

la(t) l2 +

Is(t) l ÷ i

that

t.

so that

~(h(s-lt))u

that if one

the argument will carry

of the matrix

= w(t)w(-l).

and that

W e then write

and note that

the transpose

and if we let

u = z(w(-l))-iv

that we will introduce,

that the one-parameter

diagonal matrix with entries presentation

with the understanding

to the general case.

W e may clearly assume ~).

SL2(R)

to er-

of some more questions

to subgroups.

theory whereas

of these results

con-

These final results have

Theorems

i and 2 have a very

direct connection. Let us first introduce theory. called

Let

M

the setting

be a Borel space,

the Borel sets,

and let

G

that is, a set equipped with o-field be a locally

of the second axiom of countability. transformation

in which we are going to study ergodic

We shall @uppose

group so that we have specrified a map

f(g,m)

= g • m

of

onto itself and such that the function

M

this bijective into itself. f(g,m)

such that for fixed

map,

G

g E G, the map

of

G x M

f

of

should consult Chapter

I of

acts on

in the sense M

as a

G x M ÷ M, written

m + g • m

M

is a hijective map to each

g

in

G,

if

G

is given the o-field

is given the product Borel f-l(E)

Thus not only is varies

M

group if the map

where

G x M

is a Borel function

[3] for further

of subsets,

separable

into the group of all such maps of

into

M, but this Borel automorphism

G

of

that associates

by the open sets ancL where

A function

domain for every Borel set in the range. morphism

G

that f

is a Borel transformation

is a Borel map from

of sets generated structure.

is a homomorphism

We say that

= g • m

compact group,

is a Borel set in the m ÷ g • m

"smoothly" with

details.

g.

a Borel autoThe reader

8

For any Borel set

E

of

the transform of

E

by

g • ~

g

using the formula

of

D

G-invariant r, mny

by

g.

If

M, g • E = {g • m l m E E} ~

(or simply invariant)

is a measure on

if

M, we define the transform

g • ~(E) = ~(g-I g • ~ = ~

. E).

for all

We say that

g E G.

~

is

Unfortunately,

interesting measures arising in practice fail to have this property, but pos-

sess instead the weaker property of quasi-invariance. iant if

~

and

concretely,

on

M

g • ~

are equivalent

~(E) = 0 = ~(g • E)

If G

is a Borel set called

G

A measure

~

in the sense of absolute continuity,

for all

is the additive group of the integers,

is equivalent to

~

then

~

If

or more

g E G. then specifying an action of

is the same thing as specifying the Borel automorphism

sponding to the group element one.

is quasi-invar-

h • D = ~

is invariant

h

or

M

for some measure or if

correh •

(or respectively quasi-invariant).

The

case of a single measure preserving automorphism of a finite measure space is the classical setting for ergodic of

G

consists

theory (see

[15]).

If

in giving a one-parameter group

subject to the joint measurability

condition.

G

h(t)

is the real line, an action of Borel automorphisms

of

M

This condition is readily verifiable

in cases of interest and indeed in general it is a condition that permits us to w o r k with actions of groups If

M

that are not discrete groups.

is a compact manifold and if

X

is a

C

vector field on

M, then

the usual existence theorems for ordinary differential equations provide us with a one-parameter such that

group

h(t)

(t,m) * h(t)(m)

of diffeomorphisms is a

of

M

such that

h(t)

map and hence certainly Borel.

C~

= X(h(t))

and

Such a flow

may or may not leave invariant a measure, but if for instance

X

type, then Liouville's

Since this subject

theorem provides

an invariant measure.

is of H a m i l t o n i a n

is discussed in Kostant's article in this volume we w i l l not go into more details here (see also

[4]).

Before proceeding in our general context of which we have seen several examples above, we must impose a regularity condition on more precisely, we shall assume that

M

M

of a technical nature;

is an analytic Borel space.

referred to [3] for further exposition concerning

this condition;

a condition that is satisfied in all reasonable examples. quasi-invariant measure on

M

for

G.

respect to the measure g • E = E

for all

In other words, complements. M = M 1U measure.

M2

~, or that

g ~ G,

equivalent

One says that ~

in any case it is

Suppose that

~

is a

(In fact this is not just a property of

but rather a property of the set of all measures of a quasi-invariant measure class.)

The reader is

G

to

~,

~, so we may speak

acts ergodically on

M, with

is an ergodic measure if whenever we have

and some Borel set

E, then

~(E) = 0

or

~(M - E) = 0.

the only invariant Borel sets under the action are null sets or their

It also says that the action is indecomposable where

M1

and

M2

are disjoint

in that we cannot write

invariant Borel sets of positive

A rather natural modification whenever

~(g • E A E )

~(M - E) = 0.

Here

that is, the points ~(g • E A E) = 0

= 0

for all

g • E A E

of this definition

g E G and some Borel set

denotes

the symmetric

is clearly more restrictive

a non-trivial

result of Mackey

Suppose now that

~

of a null set.

D(E) = 0

For a general

or

of the two sets,

A set with that

The second definition

and if

G

is countable

locally compact group it is

(see [32]) that the two conditions

is a finite invariant measure

are equivalent.

for an action of the

integers.

This as we have seen is specified

formation,

and if the action is ergodic one has the Birkhoff

For

that

and one would be asserting

than the first,

they can easily be seen to be equivalent.

in assuming

E, then

difference

in one but not the other of the two sets.

might be called almost-invariant,

any such set is a null set or the complement of ergodicity

consists

by a single measure

preserving

ergodic

trans-

theorem

[15].

f E LI(~) ,

i

n

lim~-~--~ n-~o for almost all line.)

x.

u(x)

(ui(x)) = [

fd~

J

(There is a similar

If we interpret

ics where

Ei=0f

this formula

statement

for ergodic actions

in its classical

is the evolution of a state

side is the time average of a function

f

x

context of statistical

is to hold for almost all initial

difficult

to see that the validity

The question of equality

then the left-hand

(a dynamical variable)

and the right-hand

statistical

mechanics,

problem of showing

and the ergodic

that certain actions

Not only for this reason, tions in ergodic of actions available

are ergodic

(see

but for many others,

was initiated

to ergodicity.

has a long history the question

conditions

We have remarked

by Gelfand and Fomin for geodesic

theory and unitary representations

before flows,

are closely

to the

one of the fundamental

ques-

for an action or a class

Our object here is to review a general method

section.

in

[24]).

for doing this by means of unitary representations

sults of the previous

ergodic

theorem just reduces

of these two

In fact it is not

is equivalent

and phase averages

theory is to supply sufficient

to be ergodic.

The equality

states of the system.

of such a formula

of time averages

mechan-

after unit time,

side is the phase average of the same dynamical variable. averages

of the real

one has

and to apply the re-

that the present

technique

although

the observation

connected

goes back to

that

Koopman. We shall assume now that we are dealing with actions space

M

erality of

G

with a finite invariant measure that

define

(~(g)f)(x)

is a unitary that

~(M) = i.

on the Hilbert

~

= f(g-i

operator,

L2~M,~). . x).

More precisely

unitary representation

one can detect ergodicity of invariant

for

It is easy to verify

and using joint measurability

is a continuous

non-existence

~, and we may assume without

One can then define an associated

space

of a group

of

f E L2, and

that for each

loss of gen-

g E G, we g E G, ~(g)

one can show

The key observation

of an action merely by looking at sets is equivalent~othe

on a

unitary representation

of the action, G.

G

non-existence

~.

is that

Heuristically,

the

of invarimnt measurable

10

functions, existence

and since the space has finite measure of square summable

invariant

this is equivalent

to the non-

functions.

Lemma 3.1

The action of g E G

implies

that

G

on

f

M

is ergodic

if and only if

is a constant almost everywhere,

~(g)f = f

for all

and hence a constant

in

L2(M,P). There is a somewhat namely weak mixing. G

on

M.

We notice that

that the product measure M

is weakly mixing

trivially

implies

stronger notion than ergodicity which is often useful,

For this one must define the Cartesian G

acts on

~ x ~

M x M

[15] if the action of

ergodicity,

g • (m,n) = (g • m,g

~

• n)

One says that the action of

G

on

M x M

and if it is satisfied,

action is weakly mixing and ergodic. the unitary representation

by

is invariant.

square of an action of

is ergodic.

on

This condition

every Cartesian power of the

Weak mixing can also be detected

(see [15] and

and G

by looking at

[39]).

Lemma 3.2

The action of sional subspace

V

G

of

on

M

L2(M,~)

is weakly mixing

if and only if any finite dimen-

invariant under

~

consists

of constant

functions

(and hence is one-dimensional). These lemmas serve to motivate

the definitions

Section 2 above since these definitions sional subspaces R

of representations

or the integers

mixing.

of measurable

sets

of a group.

Z, an action may possess

To motivate

Finally,

and

if

G

the yet stronger

this we note that the ergodic A

of properties

concern invariant vectors

E

and

WM

in

and finite dimenis the real line

property

theorem implies

of strong

that for any pair

B lim p(h(t)A N B) = ~(A)p(B) t~

in the sense of Cesaro limits. t ~ Z.

~

[15].

is the action defined

One can find a sufficient

for this to be the case;

projection valued measure group

h(t)

for

t ~ R

One says that the action is strongly mixing if the above limit exists

usual sense tion

Here

T) corresponding

P(E) = 0

if

E

P

to

condition

indeed by Stone's

~.

We shall say that null set.

theorem

P

in the

in terms of the representa[25], there exists a

on the Borel sets of the dual of

is a Lebesgue

or

G

is absolutely

(R

or the circle

continuous

if

See [15] for the following.

Lemma 3.3

If

G

is as above,

the subrepresentation

of

~

then an action of on the orthogonal

G

is strongly mixing provided

complement

of the constant

that

function

11

has an absolutely continuous projection valued measure. It is clear that strong mixing implies weak mixing and that Theorem 2 of Section 2 is exactly the sort of result that will enable us to establish that certain actions are strongly mixing. More specifically the theorems from Section 2 will be applied in the following manner.

Suppose that we have an action of a group

invariant measure, and suppose that on

M

that

preserving H

~

G

G

H

on

M

in the action of a larger group

G.

H

on

and

which acts

on

M.

WM

H

of

In

may be "embedded"

We shall assume that the larger group

be ergodic by some method, and pick a subgroup E

H

H

M

H

G

acts

also necessarily

Equivalently we can start with an action of

Properties

with a finite

determined by the fact

ergodically, and then ask if we can conclude from this that

ergodieally.

M

coincides with the given action of

other words, we are assuming that the given action of

acts ergodically.

on

is a larger group containing

such that the action of

is a subgroup of

H

G

on

G, and ask if

M H

known to also acts

from Section 2 are immediately relevant to

this situation.

Theorem 4

Suppose that acts ergodically on

H M

has property

E

(respectively

WM) in

with a finite invariant measure.

Then

G, and that H

G

is also ergodic

(respectively weakly mixing).

Proof.

We consider the representation

an invariant function for G, and by ergodicity of

H

then by property

G, f

is a constant.

same argument we can conclude weak mixing if

~

of

G

on

L2(M,~).

If

f

is

E, it is an invariant function for Hence H

H

acts ergodically; by the

has property

WM.

The following follows immediately using Theorem i.

Corollary. and if

H

If

G

is semi-simple and acts ergodically on

is totally non-compact in If

H

is a subgroup of

G

G, then

H

M

as above,

is ergodic and weakly mixing.

isomorphic to the real line (or the integers)

we can also obtain results concerning strong mixing using Theorem 2.

Theorem 5

Let

G

be semi-simple and let

H c G

phic to the real line (or the integers). then the action of

H

If

be totally non-compact, and isomorG

acts ergodically on

M

as above,

is strongly mixing.

Although the hypothesis of the above theorems, as far as

H

is concerned,

may seem rather special, their interest lies in the observation that this hypothesis

12

of embedability satisfied theory.

in a larger group

for some classical

G

flows,

is satisfied that seemingly

One way that an action of a group

is transitive; g E G

that is if

such that

identify

m

the coset space

hypotheses

on

and

g • m = n.

6

with

M, one can conclude

of the details

given concretely by iant measure means

that

is immediately

that

M

is

G/F

for a fixed

(See

that

G

With our

and that

[30] and

M

such a measure,

can be (see

m, one can

gF + g • m.

The action of

Our assumption

also possesses F

G/F.

is if it

M, there is an element

by means of the map

of this reasoning.)

on what

have no connection with group

that this is a Borel isomorphism

g • (hF) = ghF.

severe restrictions

Indeed it is

can be seen to be ergodic

F = {glg " m = m}

M

closed so that we may as well assume exposition

G

are given points of

If w e put

G/F

in many cases.

on

G/F

G-invar-

and this places A transitive

seen to be ergodic by the first of the two definitions

is

is then

has a finite

[7] for instance).

F

[3] for an

rather action

of ergodicity

above. One of the simplest bedable

in a transitive

= {(Zl,'.',Zn) , z i numbers

rationally

tion.

and let

independent

[15].

action of

Leon Green

act on

since

by

Let

be an n-torus.

t • (Zl,...,Zn)

however,

T n = Rn/z n

[i] has established

[13] observed

that geodesic

Tn We pick real

= (exp(itan)Zl,'--,

if and only if the

a.l

are

that this action can be embedded where

a beautiful

Zn

is a lattice.

the same for horocycle

certain higher dimensional

semi-simple

(see also

of constant

actions of the group

flow,

manifolds

groups,

generalization

the same ideas

flows on surfaces

in transitive

on the same idea and the following cerning

Tn

We observe,

Rn

which again uses exactly

are embedded naturally observed

R

Izil = i}

The

proof of the result quoted above is based in its essence on this observa-

nil manifolds Fomin

action is that of a rotation on a torus.

This action is well known to be ergodic

in a transitive classical

of an action of the real line which is em-

complex numbers with

al,,.. , a n

exp(itan)Zn).

examples

and Mautner [35].

Gelfand

negative

G = SL2(R);

The proofs

of ergodicity

immediately

curvature [43]

flow on

are all based

all such results

from Theorems

to and

Parasyuk

did the same for geodesic

general result subsumes

and follows

of this result [2]).

con-

4 and 5.

Theorem 6

Let G/F

G

be a semi-simple

Lie group and let

has a finite invariant measure,

(or the integers),

then the action is strongly mixing.

tries

(see

SLn(Z), [8]).

is ergodic

F

of subgroups

the subgroup

of

F

and weakly mixing.

satisfying

SLn(R)

be a subgroup

such that

be totally non-compact.

H

instance

G/F

H

natural action of

Many examples

on

and let

If

this condition

consisting

of matrices

H

Then the

is the line

are known;

for

with integral

en-

43

We shall close this brief discussion a duality author

theorem of sorts which was noticed

(see [39]).

F

operate on

independently

since we let

H\G = {Hglg ~ G}

for any pair of closed subgroups

by

F

H

act on

H

of

H

G/F.

y • Hg = Hgy -I.

and

theory with

by G. W. Mackey and the

Above we were dealing with two subgroups

there was an assumed asymmetry let

of one aspect of ergodic

and

F

of

G

but

We can just as well

The following

fact holds

G.

Lemma 3.4

The action of H\G

H

on

G/F

is ergodic if and only if the action of

on

is ergodic. In general a coset space

G/F

or

H\G

has no invariant measure,

less a finite one, but it always has a unique quasi-invariant and it is with respect

to this measure

As an application

of this,

group of matrices with integral sisting of matrices and

F

H

is totally non-compact

principle

says that

and is in fact

Rn

F

and let

is ergodic on

Since

H\G.

is measure

linear action.

theoretically

F = SLn(Z),

H denote the subgroup

(i,0,...,0).

n ~ 2.

and let

Then H

G/F

[28],

The space

H\G

the sub-

of

G

G/F,

the duality

is easy to identify and the action of

Since a single point is a Lebesgue

the same as

Rn

con-

has finite volume

is ergodic on

minus the origin with Lebesgue measure,

this space is the natural set, HIG

G = SLn(R )

entries,

if

class

class that the above lemma applies.

let

with first column

measure

much

G

on

null

and we have the following

result.

Theorem

7

The linear action of

F = SL (Z) on R n with Lebesgue measure is ergodic. n M o r e o v e g the same is true for any F such that SLn(R)/F has a finite invariant measure. This easily stated result does not appear to be amenable

to any direct

approach.

4.

In Section

2 we studied

ting a representation decomposes.

These results

questions

certain aspects

of a semi-simple

nection was discussed additional

MORE ON RESTRICTIONS

are of immediate

in Section concerning

Theorem 2 for instance

concerned

3.

of the general problem of restric-

Lie group to a subgroup interest

in ergodic

and examining

theory and the con-

In this final section we Want to discuss

the restriction the restriction

how it

of representations of representations

some

to subgroups. to one-parameter

subgroups,

and it is natural

of representations groups.

to raise similar questions

to more general subgroups

From the general version

know that any unitary the Borel subsets function

representation

(see [33]) this projection

We are interested

H, particularly

of Stone's

of the dual group

general

of

H

H.

Together with an appropriate

leads to a projection

that

P

is absolutely

if and only if G

and if

~

H

determines

H

of

on

lutely continuous.

It is natural

G.

one may immediately

corresponding

tion

G = KAN.

rately. groups,

satisfied when

that

Any abelian

G

subgroup

A, and a part from

H = A.

We will say P(E) = 0 if

results

of

~

that if

subgroup

of

G

then the projecto

H

is abso-

is not an issue since the

is a compact

subgroup

valued measure

concerning

for

P(E) = 0

of when other abelian sub-

of any group

for a vector sub-

eigenvalues

of any operator

algebra of the Lie algebra of

is simple,

and we write

H, a

the Iwasawa decomposi-

is in some vague sense made up of a part from N, and we shall consider

the three cases sepa-

We have already noted that the question posed is trivial of

T. Sherman

the answer is affirmative,

Theorem

H

about the projection

and we turn to subgroups

sider the case

if

P

in physics.

Let us now suppose

K, a part from

H.

vectors,

with the restriction

to an element of the enveloping

of some interest

on

one-parameter

with no G-invariant

on

multiplicity

Theorem 2 above says in particular

to raise the question

conclude

valued measure

to Haar measure

We note that non-compactness

above is trivially

If one knows such results

question

G

associated

groups have this property.

H [25], we

the representation.

to Haar measure)

is equivalent

is a totally non-compact

tion valued measure

group,

P

is a Haar null set. and if

is a representation

conclusion

(with respect

E c H, and that

E

is semi-simple

continuous

abelian sub-

class of this projection valued measure

and in particular we would like to compare it to Haar measure

any Haar null set

the restriction

theorem for abelian groups

valued measure

in the equivalence

concerning

A.

Since

A

for compact

sub-

is abelian, we may as well con-

[48] has observed

at least in a special

case that

and the same holds in general one can easily see.

8

If

G

is simple and if

G-invariant

vectors,

tion of

to

~

A

~

is a unitary representation

then the projection valued measure

is equivalent

to Haar measure on

of

G

with no

associated with the restric-

A.

Moreover, the multiplicity

is uniform. We now turn to the consideration at present 2n x 2n

is a counterexample.

matrices

preserving

of subgroups

We consider

a non-degenerate

of

N, and here all we have

the symplectic

group

SPn(R)

skew bilinear

form.

Since the funda-

mental group of this group is the integers,

there is a unique double

G

of

group

N

for this group contains

SPn(R).

(This is Weil's metaplectic a normal

[51].)

abelian subgroup

V

of real

covering group

It is easy to verify isomorphic

that

to the vector

45

space of real of

V

n x n

with a group

symmetric matrices. T

0

[51] has constructed a representation tion (see also Shale [46]).

V

with

V

of rank one. P

~

V = V

Thus, if

n x n

matrices

(that

and all diagonal entries are one). of

G

~

to

V

Weil

called the metaplectic representa-

quite easily.

by means of the bilinear form

In fact if one identi-

~(a,b) = Tr(ab), this projection

is concentrated on the set of positive definite matrices n > i, this is a Haar null set, and gives an example where

is singular with respect to Haar measure.

one-parameter subgroups of non noticed for

is the semi-direct product

One may compute the projection valued measure on

associated with the restriction of

valued measure on

N

isomorphic to all strictly triangular

is, all entries above the diagonal are

fies

In fact

SL2(R)

V

This example also shows that many

have one sided spectrum and hence that the phenome-

persists in higher dimensions.

There is another point worth noticing about this situation; let denote the kth tensor power of the representation

~

with itself.

~(k)

Then it is quite

easy to check that the projection valued measure associated with the restriction of ~(k)

to

V

is concentrated on the set of positive definite matrices of rank equal

to the minimum of

k

and

lutely continuous spectrum. representation of

G

n.

Thus

~(n)

is the first tensor power to have abso-

Since it is virtually obvious that any discrete series

has a projection valued measure which is absolutely continu-

ous, any connection between discrete series and the tensor powers of to the situation for

analogous

n = i, is likely to involve large tensor powers.

We notice that the condition above for abelian groups tion valued measure on the dual group lutely continuous,

~

H

H

that the projec-

associated to a representation be abso-

can be rephrased so as to make sense for any subgroup

H.

The

condition can be readily seen to be equivalent to the condition that the representation of

H

should be unitarily equivalent to a subrepresentation of the direct sum

of the regular representation with itself infinitely many times. for any

H, and we shall say following

This makes sense

[33], that a representation of

H

fying this condition is quasi-contained in the regular representation. happens,

satis(If as often

the regular representation is equivalent to the infinite direct sum of it-

self, the condition is simply that the given representation is a subrepresentation of the regular representation.) We do not have any general theorems concerning this situation, but there is one case of special interest. G = KAN, we let knows that subgroups of

B

B

If

G

be the centralizer of

is semi-simple with Iwasawa decomposition A

in

K, and define

B = MAN.

One

is a group [6], and in some sense it is one of the most important G.

representations and

M

The principal series representations of of

G

G

consist simply of the

induced by the finite dimensional representations

plays a key role in the structure of

is simply the subgroup of triangular matrices.

G.

When

G

is

of

B

SLn(R), then

B

[6],

~6

Our interest here will be the study of restrictions of

G

to

B

something

the regular addition

in the spirit indicated

about the structure representation

to its obvious

and for this we shall need to know

of the regular representation

of

B

is rather striking.

one-dimensional

ducible representations, simple

above,

all infinite

dimensional

is the discrete

sentation ducible

of

B

irreducible

G = SLn(R )

B

has in

~i' i = i, 2, 3, 4. ~

representations,

for

n ~ 3

as a continuous

dimensional

is the discrete

finitely often. forthcoming

[33], say

representation

of

B

A

is the ThUS

with in fact only a

or do we find the same phenomenon

On the other hand for

infinite

presentation

for

decomposes

summands.

G = SL2(R) ,

It is easy to see

G = SL2(C).

Is this an accident A calculation

When

exactly four other irre-

types entering into the decomposition.

that the same is true for

groups?

B.

often, ~ = ~(~i @ ~2 @ 73 @ ~4 )"

~. each taken infinitely l direct sum of irreducible

finite number of distinct

of

It is known that

representations,

calculation will show that the regular

direct sum of the

of representations

reveals

that the regular repre-

direct integral

G = SPn(R),

representations

of

for other semi-simple

and contains

B

4n

such that the regular re-

direct sum of these representations,

We shall now state a general

no irre-

one can find exactly

each taken in-

criterion which will appear in a

paper.

As usual we consider W = MI/M

is a finite group,

morphisms

of

the normalizer

MI

of

A

A, and hence also on the Lie algebra of

not contain the automorphism the Lie algebra of

A).

in

K, and note that

called the Weyl group, which acts as a group of auto-

a ÷ a -I

of

A

A.

The group

(or equivalently

If it does we shall say that

-i

W

the map

belongs

may or may Y ÷ -Y

to

of

W.

Theorem 9

If

G

is semi-simple,

either decomposes

as a discrete

then the regular

representation

direct sum of irreducible

(with a finite or countable number of inequivalent continuous

direct integral with no irreducible

occurs if and only if If irreducible "discrete

-i

belongs

representations

series",

square integrable

G

since

B

of

series,

of the r e g u l a r

B

which

and sufficient

are in "general posiare

Finally we note condition

that is, there are irreducible representation.

as

since they

case that the matrix entries

is not unimodular.

[21] has given a necessary

have a discrete

which are summands

representations

B

as a

number of

of the regular representation

as in the semi-simple on

B

B

W.

which are summands

functions

that Harish Chandra group

or it decomposes

but we prefer to call them generic representations

One may not conclude

of

The first possibility

-i E W, we may think of the finite or countably infinite

are in a sense those irreducible tion".

to

summands)

summands.

of its subgroup

representations

that the

representations

It is interesting

to note that

47

his condition hence if

G

implies

that

-i E W

has a discrete

(and in fact is almost equivalent

series

so does

to it) and

B.

Once we have this kind of control over at least some of the representations of

B, it is natural

tion

~

of

G

to

to raise the question of what the restriction B

looks like.

This technique

G = SL2, and Stein in his lectures with

B

replaced

is exceedingly

in this volume uses a similar

which has no G-invariant regular representation

series

of

quasi-contained

above question

symmetric

and

N

in general, ~

matrices.

representation

restriction

of

know is false.

V

of

~ B

of a simple group quasi-contained

G

in the

of

H.

a normal

covering

subgroup

If the restriction of

B, it follows

is quasi-contained

G

of

of

~

to

is our friend

from

the We

space of

is quasi-contained

in

above that its further

in the regular representation

Again we do not know what the best possible

H

to the

to the vector

B

by the comment

case follows

the answer

SPn(R) , n ~ 2.

isomorphic

con-

to any subgroup

The general

Unfortunately

and the counterexample

of the double

formula.

that any representation

G, has its restriction

representation

contains

the regular

to

from the observation

induced representations.

is negative

n × n

except

This is true for all the series of representations

in the regular

representation

B = MAN

for

[20] which are used to obtain the Plancherel

representation

known facts concerning

have

B.

this follows

tained in the regular

metaplectic

vectors has its restriction

by Harish-Chandra

For discrete

fruitful technique

by an even larger subgroup.

One may ask if it is true that any representation

constructed

of a representa-

of

theorems

V

which we

are in the gen-

eral case.

PART II.

5.

GROUP EXTENSIONS

STRUCTURE

AND GROUP COHOMOLOGY

OF LOCALLY

COMPACT GROUPS

In this second part we shall take up a rather representations, structure

and indeed here the major

of locally

compact groups.

sions comes from the phenomenon ever,

to treat these questions

problem,

different

considerations

The motivation

of ray or projective

will concern more the

for the study of group extenrepresentations

properly, we feel it is better

and then come back to the original

questions

aspect of group

of groups;

to first widen

using the general

how-

the

techniques

which we shall develop. We shall suppose and that

G

operates

More precisely,

p(g),

that A

G

and

A

g ~ G, the map

and further

are topological

is a topological

we are given a continuous

such that for fixed denote by

on

that

groups with

transformation

map of

G × A ÷ A written

a ÷ g • a is an automorphism p

is a homomorphism

of

A

abelian,

group of automorphisms.

G

of

(g,a) ÷ g • a A, which we

into the group of

18

automorphisms that

p(g)

of

A.

The hypothesis

is an automorphism

"smoothly"

with

g.

logical G-module

If

G

and

A

satisfy

A

is a G-module

space,

includes

G A

on

A

and if

A

If

~(g)az(g) -I 6 A

Va ~ A, then A.

~

operators

A

defines

is a G-module.

Finally,

as a G-module.

Such modules will be called of a given

first of all an exact sequence

A

is

unitary represen-

that of

(g • a) = ~(g)(a) G

on a Hilbert space such that

for each

is any topological

G

is a topo-

g 6 G

an

A, equipped with the strong operator

topology,

A group extension

if

varies

If for instance

on this Hilbert

fines

A

A

is a continuous

then one may verify

that

p(g)

not only

[37].

g • a = ~(g)a~(g) -I

It may be verified

assures

A, but that

is again a representation

is some group of unitary

automorphism

of

~

group

g • a

of examples.

and if

in the usual sense,

as a G-module.

of

the above, we say that

a wide variety

space with its norm topology,

tation of defines

of the topological

or simply that

This definition a Hilbert

of joint continuity

group

g • a = a

trivial topological

by a given topological

de-

G-modules.

G-module will be

of groups i I÷A÷E÷G+I

where

i

is an injection

and where

the kernel of

is continuous, that

~ is

A

into

E, and

is exactly

E, and that

and open.

This means

but as topological E/i(A)

and

G

first of these having the quotient

phism of

A

of

E

above

A

with

which depends

this automorphism a G-module.

uity.

i.

topology. G

A

and i(A)

only on

A.

g, and not on the choice of a ÷ g • a

for defining The reader

G-modules

is referred

One of the simplest sion of the circle the real line

T

group.

examples

Z

G

i

topology the

g E G

assump-

and if

g'

is an a u t o m o r g'.

We demand

by

of

that

A

as

A, then by the

G x A ÷ A

following

is one of the main moti-

the condition

of joint contin-

[38] for more details.

of a topological

by the integers

of

of the map

as we did by imposing

groups,

in the ~efinition

This observation

to [37] and

G,

and we assume

has the relative

We note that if

the joint continuity

onto

we impose an algebraic

We note that whenever we have an extension a G-module,

E

can be identified

a ÷ i-l(g'i(a)(g') -I)

be the given automorphism

becomes

i(A)

as topological

Finally,

on

of

We assume not only that

onto its range,

that

groups where

~(g') = g, then

from the axioms for a topological vations

is a surjection

may be identified

tion to take account of the action of is an element

~

the range of

but also that it is a homeomorphism

continuous

not only as groups, from

of ~

group extension

is the exten-

(viewed as a trivial module),

defined

by

R, namely 1 ÷ Z ÷ R ÷ T + 1 .

Another

example which is of more significance

follows.

Let

operators

on

viewed

H H

be a Hilbert

space,

and let

with the strong operator

as scalar multiples

of the identity

particularly U(H)

topology. operator

in quantum physics

is as

be the group of all unitary Then the circle group is a normal subgroup,

T and

19

U(H)/T = PU(H)

is called the projective

unitary group.

Then

i + r ÷ U(H) ÷ PU(H) + i is an extension of

PU(H)

set of one-dimensional on

P(H)

classic

by

r ÷ a • r

of

being trivial module.

Indeed if

H, then any

defines

a • r

= l(Ul,U2) l

We observe

a = tb

naturally

and if

G

chanics

say that we have a homomorphism

If

H

P(H)

that the projective

morphic

with

in this heuristic

assume that let

space associated

of

discussion).

then if

p

G

of

corresponding

Thus

A

trans-

the func-

to

a

U(H)/T = PU(H)

and is iso-

system, into

system,

the axioms of quantum me-

PU(H)

(except for those sym-

but this will not change anything

[5].

is the projection

is precisely what is

For the moment let us

from

U(H)

onto

PU(H), we

and then

a group extension

a subgroup

a.

by picking unit vectors

Such a homomorphism

I÷T+Gr becomes

under

with a quantum mechanical

or ray unitary representation

G c PU(H);

G r = p-l(G),

r i ~ P(H)

t ~ T.

metries which we would want to be anti-unitary, essential

is the

transformations.

is a symmetry group of this physical

known as a projective

r 6 P(H)

onto itself which preserve for

transformations

to a group of projective

is the Hilbert

P(H)

a collineation

[5]) says that except for anti-unitary

which is defined

agree if and only if

a 6 U(H)

is the transform of

(see [52] or

these are the only maps of

f(rl,r2)

u. 6 r.. I l b ~ U(H)

T, T

where

theorem of Wigner

formations, tion

by

subspaces

of

G

by

+G÷I

T.

Even when we do not want to identify

PU(H), we Shall see that we can still construct

G

as

a group extension

I÷T÷E+G+I where

E

has a homomorphism

The fact that projective of a suitable

into

U(H),

or in other words,

representations

group extension

can be viewed

a unitary representation.

as ordinary

is a well known and fundamental

representations

fact.

In these notes we want to present a brief outline of a systematic group extensions mately related

and more generally

to the initial

and to the references

G. W. Mackey

[29],

sions.

A large part of the contents

These results

be equivalent

G

to [37] and

in particular

[38] for

the pioneering

this point of view concerning

work of

group exten-

of these notes will be the subject of a forth-

and we will try to summarize

extend and generalize

One of the most important of a given group

We refer the reader

cited there,

[31] who originated

coming paper of the author, volved.

of a theory of group cohomology which is inti-

problem.

more details

theory of

problems

by a given topological

if there is a commutative

the major new points

those in [38] and is to classify G-module

A.

in-

[39].

the set of all extensions Two extensions

are said to

diagram

I÷A÷E+G÷I +

+

+

1 ÷ A ÷ E'÷ G ÷ 1 of continuous

maps where the end vertical maps are the identity maps and where the

20

middle vertical is an isomorphism of topological groups. is not sufficient to assume that

E

and

E'

It should be noted that it

are isomorphic as topological groups

to have equivalent extensions, but rather there must be a particular isomorphism which respects the data of a group extension. set of equivalence classes of extensions of

One of the first facts is that the G

by

A

forms a group

Ext(G,A)

by

means of the Baer product (ef. [16]), and at least in many cases this group is given as a two-dimensional cohomology group the other cohomology groups

H2(G,A).

It turns out to be useful to study

Hn(G,A), both to gain a better understanding of exten-

sions, and also to have at hand general methods of computing

Ext(G,A)

in many

specific situations.

6.

G-MODULES

After this introduction we shall now proceed to some of the details. will henceforth assume that

G

second axiom of countability.

We

is locally compact and separable in the sense of the (Local compactness seems to be essential for this

treatment, although we hope in the future to be able to dispense with it; separability is an assumption of a more technical nature used to avoid certain pathologies.)

We shall also assume that

able by some complete metric.

A

is separable, metrizable and moreover metriz-

Following Bourbaki, one might call such groups polon-

ais, and we denote the family of all such groups by

P.

Since we will always be

dealing with G-modules, we consider all polonais G-modules which we denote by We note that

P(G)

contains all separable locally compact G-modules

tensions were studied in the case of locally compact

A

A.

P(G).

Group ex-

in [38] and [39], and one

of the key points in the present treatment is that we now enlarge the category of modules to

P(G).

In addition to including many important and interesting examples

which were excluded before, we also achieve more technical versatility in that the larger category will contain cohomologically trivial modules, will enable us to define induced modules in a natural way, and will allow us to construct resolutions without going outside the category. If

A,B E P(G)

a G-homomorphism

f

of

A

into

intertwining homomorphism, that is, one satisfying that

P

(resp.

B

is simply a continuous

f(g • a) = g • f(a).

We note

P(G)) is closed under the operations of countable Cartesian prod-

ucts, closed subgroups (closed submodul~s), and quotient groups (quotient modules). In addition, if we have a sequence of elements of i i ÷ A p ÷A

P

~ A" ÷ l

which is exact in the sense of Section 5, then one can show that if

A"

and

A"

are in

P.

The same is clearly true for

phisms in the sequence above are G-equivariant. and it is only assumed that

i

and

~

P(G)

Morever, if

A E P

if and only

if the homomor-

A, AP, A"

are in

are continuous, it follows by classical

P,

21

closed graph theorems

(cf. [3]) that

i

is a homeomorphism and that

~

is open,

and hence that the sequence is exact in the sense of Section 5. In addition

P

is closed under the following construction which might be

described as a sort of direct integral. portance to us. bra of

(M,~)

[0,i]

Let

be a o-finite measure space such that the measure alge-

is separable

[14].

This means that we may as well assume that

with Lebesgue measure together with a countable number of atoms.

A E P, and define A

(M,u)

This construction will be of paramount im-

U(M,A)

U(M,A)

i (the identity in

M

to

A) almost everywhere.

is then an equivalence class of measurable functions, all of

which are equal to each other almost everywhere. f-l(0)

is

Now let

to be the group of all measurable functions from

modulo the group of functions equal to

An element of

M

is measurable in

M

for every open

0

(A function in

A.)

f

is measurable if

It is clear that

U(M,A)

is a group under pointwise multiplication. We topologize cisely let

Pl

U(M,A)

by the topology of convergence in measure; more pre-

be a bounded metric on

finite measure on

M

equivalent to

~.

A, which always exists, and let We define a metric on

~

U(M,A)

be a

by

P(f'g) = I Pl(f(x)'g(x))d~(x) which is always finite since

~

is finite and

Pl

is bounded.

Lemma 6.1

U(M,A)

with

p

Pl

and

of the choice of If namely let

A = T H

f E U(M,T).

be Then

U(M,T)

P, and the topology is independent

~.

is the circle group,

U(M,T)

has a natural interpretation;

L2(M) , the space of square integrable functions on f

defines a unitary operator on

(U(f)h)(x) = f(x)h(x). so

as defined above is in

Clearly

U(f) = U(g)

H

by multiplication by

if and only if

may be viewed as a group of unitary operators on

that the topology on ogy when we view

U(M,T)

U(M,T)

M, and let

f = g H.

in

F,

U(M,T)

and

It may be verified

introduced above is exactly the strong operator topol-

as operators.

If

A = R

is the real line, U(M,R)

is a

topological vector space; in fact a Frechet space, although it is not locally convex. Finally,

if

formula

(g • F)(m) = g • (F(m)).

copies of

A ~ P(G), we can define an action of

A.

If

M

G

on

U(M,A)

by means of the

This may be thought of as a direct integral of

is an atomic measure space, the construction does give the

Cartesian product of copies of

A.

In analogy with direct integrals of representa-

tions

[33] one might hope to find a reasonable definition of a measurable map of

into

P(G), and then define a direct integral where the "fiber"

is allowed to vary instead of remaining constant as above.

A(m)

over

m ~ M

Since we have found no

use for this kind of construction as yet, we shall not proceed any further.

M

22

The group

U(M,A)

the most important

for

for our purposes

and then intuitively

m2

into

has many interesting

A.

properties,

is a "law of exponents".

a function of two variables

as a function of one variable variable

A E P

(say

on

M

We let

into

A

M = M I × M2,

can be thought of

ml) into the space of functions

Such a correspondence

and one of

of the second

holds exactly and indeed follows

from

a version of the Fubini theorem.

Lemma 6.2

There is a canonical topological

A most important M = G

any

(separable)

will be denoted by

A E P.

I(A)

U(M I x M2,A )

onto

U(MI,U(M2,A))

A E P(G)

by the map

f

G

so that

defined by

of

I(A)

as

also operates

(f(a))(s)

on

is when

In this case

is itself a G-module

act by translation:

G

U(M,A)

group with Haar measure.

and we note that

In fact we simply let

If in addition I(A)

of

special case of the construction

is a locally compact

U(M,A)

isomorphism

groups.

(g • F)(s)

for

= F(g-ls).

A

we can embed

A

into

f

is an equivariant

= s -I • a.

Lemma 6.3

If phism of

A E P, then

A

I(A)

onto a closed

is in

submodule

P(G). of

Moreover,

I(A)

isomor-

so that

1 ÷ A ÷ I(A) ÷ U(A) ÷ i is exact where

U(A)

is the quotient module.

It is clear that with coefficients cohomologically

in

A.

trivial

I(A)

is in some sense the regular

In the case of a finite group, in that

Hn(G,A)

= 0

for

n ~ i

can also be proved in the present

context will be of vital

above would then assert

A

module,

that any

[45].

we have defined

importance.

If

have regarded functions, submodule

elements

of

as a submodule

= s-lf(g)

is understood~.

I(A)

of

I(A)

as functions

duced representations defer these details

and hence is in

of

G, and if

I(A).

for almost all pairs

We have engaged

but this poses no problem.

such as inducing to our forthcoming

is

6.3

trivial algebra.

Finally,

once

A 6 P(H), we

More precisely, (s,g)

in

instead of equivalence

All of the expected

I~(A)

where

classes of is

properties

in stages hold in our context, paper.

H × G

in the usual abuse of notation and

It is easy to show that P(G).

G

it is but a short step to the notion of

is a closed subgroup of

the induced module,

I~(A) = {fI f E I(A), f(gs) Haar measure

H

Lemma

from homological

is almost never locally compact.

the regular representation,

induced representations. I~(A),

I(A)

of

I(A)

The fact that this

in a cohomologically

and this fact will allow us to use many techniques

It should also be noted that

define

may be embedded

representation

it is known that

a

closed of in-

but we shall

23

7.

Having discussed a more explicit

GROUP EXTENSIONS

the G-modules

discussion

of group extensions

analogy with the case of discrete Hn(G,A),

n ~ 0

dimensions; g E G}.

A E P(G).

namely

For

morphisms

for

H0(G,A)

groups

of

G

into

A

and group eohomology.

[ii], we shall introduce

In complete

cohomology

These groups have simple interpretations

= A G, the G-fixed points

n = i, and a trivial G-module,

classes of continuous Ext(G,A),

which will enter into our theory, we turn to

in

HI(G,A)

groups

in low

A = {alg • a = a

for all

will be the continuous

homo-

(while for a general module we will have equivalence

crossed homomorphisms).

the group of topological By way of introduction

of how one may parameterize

For

n = 2, H2(G,A)

group extensions

to the cohomology

the group extensions

of

G

by

will be

A.

we shall begin with a discussion of

G

by

A

using cocycles.

Let

I+A÷E÷G÷I be a given extension. direct product

of

G

the direct product

The identity and

of

G

element of the group

Ext(G,A)

A, and in the special case when and

A.

This extension

A

is the semi-

is a trivial G-module,

is characterized

that one may find a continuous

homomorphism

f

(~of)(g)

the following

is to compare a general

by

A

= g.

The idea behind

to the semi-direct

such that

~of

homomorphism. groups,

product.

of

It is natural

is the identity map, and compute

G

by the property

it would not be sensible

the defect of groups,

f

of

G

such that

into

it is simply a fact of life that such a continuous

E

f

extension

of

G

f

to

E

of

G

from being a

but with topological

to choose any arbitrary map

map

E

to consider a map

Since we are dealing not with abstract

want to look for a continuous

back into

f.

satisfying

Ideally one would the above,

however,

map does not always exist.

Indeed

even in the case I+Z+R÷Z÷I such a map does not exist, a general base

G

extension would and fiber

topological

space.

A

and in general imply that

E

the existence viewed

sidered

= s

for

s E G,

(see [9]).

such as those continuous

in a neighborhood Borel map

f

G x G A

f

of

G

Other choices

into

E

element of

as

him we ob-

satisfying

of an appropriate

at the identity element of

of the identity

E = G × A

this, and following

G

G, but we believe

map may be con-

or those continuous that the choice of a

leads to a theory which is in general more satisfactory.

Once we have selected sometimes

fiber bundle with

would be a trivial bundle and so in particular

Mackey has shown how to resolve

serve that one may always find a Borel map (nof)(s)

of such a continuous map for

as a principal

called)

we note that

into the subgroup

and we notice

i(A)

such a Borel function a(g,h) of

E.

f

= f(g)f(h)f(gh) -I

(or cross section as it is is a Borel function from

We view it as a function

that it is a Borel function,

from

and as a consequence

G x G

into

of the associative

24

law in

E

satisfied the "cocycle identity", a(s,t)a(st,r)

for all G x G

s, t, r, G x G x G. into

A

= (s • a(t,r))a(st,r)

We denote by

Z2(G,A)

the group of all Borel maps of

satisfying this identity, and call such functions 2-cocycles.

The

group structure is understood to be multiplication of such functions pointwise. have associated now to each element of

Ext(G,A)

an element of

depends on the selection of a Borel cross section f

by any other Borel cross section

f

only by multiplication by a 2-cocycle of the form Borel function

b

of

G

into

note the group of such by extension in

Ext(G,A)

Z2(G,A)/B2(G,A) noted by

A.

of

f', the cocycle

into

for some

B2(G,A), and notice the very important fact that to each

we can associate a unique element of the quotient group This quotient group is deG

Ext(G,A)

into

H2(G,A)

may be verified to be a homomorphism If

A

is locally compact

[30] has shown that this map is surjective as well.

Ext(G,A)

A E P(G).

Hn(G,A).

A £ P(G), we define a complex of groups

is the set of all Borel functions from 6

We are able to show (see

This construction gives a param-

in terms of a cohomology group and also motivates the in-

troduction of the general cohomology groups

a coboundary operator

with coefficients in

A.

below) that this is also true for any

If

If we replace

(s • b(t))b(s)b(st) -I

of groups, and moreover may be seen to be injective.

eterization of

E.

changes, but it changes

H2(G,A), the two dimensional cohomology group of

The map of

Z2(G,A), but this

We call such functions 2-coboundaries, and de-

which is independent of any choices.

the topological G-module

Mackey

G

a

We

from

n

Gx.-.xG

Cn(G,A)

into

(n

Cn(G,A), n ~ 0, where factors) into

cn+I(G,A)

Cn(G,A)

A, and we define

by the classical formula

[ll], (~nf)(sl,-..,Sn+l)

= s I • f(s2,s3,..-,sn+l)

f(sls2,s3,...,sn+l)

-

... ± f(sl,''',SnSn+l )

¥ f(Sl,''',S n) where we are writing if

f

define and

A

additively.

The verification that

is a Borel function is routine Zn(G,A)

Hn(G,A)

H2(G,A)

to be the kernel of

~ n

to be the quotient group

and

Thus

B0 = 0

Borel}.

and

is a Borel function ~n+16n = 0.

to be the range of

We

~n-l,

n = 2, this gives the group

A, and

For 60

n = 0, a function of

is given by

~0(a)(s)

Z 0 = H0(G,A) = A G, the G-fixed points in

A.

For

Z 1 = HI(G,A) = {flf(st) = f(s) + f(t),

By a classical theorem of Banach, every such Borel homomorphism is auto-

matically continuous phisms of

For

as defined above, so everything is compatible.

n = i, and a trivial G-module, B 1 = 0, and f

Bn(G,A)

Zn/B n.

zero variables is by convention an element of = s • a - a.

~n f

[38], as is also the formula

G

into

homomorphisms of

(cf. A.

G

[3]), so

If

into

G A

HI(G,A)

acts on

is the group of continuous homomor-

A, B 1 # 0, and

or functions satisfying

Z1

consists of Borel crossed

f(st) = s • f(t) + f(s).

Such

25

a function classes

is by the same theorem of Banach continuous

of continuous

of a coset of

Elements tations

[31].

tive unitary

of

Let

of

= {f(s) = s • a - a Z2(G,T)

p

be a continuous

f

of

PU(H)

H

and hence an element

A

consists of

where a class consists

a E A}.

in the study Of unitary ray represenof

G

into

PU(H),

as defined previously.

back into

of

HI(G,A)

into

homomorphism

space

f(p(s))f(p(t))f(p(st)) -I = a(s,t)

Z2(G,T),

G

for some

arise naturally

group of a Hilbert

Borel cross section and then

crossed homomorphisms

BI(G,A)

so

U(H)

by general

theorems

H2(G,T).

It is clear that the element of unitary represnetation

H

of

"induces"

standing

p

[31].

Thus an analysis

of when a ray representation

Even if the element Mackey's

of

H2(G,T)

H2(G,T)

z

H2(G,T) of

G

on

is crucial for an under-

"is" in fact an honest unitary representation.

is non zero we can still construct

theorem a group extension

as above,

can be seen to define an element of

is zero if and only if we may find a continuous which

the projec-

We can find a

of

G

by

according

to

T

1 ÷ T ÷ E ÷ G ÷ i and one may verify

that

which is of the form ray representation

of

E

possesses

t ÷ t • 1 G.

on

COHOMOLOGY

of cohomology

First of all we selected a particular

perhaps

to give us what we wanted somewhat artificial

by the observation definition

We defined

that ray repre-

ad hoc.

(Borel functions)

and moreover

which

we selected a

6 . One's doubts are further compounded n of Section 6 suggest a somewhat different

to be all Borel functions

the possibility

from

of replacing

group of equivalence

classes modulo null functions

(Haar measure)

A, and we denote this group by

into

or

Hn(G,A).

Cn(G,A)

but one is led to consider

comment

is grantedly very

class of functions

of

H

of a group extension.

Hn(G,A)

in low dimensions,

definition

on

the given projective

GROUPS

groups

that the constructions

of the groups

our earlier

as ordinary representations

8.

happened

and which "induces"

This makes explicit

sentations may be interpreted

The introduction

an "honest" unitary representation T

Gx---xG = G n" into

Cn(G,A)

of measurable c__n(G,A).

to verify

that 8 as above is a well defined map from n that we get cohomology groups H n(G,A) = z_n(G,A)/B_n(G,A)

by

U(Gn,A),

functions

A,

the

from

Gn

It is not difficult

C n to -where

C n+1, and hence -Z_n is the kernel

of

6 and B n is the range of 6n_ I . The cocycles in dimension zero consist of n the kernel of 60 or the elements a of A such that s • a = a for almost all s

in

G.

and hence

It is not hard to see that this implies H0(G,A)

= A G.

If

A

sion one are exactly the equivalence that

f(st)

= f(s) + f(t)

that

is a trivial G-module, classes

s • a = a

of functions

for almost all pairs

s

and

for all s 6 G,

then the cocycles f t.

from

G

Similarly,

to

in dimenA

such

in dimension

26

two we look at functions which satisfy the cocycle identity above for almost all triples

(s,t,r).

A result of Mackey in [36] suggests that such an approach is not

as outlandish as it first appears. Motivated by the above, together with the possibility of a wide variety of other choices of cohomology groups we ask if we can somehow find a set of reasonable axioms which any cohomology theory should in principle satisfy, and then prove that there is up to isomorphism only one way of satisfying these axioms. that this is the case, and moreover that the groups cocycles do satisfy these axioms.

Hn

and

Hn

We shall show

defined above by

We then will know not only that these two defini-

tions of cohomology groups agree, but also that any other attempt to define cohomology groups satisfying the axioms below must necessarily lead to the same groups. (a) each

Our first axiom is of a general algebraic nature.

A ~ P(G), G

fixed, and for each

n ~ 0, an abelian group denoted by

such that these are "functors of cohomological type". that for any G-homomorphism of

Hn(G,A)

into

(gf)n = gnfn i

Hn(G,B)

when

g

f

of

A

We assume given for

into

Hn(G,A)

More precisely, we assume

B, we have induced homomorphisms

fn

such that the law of composition is s a ~ s f i e d :

is a G-homomorphism of

denotes the identity homomorphism of

A

B

into

into

C.

Moreove~

in = 1

where

A, and we assume that for any

short exact sequence i ÷ A" ÷ A ÷ A ~ ÷ i in

P(G), we have natural coboundary operators

~ : Hn(G,A ") ÷ Hn+I(G,A ~)

such that

n

the infinite long sequence 0 ÷ H0(G,A ") ÷ H0(G,A) ÷ H0(G,A ") ÷ HI(G,A ") ÷ ... ÷ Hn(G,A) ÷ Hn(G,A ") ÷ H n+I(G,A ~) ÷ H n+I(G,A) ÷ ... is exact (see [38] and [45]). (b)

The second axiom demands

(c)

The third axiom is a vanishing axiom which is motivated by the cohomol-

ogy of finite groups; namely we demand polonais group, where

I(A)

H0(G,A) = A G

Hn(G,I(A))

for any

= (0)

for

A ~ P(G).

n ~ i, and every

is the "regular representation" as defined in Section 6.

Axiom (c) is of course the really crucial one; it asserts that certain modules are cohomologically trivial and although there is a great deal of motivation for it from the cohomology of abstract groups, it does represent a definite choice. One could conceivably select some other class of modules and assume them to be cohomologically trivial, and this would lead to a unicity theorem for some possibly different cohomology theory.

Our defense here is that the groups defined by co-

chains above do satisfy this vanishing axiom, and that the groups

I(A)

do seem to

play a natural role in analysis and group representations. The following unicity theorem follows immediately from Lermna 6.3 and standard methods of homological algebra.

2?

Theorem i0

If A ~ P(G)

H~(G,A), i = i, 2

for a fixed

G

are two assignments of cohomology groups defined for

which satisfy Axioms (a), (b), and (c) above, there are

canonical isomorphisms of

H~(G,A)

onto

H~(G,A)

for all

n

and all

A.

I

One of our major results is that the groups

Hn(G,A)

and

H n(G,A)

defined

above by Borel cochains, and equivalence classes of measurable cochains do satisfy these axioms.

Theorem ii

The groups

Hn(G,A)

hence are isomorphic. in

and

Hn(G,A)

satisfy Axioms (a), (b), and (c) and

More precisely, the map which attaches to each Borel cochain

Cn(G,A), its equivalence class in

Cn(G,A),

induces this isomorphism on co-

homology. The verification of Axioms (a) and (b) is routine in both cases (see [38]); however, the verification of Axiom (c) is non-trivial.

In fact for

n = i, this

verification is for all intents and purposes equivalent to Mackey's general version of the Stone-von Neumann theorem in [26].

A close examination of Mackey's argument

in [26] reveals that what is essentially being proved is that tually one wants to replace

T

by a unitary group

this would lead us into non-abelian cohomology details however are the same as when

H

U(H)

HI(G,I(T)) = 0.

(Ac-

on a Hilbert space, and

(see [45]~

is one-dimensional

The essential analytic so that

U(H) = T.)

Theorem ii is proved first for

n = i, and then the general case is reduced to this

case by an induction argument.

The argument follows in spirit the argument for ab-

stract groups where in fact the result is trivial; however, there are non-trivial analytical complications concerning null sets in our case. In view of Theorem ii we shall henceforth use the notations Hn(G,A)

interchangeably;

Hn(G,A)

and

our choice of notation will serve to emphasize that we are

interested in a particular facet of these groups which may be evident from one of the definitions, but not the other.

We note in particular that such results as the

above are not approachable if one stays within the category of locally compact Gmodules, and that essential use is made of non-locally compact modules. We have remarked before that we have a natural notion of induced modules which gives us for each subgroup of

G.

A ~ P(H), a module

I~(A) E P(G)

where

H

is a closed

A very useful tool for finite groups is Shapiro's lemma [45] which

relates the cohomology of

A

with that of the induced module.

Theorem 12 There are canonical isomorphisms and all

n.

Hn(H,A ) _~ Hn(G,IH(A) )r_

for all

A ~ P(H)

28

The proof is obtained by noting that both sides of the above as functors on P(H)

satisfy Axioms

that for

(a), (b), and (c), and then one applies Theorem i0.

H = (e), this is simply the vanishing theorem.

We note

Also we note for

n = i,

that this theorem is essentially Mackey's imprimitivity theorem [27].

9.

ADDITIONAL PROPERTIES

We shall now discuss some additional properties of these cohomology groups, and in particular nail down the connection with group extensions. have already seen that

H0(G,A) = A G

tinuous crossed homomorphisms of a trivial G-module, into

A.

Ext(G,A) A

and that

G

into

A

n = i, HI(G,A)

For

n = 0, we

is the group of con-

modulo principal ones, and if

A

is

it is simply the group of all continuous homomorphisms of

G

In Section 7 we constructed an injective homomorphism of the group of equivalence classes of topological group extensions into

H2(G,A).

locally compact, Maekey has shown that this map is onto, but his argument

does not extend since it makes essential use of the Haar measure on

A.

For

[30]

We have an

alternate argument which works in general and which we outline below. If

a ~ H2(G,A), we embed

image of the class

a

in

A

H2(G,I(A))

into

I(A)

by Lemma 6.3 and let

under the map given in Axiom (a).

H2(G,I(A)) = 0, a p = 0, and so there is clearly an extension of sponding to

a', namely the semi-direct product

extension of

G

by

A

corresponding to

I(A) • G.

a ~ H2(G,A)

I(A) • G

G

by

be the

Since I(A)

corre-

We wish to construct an

and on general principles we

would expect this extension, if it exists, as a subgroup of we pick a cocycle in the class

ap

I(A) • G.

In fact if

a, we can immediately construct a subgroup

and then prove that it has all the required properties.

Ep

of

(This particular

construction is virtually forced on us, again by general principles.)

Thus

Ext(G,A) = N2(G,A). The higher cohomology groups have as yet no direct interpretation, however, we certainly do expect

H3(G,A)

to contain obstructions to the construction of non-

abelian extensions as in [3], Chapter IV. When the cohomology groups are constructed via equivalence classes of measurable cochains, another interesting and significant property emerges. since

cn(G,A)

is a polonais group, and since it may be readily checked that the

coboundary operators hence in

P.

Namely,

Thus

~n

are continuous,

H__n(G,A)

it follows that

is the quotient of a group in

z_n(G,A) P

is closed and

by a subgroup and,

hence when given the quotient topology, is itself a topological group.

a priori reason for

B_n(G,A)

There is no

to be closed, and it is an unpleasant fact of life

that it is not always closed so that

Hg(G,A )

may not even be Hausdorff.

The

closure of the identity element in such a group is a closed subgroup, and upon dividing by it, we obtain a Hausdorff group which in the case of

H_n(G,A)

is simply

29 Zn(G,A)

divided by the closure of

polonais, and

Hn(G,A)

Bn(G,A).

This quotient group will again be

will satisfy all the axioms of a polonais group except with

"metric" replaced by "pseudo-metric".

Thus

Hn(G,A)

is in a class of groups one

might reluctantly call pseudo-polonais. In any case, the fact that

Hn(G,A)

and hence

Hn(G,A)

more or less reasonable topology will be quite important for us. strengthen Axiom (a) above and prove that the groups

Hn(G,A)

have a natural and In fact we can

are functors of co-

homological type taking values in the category of topological groups. n = i, and if of

G

into

pact sets. topology. H2(G,A)

A A

is a trivial

G

module, HI(G,A)

has a natural Hausdorff topology, namely that of Convergence on com-

It may be verified that the topology on

HI(G,A)

coincides with this

In [39] a great deal of effort was devoted to constructing a topology for for various

G

and

A

by rather

ad hoc methods.

that this topology coincides with the one above on exists.

Moreover, if

being continuous homomorphisms

H2(G,A)

It is not hard to show whenever the former

Details of this will appear in our subsequent paper. One reason for seeking a topology on

Hn(G,A)

(aside from the esthetic one

of expecting a topological object when one starts with topological data) is so that one can hope to make sense out of the spectral sequence for the cohomology of a group extension (cf. [19]).

If

H

is a closed normal subgroup of

with finite groups leads us to hope for a spectral sequence

E p'q

G, the analogy converging to

r

H*(G,A)

with

= HP(G/H,Hq(H,A)) E P'q 2

gin to make sense, we must have topology.

(see [19]).

We observe that for this to be-

Hq(H,A) E P(G/H), and in particular it must have a

We can show that there is always a spectral sequence of this type, and

moreover that if

Hq(H,A)

the expected formula.

happens to be Hausdorff then the

E~ 'q

term is given by

The existence of such a spectral sequence is quite important

since it is an almost indispensable tool in making all but the simplest calculations of our cohomology groups. the case when

A

The reader is referred to [38] and [39] for examples in

is locally compact.

We shall close this section with one final result concerning direct integrals of G-modules.

Recall from Section 6 that if

A ~ P(G), the group

a natural structure as G-module which we called the direct integral.

U(X,A)

had

Since

Cartesian products are a special case of this, and since cohomology commutes with products, we may ask if the same is true for integrals and we have the following result.

Theorem 13

If

Hn(G,A)

is Hausdorff, we have an isomorphism of topological groups Hn(G,U(X,A))

= U(X,Hn(G,A))

.

The content of this result is that a cocycle with values in a direct integral module

U(X,A)

may be represented as a direct integral of cocycles.

If

n = 1

30 with trivial action, the side condition is satisfied and since one cocycles are homomorphisms, this result essentially gives us a new proof of the existence of direct integral decompositions of unitary representations.

i0.

EXAMPLES AND APPLICATIONS

We want to conclude with some examples, some computations, and some applications of the general theory above. Suppose that G-module.

G = G1 × G2

and suppose for simplicity that

A

is a trivial

Then either as a consequence of the spectral sequence above, or as a re-

sult of explicit computations (cf. [31]), we may obtain a structure theorem for H2(G,A)

as follows: H2(G,A) = H2(GI,A ) @ H2(G2,A) @ HI(GI,HI(G2,A))

.

The first two terms are easy enough to understand and represent the contributions of the factors

G1

and

G2

to the cohomology of

term representing the interaction of immediately compute

G1

G 2.

H2(Rn,T), H2(Zn,T), and

it is easy to verify that

H2(Tn,T)

lows readily by induction that

H2(Rn,T)

Since

n(n - i)/2, and that

H2(Zn,T)

dimension

n(n - 1)/2, and that

H2(Tn,T) = 0.

V

H__2(Rn,T) and

and torus

S.

by induction on

H2(Zn,T)

R n = Rn-I x R, and so on, it fol-

is isomorphic to a torus

H2(Rn,T)

slightly more involved statement holds for

contains a unique skew symmetric H2(Rn,T)

G.

n(n - 1)/2.

A

A similar but

is a trivial locally compact

H2(G,A) = HI(~I(G),A )

Furthermore

with the group of such

H2(G,A)

where

~I(G)

is the

is Hausdorff in its natural

topology and this topology coincides with the compact open topology on = Hom(~I(G),A)

of

H2(Zn,T).

is a semi-simple Lie group and if

G-module, it is classical [47] that

vergence.

V

again of

Moreover the topology defined above

functions which is a vector group of dimension exactly

usual fundamental group of

S

coincides with the usual topology on the

continuous bilinear function, and one may identify

G

Indeed

The isomorphism can also be implemented quite explicitly

since one may show that each class in

If

n.

by looking at the possi-

is isomorphic to a vector space

dimension

vector

This enables us for instance to

H2(R,T) = H2(Z,T) = H2(T,T) = 0

ble group extensions in these three cases.

on the groups

G, while the final term is a cross-

and

HI(~I(G),A)

which in this simple case is simply the topology of pointwise con-

This result also holds for any trivial G-module in

similar result holds for a much broader class of groups

G

P(G)

and moreover a

if one is willing to

suitably redefine and generalize the notion of the fundamental group

~I(G)

of

G

(see [40]). Using the spectral sequence of the previous section one may compute when

G

H2(G,T)

is a semi-direct product of a semi-simple group and say a vector group.

One

31

may verify

in this case known results

similar kinds of groups.

We refer the reader

Another application cerning non-locally

for the inhomogeneous

of this material,

compact G-modules

be a finite or infinite

to [38] and

dimensional

concerns

Lorentz

and

[39] for more details.

and especially the following

torus where

group,

of our results situation.

con-

Let

A = Tn

n = i, 2, -.., ~, and suppose

that

i ÷ A ÷ E ÷ G ÷ i is a group extension

of

G

by

A

where

E

is locally compact and abelian.

a trivial and well known consequence

of the duality

groups

(that is, represents

that such an extension

Ext(G,A))

and so

E = A @ G

With this result

splits

is a direct

sum of

in mind for Cartesian

to ask if a similar result holds for direct

A

It is

theory of locally compact abelian

and

products

integrals

the identity G

element

as topological

of circles,

of

groups.

it is natural

of the circle group and the

answer is affirmative.

Theorem 14

If

1 ÷ U(X,T) ÷ E ÷ G ÷ i

abelian and

G

locally compact,

as topological

is an extension of

then the extension

that

E

and so Theorem 13 is applicable. is assumed

case when useful

U(X,T)

following

to be abelian,

= T

in settling

topological

by

U(X,T)

is split so that

with

E

E = U(X,T) + G

groups.

The idea of the proof is quite simple; Hausdorff

G

where the result

Moreover

H2(G,T)

After some extra argument

using

is the fact

the problem is thrown back using Theorem 13 to

certain questions

groups.

one may verify that

is known.

concerning

Theorem 14 is found to be quite

the structure

exactly the same technique

of non-locally

compact

allows us to establish

the

result.

Theorem 15

If

H2(G,T)

locally compact

= (0)

and

then any extension

U(X,T)

a trivial G-module

i ÷ U(X,T) ÷ E ÷ G ÷ i

This final result leads to a very useful theorem concerning groups of von Neumann algebras which will have some applications theory.

Suppose

Hilbert phism

that

B

space and that f

of

G

u(g)

in

set down in [22]. B

such that

question we raise is whether a continuous

Neumann algebra of operators

is a locally compact

group.

Thus for each

f(g)(b)

unitary representation

field theory when for instance

B

of

G.

automorphism

in quantum field on a separable

of

B

satisfying

the con-

g E G, we have a unitary

= u(g)b u(g) -I

one can choose

G

We suppose given a homomor-

into the group of inner *-automorphisms

tinuity requirements operator

is a v o n G

with

splits.

the operators This question

for all u(g)

b ~ B.

is relevant

is some algebra of observables

The

so that they form

and

in quantum G

is some

32 symmetry group of the physical system.

If

B

is the algebra of all bounded opera-

tors on Hilbert space, a moment's reflection will show that we are raising exactly the question of when a projective or ray representation of

G

can be converted into

an ordinary representation since the group of *-inner automorphisms of

B

is

PU(H).

It follows from our general discussion of group extensions that we can do this for projective representations if of

G

H2(G,T) = 0, or equivalently if every group extension

by the circle group splits as a product.

The theorem to follow asserts that

the same is true in the general context described above.

Theorem 16

If of a v o n

f

is any homomorphism of

Neumann algebra

described in [22], and if of

G

with

~(g) 6 B

B

G

into the group of *-inner automorphisms

on a separable Hilbert space, continuous in the sense H2(G,T) = 0, then there is a unitary representation

such that

f(g)(b) = ~(g)bz(g) -I

The proof is almost immediate for the map cohomology class center of

a

in

B, such that

H2(G,W) a = 0

where

W

f

for

b 6 B.

immediately gives rise to a

is the group of unitary operators in the

if and only if a representation

~

as described in

the theorem exists.

However, by the structure theory of von Neumann algebras

of the form

and the result follows by Theorem 15.

When

U(X,T)

G = R, Kadison in [22] established a special case of this.

R. Kallman [23] has obtained a far more general result.

W

Recently

For the Poincar~ group,

another case of physical interest, L. Michel has already obtained the above result by rather different methods [36].

is

33

REFERENCES

[i] Auslander, L., et al.

"Flows on Homogeneous Spaces", Annals of Mathematics Studies, No. 53, Princeton (1963).

[2]

Auslander, L., and Green, L. (1966).

[3]

Auslander, L., and Moore, C. C. "Unitary Representations Groups", Mem. Am. Math. Soc., No. 62 (1966).

[4]

Avez, A. "Ergodic Theory of Dynamical Systems", Notes, University of Minnesota (1966).

[5]

Bargmann, V. "On Unitary Ray Representations of Continuous Groups", Ann. Math., 59, 1-46 (1954).

[6]

Bruhat, F. "Sur les Representations Math. France, 8-4, 97-205 (1956).

[7]

Borel, A. "Density Properties for Certain Subgroups of Semi-simple Groups Without Compact Components", Ann. Math., 72, 179-188 (1960).

[8]

Borel, A., and Harish-Chandra. Ann. Math., 75, 485-535 (1962).

[9]

Dixmier, J. "Dual et Quasi-dual d'une Algebre de Banach Involutiv", Trans. Am. Math. Soc., 104, 278-283 (1962).

"G-induced Flows", Am. J. Math., 8-8, 43-60 of Solvable Lie

Induites des Groupes de Lie", Bull. Soc.

"Arithmetic Subgroups of Algebraic Groups",

[10]

Dixmier, J. Les Paris (1964).

[11]

Eilenberg, S., and MacLane, S. Ann. Math., 48, 51-78 (1947).

[12]

Eilenberg, S. "Cohomology Theory in Abstract Groups, II", Ann. Math., 48, 326-341 (1947).

[13]

Gelfand, I., and Fomin, S. "Geodesic Flows on Manifolds of Constant Negative Curvature", Uspehi Mat. Nauk, Z, 118-137 (1952).

[14]

Halmos, P.

[15]

Halmos, P. Lectures on Ergodic Theory, Publications of the Mathematical Society of Japan, No. 3 (1956).

[16]

Hochschild, G. (1951).

"Group Extensions of Lie Groups I", Ann. Math., 54, 96-109

[17]

Hochschild,

The Structure of Lie Groups, Holden Day, San Francisco (1965).

[18]

Hochschild, G., and Mostow, G. D. Math., 6, 367-401 (1962).

[19]

Hochschild, G., and Serre, J. P. "Cohomology of Group Extensions", Trans. Am. Math. Soc., 74, 110-134 (1953).

[20]

Harish-Chandra. "Representations Acad. S c i . , 40, 1076-1077 (1954).

C*

Algebres et Leur Representations, Gauthier-Villars, "Cohomology Theory in Abstract Groups, I",

Measure Theory, Van Nostrand, New York (1950).

G.

"Cohomology of Lie Groups", Illinois J.

of Semi-simple Lie Groups V", Proc. Nat.

34

[21]

Harish-Chandra. "Discrete Series for Semi-simple Lie Groups", Acta Math., 113, 242-318 (1965).

[22]

Kadison, R. V. "Transformations of States in Operator Theory and Dynamics", Topology, ~, 177-198 (1965).

[23]

t" Kallman, R. " Spa_tally Induced Groups of Automorphisms of Certain von Neumann Algebras" (to appear).

[24]

Khinchin, A. Mathematical Foundations of Statistical Mechanics, Dover, New York (1949),

[25]

Loomis, L. An Introduction to Abstract Harmonic Analysis, Van Nostrand, New York (1953).

[26]

Mackey, G. W. 326 (1949).

"A Theorem of Stone and von Neumann", Duke Math. J., 16, 313-

[27]

Mackey, G. W. 593 (1951).

"On Induced Representations of Groups", Am. J. Math., 73, 576-

[28] Mackey, G. W.

"Induced Representations of Locally Compact Groups I", Ann. of

Math., 55, 101-139 (1952). [29]

Mackey, G. W.

Les Ensembles Boreliens et Les Extensions des Groupes", J.

Math. Pures Appl., 36, 171-178 (1957). [30]

Mackey, G. W. "Borel Structures in Groups and Their Duals", Trans. Am. Math. Soc., 85, 134-165 (1957).

[31]

Mackey, G . W . "Unitary Representations of Group Extensions", Acta Math., 99, 265-311 (1958).

[32]

Mackey, G. W.

"Point Realizations of Transformation Groups", Illinois J.

Math., k, 327-335 (1962). [33]

[34]

Mackey, G . W . "The Theory of Group Representations", mimeographed notes, University of Chicago (1955). Mackey, G. W.

"Infinite Dimensional Group Representations", Bull. Amer. Math.

Soc., 69, 628-686 (1963). [35]

Mautner, F. I. "Geodesic Flows on Symmetric Riemann Spaces", Ann. Math., 65, 416-431 (1957).

[36]

Michel, L. "Sur les Extensions Centrales du Groupe de Lorentz Inhomogene Connexe", Nucl. Phys., 57, 356-385 (1964).

[37]

Moore, C. C. "Extensions and Low Dimensional Cohomology Theory of Locally Compact Groups, I", Trans. Am. Math. Soc., 113, 40-63 (1964).

[38]

Moore, C. C. "Extensions and Low Dimensional Cohomology Theory of Locally Compact Groups, II", Trans. Am. Math. Soc., 313, 63-86 (1964).

[39]

Moore, C. C. "Ergodicity of Flows on Homogeneous Spaces", Am. J. Math., 88, 154-178 (1966).

[40]

Moore, C. C.

"Group Extensions of P-adic and Adelic Linear Groups", Inst.

Hautes ~tudes Sci. Publ. Math., (35), 5-74 (1968). [41]

Mostow, G. D.

"Cohomology of Topological Groups and Solvmanifolds", Ann.

Math., 73, 20-48 (1961).

35

[42]

O'Raifeartaigh,

L.

"Mass Differences and Lie Algebras of Finite Order", Phys.

Rev. Lett., 14, 575-577 (1965). [43]

Parasyuk, O.

"Horocycle Flows on Surfaces of Negative Curvature", Uspehi Mat.

Nauk, 8, 125-26 (1953). [44]

Segal, I.

"An Extension of a Theorem of L. O'Raifeartaigh",

J. Functional

Analysis, ~, 1-21 (1967). Cohomologie Galoisienne,

[45]

Serre, J. P.

Berlin, Springer (1964).

[46]

Shale, D. "Linear Isometries of Free Boson Fields", Trans. Am. Math. Soc., 103, 149-167 (1962).

[47]

Shapiro, A. 585 (1949).

[48]

Sherman, T. "A Weight Theory for Unitary Representations", 18, 159-168 (1966).

[49]

Seminaire "Sophus Lie", Paris (1954).

[50]

Weil, W. L'Integration dans les Groupes Topologiques et ses Applications, Hermann, Paris (1940).

[51]

Weil, A. "Sur Certains Groupes D'Operateurs Unitaires", Acta Math., iii, 143211 (1964).

[52]

Wigner, E.

"Group Extensions of Compact Lie Groups", Ann. Math., 50, 581-

Group Theory, Academic Press, New York (1959).

Canad. J. Math.,

APPLICATIONS

OF GROUP T H E O R Y TO Q U A N T U M P H Y S I C S ALGEBRAIC

ASPECTS

by Louis Michel#

T A B L E OF C O N T E N T S

0.

INTRODUCTION

i.

COVARIANCE

IN Q U A N T U M T H E O R Y AND ITS M A T H E M A T I C A L

i.i.

What

i. 2.

Group

i. 3.

G-Vector

Is Q u a n t u m M e c h a n i c s Invariance Spaces

i. 4.

Unitary

i. 5.

More Algebras

I. 6.

M o r e on

of

2.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Groups

SU(n)

47

and P e r m u t a t i o n

Groups

Operators.

S(n) . . . . . . . . .

Pseudo

and its T e n s o r O p e r a t o r s PHYSICS

Group Theory

2. 2.

The C o r r e s p o n d e n c e

2. 3.

Particle

2. 4.

The H y d r o g e n A t o m

2. 5.

The H e l i u m A t o m

of M a s s

P a u l i Principle. Atomic

Shell States

Principle m

. . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . .

and A t o m i c P h y s i c s

62

. . . . . . . . . . . . . . . . . .

62

. . . . . . . . . . . . . . . . . . .

62

in a S p h e r i c a l l y

Symmetric

Potential

.....

64

. . . . . . . . . . . . . . . . . . . . . . . . . .

71

The E l e c t r o n

Spin

Structure -Peri0dic

Atomic

Spin and E u c l i d e a n

2.10.

Molecules

2.11.

Measurement

. . . . . . . . . . . . . . . .

Table

. . . . . . . . . . . . . .

in a G i v e n S h e l l - Spin O r b i t or G a l i l e a n

Invariance

Coupling

........

. . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . of Spin and S t a t i s t i c s Spectra

3. i.

T h e Set of K n o w n N u c l e i

3. 2.

Isospin

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Scientifiques,

73 74 78 79 81

of N u c l e i by the Study

S T R O N G AND W E A K I N T E R A C T I O N S

Etudes

57

65

2. 8.

des Hautes

54

. . . . . . . . . . . . . . . . . . . . . . . . .

2. 9.

PHYSICS:

51

Roots

. . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2. 7.

43

. . . . . . . . . . . . . . . . . . . . . . . . . . U(n)

2. 6.

43

45

2. I.

# Institut

. . . . . . . . . . . . . . . . . . . . .

and M o r e Tensor

SU(2)

A T O M I C AND M O L E C U L A R

NUCLEAR

. . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . .

of D i a t o m i c - M o l e c u l e 3.

TOOLS

38

91 - B U R E S - S U R - Y V E T T E

- France.

82 84 84 86

57

3. 3.

4.

. . . . . . . . . . . . . . . . . . . . . . . . .

89

3. 4.

Shell M o d e l . . . . . . . . . . . . . . . . . . . . . . . . . . . .

92

The Hadrons . . . . . . . . . . . . . . . . . . . . . . . . . . . .

95

3. 6.

The Other Particles

98

RELATIVISTIC

IN-VARIANCE.

and the O t h e r THE D I S C R E T E

4. i.

The P o i n c a r ~

Group

4. 2.

Relativistic

Invariance

SYMMETRIES

C, P, T . . . . . . . .

its A u t o m o r p h i s m s ; and I n t e r n a l

. . . . . . . . . .

Zeeman Theory

i01

.....

i01

Symmetries . . . . . . . . . .

102

Irrep

4. 4.

Particle

States

4. 5.

Particle

Polarization . . . . . . . . . . . . . . . . . . . . . . .

4. 6.

Invariance

4. 7.

How

to O b s e r v e V i o l a t i o n . . . . . . . . . . . . . . . . . . . . .

115

a.

Action

115

b.

Parity Violation

5.3.

105 and Irreps

Under

of

P, T, C

Time R e v e r s a l Violation

T0 . . . . . . . . . . . . . . . . .

on O b s e r v a b l e s

. . . . . . . . . . . . .

. . . . . . . . . . . . . .

. . . . . . . . . . . . .

Invariance

, . . . . . . . . .

. . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . .

SYMMETRIES

SU(3)

of

P × Z 2 ( C ) ; PCT T h e o r e m

c.

a.

2.

TO

CP

THE I N T E R N A L

5.

of

and

Interactions

4. 3.

5. i.

6.

Invariance

3. 5.

4. 8.

5.

u(4)

OF HADRONS . . . . . . . . . . . . . . . . . . . .

Symmetry . . . . . . . . . . . . . . . . . . . . . . . . . .

The O c t e t s

. . . . . . . . . . . . . . . . . . . . . . . . . .

108 ii0 113

116 118 119 121 121 121

b.

The M a s s O p e r a t o r . . . . . . . . . . . . . . . . . . . . . . .

122

C.

The First B a r y o n D e c u p l e t . . . . . . . . . . . . . . . . . . .

124

d.

Other

e.

Cross-Sections

Geometry

SU(3)

on the

Electromagnetic

Multiplets and Decays SU(3) - O c t e t

Electromagnetic

b.

Weak

Critical

5. 5.

SU(3)

5. 6.

SU(6),

125

. . . . . . . . . . . . . . . . . . .

125

Interaction

Interaction

Quarks,

in

SU(3) . . . . . . . . . .

. . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . .

Orbits of a G - I n v a r i a n t

× SU(3)

ACKNOWLEDGMENTS

124

of R e s o n a n c e s . . . . . . . . . . . .

and W e a k I n t e r a c t i o n s

a.

5. 4,

. . . . . . . . . . . . . . . . . . .

F u n c t i o n on a M a n i f o l d

M .....

128 128 129 131

Symmetry . . . . . . . . . . . . . . . . . . . . . .

134

Current Algebra,

138

. . . . . . . . .

Boot-Strap,

Etc . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . .

143

38

0.

INTRODUCTION

Since you mathematicians and we physicists came here to meet together, there is no need to emphasize that we both believe that the progress of physics requires for its theoretical formulation more and more advanced mathematics.

I

thought fit however to give you the opportunity to read what Dirac wrote on this subject, 38 years ago, as an introduction to the very paper where he predicted the existence of the "antielectron", which we now call positron. Notwithstanding Dirac's prediction, when positrons were observed one year later by Blackett and Occhialini, and by the Joliot-Curies, they were not immediately recognized.

And Anderson who was the first to identify a positron (in cosmic rays)

did not know Dirac's paper.

This illustrates the communication difficulties which

existed and still exist between theoretical and experimental physicists. also expect them between mathematicians and physicists. culties due to my use of English.)

You should

(Not to speak of the diffi-

You and I are here determined to overcome them,

but I beg your patience in advance. Extract from

Proc. Roy. Soc., Ser. A, 130, 60 (1930):

Quantised Singularities in the Electromagnetic Field By P. A. M. DIRAC, F. R. S., St. John's College, Cambridge § i.

Introduction

The steady progress of physics requires for its theoretical formulation a mathematics that gets continually more advanced. This is only natural and to be expected. What, however, was not expected by the scientific workers of the last century was the particular form that the line of advancement of the mathematics would take, namely, it was expected that the mathematics would get more and more complicated, but would rest on a permanent basis of axioms and definitions, while actually the modern physical developments have required a mathematics that continually shifts its foundations and gets more abstract. Non-euclidean geometry and non-commutative algebra, which were at one time considered to be purely fictions of the mind and pastimes for logical thinkers, have now been found to be very necessary for the description of general facts of the physical world. It seems likely that this process of increasing abstraction will continue in the future and that advance in physics is to be associated with a continual modification and generalization of the axioms at the base of the mathematics rather than with a logical development of any one mathematical scheme on a fixed foundation. There are at present fundamental problems in theoretical physics awaiting solution, e.g. , the relativistic formulation of quantum mechanics and the nature of atomic nuclei (to be followed by more difficult ones such as the problem of life), the solution of which problems will presumably require a more drastic revision of our fundamental concepts than any that have gone before. Quite likely these changes will be so great that it will be beyond the power of human intelligence to get the necessary new ideas by direct attempts to formulate the experimental data in mathematical terms. The theoretical worker in the future will therefore have to proceed in a more indirect way. The most powerful method of advance that can be suggested at present is to employ all the resources of pure mathematics in attempts to perfect and generalize the mathematical formalism that forms the existing basis of theoretical physics and after each success in this direction, to try to interpret the new mathematical

39 features in terms of physical entities (by a process like Eddington's Principle of Identification). A recent paper by the author# may possibly be regarded as a small step according to this general scheme of advance. The mathematical formalism at that time involved a serious difficulty through its prediction of negative kinetic energy values for an electron. It was proposed to get over this difficulty, making use of Pauli's Exclusion Principle which does not allow more than one electron in any state, by saying that in the physical world almost all the negative-energy states are already occupied, so that our ordinary electrons of positive energy cannot fall into them. The question then arises as to the physical interpretation of the negative-energy states, which on this view really exist. We should expect the uniformly filled distribution of negativeenergy states to be completely unobservable to us, but an unoccupied one of these states, being something exceptional, would make its presence felt as a kind of hole. It was shown that one of these holes would appear to us as a particle with a positive energy and a positive charge and it was suggested that this particle should be identified with a proton. Subsequent investigations, however, have shown that this particle necessarily has the same mass as an electrontt and also that, if it collides with an electron, the two will have a chance of annihilating one another much too great to be consistent with the known stability of matter.+tt It thus appears that we must abandon the identification of the holes with protons and must find some other interpretation for them. Following Oppenheimer,tttt we can assume that in the world as we know it, all, and not merely nearly all, of the negative-energy states for electrons are occupied. A hole, if there were one, would be a new kind of particle, unknown to experimental physics, having the same mass and opposite charge to an electron. We may call such a particle an antielectron. We should not expect to find any of them in nature, on account of their rapid rate of recombination with electrons, but if they could be produced experimentally in high vacuum they would be quite stable and amenable to observation. An encounter between two hard x-rays (of energy at least half a million volts) could lead to the creation simultaneously of an electron and antielectron, the probability of occurence of this process being of the same order of magnitude as that of the collision of the two y-rays on the assumption that they are spheres of the same size as classical electrons. This probability is negligible, however, with the intensities of y-rays at present available. The protons on the above view are quite unconnected with electrons. Presumably the protons will have their own negative-energy states, all of which normally are occupied, an unoccupied one appearing as an antiproton. Let me just remind you that antiprotons were first observed twenty-four years later (1955). There will be many advanced seminars on the applications of group theory to quantum physics.

So I believe that these lectures must be introductory~ and that I

have to present concepts that will be used by all physicists here.

That will be

Part i.

t Proc. Roy. Soc., Ser. A, 126, 360 (1930). #t H. Weyl, Gruppentheorie und Quantenmechanik,

2nd ed., p. 234 (1931).

tit I. Tamm, Z Physik, 62, 545 (1930); J. R. Oppenheimer, Phys. Rev., 35, 939 (1930); P. Dirac, Proc. Cconb. Philos. Soc., 26, 361 (1930). tttt J. R. Oppenheimer, Phys. Rev.,35,

562 (1930).

40

The ultimate goal of these lectures will be to bring you to the present problems on the subject, mainly in the field of the fundamental particle physics. Then there might be some overlap with Professor O'Raifeartaigh's

lectures, but there

should be no inconvenience to see some aspects of physics from probably two different points of view.

We have to face the f a c t t h a t

fruitful discussions of frontier

problems of physics between mathematicians and physicists are difficult, because these problems often cannot be presented in a formalized language, but only through some physical analogy.

So, obviously, to understand what is the problem, one must

know some physics! In these lectures I will therefore present a quick survey of applications of group theory to atomic, molecular and nuclear physics.

Often, I will even

follow an historical approach.

Indeed, physicist minds are partly conditioned by

the recent history of physics.

But I also hope to use the power of your language,

mathematics,

to convey to you a maximum of physics in a minimum of time.

Of course

I shall have succeeded only if I have also been able to convince you that physics is fascinating~ It is fit to end this introduction by the history of the birth of our subject.

Less than three years after the first paper on quantum mechanics

(W. Heisenherg, Z. Phys., 33, 879 (1925), there appeared the first two papers devoted to the application of group theory to quantum mechanics: •

E. P. Wigner, "Einige Folgerungen aus der Schrodingerschen Theorie fur die Termstrukturen",

•

Z. Phys., 43, 624 (1927).

F. Hund, "Symmetriecharaktere von Termen bei Systemen mit gleichen Partikeln in der Quantenmechanik",

Z. Phys., 43,

788 (1927). Wigner will surely be the most quoted author on our subject.

Let us just say that,

with J. von Neumann, he applied group theory to atomic spectra ("Zur Erkl~rung einiger Eigenshaften der Spektren aus der Quantenmechanik des Drehelektrons I., II., III., Z. Phys., 47, 203; 49, 73; 51, 844 (1928)), and published a self-contained book on this question:

E. P. Wigner, Gruppentheorie und ihre Anwendung auf die

Quantenmechanik der Atomspektren, Vieweg, Braunschweig (1931). It is remarkable that two famous mathematicians, Hermann Weyl and Van der Waerden, also published very early books on our subject: •

H. Weyl, Gruppentheorie und Quantenmechanik,

•

Van der Waerden, Die Gruppentheoretische Methode in der Quanten-

Hirzel, Leipzig

(1928).

meehanik, Springer, Berlin (1932). Then the excellent, but more elementary book, by E. Bauer, In~oduction &

la Th~orie de8 Groupes et 8es Applioation ~ la Physique Quantique, Presses Universitaires de France, Paris (1933), continued a list of books which, today, may have reached several dozen.

41

For the interested

mathematicians

first books, but in their second,

revised

I would still recon~nend the two very and enlarged

edition:

of Groups and Quantum Mechanics,

Methuen,

New York

by J. J. Griffin of Wigner's

(1949);

the translation

London

(1931); Paper Back reprint, book,

and Its Application to the Quantum Mechanics of Atomic Spectra, New York

Dover,

Group ~eory

Academic Press,

(1959).

For The Mathematician

Readers

Physics will be injected venient

Theory

H. Weyl, The

in these notes as needed.

to gather here some information

on physical

However,

it seems con-

constants which might be useful

at any time. We will study quantum phenomena. physics,

the key number

In atomic,

to pass from macroscopic

nuclear,

fundamental

particle

scale is the Avogadro number:

a = 6.0228 × 1023 ~ 6.03 × 1023 It is the number

of atoms in a mass of one gram of hydrogen.

A hydrogen atom is m a d e of one proton (mass

(0.i)

(mass

mp)

and one electron

ms). m

-J~ = 1836.5 m

These two particles

are electrically

+ p , ~ , the absolute value of this

charged,

charge is e =

The most

convenient

1 Faraday

96,600

=

6.03 × 1023

unit systems,

for us, will use

h = (elanck constant) c = (velocity In this system

and a nucleus which A = Z + N topes

number.

= i

contains

Z

protons

and

N

is nearly

A = 3, tritium isotope is

are made of

neutrons;

e.g., for hydrogen

Z = 92; the most abundant

charge and its mass

i 137.04

Atoms of the other elements

N = 0, A = i; A = 2, deuterium;

electric

i

e = (137.04) -1/2 , indeed

the atomic mass number,

For uranium

x (27)-1=

of light)

2 e hc

is a dimensionless

Coulombs

a

Z

Z

electrons

is the atomic number,

Z = i; hydrogen has 3 iso(unstable,

A = 238.

lifetime

12 years).

The neutron has no

equal to that of the proton,

42

m n

--= m So the atom mass is practically

1839.0

concentrated

in its nucleus.

Many more particles will be introduced, different

neutrinos

all with

We remind

the reader

mass is a form of energy. E = me2(l_(v/c)2)-I/2.

zero rest-mass

e.g.,

and zero charge,

that in relativistic

The energy of a particle

The rest energy

the photon

(v = 0)

X, and the four

etc..,

see 3.5 and 3.6.

physics mass is not conserved; of mass

is

m, velocity

We need to choose another unit to complete our unit systems. choice for atomic physics

is the electron mass

momentum

However,

the universal

length

2 m c g h/m c

time

him c 2 = i . 2 8

difference

is the electron volt charge

3.86

=

10 - 1 1 em x 10 - 2 1 s e c .

e

the most conxnon energy unit

It is the energy that a particle with

gains by traversing

an electric field of potential

one volt.

m c s m c P

I

x

(eV).

The conversion with the preceeding

Note that

The best

m . ° Then the other units are:

due to the nature of its measurement,

electric

is

m ge

energy

used for particles

v

mc 2 .

2 2

unit system is

= 0.511 × 106 eV ~'i/2 MeV = 938.256 MeV ~ 1 GeV = 109 eV

1 = ~ ~ (10 -13 cm) × (200 MeV) Before 1932,

electromagnetism. by proportional

only two kinds of interactions

In the static approximation potentials

K/r

where

of the (attractive)

gravitational

between

is independent

two protons

r

were known,

the two interactions

is the distance.

energy to the (repulsive)

gravitation

and

can be described

So the absolute ratio electrostatic

energy

of their distance.

It is Gem 2 P = I= 2 ~ e Thus, gravitation will be completely

137 x 10 - 3 6 175 neglected

in these lectures.+

t The gravitational energy of a system increases roughly as the square of the number N of nucleons while, in neutral matter, the electrostatic energy is roughly proportional to N. So gravitation becomes important only for masses as large as that of asteroids, planets (we know it on the earth!) or stars. It is not a coincidence that most stars have a number of nucleons ~ ~{-3/2 = IOS7 (see for instance E. E. Salpeter, "Dimensionless Ratio and Stellar Structure", in Frontier in Physics, Bethe Festschrift, p. 463, R. Marshak Editor).

43

The binding energy of atoms, molecules, solids, etc., is of electromagnetic origin.

This energy can be released in chemical form, with an order of

magnitude: a × i eV = 23 cal/mole which is ~ 10 -9 to i0 -II the rest-mass energy.

i

If the energy we receive from the

sun came from chemical reactions, the sun would produce it for less than 105 years! There are two other known kinds of interactions:

the nuclear interaction

(see part 3 and 5), stronger than the electromagnetic interaction at distance smaller than 10 -13 cm; the Fermi or "weak interaction" (see 3.6 and 5), which is very short range.

Both interactions are important in stars and nuclear reactions

and can yield an energy up to 10 -3 the rest mass energy.

For the Physicist Reader

All mathematical terms used here are not defined. are known to physicists Salam's lectures in

196~

Of course many of them

(e.g., for the notion of root vectors of Lie algebra, see

High Energy Physics and Elementary Particles, Trieste Seminar

(International Atomic Energy Agency, Vienna (1963)).

in I) come from a modern mathematical terminology.

Some terms (used mainly

They were not absolutely neces-

sary and they are used explicitly as synonyms of other terms generally used by physicists.

Physicists should know the proper mathematical terms of the mathematical

concepts they need:

indeed their students, and even their young children know them

and physicists want to communicate with their students and their children! An excellent and elementary exposition of this modern mathematical language is given in the text

Algebra

by S. Mac Lane and G. Birkhoff, Macmillan, New York,

(1967), particularly Chapter I; note also the list of symbols, p. XVII to XIX.

i.

COVARIANCE IN QUANTUM THEORY AND ITS MATHEMATICAL TOOLS

i.i.

What Is Quantum Mechanics

Less than two years after the first paper (quoted above) of Heisenberg on quantum mechanics, J. von Neumann answered this question in three successive papers in Gottingen Nachrichten, (1927), (pp. i, matische Grundlagen der Quanten Mechanik,

245, 273) expanded in a book: (1930)

Mathe-

(English translation, Princeton

University Press (1955)).#

He later published with G. Birkhoff, "The Logic of Quantum Mechanics", Ann. 37, 935 (1936). This subject is still controversial and lively.

Math.,

of

Two early books on quantum mechanics by physicists are reedited and still very advisable reading:

P. A. M. Dirac, The

Principles of Quantum Mechanics,

Clarendon Press, Oxford Ist ed. (1930) 4th ed. (1958). Quanten Theorie",

Handbuch der Physik,

W. Pauli, "Prinzipien der

i, Springer (1958) ist ed. (1933).t

If you have not read these books it is not too late to do it, but today let us just give a mlni-description of quantum mechanics. a)

To each physical system corresponds a separable complex Hilbert space

~C.

A physical state is represented by a vector

(Normed to 1 for convenience: b)

Each physical observable

<x,x> = I.)

a

(e.g., energy, electric charge, etc.)

is represented by a self adjolnt operator of c)

A

x> 6 ~C.

is the set of possible values of

A

on

~C.

The spectrum

a.

Quantum mechanics does not predict, in general, the value of for the state

a

x>, it gives only its expectation value: <x,Ax> = Tr A P

(i.I) X

where

P

is the Hermitian projector

(Px = Px *)

onto the one dimensional space

X

spanned by

x>.

Note that unit eigen vectors of

Px

(with eigen-value i) differ

only by a scalar phase factor and describe the same state since they yield the same physical predictions.

The projectors

P

are themselves observables.

Indeed

x

TrP P xy

= I<x,y>I 2

is the probability to observe in the state system which is known to be in the state

x> y>

(1.2)

(respectively (respectively

y>) the physical x>).

Part of the art

of the quantum physicist is to code what he sees in nature into vectors of Hilbert space!

This always requires "physical approximations". When we can describe a state by a rank one projector (or a vector up to a

phase) we say that we have a pure state and that we have a complete information on it. More often our information on the state is only partial. case we know only a set of probabilities

c.

in the set of orthogonal pure states

(i.e.,

Pi

(with

In the simplest

Ec. = i) for the system to be

PiPj = ~ijPj) so the expectation

value is E.c. Tr A P. = Tr A R i

i

(1.3)

1

with R = E.c.P. i

Since

I

Tr R = E.c. = 1

I

0 ~ ci, the self adjoint operator

i

R

i

is positive and it is called the

# There are also books on the mathematical foundation of quantum mechanics by mathematicians: G. Mackey, L. Schwartz.

(1.3')

45

density matrix# of the mixture (= not pure) state of the system.

Pure states are

extremal points of the convex domain of states. This leads us to a natural generalization. B

with unit I, generated by the observables.tt

C*-algebra). is

Then a state is a linear functional

VA E B, ~(A*A) ~ O.

One defines a Banach*-algebra

(More specifically it is usually a ~

on

B

which is positive, that

For systems with a finite number of degrees of freedom this

is not an essential generalization.

It becomes so for infinite degrees of freedom

as in quantum field theory and statistical mechanics.

Classical statistical

mechanics can also be put in the same mathematical mould with an abelian algebra.ttt

1.2

Group Invariance

We assume that there is a relativity group considered here.

That is

G

phism between the physics of observables by

K E G

acts on a physical system S

(e.g.,

~

for every physical theory S, and

(its Hilbert space of states

B c L(~)tttt, etc...)

view of G-invariance.

G

and the physics of

can be a rotation).

there is an isomor-

~, its algebra of

~(S), the transform of

S,

This will be called the "active" point of

The "passive" point of view for a transformation group is

simply the isomorphism between the physical description of the same system

S

by

two observers choosing different coordinate frames, G-transforms of each other. For any

g E G, we denote by

P

the transformed of the state gx i

say that

G

P

.

To

xi

is an invariance group is equivalent to saying that all probabilities

of Equation (1.2) are invariant x> E ~,

t

g E G, TrP

P = TrP P gx I gx 2 xI x2

(1.4)

J. von Neumann introduced the density-matrix in 1927 in the papers quoted above.

tt Quite early physicists also considered non-associative algebras formed by the observables and introduced Jordan algebras The first fundamental paper on those algebras is by P. Jordan, J. von Neumann and E. Wigner; "On an Algebraic Generalization of the Quantum Mechanical Formalism", Ann. of Math., 35, 29 (1934). it* I. E. Segal advocated twenty years ago the use of C*-algebra for quantum physics. The fundamental paper showing the benefits from this choice (physical approximation and Feld's s-equivalence; introduction of super-selection rules) is that of R. Haag and D. Kastler, "An Algebraic Approach to Quantum Field Theory", J. Math. Phys., Supplement 848 (1964). Most of the C*-algebra physics is published in the journal: Communications in Mathematical Physics, and is written in a rigorous mathematical style. For statistical mechanics, see D. Ruelle, Statistical Mechanics, Benjamin, New York (1969). Soon there will appear in the collection of C. N. R. S. Colloquia (France) "Rigorous Results on Interacting Systems with Infinite Degrees of Freedom". tttt We denote by

i(~)

the space of linear operators on

~.

46

or

1 12= I<Xl,X2> [2 This means that

G

acts on

(1.4')

~C by isometrics.

Wigner proved in his book (Appendix to Chapter 20) # that either a unitary operator

U(g)

that an antiunitary operator

or an antiunitary operator

V

V(g)

x> ~+ gx> on

~C.

has the characteristic properties

(1.5)

x>, y> E ~, V(~x> + ~y>) = EVx> + ~Vy>

(1.51 )

= <x,y> =

(1.5")

it has an inverse Given an isometry on

~, there is a simple criterion ## for deciding whether it is

realized by a unitary operator U

or

U

or an antiunitamy operator

is defined up to a scalar phase factor.

V

is

We recall

V.

In either case

The product of two antiunitary

operators is a unitary operator. Let U00

V(~)

be the group of unitary and antiunitary operators on

the subgroup of unitary operators.

since it is a subgroup of index two. i.e., no other element than for

g 6 G

group

UI

i 6 G

generate a subgroup

U~)

is an invariant subgroup of

We assume that acts trivially on

E(G)

of

$C and

G ~.

The

U(g)'s

or

V($C) which is an extension of

(phase multiplication of the vectors of

V(~)

acts effectively on

~C, V(g)'s

G

by the

X, leaving invariant the states)

with the action f G --+Aut U I where

Ker f

is the invariant subgroup of index two

transformations and the non-trivial element of -i ~--+ ~ = ~ E U(1). We can also say that

G+

Im f

G+ c G

which acts by unitary

is the complex conjugation

acts by a linear unitary projective representa-

tion and Wigner has coined the word projective "corepresentation" for the action of G

(when

G

is strictly larger than

G+).

Wigner also showed from physical arguments that antiunitary operators are to be used with transformations which reverse the direction of time, this in order that energy be positive: indeed, the time translation t is represented by the iHt iHt e ; if t ÷ -t, i has to go to -i in order that both H and e

operator

be invariant.

#

A more explicit proof of Wigner's theorem has been given by V. Bargmann, J. Math. Uhlhorn, Neumann axiomatics, the equivalent theorem has been proven by G. Emch and C. Piron, J. Math. Phys., i, 469 (1963).

Phys., 5, 862 (1964). See also proofs of slight generalizations by U. Ark~v for Fys~k, 23, 307 (1963). In the framework of Birkhoff and von

## See Bargmann:# for any triplet of vectors x>, y>, z>, <x,y> is invariant under a unitary transformation U and is transformed into its complex conjugate under the antiunitary transformation V.

47

Continuous projective linear unitary representation of finite groups or Lie groups are well known. SO(3,R)

For instance, for the three-dimensional rotation group

these projective representations are in a one to one correspondence with

the "linear irreducible unitary representations" (= irrep lectures) of

SU(2)

the universal covering of

S0(3,R).

through all these This justifies the intro-

duction of spinors in quantum physics. In Part 4 we will study invariance under the relativity groups of nonrelativistic (= Newtonian) mechanics and of special relativity theory. are other invariance groups in physics. acting on

n

But there

For instance the permutation group

identical particles (as the electrons of an atom).

S(n)

In nuclear physics

and fundamental particle physics we shall meet many "approximate invariances". corresponding invariance group is most often a unitary

n × n

matrices, with determinant

1

U(n) for

or

SU(n)

The

group (group of

SU(n)) with

n = i, 2, 3, 4, 6.

We shall have more to say for the word "approximate" symmetry. We will also have to study invariance under a group

G

when

G

is a

symmetry group for a physical system, e.g., the symmetry group (one of the crystallographic group) of a crystal. invariance in physics.

This example raises a fascinating question about group

Surely the interaction between atoms are translation in-

variant (and may be invariant under a larger transformation group).

How is it

possible that atoms aggregate to form a crystal whose lattice is invariant only under a subgroup of a translation group?

When such a phenomenon occurs, i.e., when

a stable state has a lesser symmetry than that of the physical laws we will say that we have a broken symmetry.+ We will continue this Part i by introducing some mathematical tools that we will use quite frequently.

1.3

Let

G

G-Vector Spaces

be a given group.

If you like you can say that we consider a

category whose objects are vector spaces action of

G

on

endomorphisms of

E

(i.e.,

E, with

G D x ~

E

(over a given field

g(x) E L(E), where

L(E)

K) with a linear is the algebra of

xy ~-+ g(x)g(y) = g(xy).

The morphisms of the category are the vector space homomorphisms

E

f~ E ~

compatible with the group action, i.e., they are the commutative diagrams for every x E G, of vector space homomorphisms.

We will call these morphisms G-homomorphisms

of G-vector spaces.

+ This short section on group invariance is too sketchy. Much more should be said of the symmetry of physical laws (e.g., E. P. Wigner, "Symmetry and Conversation Laws", ~oc. Nat. Ac~. Sci., U.S.A.j 51, 956 (1964)) without which symmetries of states, that we have considered, would not last. Of course much more will be said in these lectures.

~8

E

~ (x)

~ E

E'

g' (x)

~ E'

L

Diagram i. Of course we could have also said that we study bimodules

(G- and K-

modules) or even more simply that we are interested in the linear representations of G; and G-homomorphisms morphisms from

E1

are also called "intertwining" operators.

to

E2

form a vector space that we denote

Indeed, it is the subspace of the invariant vectors of intertwining operators for the two representations of Given G-vector spaces,

Note that the GHom(El, E2)G.

Horn(El, E2); they are the G

on

E1

and

E 2.

El, E 2 ... all vector spaces which can be formed

functorially from them are also G-vector spaces, e.g.,

E 1 0 E2, Hom(El, E2) , L(E)

= Hom(E, E), the vector space of the tensor algebra on

E:

E (n) = E 8 E 8 ... 8 E, n

factors,

(and

Given a physical system, let Assume that

3{ is a G-vector space.

T(E) = n@-~--OE(n)

when

E (0) = K), etc. ~C be the Hilbert space of its state vectors.

So is

LOC).

We are then led to the study of

the objects (of the category of G-vector spaces) "above" the physical literature "Tensor operators on

$C".

LOC).

They are called in

(A notable exception is the book

devoted to this subject, Irreduoible Tensoz~al Sets, by V. Fano and G. Racah,

By definition, for physicists, an "E.-tensor

Academic Press, New York (1959)).

l

operator" is a G-morphism (or intertwining operator)from

E.i to

LOC).

If the

representation of

G on E.i is irreducible, then the corresponding G-morphism is called in physics an "irreducible tensor operator". If G acts trivially on E.i then we have "scalar tensor operator". (Just try to remember that tensor operators

on

~C are not operators on

~C!)t

It is time to specify the field of complex numbers since essential in physics.

5£

So often

tor" is a G-homomorphism

T

of self-adjoint operators on the field from IR When

G

to

K.

Generally, of course, it is the field

is a complex Hilbert space. E

is a real vector space and the "E-tensor opera-

of real vector spaces from ~.

However, reality is also

E

to the real vector space

Of course it is always possible later to enlarge

~.

is a Lie group we consider, of course, only continuous differ-

entiable representations so a G-vector space is also a ~-module for the Lie algebra ~_ of L~) ~l on

G.

We denote

G

the vector space of

there is a particular one 5£.

F'

When the representation of

~.

Among the G-tensor operators on

which is also the Lie algebra representation of G

on

~C

is unitary, then

adjoint operators for images which satisfy

# "Scalar" is often used by physicists for "invariant"!

F = iF s

has self-

~9 [F(a),F(b)] When translations,

G

(F(a)F(b) - F(b)F(a)) = iF(a A b)

(1.6)

is respectively the group of rotations, space translations,

etc., F

momentum, energy,

=

...

corresponds respectively to the observables;

time-

angular-momentum,

In the technical sense of i.i, what we called observables

there, are the elements of the image of

F, i.e., for instance, the component of the

angular momentum or of the momentum, in a given direction. clear that G-morphisms on

L~)

But I hope it is by now

are what correspond to the physical concepts with

a tensorial character with respect to a group

G

(other examples:

velocity, mag-

netic moment, electric quadrupole moment, energy-momentum tensor, tensor of inertia, etc. ). Let E

and on

R

3£.

and

U

(unitary) be the representations

By definition of the E-tensor operator Vx 6 E, Vg 6 G,

If

D

and

F' = iF

U(g)T(x)u-l(g)

of

respectively on

= T(R(g)x)

are the corresponding representations D(a) = ~d

G

T

R(e ~a) ~-0 ; iF = ~d

(1.7)

of the Lie algebra

U(e ~a)

-

~=0

then an equivalent definition of the E-tensor operator

T

(1.8)

is

Vx 6 E, Va 6 ~, [F(a),T(x)] = iT(D(a)x)

(1.9)

In a nutshell, I would say that much of the application of group theory to quantum physics consist in the study of the "tensor-operators" (Hilbert) space T1

and

T2 E1

~

of a physical system.

be respectively

E1

and E2-tensor operators on

E 2 9 x @ y q--+ Tl(X) + T2(Y )

E1 0 E2 ~ x 0 y ~

Tl(X)T2(Y)

that we denote respectively

TI ~ T 2

defines a

defines and

on the G-vector

They form a ring~ (and an algebra).

a

Let

~, then

E 1 • E2-tensor operator E1 0 E 2 - t e n s o r o p e r a t o r

T l O T 2.

The latter is generally re-

ducible and can be decomposed into a direct sum of irreducible "tensor operators". I believe that many problems arise which have not been systematically studied by physicists although they work very much with this ring (for fixed and action of

G

on

G,

~).

For instance, if Yx, y 6 G, [T(x),T(y)] = 0.

G

is simple, and

T

is a G-tensor operator and

I believe this implies

dim Hom(G,~) G is infinite.%+

Of course the subalgebra generated by an element is well known; given an A T there is a functorial G-morphism T from the tensor algebra

E-tensor operator

For infinite dimensional ~, the operators T(x) are not bounded so their product is not always well defined. I will forget here this difficulty which has to be faced in quantum mechanics and is considered in O'Raifeartaigh's lectures. %% C. Moore proved it during the Rencontres.

5O

T(E)

on

E

LO0,

to

onical injection of

E

which is moreover an algebra homomorphism. into

If

i

is the can-

T(E)(Im i = E(1)), then the Diagram 2 is commutative.

T-(E)

E

~

~ L('~)

Diagram 2. in the-particular case where i, (see Equation

U(G),

(1.6)) of

~

on

T

is the representation

F

(up to a factor

~, then it appears also in the representation of

the universal enveloping algebra of

T(G)

s

G

> U(G)

~ L(SC)

F

Diagram 3. A F

(in diagram 2) = F • s

A remarkable "scalar tensor operator" is the Casimir# operator. Let

G

be a semi-simple Lie group.

resentation of the Lie algebra

~

D(a)b = a A b,

Let

a ~

on its vector space

D(a)

the adjoint rep-

G

[D(a),D(D)] = D(a A b)

(i.i0)

The symmetrical bilinear Caftan-Killing form

(1.11)

B(a,b) = TrD(a)D(D) is non-degenerate. dual

Gt .

Therefore, it defines a G-isomorphism

This also defines the isomorphism G O G
i.

Of

I 1

t h e Young d i a g r a m of

For

with the integers

X I ~ h 2 ~ 0. The diagram of the contragredient representation of

with

of

(= f u n d a m e n t a l ) r e p r e s e n t a t i o n

to the trivial (= one-dimensional)

example, the representations

n

[Xi,]

i s t h e complement ( u p -

~°

s i d e down) of t h a t

SU(n) group

SU(k)

of

[Xil]

in the rectangle

Representations.

of

U(k)

of

k

lines

of

h1

The restriction of an irrep of

is an irrep of

SU(k).

Irreps of

U(k)

squares.

U(k)

whose Young dia-

grams differ only on the left by a rectangular block of columns of length by restriction equivalent irreps of unambiguously label irreps of inequivalent irreps of

Example.

SU(k)

SU(k).

by Young diagrams.

Moreover, this yields all

[%l,h2]

of

U(2)

SU(2)

obtained by re-

are given by the value of the integer

h I - h 2.

So their Young diagram can be written as a horizontal line of

squares.

For

with

i j = ~(h I - h 2)

is called the spin of the representation.

Casimir operator of

hI - h 2

irreps it is customary to use the symbol D. 3

j

yield

SU(k).

striction of the irreps

where

k

Taking into account this remark, one

The equivalence classes of irrep of

SU(2)

to the sub-

D., 3

that defined by (1.9).

(1.16) 2j + I

as normalized by physicists,

is

is its dimension.

The

j(j + i)I, which is twice

(Indeed, physicists take as Cartan-Killing form

1/2 TrD(a)D(D)). We also recall the well known decomposition Jl + J2 D. @ D. = , ~ 31 32 @J=l Jl-J 21Dj Note that all representations Representations Zk, the cyclic group of

of

SU(2)

are self eontragredient.

of the Adjoint Groups k

(1.17)

SU(n)/Z n.

The center of

elements, so the adjoint group of

SU(k)

is

SU(k)

is

SU(k)/Z k.

54 The representations

of this group are those of

number of squares multiple of

k.

tions are

D. with integral j. 3 even number (= 2j) of squares•

For example,

equivalent

whose Young diagram has a

SU(2)/Z 2 = S0(3).

Their Young diagrams

The adjoint representation space of its Lie algebra,

SU(k)

Its representa-

contain only one line of an

of

SU(n) or its adjoint group is that on the it has dimension n 2 - i and label [2, In-2]; it is

to its contragredient.

Remark For Any Group. linear unitary representation As we saw

S(n)

resentations

and

of

G

S(n)

For any group

G, let

~C(I)

(it may be reducible and

acts on

~C(n) = ~ C (I) •

may be infinite).

~(n) o~[ ]~

Subspaces

are not in general subspaces

be the space of a

dim ~C(I)

of primary rep-

of primary representation

Methods for knowing the nature of the G-representation

of the different

of

G.

~¢(n)

~[ ]h would be interesting

for the physicists

(bosons) and

(fermions).

~(n) [i n]

especially in some case, for %~(n) , W[n ] Here is an example of a result, proven by A. Bohr,

Mat. Fys. Medd. Dan. Vid. Selsk, 2 6 (No. 14), 16. G

is

resentation not contain with spin

S0(3), ~C(I)

D 2.

For any

DI 0

n

is the five-dimensional the representation

in its reduction

of

Hilbert-space S0(3)

on

into direct sum of irreps.

ground state has no spin

1

of the rep-

~(n) = ~ ~C(1) "~[n] (Physically,

state corresponding

does

a nucleus

to collective

excitations.) Of course we also can add that if an irrep of once, then it acts either on 1¢(n) "'[n] or on ~C(n) [in ]

1.5.

More Algebras

and More Tensor Operators•

An algebra on the vector space algebra is symmetrical, Hom(E V E, E), (resp. , metrical, or

(respectively, Hom(E A E,E)).

antisymmetrical

E

G

appears

on

Pseudo Roots of

is an element of

antisymmetrical)

X (n)

SU(n)

Hom(E @ E, E).

The

if it is an element of

Similarly we can define a co-algebra,

co-algebra as an element of

only

sym-

Horn(E, E • E), Hom(E,E V E)

Hom(E, E A E). If

E

is the space of a linear representation

Hom(E 8 E,E) G, resp., of automorphisms When

contains G

for the space

are algebras, resp.,

on

E

of the adjoint representation,

for all simple compact Lie groups and the corresponding the Lie algebra itself.

co-algebras,

compact Lie group, a necessary

is that the representation G

G, elements

of whose group

G.

is a semi-simple

dim Hom(E @ E, E) G > 0 example,

Hom(E,E 8 E) G

of

condition for

has a null weight.

For

dim Hom(G A G,G) G = I antisymmetrical

algebra is

55

In

V

we shall see two examples

on a real irrep space

E

of

the adjoint representation or

i.

G = SU(3)

ture of elementary

algebras uniquely

algebra of particles,

SU(n),

n > 2.

Let us give some properties

since it has been used in the physics

after its introduction

For

dim Hom(G V G,G) G = 0

of a simple compact Lie algebra SU(n),

defined

dim Hom(E V E,E) G = i.

x SU(3), with

It has the latter value for the

of this symmetrical

of symmetrical

by Gell-Mann.

litera-

What follows

is

extracted

from a preprint written in collaboration with L. A. Radicati. 2 G be the n - 1 real vector space of the n × n traceless hermin tian matrices x. The action of u 6 SU(n) on G (vector space of the Lie n algebra) is x u~+ uxu-i = uxu*. The euclidean scalar product Let

1 (x,y) = ~ is invariant

(= i/n

the Cartan-Killing

trace xy bilinear

(1.18)

form).

The

SU(n)

Lie algebra law

is i i x A y = - ~(xy - yx) ~ - ~[x,y] and the symmetrical

algebra law ist

I x V y = ~{x,y} Note that for

n = 2

fijk~ dijk

2 - ~(x,y)~

it is trivial:

In the physics normal basis

(1.19)

literature

where

(1.19')

x V y = 0. (mainly for 2 - i)

(ei,e j) = 6ij(i, j = l,...,n for the structure

{x,y} = xy + yx

n = 3) one introduces and uses traditionally

an orthothe notation

constants

e i A e . 3 = Ekfij kek' e i V e j = Ekdijkek Let us use

F(a),

D(a)

for the linear mappings

of

G

(1.2o)

F(a)x = a A x, D(a)x = a V x (the matrices

are

F(ej)ik = fijk' D(ej)ik = dijk)"

With the scalar product symmetric.

D

and

F

of dimension

D(a) = 0.

of an element

m n - i.

the Caftan subalgebra

F(a)

As is well known,

x, i.e.,

the set

When its dimension C

x

of

x.

is antisymmetric

6 Hom(G,L(G)) SU(n)

are tensor-operators

at the end of 1.3, trace the centralizer

(1.18),

is

D(a)

is

so from the theorem

in the Lie algebra

{y,y A x = O} n - 1

and

SU(n),

is a Lie subalgebra

it is abelian and it is called

(All Cartan subalgebras

are transformed

into each

other by the group.) x, x V x,

C is spanned by the n - 1 linearly independent vectors x (x V x) V x = x V (x V x), ((x V x) V x) V x, etc., up to n - 1 factors

and C is also a subalgebra for the law " V ". The roots of SU(n) are solutions x of the equation r n - (r,r)r n-2 = 0. We shall normaliz e them by (r,r) = i. In a

2 * This is not a Jordan algebra. However, one could have started from the n The cordimensional representation realized by the n x n hermitian matrices. responding symmetrical algebra is a Jordan algebra.

56

Cartan algebra

C, there are

is also one), for every eigen space

C

n(n - i)

a 6 C, the spectrum of

and on the orthogonal Spectrum

Define

(for

normalized roots

space

F(a)

rk, (if

has

n - i

r

is a root,

-r

zeros for the

the spectrum is the set

C~

F(a) l = {i(a,rk)} Cm

(1.21)

n > 2) n-

2

7--~

qk = rk V r k = (-rk) V (-rk)

(1.22)

then (i. 22' )

(qk,qk) = i and they are idempotents

of the V-algebra

n - 4 - qk # n ( n - 2)

qk V qk

We will call them "pseudo roots" every

(they are weights

of

(1.23)

SU(n)) for they satisfy for

a 6 C Spectrum

D(a) l ~ C

=

{n -

~tqk

2,

a"

, ) = (a,r k V rk)}

(all the eigen values have at least multiplicity Let us denote by

(1.24)

2).

I 6 Hom(G A G,G) SU(n)" , v 6 Hom(G V

G,G) SU(n)"" the vector

space homomorphisms %(x @ y) = x A y, ~(x @ y) = x V y and consider

their right inverse o ~' = Identity on and

Note that %'

%'

G, v o ~' = Identity in

G

(1.25)

can be defined for any semi-simple Lie algebra.

As we said

~' define co-algebras on G. If T is a G-tensor operator, using the A T of Diagram 2, one can define the G-tensor operators

and

mapping

A T A T = T o %a

and

A T V T = T o i

(1.26)

and by recursion (...(T

T) ~i

where

"

7. 11

is either

|,

A

11

or

" V ".

T)...) 72

T

(1.26')

~k For physicists who need to see coordi-

i

nates, in

SU(3)

octet space

i g e k, , (e i) = Zjk ~3 dijkej 8 ek l' (ei) = Zj,k - ~ fijkej

% The

fijk

and

dij k

are the structure constants

.%

introduced by Gell-Mann.

57

If we set

T(e i) = Ti, then 1

(T A T)i = Ej, k - ~ fijkTjTk , Note of course that we can define operator when

G

T A T, T V T

dean scalar product. (Ker %)~

G

on

Indeed,

and

E

%

E or

i

=

3

Ejk 5 dijkTjTk

for any real irreducible E-tensor

dim Hom(ETE,E) G = 1

is a compact group when

since there is the irrep of

between

(T V T)

(~

is

A

or

V)

is orthogonal and leaves invariant a eucliv

are then surjective and are isomorphisms

(reap., (Ker ~)~

and

E) so we can define their right

inverse. Let us consider the more particular case when the G-morphism

T

is

F

itself (see Equation (1.6)), i.e., the representation (up to the factor i) of the Lie algebra on F V F

3£.

Then

F A F = iF.

When

SU(3)

is used for elementary particles,

is often called the D-coupling operator (see 5.1b).

For

SU(2), in order to

follow the tradition started in elementary school, we denote by

×

the Lie algebra

law (= vector product) [F(a),F(b)] = iF(a × ~) and by

eijk

(1.27)

the structure constants e i × ej = Ekeijke k

So if

A

is a vector operator (with

(1.28)

A(ei) = A i

+ ÷ i E s..,[A. (A × A) i = jkE£.I3~.,A.A,3 K = ~ jk 13K 3'Ak]

Remark.

Given two G-tensor operators AV

and in particular

A x B.

1.6.

For is trivial,

B =A

SU(2)

O B

o

t,

AA

A

and

B =A

@ B

B, we can also define

o ~

This reduces to Equation (1.26) when

More on

SU(2)

the symmetrical algebra

V

on the adjoint representation

G

dim(G 0 G,G) SU[2)" = dim(G A G,G) SU(2)" = i

Ejl ~ Ej2, Ej3,

dim Hom (Ej 1 8 E32'Ej3 )G = A(Jl'J2'J3) = 0 A(Jl,J2,J3) = 1

otherwise.

A = B.

and its Tensor Operators

Much more generally, given any three irrep on

where

(1.29)

if

lJl - J21 ~ J3 ~; Jl + J2

or

1

(1.30)

(triangular relation), 0

This property, under an equivalent formulation, is called the Wisner-

Eekart theorem by physicists, and groups with the property (1.30) have been called simply reducible by Wigner.

58

Let us give here two references that we shall quote often in this section.

A.

Quantum Theory of Angular Momentum - a collection of reprints and original papers edited by L. C. Biedenharn and H. Van Dam, Academic Press, New York (1965).

B. Spectroscopic and Group Theoretical Methods in Physics, Racah Memorial Volume, North Holland, Amsterdam (1968). In

B

p. 131-136, Wigner proves the following theorem for finite groups.

Theorem

Let

G

be a finite group and

H

a subgroup.

The following conditions

are equivalent a)

The restriction to

H

of any irrep of

decomposed into irrep of b)

H;

G

is multiplicity free when

/

The ring of conjugation classes by

H

of elements of

G

is abelian.

Let us explain a) and b) in more detail. a)

Given an irrep of

G

on

is generally reducible.

~, its restriction as a representation of To say that it is multiplicity free means

that in its decomposition into irreps of more than once.

H

H

no such irrep appears

Equivalently, one can say that the commutant of the

representation of

H

(i.e., the set of all bounded elements of

which commute with every operator of the representation of set is an algebra) is an abelian algebra.

[~)

N; this

That last condition can be

used as definition of multiplicity free for any linear representation of any group. b)

Given

a 6 G, the conjugation class of

A = {hah -I, Vh 6 H}. set

A • B = B • A.

compact groups. this theorem:

by

H

is the set

Given two such classes we define as

{ab, a E A, b 6 B}.

classes,

a

A • B

the

Condition b) states that for any pair of It seems easy to extend Wigner's proof to

Examples of pairs of group and subgroup which satisfy S(n)

and

S(n-

i), U(n)

and

U(n-

From the group law one verifies that the direct product

l).t SU(2) × SU(2)

and its diagonal subgroup satisfies b); by Wigner's theorem this implies (1.30). would be interesting to extend, if possible, Wigner's proof to type

I

It

locally

compact groups.tt

t

After the lecture, Professor G. Mackey gave a proof for compact groups, using his theory of induced representations.

tt Wigner proved in 1941 (paper reproduced in reference A., see 1.6) for finite groups another property equivalent to a) and b). Let ~(g) the number of square roots of g in the finite group G, and v(g) the number of elements of G which commute with g. In a finite group ~ £ ( g ) 2 _ ~(g)3 ~ 0. The equality g occurs if and only if G is simply reducible.

59 Another property of SU(2)

SU(2)

that we have mentioned is that any irrep of

is equivalent to its contragredient.

fines an isomorphism

C:

E. ~ E~ 3 3

For any irrep

between the

with the canonical property

CT

of

C.

space of

of D. J

SU(2)

this de-

and its dual

~ C T = (-l)2Jc E3;

Ej for the transposed

E. 3

D. 3

E~ 3 (1.31)

Physicists normalize

C

by

cTc = 1

(1.32)

We are by now ready to give you a glimpse of the calculus developed independently by Wigner and Racah for the necessity of physics in order to exploit completely rotational invariance in atomic physics (and later on in nuclear physics and many other branches of quantum physics).

Many of the numbers observed in atomic

spectra (sp~cing between neighbors in a family of lines, relative intensity of these lines, etc.> turn out to be algebraic functions of the coefficients defined by I

Wigner and ~acah. Since they are so useful, the literature on Wigner and Racah co! efficients is abundant. They have been found to possess unexpected symmetries, there are u~proven conjectures on them.

However, the language of this physics folk-

i

lore seems to be unknown to the mathematically minded ethnographer. I hope there will be much discussion on this subject in this Rencontres. The rest of this section should help to start it. To write Wigner's "three

j" coefficients,

physicists choose a base in

each Hilbert space

E.s carrier of the irrep Dj, for every j. It is formed by J the eigen vectors of a chosen U(1) (Cartan subgroup of SU(2)) ordered in terms of decreasing eigen value

~

(going from

j

to

-j

by integer steps).

It is obvious

that most of their properties are base independent. Consider an element of the one-dimensional vector space (E~31 8 E~32 8 E.j3)G = Hom(Ej I @ E.32,Ej3)G

(1.33)

and denote it o

o

(jlj 2J3) The isomorphism

C

(1.34

and its inverse, defined in (1.31), (1.32), transform

the tensor (1.20) into its following images

and so on.

(313233) ~ (Eli ~ E I. @ E I. )G = Hom(Ej I 8 E. @ E. ,~)G i J2 33 32 33

(1.35)

(jlj?jq) E (Ej 8 E. 8 E. )G = Hom(~,Ej ~9 E. ~9 E. )G ...... 1 32 33 i 32 33

(1.36)

o @ E. @ E. )G = Hom(EjI,Ej 2 @ E. )O (jlJ2J3) E (Eli 32 33 33

(1.37)

60 o

Equations (1.35), (1.36) show that

o

o

(jlJ2J3) (resp.,

(jolJ2J3)) belong to

a one-dimensional representation of the permutation group of the three factor spaces labeled by

31' J2' J3"

Computation shows that the representation is

,,,, (symmetrical)

if

Jl + J2 + J3

is even

~ (antisymmetrical)

if

Jl + J2 + J3

is odd

(1.38)

The composition of the two homomorphisms o

E.

0 E.

31

32

o

o

(JlJ2J 3)

0 E.

(JlJgJq)

~. ~

....

o~

@ E. 0 E. 32 33

> E.

33

31

(1.39) o

is an element of

Hom(E.31 O E.32 0 Ej3 , E.31 0 E.32 0 E.33)G that we denote

o

o

(jlJ2J3)

q1 2 3) Wigner proved (see reference A and Equation (24.18b) of his book quoted in the Introduction) f

where

dp(g)

o

o

o

D. (g) 0 Dj2(g) 0 Dj (g)d~(g) = (jljvjq)(jljgj~) SU(2) 31 3 . . . . . . . . .

is the invariant

SU(2)

measure of mass

(1.40)

[ dp(g) = i. J SU(2)

This also defines for ~ou, up to a sign, which element of the onedimensional vector space o

o

(E1 8 E 2 0 E3)G

has been chosen by physicists for

o

(31j 2j 3 )" Of course tensors can have their indices contracted (notation stance

oo c

XOO

(abx)(cpq)

is the composed homomorphism ooc

(abo)

E

x); for in-

0 I

0 Eb o E 0 E a p q

ooo

0 I

P

q~ E ~ E

c

P

oE

(cpq)~ q

Wigner's notation is v@ry handy! Note that equation (1.40) yields I

where

Xj

x x x jlJ2J3 SU(2)XjI(g)xj2(g)xj3(g)dD(g ) = (jlJ2J3)(x x x ) = A(jl,J2,J3)

is the character of D.. 3 Approximate expression, asymptotic expression, for large

j's

(1.41)

exist for

the components of those tensors in the basis described above (see ~he thorough bibliography of reference A (see 1.6)). found a 72 elements group of symmetry

Regge (paper reproduced in A (see 1.6)) has ~ Aut(S(3) x S(3))

for the set of components

61

~i~2~3 (jlJ2J3)

o

of

o

o

(jlJ2J3).T

In 1941 Racah and Wigner (both papers reproduced in A (see 1.6)) introduced a "six j" symbol (numerical function of six irrep of recoupling coefficient.

Consider the sequence of

Ee

SU(2)-homomorphisms

(ofa) If 8 ~aoo) coo ~ Ef 8 Ea ~ Ef ~ Eb 8 E c

Ed @ Ec Since

SU(2)), also known as

It is canonical.

c

(o~e~ a=o)~ E e

(1.42)

E

is the carrier of an irrep, this SU(2)-homomorphism must be a e multiple of the identity operator on E . e Its trace defines (up to a sign that I do not guarantee here) the six j's symbol l abc I

- -b+c-d+e+f x xx xx = (-i) (efxa) (~bc) (fbd) (ace)

(1.43)

i defJ Wigner has shown that for given dices

c, f.

def,

labc I" is an orthogonal matrix with ina, b, d , e, 'def

He also proved the relation (in his book, Chapter 24)

= ;If Xa(r)Xb(s)Xc(t)Xd(st-l)Xe(tr-l)Xf(rs-l)d~(r)d~(s)d~(t)

Asymptotically its value is a rapidly oscillating function of some variables, but an average over some range of one argument yields, when

a, b, c, d, e, f

form the

edges of a tetrahadron, the asymptotic value

abcI2 + (24~V) -I def2 where

V

is the volume of the tetrahedron. G. Ponzano and T. Regge (in reference B (see 1.6), first paper) have con-

jectured precise asymptotic formulae for

{abcl whether or not the valu~ of the def arguments can be the length of the edges of a tetrahedron. Furthermore, Regge has found (paper reproduced in A (see 1.6)) the largest

linear group acting on the Z-module generated by the symbols a/2, b/2, c/2, d/2, ~abc e/2, f/2 and having idef I as invariant. It is the group S(3) × S(4) which includes the permutation group of the columns. T Most of these symmetries appear naturally (see Bargmann's paper in A (see 1.6)); for the others see G. Flamand, Ann. Inst. H. P o i n c ~ , ~, 353 (1967).

62

Neatest and very symmetrical expressions for the

(jlJ2J3)

and

{

JlJ2J.3~ symbols can be found in the paper of V. Bargmann (last paper reproduced J4J536~ in book A (see 1.6)) who uses Hilbert spaces of analytic functions as spaces of SU-2

irreps.

2.

ATOMIC AND MOLECULAR PHYSICS

2.1.

Group Theory and Atomic Physics

The application of group theory to atomic physics is essentially of this ÷

type; only the even part ÷

part

÷

÷

-~

f+(r) = i/2(f(r) + f(-r) ÷

÷

÷

-->

f+(rl,r2) -- i/2(f(rl,r2) + f(r2,rl))

to the integral over the whole space

(respectively, the symmetric

÷

of

If(~)d3~r

f(~)

(resp.,

(resp.,

÷

f(rl,r2)) contributes

Ifkrl,r2)a "÷ ÷'13+ rla"3÷" r2).

This

is the explanation of two empirically known facts (before 1926), the Laporte selection rule for atomic spectra and the partition of the helium spectrum into two independent subsets (attributed to ortho and parahelium).

Of course, these examples

are the simplest because they are based on invariance under a two element group (Z2).

We will have to use invariance under

invariance under subgroups of

2.2.

SO(3)

SO(3), S(n)

and

U(2)

for atoms and

for molecules.

The Correspondence Principle

We had a general description of quantum mechanics, but now we have to, know how to study a given physical system.

There does not seem to exist an axiomatic

formulation of the question, so here again, physics is still an art!

However, when

the system under consideration, with a finite number of degrees of freedom can be described by classical Hamiltonian mechanics, the "correspondence principle" tells physicists how to treat it quantum-wise. Let

h(Pk,qe)

be the classical Hemiltonian and dPk

~

dt ..... -k the Hamiltonian equations.

~h ~ = - ~qk'

~h = ~P£

The corresponding observables

pz, Q£

in quantum

mechanics form the abstract algebra with unit PkQz - QzPk = [Pk'Q£ ] = - i ~ 6 k ~ [Pk,Pz] = 0 = [Qk,Qz] where

2~h

is Planck's constant.

(2.1)

63

In the cases we shall study, function of the where

Ht

and

p's H ~t

and

htt

h

is a sum

h = h t + h It

is a function of the

q's.

Then

are the same functions respectively of the

where

ht

is a

H = H t + Hte P's

and the

Q's.

There is yet no synthetic formulation of quantum mechanics as there is for classical mechanics by symplectic manifolds Souriau).

(see, however, work in progress by Kostant,

We also know that the relations between the classical and quantum treat-

ment of the same problem are not simple (see e.g., Van Hove's work in 1951 comparing the two automorphism groups). The Hamiltonian operator is the generator of the group of time translations [H,Qk] = i~Qk , [H,P~] = i ~

.

(2.2)

A representation of the algebra defined by (2.1) and (2.2) was obtained, independently of Heisenberg's work by Schrodinger, using the concept of de Broglie's waves. Indeed, the algebra (2.1) is realized by self-adjoint operators of is the Hilbert space of square integrable functions

~(qi ).

Qk P = qk ~, P£P = ~i ~q% ~ P The

~

are also functions of the time

(t)

where

Then

.

(2.3)

and the SchrDdinger equation is

H = i~ 9-~

.

(2.4)

This representation raises some analysis problems. von Neumann's theorem (J. von Neumann:

LOC)

On the other hand,

"Die Eindeutigkeit der Schr~dingerschen

Operatoren", Math. An~le~,

104, 5]0 (1937)) tells us that all irreducible represeniP k iQ~ tations of the algebra defined by Equation (2.1) are equivalent when e , e are realized by unitary operators.t Quantum mechanics was also discovered by Dirac who gave the neatest for-

mulation of the "correspondence principle".##

In classical Hamiltonian mechanics

one has also a Lie algebra, that of the Poisson brackets functions of the

p's

and the

(P.B).

Let

f, g

be two

q's,

P.B.(f,g) = E% ~f ~g ~q~ ~q~

~f ~$ ~q% ~P~

(2.5)

For systems with an infinite number of degree of freedom, as they appear in statistical mechanics and field theory, this is far from true. Infinities of irreducible representations of (2.2) have been given first by Friedrichs, Van Hove, Ggrding and Wightman, Segal and several other physicists and mathematicians. An excellent thin book on the subject is by A. Guichardet, Alg~bres d'Observables Associ~es aux Relations de Commutation, Armand Colin, Paris, (1969). (See also G. Mackey, Duke Math. J., 16, 313 (1949)). ## Historically, meaning.

the expression "correspondence principle" had a more restricted

•6 4

The Lie algebra of the corresponding quantum observable is [F,G] = i~ As you surely know

quantum observable of l~]2~dqk , where

~

P.B.(f,g)

(2.5 t)

is a solution of Schr~dinger

Equation (2.4), is the density of ~robability to find the system at the coordinate {qk }.

This of course appeals very much to physicists.

like just as well to work with the abstract algebra.

As mathematicians you will As a short, but fundamental

illustration of the use of that algebra, let us prove the Heisenberg uncertainty relations. Let x>

A, B

be the self adjoint operators corresponding to the observables

a, b.

If

<xAx>

is the expectation value of

is a given state of the physical system we study, we have seen that "a"

for

x>

and the mean square dispersion of

probability is given by (Aa) x = l<x,(A - <x,Ax>)2x>l I/2 = I<x,12x>l I/2 = II~II

(2.6)

= A - I<xAx>

(2.6 t)

where

By Schwarz' inequality (Aa)x(Ab) x = If

A

and

B

l li~ll

i< l l'~xll ~ ll ~ 71

•

x'[A'B]x>l

satisfy the same canonical relations as the

P's

and

(2.7) Q's

we do

obtain i (Aa)x(Ah)x ~ ~ ~

2.3

Particle of Mass

Let

V(r)

m

(2.8)

in a Spherically Symmetric Potential

be a spherical symmetric potential, where

r

denotes

I~I.

The Hamiltonian of the particle is 1 ~2 H = ~m + V(r)

which is invariant under the orthogonal group H, ~2, V(r)

are "scalar operators";

~, R

0(3).

and

(2.9)

,

Using the vocabulary of 1.3,

R x ~ = ~ ÷

vector operators.

(So we put an arrow on .them!)

the three-dimensional vector space

E3

÷ ÷ ,Q(D)] = i~

where the Cartan-Killing form

~

a, b,

etc., are vectors of

of the adjoint representation of

should write the canonical commutation relations E~(~)

If,

are (polar and axial) ÷

0 (3), w e

(2.1)

÷ ÷ ÷ ÷ ~1 B(a,b) = i~a,bX

,

has been defined in (i.ii).#

# See also the Appendix on commutation relations at the end of 2.

(2.10)

65

From (2.10) and the definition by the correspondence angular momentum operator

(see end of 1.5), ÷

÷

÷

principle of the

~ = R x ~, we obtain

÷

[L(a),L(b)]

= i~(a

A b)

,

(2.11)

which confirms

that the vector-operator

representing

representation

(up to

Lie algebra on the Hilbert space of our prob-

i) of the

0(3)

lem. Some physicists write prised if in all physics ÷

Pj, L k

are used for

~ • ~, ~ • ~

~(

~) , ÷e(n). ÷

is the

But do not be sur-

text books an orthonormal basis of vectors notations ÷

÷

÷

÷

corresponding

motion generate the algebra

Qi'

÷

Q(ei), P(ej), L(ek),

The operators

for

the angular momentum

etc.

to the observables which are constants

{H} ~ , the commutant of

one deduces from (2.10) and the definition

H.

Hence,

of

the equation that

of

÷

÷

E E 3, [L(a),H] = 0 means both that the Hamiltonian

or symbolically

[L,H] = 0

is invariant under rotations

(2.12)

and that the angular

momentum is a constant of motion. ~2

The Casimir operator (with the physicists' normalization) of 0(3) is 3 2 = Ei=IL i. As is well known, its values for irreducible representations of SU(2)

are

j(j + i)~ 2

representation.

where

2j

is an integer

Only integer values of

state vector is an eigenvector that the corresponding

of

~2

Consider

two particles

positive integer).

where

r =

is

of mass

with

r =

tot.

is the dimension of the S0(3)

irrep,

when the

j(j + I)~ 2, we say shortly

jh.

The Hydrogen Atom

ml, m 2

The total Hamiltonian

h

2j + i

appear in the

with eigenvalue

angular momentum

2.4.

~ 0; and j

+2 Pl -2m I

electric charge

Ze, -e

for this system of two particles +2 P2

(Z

is a

is

Ze 2

+ - -

,

2m 2

(2.13)

r

- r I.

Introduce the center of mass ÷r 0 = (mlr÷ I + m2r÷ 2)(m I + m2 ) -i ÷

and

r

÷

as new variables

conjugate variables.

instead of

rI

and

÷

÷

r2; let

P0

(2.14) ÷

and

p

the

Then +2 PO

htot.

2(m I + m2)

+

*2 ~--Ze2" h +h (2m r ) = cm

(2.15)

where m = mlm2(m I + m2 )-I

(2.15 I)

66

The motion of the center of mass is described by

while h corresponds to the cm So quantum-wise, we have to study the spectrum of

internal energy of the system.

~2 2m

H

Ze 2 R

h

'

(2.16)

for obtaining the energy of the hydrogen atom levels.

The first quantum study of

the hydrogen atom was made by Pauli, Z. Phys. , 36, 336 (1926) before SchrSdinger's ÷

equation was published.

Pauli did study the abstract algebra generated by

and Equations (2.1), (2.2), and (2.15). stant of motion.

The angular momentum

~ = ~ x ~

÷

R, P, H is a con-

Another constant of motion is the Runge-Lenz vector I = y(Ll + x ~ - ~ x ~) + ~% ~

with

X = mZe 2

(2.17)

Note that ~(LI ÷ x ~ . P ×. I) . (R . ~)~.

~(~2) . . i ~.

P(P

R)

(~2)R + i ~

(2.18)

so we can check that [~,H] = 0, [~,H] = 0 We recall that dices are equal.

(2.19)

.123) sij k = sign of the permutation ~ijk

or

0

if two in-

From now on we will use the Einstein summation convention, i.e.,

summation of repeated indices is implied, and we find (2.20)

[Li,Lj] = i~gijkLk, [Li,Aj] = i~E...~13K~

(2.21)

[Ai,Ak] = -i~2mHSijkL k

~.;=;.I=o

(2.22)

~2 _ 2mH(~2 + ~2) = (Ze2m)21

(2.23)

Let us just consider the bound states of the hydrogen atom. spectrum of

H < 0.

Let

P_

be the projector on the bound states. ÷

write

X- -- XP .

From (2.19) when

X

P XP Furthermore,

-2mH-

They correspond to the

= XP

is

~(~)

or

For any

X

÷

A(b),

= P X = X-

is an inversible positive operator.

positive square root of its inverse and define

Let

(-2mH-) -I/2

K~ = A-(2mH-)-I/21"

be the

Then Equations

(2.20 i) to (2.23) read ILZ i i IL: i i [~ ,~ Lj] = ~ ~ijkLk , [~ ,~ Kj] = % S i j k


J

I

-~

÷

= = se

(2.47 ~)

As we announced in 2.1, this shows that the helium levels are to be divided in two sets according to their symmetry characters, and eledtric dipole transitions occur only within each set.

Let me remind you that helium got its name because it was

observed in the sun before being observed on earth. posed of two spectra, one for orthohelium

Its spectrum appears to be com-

(e = +i), one for parahelium

This was a complete mystery before quantum mechanics.

(e = -i).

The explanation was given by

Heisenberg in 1926, "(Uber die ~pektra von Atomsystem mit zwei Elektronen", 39, 499 (1926)). the type

x @ x, as for instance the lowest level (n = I, ~ = 0

It is observed that corresponding rays (e.g., transitions V (n e,ge)

and

Z. Phys~k,

It also explained that the orthohelium has more levels; those of

(i,0) A (n,~) ÷ (i,0) A (n a,gl)

with

for each electron).

(i,0) V (n,~) ÷ (i,0)

n I ~ i,£ S ~ 0) of parahelium

are about three times more intense than those of orthohelium.

To explain it, the

electron spin will have to be taken into account (see also 2.9).

73 2.6.

Pauli Principle.

The Electron Spin

We want to pass now to the case of n-electron atoms.

The internal energy

Hamiltonian is t

~n)

Of course trons.

%

2 e E R.. l~i<j~n 13

n = Ei=IH i +

with

H.

P~l

Ze 2

2m

R.

l

is invariant under the permutation g r o u p

H (n)

S(n)

It is also the case for the electric dipole operator

observables.

(2.48)

l of the

e(EiRi)

n

elec-

and for all

Identical particles cannot be distinguished from each other and every

prediction of the theory must be invariant under When we consider states of

Z

(or

S(n).

n) distinsuishable particles, we con-

sidered (with success for the helium atom) the Hilbert space tensor product of the $C

for each particle. i

S(n)

acts on

Consider again

~C(n) = ~ $C[I)" "

for

~C~n)

by the representation

s ÷ S(s).

of operators.

which represents a state.

{S(s)}'

S(n)

Vs, S(S)PxS(S) = Px

This requires that the vector

x>

(completely symmetrical)

or to

symmetrical).

The othe r spaces

of the other factorial representations

(completely antiof

are excluded as space of physical states. We have used both

use of

$C[ ]~

~C[in] = ~ C (l)

for any vector

belonss either to

~[n] = ~$C(I)

S(n)

of

of the set

As we saw the rank one projectors wich represent

physical state are observables of the theory so x

identical particles.

Invariance under

all observables requires that they are in the commutant {S(s),s 6 S(n)}

n

%

X[n](n > 2)

$C[2 ]

and

~C[12]

for the helium atom.

for atoms does not represent nature.

However, the

Indeed, the ground state

of any atom would have all electrons with the same binding energy (of the order of (Z~)2/2).

Experimentally,

trum for Z-large enough).

only two electrons have this binding energy (X-ray specThe necessary energy (called ionization energy) for re-

moving a first, a second, a

kth ..., the crease irregularly from a fraction of e 2

vectors of some other

~(n)

do appearX

Zth to

electrons of any neutral atom inZ(e) 2/2 . Moreover, as we shall see,

The solution to this puzzle is that

~C[I)" "

is not the Hilbert space of the bound states of one electron in a constant potential. The electron has another degree of freedom~ the spin and the Hilbert space of its states has to be changed into a new X(1) = /(i) @ K( I )

where

i (1)

is the

L2(R3,t)

dimensional Hilbert space.

previously called

3C( I )

and

K(1)

is a two-

Pauli was the first in 1924 to introduce the spin as an

intrinsic angular momentum and magnetic moment for the nuclei, but it was Goudsmit and Uhlenbeck who introduced in 1925 the spin as an intrinsic angular momentum for the electron.

h/2

This explained the number of energy levels which appear in the

Zeeman effect, but it did not explain the magnitude of their splitting. magnetic moment produced by an electric charge

e

Indeed, the

moving with an angular momentum

74

j

(Ampere's law!) is ~ /2-~c

where

e~/2mc

is the Bohr magneton.

g = i, but for the spin

(2.49)

For the orbital momentum, g(l~/~l

lJ/hl = 1/2, g

solved by Thomas in 1925.

gJ/

appeared to be

2.

integral)

This was a mystery

It is a relativistic effect.

It is an experimental fact that we have to use Fermi Statistics for electrons, i.e., the Hilbert space of electronic states of an n-electron atom is

• [in] (= ~ ( i ) n C where

~(i)

(2.50)

is the (new) one electron Hilbert space defined in (2.48).

Since length

)

. ,,(n) @ [(n) wh£L[ ]h _[ ]~.

dim K (I) = 2, the Young diagram of

h I ~ h 2 a 0.

Of course

c [ ]h

has only two lines of

h I + h 2 = n; we will show that

hI - h2

is the

chemical valence. The diagram of It has two columns and

hI - h2

[ ]h

in

h I ~ h 2 ~ 0.

lines of length

l(n) -[ ]h

is the one symmetric through the diagonal.

In other words, it has

i.

h2

lines of length

2

That means that it cannot be completely symmetri-

cal in more than two electrons, i.e., there can be only two electrons at most in each orbital state; then two electrons must have "different spin states", or more exactly, their spin-state has to be antisymmetrical. discovered by Pauli

(Z. Physik,

FIGURE 2.2.

YOUNG DIAGRAM OF THE ORBITAL PART OF n-ELECTRON STATE:

2.7.

This is the Pauli principle,

31, 765 (1925)).

l(n) ~[ ]~

HERE

n = 21.

Atomic Shell Structure - Periodic Table

We can now clearly describe the lowest state of an n-electron atom. orbital part is a vector of

~(n)

The

obtained by filling all the lowest energy states,

putting only two electrons in each electronic orbital state.

Of course, in atoms

75

with

n > i

appear.

electrons,#

the two specific properties of the

The number of bounded states is finite, and the

exists, i.e., states with the same

n

and different

%

I/R

S0(4)

potential disdegeneracy no longer

have different energy.

The

observed order of increasing energy for the states is given in Table 2.1.

TABLE 2.1.

THE ELECTRON STATES ARE LISTED IN ORDER OF INCREASING ENERGY

n

i

2

2

3

3

4

3

4

5

4

5

6

5

4

6

7

...

%

0

0

i

0

i

0

2

i

0

2

i

0

2

3

i

0

...

Spectroscopist Notation

is

2s

2p

3s

3p

4s

3d

4p

5s

4d

5p

6s

5d

4f

6p

7s ...

2(2% + i)

2

2

6

2

6

2

I0

6

2

i0

6

2

i0

14

6

2

Total

2

4

i0

12

18

20

30

36

38

48

54

56

66

80

86

88 ...

Note that for a given

n, the energy increases with

predicted by computation and can be qualitatively understood. charge tial

Ze

and

k

electrons.

%.

...

This is of course

Consider a nucleus of

At infinity an electron will feel the coulomb poten-

(Z - k)e/r, but when it gets near, the probability of finding some of the

electrons at the distance

r ~ n(Z~) -I ~(mc) -I

is not negligible,

electron feels a potential above the Coulomb potential. ence if it is in a than in a

~ = 0

(k + i) th

It feels less the differ-

state (more concentrated inside the sphere of radius

r)

% = i state, ~ ~ 2 state, etc., where the concentration of probability is

more and more on the surface of the sphere. (% = i)

the

k

Note also from Table 2.1 that "p shells"

are filled by the 10th, 18th, 36th, 54th, 86th electron and this is just the

atomic number of the "rare gas" elements, very inactive chemically, Neon, Argon, Krypton, Xenon, and Radon.

We can even give the Mendeleev's periodic table, in

terms of electron states, if we know the order of increasing binding energy of the states

(n,%).

Using the spectroscopic notation %=0

1 23

4 5 6

s p d f g h i This order is, from Table 2.1, (n,%): 6s, 5d ~ 4f, etc.

The sign

~

(++) sign indicates

i

is, 2s, 2p, 3s, 3p, 4s ~ 3d, 4p, 5s ~ 4d, 5p,

indicates roughly the same energy so that the two

shells are filled simultaneously. +

(2.51)

Indeed, for the element of

% = 2

(d-shell) the

(or 2) more electrons in the d-shell, taken from the

# Because of the electrostatic repulsion among the electrons.

s

76

shell.

We can now construct a periodic Table.

TABLE 2.2.

%=if2(2%+i) = 6

~=0 2 electron s state I 2

n=l

~

n=2

Li

n=4 n=5 n=6

PERIODIC TABLE (UP TO ELEMENT 56)

~=3~2(2~+I)=14

%=212(2%+i) = io

P

.#

1

2

3

4

5

6

i!i C6 N7 Si

O8

F9

Ne I0

1

lI~ Ca 20

2

3

4

5

6

7

8

q-

9

i0

+

Sc Ti V Cr Mn Fe Co Ni Cu Zn 21 22 23 24 25 26 27 28 29 ~30

Ge As Se Br Kr 32 33 34 35 36

14 rare earths 58 to 71 I

~9 Zr ~a 40 %" 41 ~42 ~c 43 ~u 44 ~h 45 ~46 ~ 48

Rb Sr In Sn Sb Te I Xe La~Hf 37 38 49 50 51 52 53 54 57[72 Cs Ba 55 56

In d-shell, + (or ++) means (n + l),s) state).

1

(or

[ ~--|

2) more electrons in

n-d

(coming from the

The atomic state of an atom is labeled by the filled states, e.g., Oxygen: (is)2(2s)2(2p) 4, i.e., 8 electrons.

In general, the electrons fill up all states of

lower energy and fill incompletely the last "shell"; for example, in the case of Oxygen, we can add 2 more electrons in the

2p

state.

The question which arises is

which state is the atom ground state for an incomplete shell?

We can label this

state by a Young diagram (let us do it for the first elements). TABLE 2.3.

Z

=

Name

Y~iag of orbital

YOUNG DIAGRAM OF FUNDAMENTAL STATES

I

2

3

4

5

6

7

H

He

Li

Be

B

C

N

is F7

is~

1(z)

Y-diag of spin sta~e

(z)

[]

B

is 2s

is[] 2s

EF

2s 2p

FFP

2s 2p

2s I 2pl~

FFF

??

Now w e can generalize what we say for helium. energy state

(i.e.,

configuration

is the most antisymmetrical

sional Hilbert

electrons

in the coordinates

for the spin coordinates

in Table 2.3 for the 2p-electrons

is not possible

k

in the same

k ~ 2(2~ + i)), the lowest energy state of this

so it is the most symmetrical is illustrated

Given

to have a completely space,

for

antisymmetric

so w e can give the successive

TABLE

2.4.

nb of electrons k = 1 filled shells

np-SHELL

2

k

electron

(k = 2, k = 3, ...),

(k = 2, k = 3, etc.). f = i, 2, 3.

tensor

on a

When

k = 4, it

2~ + 1 = 3

atomic states

of

p

This

dimen-

shells.

STATES

3

4

5

6

,a...

spin statefs~ll~

valence

I

i (or 3)

2(or 4)

3

2

i

O

n = 2

B

C

N

O

F

Ne

n = 3

AI

Si

P

S

CI

A

n = 4

Ga

Ge

As

Se

Br

Kr

n = 5

In

Sn

Sb

Te

I

Xe

The ionization

energy

increasing with

(energy necessary

k, as w e expect,

tron state, because Although

to extract one electron from the p-shell)

along a given p-shell,

it is the first one not completely the energy of ns-states

excited state of atoms with that is, an ns-electron

k = i

or

2

is lower is

goes to an np-state.

is

except for the four elec-

antisymmetric.

than that of the np-state,

k = i; (ns)(np)2; This increases

a low

k = 2; (ns)(np) 3,

the valence by two and

gives more bounded molecules. There would be a lot more to say, even from group theory, Mendeleev

Table.

For instance did you notice that the ferromagnetic

Co, Fe and also M n in alloys shall stop here.

about the

are together

in an incomplete

3-d shell,

elements

(Ni,

etc.?

But we

78

2.8.

Atomic States in a Given Shell - Spin Orbit Coupling

The Pauli principle, expression of the Fermi statistics, simplifies the study of atoms.

Instead of studying an n-electron system, within a good approxi-

mation we can (for not too highly excited states) consider

k

electrons outside a

closed shell which has angular momentum zero, electric charge distribution depending on the electron wave function).

(n - k)e

(with a

This closed shell can be

considered as a spherical potential and the Hilbert space of states for trons in an

n - £

k

elec-

shell is k ~(k) = A (~2%+i @ K2)

to a good approximation, metric tensor,

x A y A z ...

Example. = (~) of

for

~(k)

an electron state is a (k

(I) = d i m ~

using a complete set of observable

TABLE 2.5.

Spectrocopist notation

THE

@ K 2 = 6; and

dim~C (2) = 15

Each decomposable tensor

That is what the spectroscopist does

~2, ~2, ~2 = (~ + ~)2, Jz' that is the orbital

~, the spin angular momentum

and its projection on axis

order decomposable antisym-

(complete shell again).

can be given a name or a label.

angular momentum

kth

factors).

% = i, k = ~ d i m ~

k = 6, d i m ~ (6) = 1

(2.52)

s, the total angular momentum

J3"

IS(np)2-STATES IN ORDER OF INCREASING ENERGY

3P 0

3P I

3P2

iS

ID2

0

L and space symmetry

i ~

i ~

I ~

2 [-~

0 ~I]

S and spin symmetry

I b J-J

I t~--I

I

O

0

J = total ang. morn. O 2J+l=nb of states I

~-

-2

~-]

1 3

~-~

+

-I

Remark on Table 2.5. states) are below the

j,

2 5

+

i

~

2 5 O

Note that the space-antisymmetric

states as we already emphasized.

+

~

0 i =15 0

F-l H states

(P-

For the symmetric

states, the S-state which feels more the repulsion than the D-states, is above them. Why do the P-states appear in order of increasing effect that we can explain in th~ following way.

J?

This is the small spin-orbit

?9

The orbital state of angular momentum (e~/2mc)~,

L~

while the spin state of angular momentum

g(e~/2mc)~

with

g = 2

(see (2.49)).

produces ~h

a magnetic moment

produces

a magnetic moment

The interaction between the two magnetic

moments has for matrix element# e2c_~_)2 ~ • - 2 "mc < R3 For a state

lj,%,s>

(2.53)

>

the expectation value of

~ • ~

is easy to compute from

~2 = (~ + ~)2 = 12 + 21 • g + ~2 And the expectation value for state

lj,~,S,jz>

(2.54)

(when

Jz

is the eigenvalue of

Jz ) is ->

= ~(j(j i + i) - ~(% + i) - s(s + i))

(2.55)

where I% - s I ~ j ~ % + s~, j + s This explains

the value of



integers

m 0

(2.56)

in Table 2.5.

We have seen that for hydrogen

the

~ (Z~/n) 3, so the expecta-

tion value of the spin orbit term is (Z~)2 Z~2 ~ -2n 2 h

.

This is an order of magnitude. trum

[(is)2(2s)2(2p)6](3s)

largest splitting,

~

n

n

Z~ 2 ~

10 -4

n

In the sodium atom (alcaline = hydrogen (fundamental

like) spec-

state) and the

[ ](3p)

state has the

i.e., nearly 10 -3 , so the very bright

3p - 3s

(yellow) ha-line

is a doublet.

2.9.

In Chapter

Spin and Euclidean or Galilean Invariance

2, Sections

2.7 and 2.8, we have mainly used spin as a new

degree of freedom for the electron. values

This new degree of freedom has two discrete

(often called "up" and "down" in the physics manual)

Hilbert space

K2, of complex valued functions

dimension

The atom Hamiltonian

2.

freedom,

i.e. , it is of the form

Equation

(2.50).

operators

The permutations

of the form

have well defined

I @ S(s)

c [ ]~.

# This is a short for

(2.48) is independent

of the

illustration

applied to

@%(L (n)

[Ix

~ K (n)

of

[]~"

electron spins are represented by

which commute with

The simplest

~ ~ ~

n

set, has

of this spin degree of

acting on the space

H @ I

so the corresponding

defined in a 2-element

~. %(k) +i[

.

]%

H.

So eigenstates

is the helium atom

@

K(k)

[ ]~"

of

H

can

(n = 2).

80

C [ ]X = [2] =

The set of states with =

~

was called orthohelium.

~

Since

dim K (2) = 3

and

dim

K (2) = i, helium states

m (which are tensor products

x A y

or

C = [12] [ ]k

was called parahelium, with

8

x V y

of different

(x # y

hydrogen states)

have the statistical weight (for instance in the Boltzman distribution of thermodynamic equilibrium) of 3 for parahelium and i for orthohelium; this explains that spectral lines of the former are three times more intense than the corresponding lines of the latter. Using the considerations of Chapter i, Section 1.4, on the relations between the unitary groups and the permutation groups, we could also consider the action of the group U(2) on the two-dimensional space K~I)." Its action ~(n) c on .,[ ]~ is through the factorial representation [ ]I' and this action is on ,(n)]~ @ ..[ ~(n)]~ ~(n) C c[ Hamiltonian

of the type

c) = I @ (~Ds) ; it commutes with the I @ (@D[ ]I

H @ I.

These two points of view are formally equivalent for the classification of quantum states, but the

SU(2) c U(2)

Euclidean or Galilean invariance.

Let

has a deeper meaning. G

It is related to the

be either group, and

covering, i.e., there is a surjective homomorphism

G % G.

G

its universal

(As we have seen in

Chapter i, Section 1.2, and will see again in Chapter 4, it is an extension of the relativity group which acts through a linear representation on the Hilbert space of states.

See also O'Raifeartaigh.)

~ SU(2).

product, where action of

There is also a surjective homomorphism

(In the Euclidean case for instance

G

SU(2) on

G = R 3 [] SU(2); D = semi-direct

is the covering of the rotation group.)

~(i); the one particle-state

This gives us the

+ ÷ ~(x,t;o) 6 L2(x,t) @ KO

transfoms

into (U(g)~)(~t,o) = p(~(~)-i • (x,t); ~(~)-i • o) Often physicists prefer to write equivalently integrable function K . o

P

of

x,t

~(I)

(2.57)

as the Hilbert space of square

with value in the two-dimensional Hilbert space

Then (2.56) reads --

÷

(u(g)~)(x,t) --

l

E=I,2DI/2(~0(g))oj~j- (~(~)-i • (x,t)÷)

To s u m m a r i z e , t h e s p i n i s r e l a t e d

to (essentially

(2.58)

the rotation

part of)

Euclidean (and a portion of the larger Galilean) invariance; and it is an intrinsic angular momentum for the electron.

We will study it in 4.5.

The value

g = 2

for

the corresponding electron intrinsic magnetic moment is, however, a relativistic effect (see Figure 2.5). Conservation of angular momentum implies only that + spin angular momentum) be a constant of motion.

In atoms

rately conserved to a good approximation only because Equation (2.48).

H

~ = ~ + S ~

and

~

(orbital are sepa-

is spin-independent (see

81 2.10.

Molecules

The interaction which binds neutral molecule (= coulomb)

atomic nuclei and

(or a charged molecular

interaction.

n

ion) is essentially

Instead of treating directly a

uses the Born-Oppenheimer fixed.

N

electrons

into a

the electrostatic

N + n

body problem,

one

approximation where the (heavy) nuclei are considered

Take for example the Hamiltonian

II for the two protons,

i = i, 2

of the hydrogen molecule

for the two electrons,

i ,+2 ÷2 1 ,÷2 ÷2 H = -~-~tPl + Pll) + ~-~tPl + p2 ) - e2( .

(subscript A = I,

rA, i = I÷r A - ~il , etc.)

~

)

E A=I,II rA,i i=1,2

2 2 e e + ~- + r12

(2.59)

÷

where

d = Ir I - rii I

a parameter

the distance between

in the Born-Oppenheimer

nuclei will be neglected).

When

the two hydrogen nuclei is considered

approximation

d

where

L~ 2) (r,r) ÷

x, y

However,

and the two-dimensional

are hydrogen states.

in this basis, for

diagonal when

d

other nucleus,

so

Hlh When

2Ipl.

- IPl ÷ ~. d

are

state

operator

HIh

s = s + s . x y

is not exactly

d

is

for benzene),

etc.

the two eigen-

When

d ÷ 0, e2/d

e2/d - IPl < 0, and a value of

spins form an antisymmetrical

x @ x, and

state.

Hydro-

(covalent bond) with electrons

in order to form a closed X1 - X2

qualitatively

of

Z. Phys~k, 44, 455 (1927)).

H

"spin" shell, as was

is the "valence"

and quantitatively

H 2 , H2S , H2Se are of the form

(the repulsion of the two

H~C H7

Hence,

and they have an energy difference

for which

type of liaison

XI - X2

(W. Heitler and F. London,

C2H 4 is flat

c ± IPI.

The number of atoms which can be bound to an atomic (spin)

Quantum mechanics has explained

H2S , H20 (= 108°).

are

The ground state is of the type

empirically before 1920, and

why the molecules

(2.60)

i/2(x V y)

is minimum.

form a similar

Xll I I I II X [ I [ I

cp (~¢)

and so does e2/d - IPI"

the two-electron

shell of atoms.

2 discovered

and

IPI ÷ 0

e2/d - IPl

gen (or alcaline)

=

and its eigenvalues

i/2(x A y)

d ÷ ~,

from Fermi statistics

unfilled

Consider first the space h = (x 0 y) ~ (y @ x)

These two states have the same energy

But there is a domain for

for which

~(2),

is finite because each electron feels also the attraction of the

(since it is Hermitian),

of

spaces. subspace

h, the Hamiltonian

Hih states of

(and the kinetic energy of the

is very large, a state of (2.59) is in

the tensor product of two hydrogen atom-Hilbert dependence

as

H --

of the atom.

the covalent bond

It explains, 0~

for instance,

with an angle

> 90 ° H atoms makes the angle increasing from 90 ° for H2Se ,

It explains why NH 3 is a trihedron and CH 4 a tetrahedron, /H

(d

and

~

electrons).

It explains mesomery

why

(e.g.,

--C~H Group theory is so useful for explaining molecular

spectral

We

82

have to skip this subject for now and simply refer to an elementary but elegant textbook, ~ a n ~

Chemistry, by Eyring, Walter, and Kimball, Wiley, New York (1944).

The symmetry group of a molecule is a subgroup of dimensional orthogonal group. group

G

is known.

0(3), the three-

When its shape is known experimentally,

Let us refer to Wigner's paper (GDttingen,

its symmetry

(March 1930),

p. 133) on the characteristic elastic vibration modes of molecule (given by the equivalent classes of

G), as examples of the application of group theory.

studied CH 4 (whose group

G

is

S(4)) as an illustration.

Wigner

The H. A. Jahn, E.

Teller theorem (~oc. Roy. Soc. Ser. A, 161, 220 (1932)) proves that the electron orbital state of "non-straight" molecules cannot transform as an irrep of dimension

> i.

(The irrep has dimension

2

G

of

for molecules whose atoms are on a

straight line.) We will study here only one very important example.

2.11.

Measurement of Spin and Statistics of Nuclei by the Study of Diatomic-Molecule Spectra

The Hamiltonian

H

of a diatomic molecule can be divided into

H = Helectronic + Hvibratio n + Hrotation + H ~ where, to a good approximation, H ~ tronic states of the molecule;

can be neglected.

two nuclei) which minimizes the energy. two nuclei are identical, a fraction of

2

Helectronic

each such state yields a distance

0(2) × Z 2.

gives the elecd

The invariance group is

(between the

0(2)

or if the

Binding energy for such states are typically

(few electron volts).

H

vibration is essentially the harmonic

oscillator Hamiltonian for small oscillations around the equilibrium position fixed by the distance d. 2 to a , and the H to

%(~ + i), ~

The equidistant spacing of the vibration level is small compared rotation yields for each

integer

d

also rotation energies proportional

m 0, and small compared to the vibrational energies

tional bands; in spectrum).

is the symmetry of the molecular state for the permutation group nuclei? its

S(2)

The symmetry depends only on the spin state of the nuclei,

SU(2)

(rota-

If the two nuclei of the molecule are identical, which of these two

(each of spin

j)

irrep is Ds, 0 ~ s ~ 2j;

[-I--] s = 2j, 2j - 2, 2j - 4 .... S

and the rotational state of the system, Since

H

2j - i, 2j [7-]

for

3, ... ~

even,

H

for

~

odd.

is independent of the nuclear spin (to a very good approximatio~

the symmetry character of the nuclear spin state is a constant of motion (with often a lifetime of weeks) and is, as for Nelium, called ortho or para. "statistics", motion.

Because of

the symmetry character of the rotational state is also a constant of

So the rotational spectrum of the molecule divides into two independent

83

sets of transitions - those between even

%, and those between odd

tions occur in both states as quadrupolar energy

~ (~ + 2)(~ + ~ -

The number of

(2j + l)(2j + 2)/2 = (j + l)(2j + i).

states is

(2j + i)(2j)/2 = j(2j + i). (rotational)

lines is

The transi-

~ + 2 + ~, with a (radio-wave) photon

~(~ + i) = 4~ + 6.

states is

sity of spectral

~.

~

nuclear spin

The number of

nuclear spin

~

So if for the molecule the relative inten(2j integer

~ 0), j/j + i

% even/% odd, h r the nuclear spin is j, the statistics of the nuclei is I H x ~I/I~-~ x HI ~L_a ;.Jj ) I = Fermi; if it is j/j + i for ~ odd/% even, the nuclear spin is j the

Ia mentally,

for

xper -

=

only Fermi statistics is found for half odd integral

tron) and Bose statistics for integral

j.

j

(as for the elec-

We will summarize this important experi-

mental fact by statistics = (-i) 2j For instance when only even

~

(2.61)

rotational states exist, we conclude that

j = 0,

and the statistics has to be Bose. Historically, 598 (19309 was

the first nuclear spin measured

(F. Rasetti,

(in 1929) that of N14 (nitrogen molecule N-N).

Z. Physik, 6__!1,

Rasetti found

j = 1

and Bose statistics. But it was then believed that the universe was made of protons + p , electrons e , and photons y, (the only particles then known, and that the nucleus N14 of charge 7e, contained 14 protons and 7 electrons,

thus, half integral

spin and Fermi statistics were expected. This measurement started a crisis in physics. Appendix.

On Commutation Relations

Professor Bargmann pointed out to me that I have spoken of the invariance group of the commutation relations only in the context of rotational invariance Equation

(2.10)).

Surely it is worth mentioning

the general case:

(see

consider the

relations [Pi,Qj ] = ih6ij~ (i,j = 1

to

n).

Let

operator notations

(2.61)

a = (al...an) , b = (bl...b n) E Rn; we can use the tensor

P(a) = EiaiPi, Q(b) = EjbjQj.

Equation (2.61) defines a

dimensional Lie algebra which is a central non-abelian extension R1

(center of

on

R 2n

where

=

Rn

~

~).

~

of

2n + 1

R 2n

by

This extension is defined by the antisymmetrical bilinear form

Rn

o(a @ b, a e • b e ) = a • b t - b • a I

a • b = E.a.b..

The symplectic group

Sp(n)

(2.62)

which leaves this form invari-

i i i

ant is a group of automorphism of

~.

The corresponding simply connected group G has, up to an equivalence, a unique unitary irrep (von Neumann's theorem. Its Schr~dinger realization as operators on the space L 2 of functions of n variables: x = (x I ....x n) is U a = e iP(a) with

(Uaf)(x) = f(x + a); V b = e iQ(b)

a E En, b E E'n dual of the rotation group

SO(3)

En.

with

(Vbf)(x) = ei~b'Xf(x).

In the case of Equation

(2.10)

n = 3.

Here

Furthermore,

leaves invariant the symmetrical linear form

and w e used the corresponding identification of

E3

and its dual.

x,

@

on

E3

8#

3.

N U C L E A R PHYSICS:

3.1.

The n u c l e i have

protons

charge and

N

unstable

m

p

+ e.

excited

and the n u c l e u s

the f o l l o w i n g

which

may transform

B+-decay

÷

it b y

(Z,N);

~-decay*:

spontaneously

two p a r t i c l e s

A = Z + N nucleon~

The g r o u n d state

itself m a y be

into a n o t h e r u u c l e u s

seconds

+ ~(v = a n t i n e u t r i n o ) ;

(Z - I,N + I) + e + + ~

T

can vary

(Z,N) ÷

(Z,N) ÷

which

b y one of

(Z + I , N - i) + e

competes w i t h

(which r e q u i r e s

+

e--capture

less energy).

from 10 -3 sec to 1020 years.

(Z - 2,N - 2) +

(2,2)

for

A > 140

nuclei,

T

from

to 1020 years.

A n d two m u c h rarer

types:

c)

neutron

(Z,N) +

d)

spontaneous

emission: fission

into

Let us call n u c l e i 274 stable n u c l e i

(Z,N - i) + n

rare,

two s m a l l e r n u c l e i

r < few seconds,

(Z,N) ÷

(ZI,NI) +

if they. have a h a l f life of d e c a y

stable

(Z2,N2). r > 1020 years.

are known.

even

N

even

165

A even

Z

even

N

odd

55

Z

odd

N

even

50

A odd Z

odd

N

The m u c h in Figure

odd

greater

3.1 w h i c h

and for a g i v e n

N

have definitely

This is also

distribution

(Z = N = 1,3,5,7)

abundance gives

of

to n o t e

even,

N

even n u c l e i

of stable n u c l e i

from Figure

m o r e isotopes

is

Pb208,

of stable n u c l e i

and

Z

the n u m b e r

true, but less strikingly,

stable n u c l e u s

N - Z m 0

4

is s t r i k i n g l y

for g i v e n

Z

illus-

(isotopes)

(isotones).

It is w o r t h w h i l e (N = 82)

heaviest

These

it contains

are unstable.

(Z - I,N + i) + ~

The m e a n life

trated

n.

--

+ e

(Z,N) ÷

(Z,N) + e

Z

and n e u t r o n s

types of decay.

B--decay n ÷ p

b)

and d e n o t e

states,

+

a)

p

= 1 8 3 6 . 1 0 m = 938.25 MeV, m = 939.55 MeV. The pro~ton has e n Both have spin 1/2. We d e f i n e a n u c l e u s by its n u m b e r Z of

of neutrons,

Nuclei have bound

The Set of K n o w n N u c l e i

are m a d e of p r o t o n s

similar masses

electric

S T R O N G AND W E A K I N T E R A C T I O N S

N - Z

for

N - Z = -i

for the p r o t o n

N - Z = 0

for 13 nuclei;

N - Z = 1

for 16 nuclei;

N - Z

increases

of

Z

increasing (Z = i)

Z = 20, Z = 50,

(and also

Another

and

function

and

for

than their e v e n - n e i g h b o r s .

N = 20, N = 50

Z = 82, N = 126.

in f u n c t i o n

is a slowly

3.1 that n u c l e i

(or isotones)

N

striking

is that w i t h of

N = 28). feature

two e x c e p t i o n s

A = N + Z:

H e 3 ( Z = 2)

#

# What was (2,2).

first

called

with

A

an m - p a r t i c l e

on the a v e r a g e

(N - Z) N 6.10 -3 A 5/3.

has b e e n i d e n t i f i e d

with

The

in the

a H e l i u m nucleus:

85

50

20 71

Number of isotones

82

vN

6 5 4 3 2 i 0 I i0

I 20

I 30

I 40

I 50

I 70

I 80

I 90

I i00

! Ii0

i 120

." N 130

50

20 i0

] 60

Number of isotopes

~Z

9 8

7 6

even z

5 4 3 2 odd Z i 0 I

I

I

I

I

I

I

I

I

i0

20

30

40

50

60

70

80

90

FIGURE 3.1.

NUMBER

vN

AND

~Z

OF STABLE NUCLEI

(Z,N)

Note that there are no stable nuclei for Z N = 19, 21, 35, 39, 45, 61, 89, 115, 123, > A = N + Z = 5, 8, 147, m 208. The heaviest 126 82 Pb208 with Z = 82, N = 126. The most Figure 3.1 is that

vZ

and

vN

I

I

I

i00

ii0

120

AS FUNCTIONS OF = 43, 61, > 83, for 126 and none for stable nucleus is striking feature of

are mainly i, sometimes 2 or 0

for odd Z or odd N. Their value is more irregular for even Z or even N, there are relative maxima for Z = 20 = N, Z = 50 = N, N = 82 and also N = 28.

Z

>

Z

AND

N

86

Nuclear

forces are the most intense

208 nucleons

together#.

in nature,

however,

they do not bind more than

The reason is that they have short range and also that nu-

cleons obey Fermi statistics

which,

for condensed

states,

yield effects

similar

to

repulsion.

lifetime

More than one thousand different unstable nuclei

are known.

T > .i

are found in nature,

(age of u n i v e r s ~ and their decay products

are still produced have Z-even,

in nature

N-even.

(C14).

There exists

All other are man made; more a semi-empirical

formula

and

nuclei

giving

in function of

A = Z + N B(Z,N)

= Zm

p

+ Nm

n

- m(Z,N)

= U A - U Z(Z

~

c

- I)A -I/3

- U A 2/3 - U -(Z -- N) 2 + U (-l)Z + (-I)N A -3/4 s t A p 2 the values of the constants U

U

= 14.0 MeV,

corresponds

sponds

U

to the Coulomb

N

c

=

are in

.61 M e V ,

A I/3" .

to pairing

U

s

among

= 14.0 MeV,

Z

The term

effects

protons U

MeV,

Ut

p

=

favors

34 M e V

The term

U

corre-

c

in a sphere of

to a surface effect which

a minimum

for

IZ - N I

while

As we saw, nuclei with even

than those with odd Z

U

equally distributed

in like nucleons.

is that all known

even, N

Z

and/or odd

N.

Z A

even nuclei have zero spin

at rest).

The distribution

of nuclear spin for odd

A

nuclei

is discussed

in

3.4.

3.2.

Isospin

As soon as the neutron was discovered language

U t = 84.2

corresponds

s

forces;

are more stable and more numerous

(= angular momentum

(3.1)

MeV

average binding energy by nucleon.

repulsion

to

rule without exception

Section

U

a short range for nuclear

corresponds

P and

U

to maximal

radius proportional suggests

some

than half of those

(Weizs~cker's)

the binding energy of the lowest state of (stable or unstable) Z, N

Those with a

for the study of nuclei.

particles,

the nucleons,

Neutrons

(1932),

and protons

Heisenberg

created

are considered

which have five degrees of freedom:

3 continuous

(~)~ a two valued one, o, for the spin and a new one that Heisenberg the fifth degree of freedom,

# The existence

T, and which distinguishes

neutrons

a formal

as the same in space

simply

called

and protons

of neutron stars with a radius of i0 km to i00 km and containing

N 1057 neutrons has been postulated. These stars seem to be observed now as "pulsars". They are indeed gigantic nuclei, but the binding energy is due both to nuclear and gravitational forces.

S?

(Z. Phys.j 77, 1 (1932)); since, like the spin, it is two-valued,

it is now called

isospin.# This Heisenberg

convention has revealed itself more than useful.

it was quickly established and neutrons:

reasonable

that nuclear forces did not distinguish between protons

their differences

small mass difference)

(different electric charge and magnetic moment,

are attributed mainly to electromagnetic

approximation

Indeed

to neglect

effects and it is a

them.

If we denote the Hilbert space of our nucleon states by

~(i) that of a number

A

of nucleons

~(A)

where

P[1A]

(3.2)

is

(i) = ~ [ i A] = e[l A]

is the projector on

@ ((L 2 ~ K ) (A) % [1]

~(i)

.

0

A convenient

~(A) c) "T[%]

(3.3)

approximation

for the study

[1 A]

of a nucleus of

A

nucleons

an average potential potential,

÷

= L2(x,t ) @ Ko @ KT

is to replace the sum of 2-particle

(= sum of 1-particle Hamiltonians)

which is still attractive.

allows us to draw qualitative

interactions,it

by

plus a residual 2-particle

Then the analogy with the study of atomsttt

conclusion.

Using the same type of argument as in

Section 2.6 for atoms, but here with the opposite sign, we know that for the ground state

[%]

in Equation

antisymmetrical

(3.3) should be as syrmnetrical as possible,

as it can be with the restriction

so

[%]c

that it has only two lines.

is as This

implies that the two lines are as nearly equal as possible %i ~ 0; [%]c = [%1,%210 ~ %1 - %2 = I If the nucleus has

Z

pletely symmetrical

in

protons and sup{Z,N}

N

neutrons

particles,

minimal;

(Z + N = A)

%1 + %2 = A

(3.4)

its states are com-

so

(3.5)

%1 ~ sup{Z,N} and

IZ - NI ~ %1 - %2

'

(3.6)

Called isotopic spin since 1936, the name isobaric spin would have been more proper. Anyway it has been shortened into isospin by the natural evolution of language. tt

In fact physicists are more sophisticated: when a sum of 2-particle interaction does not yield a good enough approximation, one adds also the sum of all kparticle (2 < k ~ A) interactions, mainly for k = A (collective effects).

#t% There is still a difference. Atoms of n electrons consist of n + i particles and as we have seen, the elimination of the center of mass motion is easy: one singles out the nucleus, and the electrons are all treated on the same footing. This elimination is still clumsily carried out in nuclear physics.

88

So (3.4) can be translated possible.

As we have seen,

tromagnetic account

repulsion

0 < N - Z

into:

the most stable nuclei have as small

this is well verified

of protons

in negligible;

In the same approximation bars

(nuclei with the same number

Consider

in which A = Z + N

for light nuclei, where

when this repulsion

has to be a slowly increasing n, p

]Z - N]

function of

the elec-

is taken into

A = N + Z.

are considered

of nucleons)

as

identical,

iso-

should be identical.

Figure 3.2; it gives the energy spectrum of the known states

for

A = 15,

/2

>=

unobserved

.P 3

C 15

NI5

015

FI5

Z=6

7

8

9

N=9

8

7

6

is the spin

(j, 1/2 integer

FIGURE 3.2.

> 0)

and the parity

(+-)

of the state.

SPECTRUM OF STATES OF ISOBARS 15

States of C 15 have isospin m 3/2. Another conventional notation for nuclei is to use the chemical symbol of the corresponding atom (this gives implicitly the number Z) and write the number of nucleons A = Z + N in superscript.

89

and the known value of spin and parity of these states. spectra (at least for the low lying part) for nuclei) is striking. N 3 MeV

for

015

The similarity of the

1/21Z - N I = 1/2

(i.e. N 15

and

015

The essential difference is a shift upward in energy of

which has one more proton than

N 15.

The pairs of corresponding

states are called "doublets" of isospin 1/2 states. Let us explain in detail this point of view, which exploits the relations between the permutation groups

S(n)

and the unitary groups

U(k)

that we have

recalled in 1.4 and used in 2.9. Nuclear interaction does not distinguish between protons and neutrons. For a nucleus this implies a property of invariance under the permutations of its nucleons.

(E S(A))

We could also have translated this property as follows:

All nuclear physics observables

0

acting on

~C(I)

(space of the one

particle states for the nucleon) in Equation (3.3) are of the form (3.7), where ~(i)

= L2(x,t ) @ K

0 K

(3.2)

T

0 =

N

@ I

(3.7)

U(2) =

I

@ U(2)

(3.8)

i.e., they correspond to a trivial action on

K , the factor in this tensor product

which corresponds to Heisenberg's fifth degree of freedom "proton - neutron". The action of the group with every observable: observables".

The action of this

Nilbert space of the

U(2)

on

~(i), as defined by (3.8) , commutes

U(2) c {0} S , the commutant of the algebra of "one particle

A

U(2)

particle states.

can be extended to every

Therefore, in nuclear physics, when the

non-nuclear interactions are neglected, this group. S(A)

$C(A) and

$C(A) , (A ~ 0),

U(2)

is a subgroup of the invariance

has the same decomposition into spaces of factorial representation for

U(2)

and we use the same symbols (Young diagrams) for the corresponding

representations. Since Coulomb repulsion of the protons can be neglected only in light nuclei, it was not~expected that isospin conservation could be an interesting concept for heavier nuclei.

However, the progress of nuclear physics in the last five

years has shown that for nuclei with cept.

A

up to i00, isospin is indeed a useful con-

For a non-technical review of this question, see W. R. Coker and C. F. Moore,

"Isobaric Analog Resonances",

Physics Today, 3.3.

U(4)

22, no. 4, 53 (1969).

Invariance

In 1936 Wigner, in his paper "On the consequence of the Symmetry of the nuclear Hamiltonian on the Spectroscopy of Nuclei",

# Reproduced in Dyson's anthology: Benjamin, New York (1966).

Phys. Rev.,

51-106 (1937)t

Symmetry Groups in Nuclear and Particle Physics,

9o studied the approximation where not only isospin dependence of nuclear forces is neglected but also the spin dependence.

Then Equation

(3.7) and (3.8) can be re-

placed by 34(I) = [2(x) 8 K

In this approximation, U(4)

A

(3.2)

@ I

(3.9)

0 =

N

8 I

U(4) =

I

@ U(4)

(3.10)

nuclear theory is also invariant under the group

acting on the four dimensional

Hilbert space of

8 KT

space

K = K

@ K

and Equation

T

(3.3), for the

nucleon states can be replaced by ~(A) = $C(I) = p ~% ® K [iA] [iA] (L21%] [%]c )

where the

[%]

are representations

of

For the most stable states, two-nucleon possible,

force to be attractive

so

[%]c

U(4)

(acting on

as possible,

lowest lying states is

0, isospin 1.

%1 + %2 + %3 + %4 = A) as %1 = %2 = %3 = %4 = A/4. of

(Z < 17)

For nuclei with

A = 4n + 2, the

SU(2) × SU(2)

representations:

[%]c

representation

(spin i) and

He 6

decomposes

into the direct sum of

n = i) this gives correctly the spin of the and

Be 6

(spin = 0).

These last two

(spin 0 +) of

Li 6.

levels Whose spin are marked in Figure 3.3 belong to another equivalent U(4)

with an angular orbital momentum

degree of freedom) j = Z = 2

and

Z - s ~ j ~ Z + s Li 6

i.e., = j

with no correspon-

Be 6.

States belonging literature.

representa-

(i.e., angular momentum of the space

for the spin i, isospin 0 states i.e., those of He 6

The other

So the total angular momentum has the possible value

for the spin 0, isospin i states and

= 3, 2, i dents in

% = 2.

of

(~) = 6.

one of spin i, isospin 0, the other of spin

levels form an isospin triplet with the third level

tion of

U(4)

where Coulomb repulsion of

%1 - i = %2 - i = %3 = %4 = n; it has dimension

to the subgroup

Li 6

i.e., its Young diagram has

SU(2) × SU(2), yields a spin 0 and isospin 0

In Figure 3.3 (for which

lowest state of

as

As we have seen, the former result is observed for all such

protons is not too large.

two three-dimensional

(with

is as symmetrical

A/4 = integer this implies

the latter only for light nuclei

Its restriction

[%]

The restriction of this representation

Ko @ KT) to the subgroup

for the ground state. nuclei,

For

irrep has dim. i.

(used in 3.2) of the "residual"

implies now that

%1 a %2 a %3 z %4 a 0

nearly equal as possible. This

U(4). the property

is as antisymmetrical

its four lines of length

(3.11)

to a

U(4)

irrep are called supermultiplets

The study of Galilean invariance

in physics

of the theory of supermultiplets

very similar to that made in 2.9 for atomic physics.

is

94

2+

1+ 2+

0+

2+

,

2+~'"

0+

0+

3+

5 MeV

i+

He 6 Z=2

Li 6

N=4 jP

Z=3

N=

Be 6 3

Z=4

N=

2

is the spin (j integer a 0) and the parity p(!) FIGURE 3.3.

of the state.

SPECTRUM OF LEVELS OF THE NUCLEI WITH 6 NUCLEONS

7[

of the Galilean group

One has to consider the covering

G, ~ ÷ G

and

also the homomorphism

SU(2) x 1 c U(2)

÷

x

U(2) c U(4)

The invariance group of the theory is the direct product subgroup

G~

G × U(4)

with

G x U(4)

and

G

is the

i(g) = (~(g),~(g)).

The approximation of spin and isospin independence for nuclear forces leading to

U(4)

invariance is crude and could not be expected to be very useful

for nuclei with a number

A

of nucleons not very small.

approximation of isospin conservation, nuclei with

A

ground state: plet model",

U(4)

However, as for the better

invariance has been usefully applied to

up to i00 as shown by the statistical study of the energy of their P. Franzini and L. A. Radicati, "On the validity of the supermulti-

Phys. LGt%., 6,

in 3.3 and 4.)

322 (1963).

(Reproduced in Dyson's anthology, quoted

92

3.4.

Shell Model

We have seen that nuclei with seem more stable.

Z

or

Many properties of nuclei

N = 8, 20, 50, 82, and

(e.g. binding energy per nucleon, mag-

netic and quadripole moment) single out these numbers. shows that for

A survey of nuclear tables

A = odd nuclei, the parity and spin of the ground state vary in a

regular pattern which suggests very much the filling of shells physics).

Ground state, spin

number

or

Z

N = 126

N

j

and parity

±

(as in atomic

depend only on the value of the odd

so the order of the levels, with increasing energy, is the same

for protons and neutrons.

The order obtained can be deduced from the three-

dimensional harmonic oscillator spectrum

E

n

= n~

+ Eo

(where

~

is a constant)

with some modifications. Consider the set of nine operators Qj (i = 1,2,3)

Tij = PiPj + QiQj

where the

P'l

and

satisfy the canonical commutation relation [Pi,Qj] = ih~ij

Its use for the computation of the commutators form a representation up to

ih

[Tij,Ti,j,]

of the Lie algebra

U(3).

shows that the

Tij

The center

H = TrTij = TII + T22 + T33 is the harmonic oscillator Hamiltonian deduce that there is a tor similar to the spectrum of

H

U(3)

S0(4)

n

is

From

[H,Tij] = 0

we

invariance for the three-dimensional harmonic oscilla-

invariance of the hydrogen atom, 2.4, and we can find the

by a similar method:

The ground state E

(in convenient units).

n = 0

i/2(n + l)(n + 2).

= (-i) n, 0 ~ Z ~ n.

has energy

Eo.

The number of states of energy

Their orbital angular momentum

~

satisfies

This corresponds to the first column of Figure 3.4.

(-i) £ The second

column gives the spectrum of the Hamiltonian

~-lH' with

m, J , m tt

s = 1/2 > 0

and when

= ~H - , ~ 2

positive constants, w > ~ s

and

~ > 0, j = ~ + c 1/2, ~ = ±i

En, j ~ - E 1 = n~ - t ( ~ ( ~ ' 0 '7' 0

= 0 This Hamiltonian

_ a~''~

• ~

jt .

(3.12) Using Equation (2.55) for

one obtains the energy spectrum

1 + i) - ~ it(~ + 7) , s = sign(j - ~)

= nw Ht

(3.13 t)

is the one-nucleon Hamiltonian in the average potential pro-

duced by the whole nucleus. protons and neutrons.

(3.13)

As in 2.7 we can now "fill the successive shells" for

Such shell-model for nuclei was proposed in 1949 (see

M. Goeppert Mayer and J. H. D. Jensen, Wiley, New York (1955)).

Elementary Theory of Nuclear Shell Structure,

It is very successful in explaining the properties of the

95

n

Z

Enj ~ - Eo Z

6

5

4

or

~

11/2 +

6

i"~s

g/2 +

1 3

126 ~ ~.m k

y = 0, b = i (E)

The Other Particles

columns

the highest of these lowest states

The

E

is stable only because

were satisfied,

and that

(See

does not seem so fundamental!

the lowest states of two neighboring

were

m A - mN < m

3.6.

Neither

of the column

is due to the fact that both P, see 4.7a)

in Table 3.2 with the excep-

are the lowest states of the

or unstable

for strong interaction.

< m ; if for instance The stability

which is

They are often called resonances

in Table 3.2 and the first excited states

b = 0 (n).

D Z2(C)

Most of them are even unstable by the strong

instead of particles The particles

S0(3)

x Z2(C ).

that all particles

(and the antiproton) are indicated.

of

the

A

m E - mA

would also be stable.

~

and

~

have spin 0, parity -

q-isoparity

= +

while

~-isoparity

= -.

and the Other Interactions

There are only nine known "elementary"

particles which are not hadrons,

i.e., have no strong interaction. The photon,

y, with mass zero, spin i (see 4.4). + + The 8 leptons ~ , ~-; e , e (electrons) and their associated

neutrinos

~ +~ _~ +~ ; they all have spin 1/2. ~ e e ±

+ ÷e-+~

Only the

± +~

~

zero mass

are unstable.

+

e because

m

= 207 m . e All particles

electric

charge

Amp~re's

hypothesist

electromagnetic

(e.g.

have electromagnetic

interaction

even when they have no

q = 0; baryons have magnetic moments) that the whole electromagnetic

current

j~(x)

is well verified;

but it seems that

interaction

is through the

the interaction

Hamiltonian

is (3.17)

Hem = e ~ A~(x)j~(x)d3~ where

A~(x)

is the photon

field

which

h = c = i, the universal

2.4).

Electromagnetic

(electromagnetic

constant

interaction

is about i00 times weaker

This is also the order of magnitude

t Called nowadays

"minimal

e

potential vector). In units for 2 is given by e = ~ = 1/137.039 (see

coupling"

of mass difference

in the jargon.

than strong interaction.

in an isospin multiplet.

99 There is another universal interaction,

shared by all particles

the photon), characterized by a universal constant

(except

G, introduced by Fermi#, whose

value is G = 1.01 x 10 -5 x m 2 /~ P

(3.18)

Since it is much weaker than the electromagnetic interaction, action is simply called the "weak" interaction.

this inter-

Fermi postulated as early Ns 1934

that this interaction has some analogy with the electromagnetic interaction. instance there are four electrically charged

(!)

For

"weak" hadronic currents, respec-

tively vectors and axial vectors for the Lorentz group, v (+~(x),, a (±)(x) -interacts with the leptons through a leptonic current

which

~+)(x),~_ and the interaction

Hamiltonian being H w

= G /~

~ j" ~ ( E ) ( x ) h ~=±i

(~)(x)d3~

(3.19)

with h (c)(x) = v (s)(x) - a (E)(x), (s = ±i) Equation (3.19) has indeed some similarity with (3.17).

(3.20)

The fact that

h

is a

linear combination of a vector and an axial vector will explain the parity violation of the weak interaction

(see 4.7.b).

R. P. Feynman and M. Gell-Mann (Phys. Rev., 109, 193 (1958)) have found a very deep relation among the three interactions. U(2)

on

From the unitary representation of

~, the Hilbert space of hadrons, one obtains the representation

U(2)-Lie algebra on

~.

The operators corresponding to the observables

F y

of the and

t3

are the self-adjoint operators Y = F(y) Since

q = t 3 + 1/2 y

and

T 3 = F(t 3)

(3.21)

(Equation 3.16) for all hadronic states, this relation has

also to be true for the self adjoint operators representing these observables, i F(q) = Q = ~ j°(x)d3~ = T 3 + ~ y Note that

~ jP(x) = 0

~ Q

is time independent

~=~ [H,Q] = 0.

so (3.22)

However, Q

here

is the total electric charge of the hadronic part of the world, it is not conserved since weak interaction can t r ~ s f e r

it to the leptonic part of the world.

It is

conserved only in the approximation which neglects weak interaction. The beauty of the discovery by Feynmann and Gell-Mann is that, when electromagnetic and weak interactions are neglected, hadronic currents

vt(~)(x)p

the vector part of the weak

(Equation 3.20) and the electric current

j (x)

t E. Fermi, "Versuch einer Theorie der B-strahlen", Z. Physik, 88, 61 (1934).

of the

JOO

hadrons are images of the same tensor operator for the strong interaction for, respectively, of the complexified

the vectors

t±

U(2) and

group of invariance of q

of the vector space

U(2)-Lie algebras t± A y = 0 = y A t 3, t± A t 3 = ±t±

(3.23)

T+ = F(t+) = f v'0(_+)(x)d3~

(3.24)

This implies that

The isospin group, which was introduced in 3.2 in such a formal and abstract f a s h i o ~ becomes a physical reality since it is generated by the space integral of the weakly interacting hadronic currents! full

U (2)

3~v~(x)

group.

as well as

becomes:

i)

The addition of the electric charge generates the

When electromagnetic and weak interactions are not neglected, 3 j~(x)

do not vanish and the representation of

time-dependent for the physicists

any time with time-dependent

P, Q); 2)

Coleman and other physicists have shown). in Equation

(3.24)?

(just as

[P,Q] = i h l

Have you noticed the

U(2)

vt

on

is true at

undefined for the mathematicians

I have shortened a long story.

esis really needs the enlargement of the

U(2)

(as

instead of

v

The Feynman-Gell-Mann hypoth-

group to

SU(3)

as we will explain

in 5.1 and 5.3. To stay inside

U(2), one has to decompose

h (e)(x) = h~(c)(x) where

hi(e)

has hypercharge

Cabibbo angle #. the

a (s)(x)

IAyi = 1

y = 0

and

h

of Equation (3.19)

cos e + hlt(e)(x) hSe(s)

has

sin e

y = s

(3.25)

and

The same decomposition appears separately for the part of

h

(Equation (3.20)).

The angle

weak transitions are slower than those with

0

s

and

is a vector-operator for the h~.

SU(2)

isospin group.

is the and

has a value 15 ° so the

iAyl = 0

by a factor

They have also a different "selection rule" for isospin. v

e

v (E)(x)

tg20.

As we just said,

This is also true for

as

Hence weak transitions

IA i = 0

satisfy

I Ti = 0

while weak transitions

iAYI = 1

satisfy

[ATI : 1/2

i.e., het , v It , a el

are

or

1

SU(2)-spin0r operators.

We have also to mention two other charges conserved by all known interactions

(as the baryonic and electric charges).

which seem separately conserved: and zero for e zero for

They are the two leptonic charges

that of the e-type with value

~, ~ ; that of the ~-type:

e, ~ . e

# N. Cabibbo, Phys. Rev. Lett., iO, 531 (1963).

with value

~ = ±i

~ = ±i for

for ~ , ~

e s, and

4.

RELATIVISTIC INVAPXANCE.

4.1.

THE DISCRETE SYMMETRIES C. P. T.

The Poincar& Group and its Automorphisms;

Zeeman Theorem

Physicists call Poincar& group the inhomogeneous Lorentz group#. denote its connected component by connected Lorentz group

L0

PO"

by the translation group

It is a i0 parameter real Lie group. product

The group law of

We call flection

T.

T.

T [] L

P

P0

~0

is the semi-direct

T(r,t) = (-r,t), D

We denote by

P(r,t) = (-r,t)

the group of dilations

P, P+, F +, F

Z2(P ) × Z2(T) , Z2(P) , Z2(P)

and

T

the time re-

{a > 0,~(r,t) = (ar,~t)}.

the group generated by respectively

the groups generated by

and

m = "the

is given explicitly in Equation (4.10).

the space reflection

Z2(P), Z2(T), Z2(P) × Z2(T)

of the

It has a trivial center.

Its universal covering

T[]SL(2,C), whose center is a two element group generated by

rotation by 2~".

call

It is the semi-direct product

We will

D, Z2(P) × Z2(T)

and

We

P, T, P

P0

and respectively

D.

We call

P

and

the full

Poincar& group. It can be provent# that all automorphisms of these groups are oontinuous and, if

Aut G

is the automorphism group of

G, (4.1)

Aut PO = Aut P+ = Aut P = Aut F + = Aut F = F Given any group Out G that

G, we denote by

the quotient F

In.Aut G

the group of inner automorphisms and by

Out G = Aut G/In.Aut G.

Note that here

PO = In.Aut PO

and

is the semi-direct product F = PO [] (Z2 × Z 2 × D) = P0 [] Out P0

Binary Relation on Space Time

Given

x ~ y

(4.2)

E

two distinct points of

E, we define the notations:

x T y = (y

is inside the light cone of

s L y = (y

is on the light cone of

x)

x)

Lorentz transformations were introduced by Vogt in 1882 and applied by Lorentz to electromagnetism. H. Poincar& (C. R. Acad. Sci., Paris, 140, 1504 (1905)) required that they form a group with the rotation group and, from it deduced physical consequences. In Rend. Circ. Mat. Palermo, 21, 129 (1906) he included the translations and studied ~hysical implications of invariance under the group we call here Poincar& group. ~# L. Michel, "Relations entre sym&tries internes et invariance relativiste", lectures published in Application of Mathematics to Problems in Theoretical Physics, Carg~se 1965, Lur~at editor, Gordon and Breach (1967) referred to as LM III. We will also refer to my lectures in Istanbul (1962) and Brandeis (1965) as LM I and LM II. They are both published by Gordon and Breach (Book of the lectures, for each school).

'102

x S y = (y

is outside the light cone of

x < y = (y

is inside the future light cone of

x < • y = (y

is on the future light cone of

Given a permutation binary relations

x R y

x)

if

f

of the points of

x) x).

E, it is said to preserve the

x R y ~ f(x) R f(y).

E. C. Zeeman (J. Math. Phys., ~,

490 (1964)) proved the following theorem:

Theorem i.

The necessary and sufficient condition that E, preserve the relation

x < y

or the relation

f

and

f-l, permutations of

x < • y, is

f E F +.

Zeeman also established the corollary (proof published in LM II, p. 297):

Corollary i.

The necessary and sufficient condition that serves the three relations

4.2.

x T y, x L y, x S y

is

f, permutation of f E F.

Relativistic Invariance and Internal Symmetries#

A physical theory is relativistic if its automorphism group P0"

We are also interested in other s y m e t r y

internal symmetry groups. tions

P0

groups, subgroups of

G

contains

G, and called

Note that if we consider "passive" invariance,

the dila-

D c G. If

of

E, pre-

P0

is a subgroup of

in

G = {g E G, p C P0' gPg -i G = {g E G, p E P0, gpg E PO}. That

Aut ~ = F

G, one can consider C = CG(P0) , the centralizer -i = p}' N = NG(P0) , the normalizer of P0 in

is the semi-direct product

(2) and that

P

has no center

imply that

N = Po = (NJ

o)

(4.3)

,

and there is a canonical homomorphism f (4.4)

N/P 0 * Out P0 = Z2(P) × Z2(T) × D And for instance also see that

Z2(P ) c Im f

D N Im f will give information on the mass spectrum.

theory of mass zero particle has if there is a particle of mass ties and any

means that parity is preserved in the theory.

m > 0

D # 0

in its automorphism group.

If

We

Indeed, a ~ c Im f

and

then there are particles with the same proper-

for the mass value.

# We also refer the reader to the paper with the same title: 137B, 405 (1965).

L. Michel, Phys. Rev.,

~03

O'Raifeartaigh (Phys. Rev. Lett., 14, 519 (1965)) has proven the following theorem when

G

is a connected Lie group:

Theorem 2.

If the restriction of an irrep (= unitary irreducible representation) of G

to

PO

has an isolated point in the mass spectrum, it is the whole mass spectrun~ There have been too many papers written by physicists proving "theorems"

much weaker than the following trivial lemma (LM III, p. 450).

Lamina.

Let that

P0

be a subgroup of

Vg E G, gpg-i ~ PO" then

the homomorphism

f, G i

gP0 ~

G

xgP0

of

G.

If there exists

(G/PO)

on its homogenous space PO

p E PO' p ~ T c PO' such

is an invariant subgroup of

permutations of

is an invariant subgroup of implies

P0

containing

G.

giving the action

G/P 0.

Then

p; it is

p E Ker f

PO

and

Indeed consider Vx, g E G, so

PO N K e r f

PO c Ker f; that

Vq E P0' Yg E G, qg = gPo" In my opinion, the preceding considerations are physically very poor,

indeed

P0

acts on space time so if

G ~ PO

is an automorphism group of the theory,

Zeeman's theorem implies that in order to preserve causality, G time only through a quotient subgroup either of This led us to consider

G

as an extension of

F

A

P

if we forget dilations.

P.

We are interested in quantum mechanics. the *-algebra

or

can act on space

So we must use the existence of

of observables.

We refer the reader to the remarkable paper of Haag and Kastler, "An Algebraic Approach to Quantum Field Theory", J. Hath. Phys., ~, 848 (1964); there are physical arguments for

A

to be a C*-algebra. t

Let

A

(obtained by a Gelfand-Segal construction) by operators on states, A t

its commutant, Att

The spectral resolution of spectrum is d i s c r e t e , ~ =

Z

the enveloping W*-algebra, Z = A ~ A Act yields superselection rules t#.

@h~h

are those belonging to one of the Assume that

P0

be its representation ~, the Hilbert space of

a n d the only vectors of ~ ~h"

The

is a subgroup of

its elements can be realized by operators of

~%

its center.

For instance, if the which represent states

are called superselection sectors.

Aut A, which is implementable (i.e., L 0 0).

This proposition was made by I. E. Segal, more than ten years earlier. *~ Concept introduced by G. C. Wick, A. S. Wightman, E. P. Wigner, Phys. Ray., 88, i01 (1952). See the preprint~of Doplicher, Haag and Roberts for the most recent study of this question.

10z~

Let U(p) E L O 0 .

U(p)

be a realization of the automorphism

tion

p.

by an operator

It has to be unitary in order to be an automorphism:

(UAU-I)* = (UA*U-I). operators of

p 6 PO

If

A', U(p)V

V

is any element of the group

U(A')

VA E A;

of the unitary

is just as good for representing the Poincar4 transforma-

So the set:

E = {U(p)V, p E PO' V 6 U(A')}

(4.5)

,

forms a group of unitary operators which is a "central extension" of

PO

by

U(A')

i.e.,

E/U(A') = PO' quotient group

(4.6)

and

W

6 U(As), FU 6 E, V ~+ UVU -I

is an inner automorphism of

U(A') •

(4.6')

One can prove (see Moore's lectures), that any Polish topological group satisfying (4.6) and (4.6') is either the direct product

U(A') × PO

E

or are of the

form (see also LM II):

Eo~ = (U(A') x P'o)/Z2(c~,~o )

(4.7)

where the two element group is generated by the element

(~,m)

group of square roots

U(Z)

of

~

e)

of the unit, in the group

U(A'). t Which is the extension

E

chosen by nature?t%

with

a 6 2U(Z), the

which is the center The answer is the

extension defined by (4.7) with iz(B+EiLi) ~=e

*

2 ,a

(4.8)

= I

Equation (4.7) implies some topology as explained in Moore's lectures. In "Sur les extensions eentrales du groupe de Lorentz inhomog~ne connexe", Nucl. Phys., 57, 356 (1964), I have studied the same problem for abstract groups: any abelian group A is the direct sum A = D • K where D is the maximal divisible subgroup and K is a reduced subgroup (no infinitely divisible elements ~ i). One has the relations:

H2(Po,A) = H2(SL(2,~),A), H2(PN,A) = H2(Lo,A) = 2K + H2(Lo,D),

H2(SL(2,~),D) Aut ~ = 0

and of course

H2(Lo,D)AUtVC = 2 D.

group of automorphisms of the complex field act on Aut LolL 0 = Aut ~).

So it acts on

H2(SL(2,~),A)

SL(2,C)

Indeed and on

Aut ~ L0

the (exactly

(through a trivial action on

A) since the group of inner automorphisms of

L_ acts trivially. Following the U usual convention, also used in Chapter i, H2(SL(2,~),A) Aut ~ is the subgroup of fixed elements. 8o if H2(SL(2,~),A) # O, the automorphisms of ~ do not pass the non-trivial extensions and the corresponding extensions are very pathological. I found this a sufficient argument for considering in physics only the extensions of Equation (4.7).

%t This was the question that Lur~at and myself asked and answered in N. C~m., 21, 57 (1965) and Comptes Rendus of the Conference of Aix-en-Provence, p. 183, C.E.A. Saclay editor, (1962).

~o5

where

B

is the baryonic charge operator and

L.

the (different) leptonic charges

i

(see 3.6).

Indeed this choice of extension implies the observed relation between

spin and charges: (-i) 2j = (-i) b+Ei~i where

j

,

(4.9)

is the angular momentum of any state and

b, %. are its baryonic and i Note that Equation (4.9) shows that the integer or half

different leptonic charges.

integer nature of spin form a superselection rule.

4.3.

Irrep of

g0

All irreps (= unitary linear irreducible representations) known. to T TO

of

PO

are

In 1937, Wignert showed, by extending Frobenius' methods for finite groups

g0' that irrep of

gO

are characterized by an orbit of

T0

on

and an irrep of the corresponding little group (= stabilizer). invariant symmetric bilinear form on

products) yields an &somorphism of

T0

To be explicit, we denote by elements of

T

and

_a in a basis of group of

2 x 2

T.

SL(2,$) = TO .

_a, b, ...

T

and

and

T = R4

scalar

TS.

A, B, ...

(aO ÷,a) = (aO,a i ,a2,a 3)

Let

the dual of

(= Minkowski pseudo-Euclidean

space between

Consider the isomorphism between

respectively the the coordinates of

and the additive

hermitian matrices

(Z 0 +

_a +--+a = As we saw, the group action on

T

TI

The non-degenerate

gO

a

a

i + ia 2

-

aO

is the semi-direct product

ia 2

a3 / T~SL(2,C)

with the

SL(2,C)

T A E SL(2,C)

,

a +--+ ~ ~-+ A A~aA* +--+ Aa

The Minkowski pseudo-Euclidean scalar product is (_a,b) = aOb 0 - alb I - a2b 2 - a3b 3 and the Minkowski "length" of a We denote by

(~,A)

2

a

is

= (a,a) = determinant

the elements of

gO

with

a ~-~ (~,i)

the canonical injection

t E. P. Wigner, Ann. of Math., 40, 149 (1939) reproduced in F. J. Dysou, Sy~etr~j Groups in Nuclear and Particle Physics, Benjamin, New York (1966). Wigner,s paper was the first one giving a complete family of irreps of a non-aompact non-semisimple Lie group.

~06

T ÷ ~0

and

A %+ (0,A)

an injectiont of

%

= SL(2,C) %÷ P0"

The

~0

group law

is (~,A)(b,B) = (a + Ab, AB) We will use the same notation for elements of

T

(4.10)

and

TS.

It is useful to introduce the notion of stratum. a set

When a group

M, all the points with conjugate stabilizers form a stratum:

G

acts on

in other words,

a stratum is the union of all orbits of the same type (i.e., isomorphic as Ghomogeneous spaces).

The action of

%,

decomposes

T

or

T"

in four strata.

See

Figure 4.1.

It

The point 0 is stratum IV. x

FIGURE 4.1.

STRATA ON

T

BY THE ACTION OF

50

# This injection is unique up to a conjugation in P0" Indeed Wigner, in his paper of 1939, showed that for the abstract groups (with the action of T 0 on T just defined), HI(T0,T) = 0.

~07

Table 4.1 lists these strata, the corresponding little groups (defined up to a conjugation in

~,) and the corresponding irreps of %/

TABLE 4.1.

Stratum

I ~

2

= m

STRATUM OF

Tt

~. U

DUE TO THE ACTION OF

Little Group 2

P II p-- = 0 # 0

~0; IRREPS OF

Irrep of

P0

g0

0 m > 0 (i.e., p > 0), 2j integer ~ 0 a 0 Ib m < 0 (i.e., p < 0), 2j integer ~ 0

SU(2)

I

E(2)

II a sign of p0, 2h integer

(2-dimensional Euclidean group)

IIb sign of p0; H positive number, ~ = 1 0 II sign of p ; Z positive number, m = -i

S~(2~P0 = S~(2,1)

III a m 2 < 0, ascendant and descendant discrete series, ±j

C

III 2

< 0

N = double covering

III b m 2 < 0, principal series io,p E R lllc m2 < 0, supplementary series, 0 ~ o ~

IV p = 0

SL(2,~) = g0

IV

Irreps of SL(2,~) (see Stein lectures) are irreps of ~0 with T trivially represented.

is the non-trivial element of the center of (-i) 2j Those of

and in II by

(-I) 2h.

i

~0; in

I, ~

is represented by

Wigner constructed the irreps of type I and II.

needed for type III were given by Bargmann, Ann. of Math.,4_~8 568

S~(2,R)

(1947) and those of

SL(2,~)

(type IV for

g0 ) were first given by I. M. Gelfand,

M. A. Naimark, Acad. Sci. USSR J. Phys., iO, 93 (1946) and Isv. Akad. Nauk SSSR Ser.

Mat., ii, 91 (1947). Wigner's method is a particular case of Mackey's theory of induced representations.

Since the (measurable) axis

t10t

and

0x

are a set of representatives

of the orbits, a theorem by Mackeyt insures that this method yields all irreps of P0"

As we will see, the only irreps of

those of mass

m ~ 0

g0

which correspond to known particles are

(I a and II a in Table 4.1).

Wigner, in his paper, has given

the following realization: An d3~/p 0 = d~.

To

invariant measure on the orbit

Consider the functions

f

~:

defined on

p 2 = m 2 ~ 0, pO > O, is ~

with value in a

2j + i

# G. W. Mackey has described his theory in a book for physicists: Induced Representations of Groups and Quantum Mechanics, Benjamin, New York (1968). The needed theorem is Theorem B, p. 43.

108

K., such that ]

dimensional Hilbert

llfl12 =

J'f2 10 -21 sec).

Strictly speaking, because of Heisenberg's uncertainty relations between energy and time, the Hilbert space of states carries the

t The

~O

invariant

%

rep~(m,j~m

where

P

is a curve

is called helicity by elementary particle physicists but was

called circular polarization by its discoverer, Fresnel, in the 1820's. It just happens that the photon is the only known particle whose space of states is the carrier of a reducible representation (m = O, % = i) • (m = O, % = -i) of PO"

109

with a mean spread of decays,

Am.

For weak decays,

gm/m < 10 -14

and for electromagnetic

Am/m < 10 -5 . It is not clear that this concept of particle

the strongly unstable deed their lifetime the p-meson)

resonances •

(most of the "hadrons"

can be usefully extended to of Table 3.2 in 3.5).

might be as short as 10 -23 sec (and

Am/m

In-

reach 0.15 for

so they do not exist isolated since the range of strong interaction

10 -13 cm (of the same order as

3 x i0 I0 x 10 -23 = ~c).

is

However, we shall here call

them particles. We have used also the word particle as a collective name for different particles with the same spin and not very different masses and similar properties, so they can be considered

approximately

as identical.

This was the case of "the

nucleon" with the isospin degree of freedom corresponding + 0 n; the T-meson with the three states ~ , ~ , ~ . There is another degeneracy is expected)

for most particles.

the two values of the corresponding

and "antiparticle". conjugated, (examples:

p%, M~V

relativistic

of

%'

Such pairs are called charge-conjugate

Particles with all charges zero are called self-charge-

Elements of the enveloping

P0

and

degree of freedom are labeled "particle"

algebra

E(P 0)

observables which we will call the kinematical

The

(m,j)

indeed there is no degeneracy under charge conjugation 0 0 0 0 0 0 T,~ ,q ,p ,~ ,~ ,x , etc.).

The

p

It has been found (or it

that they come in pairs with the same representation

but all charges are opposite within the pair. pairs,

to the two states

Lie algebra is (with

C

for them

of the Lie algebra of

observables

P0

are

of the particle.

M v = -Mvv)

[iPU,ip ~ ] = O, [iPh,iM ~v] = iP~g h~ - iPWg I~

(4.14)

[iMVV,iM p°] = iM~Pg v° + iMV°g ~p _ iM~°g vp _ iMVPg v°

(4.14 t)

are the self-adjoint angular momentum.

operators

Pauli

Nat. Acad. Sci., (1967)) considered in

on

~

(unpublished)

representing

energy momentum and

and Bargmann and Wigner

(Proc.

E(Po):

1 1 M~VP p = (*M Wh = - ~ s~h~p P~Mvp = - 2 sharp " P)I

(4.15)

which satisfies [WX,P ~] = 0, [Wh,W ~] = i ¢ % ~ P P

W v 0

,

(4.16)

[WI,M ~v] = iW~g Xv _ iWVg %~

(4.161 )

P • W = PhWh = 0

(4.17)

Note that

110

The center of

E(P 0)

is generated by

P$ = PhWx

and

2W

= WXWh.

For irrep of

~0

these operators are the following multiples of the unit: m > 0,

p2 = m 2 i, 2W

m = 0

lla

m = 0

lib

or llc

4.5.

= -m2j(j + i)

p2 = 0,

W$ = 0

W = %P, % helicity

p2 = 0,

W 2 = -E < 0

Particle Polarization

In most experiments, the energy and momenta of the particles are measured (i.e.:

monokinetic beam, target at rest, bubble chamber track curved in a magnetic

field).

The additional observables to be measured in order to have a complete

knowledge of the particle state are called "the polarization". with

P, it is the polarization operator.

mute with each other.

However, the

W

Since

W

commutes

components do not com-

Equations (4.14), (4.14'), (4.15), (4.16), (4.16') show that

(4.18)

P0' PI' P2' P3' W3' W2 generate a maximal abelian subalgebra of observables of a particle

(m,j)

E(~0).

It is easy to interpret t the

when one remarks that

~(m,j), the one-particle

space of states, is a direct integral

3C(m,j) =

iO

Kj(~)

d3~ 0

'

(4.19)

p of

2j + 1

dimensional Hilbert spaces

K.. Given 3 tetrad of vectors in the Minkowski space (i.e., in

~, introduce an orthonormal

T')

_n(0) = £m-l,~(i)(i = 1,2,3,~,B = 0,i,2,3)

(4.20)

Minkowski scalar product n(~),n (B) = g

,

(4.20')

right hand orientation e%~Pn(~)n(B)n(V)n A

~

~

(~) =

p

_saBx ~

Introduce then the self-adjoint operators on S i = _ i n (i) . W(£) m

(4.20'') Kj(~) (4.21)

--

% See for instance L. Michel, N. Cim. Suppl., 14, 99 (1959) for more details and the treatment of the case m = 0.

111

where

W(£)

is the integrand of

f

=

Of course,

Equation

~(£)

d~ 0

(4.21')

p

(4.17) implies S O = _ ! n (0) . W(£) = 0 m

The S (i)

satisfies

the commutation

relations

[s(i),s (j)] = ieijkS(k) so they generate a = (m,0)

SU(2)

(4.21'')

--

Lie algebra,

,

(4.22)

that of the little group of

(particle at rest), S (i) = M0i(~).

~.

For

This shows the relation between angu-

lar momentum and polarization. For a particle of energy momentum R(~)

is a

rrR(~) = i, hermitian

which is transformed by

~, the polarization

density matrix

R*(~) = R(~) ~ 0, (2j + i) × (2j + i)

L 6 SU(2)

(defined by Equation

matrix

(4.22)) as

Lp = p, R(p) ~-> Q(p,L)R(p)Q*(p,L)

,

(4.23)

Let

R(/!) = (2j + i) -I i +

2j Z R(~)(e)

,

(4.24)

~=i be the decomposition The

R(%)(£)

and the

R(£)(£)

R(£)

into a sum of irreducible

1

= - ~ s

covariant

,R(Z)(£) = (-i)£ s mR

is a completely

s

SU(2)-tensor

of the polarization

can be given a completely

R(1)(~)

Where

of

are called the multipoles

matrix.

operators.

Note that

R(£)

form.

al~2"''eZ

W a ~ ~2. .W ~

(4.25)

syrmnetrical %th order tensor satisfying

al...~£ (partial trace) = SaBy... a This is obtained#

= 0, ~ s B ya. . " = ~ • s = 0

from the equivalent

form of relation

(4.26) (4.21)

W(~) = mZ.S(i)n (i)

(4.27)

i

From now on, we shall consider only the dipole polarization only one existing

for a spin 1/2 particle).

Its evolution

(which is the

is given in a macroscopic

# For more details see C. Henry and E. De Rafael, Ann. Inst. H. Poincar~, 2A, 87 (1965).

112

(N slowly variable in space time) electromagnetic field = EijkBk

and

*F

the polar tensor of = ~ M • s, with

- -

and

m

F

F i = Ei,F ij

(with

F) by the equation*

M = F + P ( ~

F + g'*F)P

,

(4.28)

- -

P = I - u g u = P

= ~/m, the quadri-velocity, which satisfies the

where

¢

Lorentz equation: N~ - -

The

F • u m

(4.281 )

- -

means the proper time derivative; M

and

F

are skew symmetric tensors so

(4.28) and (4.29) correspond to infinitesimal Lorentz transformations which of course preserve the Minkowski products: 2 = i, u • s = 0, 0 ~ 6 = (_ 2)1/2 ~ 1 where

~

(4.29)

is called the degree of (dipole = vector) polarization.

moment of the particle is dipole moment. g = 2

,

~ = (g/2)e/m(h = c = i)

The magnetic

and

~

= g

e/m

(For neutral particles, write directly

~

and

~.)

is a remarkable value of

g

is its electric Note that

which simplifies Equation (4.28).

This, as

first noted by Thomas in 1926, is characteristic of the Poincar~ group and does not happen for Galilean invariance

(as we commented in 2.6 and 2.9).

Consider a reaction between particles energy momenta are exactly known. polarization states is

where all

Then the transition probability between pure

% = II 2

S(PA,PA,PC,PD...) More precisely, S

A + B ÷ C + D + ...

E Horn 0CA g 4 '

~C ® ~

where the isometry

g "" ")

(4.30)

is the restriction of a unitary operator, defined on

Hilbert space of physics and called S-matrix in the physics literature.

X

the

In the

general case of given polarization density matrices X(pA,PB;Pc,PD,p...) where

RA,B(PA,PB)

@ KjB(PB)

= TrRc,D...(pc,PD. ..)SRA,B(pA,pB)S*

(reap., RC,D(Pc,PD.. .)

one of the particles, Equations

*

are hermitian operators** on

(reap., Kjc(PC) g KjD(PD) @ ...)

(etc.) when no polarization is observed.

which reduce to

(4.30') KjA(PA)

~(2JA + I)-I(2JB + I)-I

If one observes the polarization of only

(4.24), (4.25) and (4.30')show that

V. Bargmann, L. Michel, V. Telegdi,

,

%(pA,PB;Pc,~...)

Phys. Rev. Lett., 2, 435 (1959).

*T Practically, for all experiments, there is no correlation between the states of initial particles so RA,B(PA,PB) = RA(PA ) g ~ ( p B ).

113

depends linearly on the different polarization tensors of this particle (Sa,Sa~,Sasy,...) .+

4.6.

Invariance Under

p x Z2(C); PCT Theorem

If a physical theory is invariant under a group, say form the theory by an automorphism If

~

~ E Aut P0' (replace everywhere

is an inner automorphism, by definition of

theory is equivalent.

If

~

an active invariance of physical theories

P

and

T

invariance,

P, T

by

e(~).

the transformed theory

It is obvious that dilations are not

(except when only zero masses occur).

(and their product

and also under

g E P0

the transformed

If it is, then one can enlarge the invariance

group, in order to include this automorphism.

under

P0

is not an inner automorphism,

might not be physically equivalent.

What can be said about

P0' one can trans-

PT)?

We will assume invariance

C, the charge conjugation,

and in the next section,

see if these invariances are respected in nature. It seems a reasonable assumption that mutant of (4.5)).

A, the representation on We do know the action of

an antiunitary operator operator and well as

K

~ T

P

does not act on

on

A t , because

(see 1.2), i.e., by

is a complex conjugation

T

where

V(T)

leaves

U(A)

Note that if and

U(Z)

U = (U-I) *

is unitary, so is

KUK.

U(C), the charge conjugation operator on

tion

P0

C

L00

K (as

i.e.,

,

(4.31)

We assume that

T

globally invariant, but acts as an anti-linear automorphism.

Finally, we can introduce acts trivally on

physical properties of

is a unitary

(whose choice cannot be canonical).

KXAK = ~KAK, KABK = KAK KBK, K(A + B)K = KAK + KBK K 2 = ~.

(see Equation

has to be represented by

U(T) = V(T)K

U(T)) induces an anti-linear automorphism on the algebra

since

A ° , the com-

of the algebra of observables

C

and anti-commute with all charges.

tell

tially to internal symmetry.

us

how it must act on

At

~.

By defini-

More generally,

which corresponds essen-

Let D = Z2(P) × Z2(T) × Z2(C)

,

(4.32)

Pc = P x Z2(C) = P0 o D

(4.33)

In LM I, I gave the proof kindly tailor made by J. P. Serre for us physicists, (Theorem i, p. 183). H2(Pc,U(Z))

= H2(D,U(Z)) • 2U(Z) D

(see a similar theorem in Moore's lecture), where

2U(Z) D

,

(4.34)

is the group of the

t If the polarization of more than a final particle is observed one has also to introduce polarization correlations.

square roots of the unit of U(Z) invariant under every element of D. We check (iT (B+ ~iLi) ) (-i~ (B+ EiL i) ) that e = e is such an element so relation (4.9) is preserved. What is the extension in (4.34) chosen by nature? will see in the next section because

P, C, PC

phisms of the physical laws of nature.

Probably none, as we

(and probably

T) are not automor-

However, we can consider for

D

in (4.34),

a subgroup of that of (4.32). Let us first consider parity. deduced from those of

P0"

(0,0,±)

For

Irreps of

m > O, and

P+, (m,j,±)

or

tion to

P0

is irreducible.

(0,I~ I)

because their restriction to

P+ = P0 [] Z2(P)

m = 0, ~ = 0, there are two irreps of

with opposite parity (eigenvalue of For mass zero, ~ ~ 0 P0

are easily

irrep of

U(P)) whose restricP+

are denoted by

reduces to

(o,lxl)Po (o,1~1) • =

(O,-Ik I )

(4.35)

Note that, as projective representations of P0' (m,s,+) and (m,s,-) are equiva2 lent. More generally, since g ÷ g is a surjective homomorphism of U(Z),

H2(Z2(P),U(Z)) =

0.

So to speak of the parity of a state is not a canonical state-

ment; only relative parity can be defined for states in the same superselection sector.

By convention, the parity of the vacuum is taken +i, as well as that of the

proton, the neutron, the electron, the

A 0.

Wigner in his Istanbul lectures in 1962 (same reference as LM I) has studied the projective irreps of

P

(and even

P ).

This study can be easily

C

transferred to the study of the extension of

P

by

U(Z)

(and then by

U(As), from

general results of group extension by a non-abel±an kernel, as explained in LM I). This is not the case for

P

because

U(C), as unitary operator, does not act on

C

the phase of the projective representation, but trivally on

Z.

are canonical (since E 2U(Z).

C

as charge conjugation acts non-

We just give here the following results:

U(Z)

is divisible and

U(Z) ~

g ÷ g

U(T) 2, U(CPT) 2, U(PT) 2 2

is surjective) and are

For non-zero mass states, a choice different from U(T) 2 = U(PT) 2 = U(CPT) 2 = (-i) 2j

will require that irrep of

Pc

restricted to

P~Z2(C)

,

are not irreducible.

(4.36) This

would correspond to a new degree of freedom for particles which is not observed in nature.# In usual quantum field theories, relations

(4.36) are always satisfied.

This is related also to the two following theorems:

* See Wigner discussion in his Istanbul notes and for a recent review see H. Goldberg, N. C~m.j 60, 509 (1969).

115

Theorem 3. The good connection between spin and statisticst is a consequence of the Wightman axioms:it

covariance under

P0

of finite component quantum fields,

existence of vacuum, positivity of energy and "locality".

Theorem 4.

These axioms also imply invariance under CPT.ttt

4.7.

4.7.a.

How to Observe Violation

Action of

P~ T~ C

on Observables

Let us summarize in Table 4.2 the action of the automorphisms of

P0' on invariants of this group.

cles and antiparticles.

C

P, T, PT

acts trivally on them, but exchanges parti-

The self conjugated particles are eigenstates of

instance, consider quantum electrodynamics; C

C.

For

is an automorphism of this theory.

The electromagnetic interaction Hamiltonian is: Hem ='~ j~(x)A (x)d3~ By definition of

(4.37)

C,

utc)j~(x)U(C) -I = -j~(x) i.e., the electromagnetic current changes sign.

So

(4.38)

, H

em

is invariant under

C

if

also U(C)A (x)U(C) -I = -A (x)

(4.39)

i.e., integral (resp. half integral) spin fields describe particles which satisfy Bose = Completely symmetrical (resp. Fermi = antisymmetrical) statistics. This was proven by Pauli; his last publication in the subject is "Exclusion principle, Lorentz group and Reflection of space time and charge", p. 30 i__n_n Niels Bohr and the Development of Physics, Pauli editor, Pergamon, New York (1955). There he also proves the CPT theorem, first proven by LHders and Schwinger.

tt See R. F. Streater and A. S. Wightman, PCT, Spin and Statistics and All That, Benjamin, New York (1964); R. Jost, General Theory of Quantized Fields, American Mathematical Society, Providence (1965). ttt From weaker axioms (Haag-Araki theory of local observables), H. Epstein, J. Math. Phys., 8, 750 (1967) , has proven the CPT invariance of the S matrix. For infinite component fields, neither the connection between spin and statistics, nor the CPT invariance are implied by P0 invariance. For a counter example, see e.g., I. Todorov, 8th Nobel Symposium, Wiley (1968).

116

By definition, U(C)0> = 0>

where

0>

is the vacuum.

U(C)A (x)0> = -A (x)0> i.e., a photon has charged conjugation -i. formation of the electromagnetic F For

So ,

(4.40)

We have added in Table 4.2 the trans-

field

(x) = (~AA(X)) ~ = ~ A ( x )

T, time reversal, the space part

- ~ A (x) j

(4.41)

of the e.m. current

changes sign (as a velocity) while the time component

jP(x)

(whose space integral is the

electric charge) does not. variant), of

F 0i = E i

Hence the time reversal property of A , of H .° p em (electric field) and F IJ = Sijk Bk (magnetic field).

Consider Equation (4.28). transforms under pole) Equation with both

P

t

as

d/dt mu.

The quadrivector

and

mF • u = (-B "P,÷ - P 0 ~ _ p ÷ × ~ )

Hence, except for the term in

(4.28) is invariant under

P, T, PT.

(in-

The term in

gt

(electric di-

gt

is incompatible

T.

TABLE 4.2.

COVARIANCE UNDER P,T,C, OF THE INVARIANTS OF AND THE ELECTROMAGNETIC FIELD

Physical Observable

P0

P

T

PT

C

CPT

+

+

+

+

+

~'~

+

-

+

(pi,ej ,~,e~), (~,pj,~,~_~)

-

+

+

-

_

+

+

-

+

-

pj,~

~

(~,pj ,~,~_~),

helicity

(~,~_j ,~,~_~)

+

%

+

(electric field)

-

+

-

-

+

(magnetic field)

+

-

-

-

+

(a,b,c,d)

means determinant of the four components of four vectors.

4.7.b.

Parity Violation

The consequence of invariance under Consider

two s t a t e s

S 1,

I

S1

of a physical

through an "active" plane symmetry

~, and

P

is called parity conservation.

system corresponding t

$2, S 2

to each other

two other states of the same

117

system also symmetric to each other through be

the

respective

probabilities

E.

Let

of transitions

If an experiment I

EI12 = t 1 2 ,

2t12 = t12,

it

1 , a = ~(XI2 + 112)

=

I

(4.42)

I

yields

112 ~ 112, it proves parity violation.

means t h a t

t12

=

a + b,

Since

I

t12 = a - b, where

1 , is a scalar, b = ~(XI2 - I12 )

is a pseudoscalar.

So in a two particle decay of a polarized particle

the angular distribution

I

£+

£i + ~ 2 '

(4.43) (or more

P conservation

of decay products depends only on the even polarization

s B , s Bya , . . . . % In 1957, the following experiment was performed.

Co 60

nuclei at rest

(~ = m,O)

were polarized in a magnetic field B; this gives them a dipole polar±+ zation only: ~ = (0,1B). So P is a symmetry of Co 60 state P(m,O) = (m,~), P(0,XB)

= (0,1B).

electrons

Those nuclei decay spontaneously

of energy momentum

K(q0,$)

• ~ = -IB • q = -IBq cos 8. Similarly,

(B-

radioactivity)

with an angular dependence

emitting

proportional

to

This decay proved parity violation.

in the spontaneous ±

decay of zero spin

+ +p-+~¥

into a spin 1/2 p-lepton and a massless

~±

~

mesons

(

= m ):

,

(- for neutrinos,

+ for ant±neutrinos).

The p-lepton has a polarization s (which can depend only on the observed quanti2 P tatives ~ = ~ + ~ , pv = 0; remember sp pp = 0; see LM II).

S

=

$

m

--~

m 2 mp -

m f --m2

~

~

'

(4.44)

+

depends on the sign of the (electric charge of)

where also

P

violation

(by observation

of a pseudoscalar

~-.

p~ - s p

This proves

compatible with CP invariance. 2 satisfies s = -i

Note that in T-decay,

emitted in a pure helicity state suggest that

% See Equation s, s~, s~

~_ +

has helicity

requires

% = $i. +

for both

p

Then

P0

and

are

s (given --~ invariance

that the accompanying

All observation ~

and

the p-polarization

(complete polarization).

(through angular momentum conservation)

C

in the decay).

By the same type of argumentation we verify that those experiments

by (4.44t))

~$

is

on neutrinos helicity

~ . e

(4.25) and, at the end of 4.5, the property ....

for

X

I

112 = t r ~ R 2

112

generally in a decay where only two energy momenta are observed)

multipoles

I

and

1 ÷ 2.

Parity conservation ~ 112

I

112 = trRiR 2

to be linear in

118

This shows that the set of neutrino states in not invariant under C, and it implies that all reactions with neutrinos violate neutrino-less

(in fact, non-leptonic)

and

C.

or

But

decays of hyperons also violate parity.

Example: A 0 ÷ p+ + ~-, the angular distribution depends on (since

P

P

&A " ~

= -~A "

~A " ~ A = 0).

4.7.c.

Time Reversal Invariance

It would be better to call it "velocity reversal" since Let

SI

and

S2

(~/pO =

~)

T S1,

the

T

be two states of a physical system and

T S2

÷

- -v.

0

÷ ÷ (p0 a,+ p , p a~÷ - p ,

corresponding states obtained by a T active transformation + % ~+ %, etc.). s 0 ~+ s o ' ÷s ~+ -s, Then T T ~ XI2 = X21

(4.45)

Note the reversal of time ordering for the two transitions.

A precise experiment

comparing the cross section of the two inverse reactions

and

y + d+ ++ p+ + n is in progress.

+

+

,

(4.46)

(The rates are equal for pure states; since polarization is not

observed one has to divide the rate by the dimension of the polarization space for the particles

÷ (2 1/2 + 1) 2 = $, ÷ 2 × 3 = 6.)

Consider an elastic process -

+ p

+

÷ ~

-

+

p+

K. 3

(same initial and final particles)

such as

t

and compare the final polarization

s of the proton with the --p initial polarization of the proton target in another experiment. We must have ÷

÷

÷t

X(p~,pp

÷t

÷l

+t

÷ p~,pp,Sp)

+$

÷1

= X(-pw,-pp,-S

There is an approximate condition of

T

+

÷

p

+

-p

invariance,

,-pp)

(4.47)

in perturbation theory which is

based on the following expansion of the "S-matrix". S = I +iH where H).

H

+ 0 (H 2 )

has to be a self-adjoint operator

,

(write

(4.48)

SS* = S*S = i, in first order in

In this form, we have for orthogonal states (i.e., RiR j = 0) TrR.SR.S* N TrR.HR.H = TrR.HR.H 3 x 3 x x 3

,

(4.49)

i.e., in this approximation %.. = %.. x3 31 Then in this approximation,

(4.45) reads

T ~12 = %12"

(4.50) Even in this approximation

there is no positive evidence of violation of time reversal in physics, with perhaps the exception of K0-decay

(next section).

li9 Note that in Equation (4.28), using Table 4.2, the term in dipole) is not compatible with time reversal invariance

(or with

P

g'

(electric

invariance).

So the existence of an electric dipole for an elementary particle would prove violation of both I

P and 10-12.

T.

e gneutron

Experimentally

is known to be

~i0 -9

and

gelectron Note that

PT

invariance has a simple formulation. I

For example

I

h(Pl + P2 ÷ Pl + PC2) : h(Pl + P2 -~ Pl + P2 )

'

for spinless particles or for pure states (then change ~ ÷ -~, Sal..._ak ÷ (-l)ks l...~k)).

This is known in physical literature as the "principle of

detailed balancing". 4.8.

CP

CP

Violation

violation was first observed by I. H. Christenson, C. W. Cronin,

V. L. Fitch and R. Turley, Phys. Rev. Lett., 13, 138 (1964) in K0-decay.

Many

experiments have confirmed it. The state of a Hilbert space

K0

or

~ ( 1 ) = L2(R3,t ) @ K2 where

defined on the two element set while

P

PC = CP

is of the form so

(= anti-K 0, Y = -I) can be described by the

(PC) 2 = i

K2

i s the v e c t o r s p a c e of f u n c t i o n s

(Y = i, Y = -1).

P ® I

so

PC = P @ C.

Then

C

is of the form

We assume that

of

K's).

So we can write

JC(I) :JC~ 1) ~JC(I)_ ~

~+

(i) _

=

,

(4.51)

~ (i) ~±

Now it is easy to deduce the action of

(4.511 ) CP

on states of two

are two identical self-conjugated particles hence any state of C.

p2 = i, C 2 = i,

(as we have seen in 4.6, for spin 0, another assumption will

increase the degree of freedom

with value +i for

I @ C

The tensor product of the representation

270

0.

These

is eigenstate

(m,0)

of

P+, by

itself yields

®(m,0) =

~ ~=0

2

I2m

(m,~)dm

,

(5.52)

with symmetry I-I-] for even

~, ~

for odd

~

(4.52 I)

0nly the ~I] = symmetric states are allowed by Bose statistics. essentially similar to that yielding Equation (2.40), P Equation (4.51) by multiplication by 2~ 0

are eigenstates of

C, P, CP

(-i)~

And (by an argument

acts in the space of

in each direct summand.

with eigenvalue +i.

So states of

120

For states of

~+ + ~ , one has to consider these two particles as identi-

cal in order to apply Bose statistics, but in the two different possible states of + and -). So states of ~ ~ of total spin ~, are eigenstates of C, P,

charge (+ CP

with eigenvalue

(-i) ~, (-i) ~, i.

was predicted that states of states of

~C(I)

--~!)

of the two spaces

3~ 0

--~!i) and

CP

was believed to be preserved, it

in Equation (4.51) would decay into

would decay into

eigenvalue -i (as e.g. all

When

3~

states which are eigenstates of

states).

~(i)

2~

while

CP

with

This was exactly observed and the states

were also called "short" and "long" because the

"I"

2~-decay is faster. In 1964, the above quoted experiment proved that the long lived meson also decays into

2~

(with a rate

N 106 slower than the short lived).

We do know that the universe around our galaxy is not

CP

invariant, but

the influence of this asymmetry (which could depend on the relative velocity of the K-meson with respect to the galaxy, or the earth) seems to be ruled out by more precise experiments. Must we conclude that there is a small violation ((10-6) 1/2 = 10 -3 amplitude) of

CP

in the transition

K ÷ 2~?

in

Another possibility could be that

CP

is conserved in this transition but that the two observed meson with exponential decay:

short-lived

tively a large

KS

cS

and long-lived

and a small

cL

~

are non-orthogonal states with respec-

component in

K S ÷ 2~0 bS =

and

b~

(Ks ÷ 2~+ + ~-)

~il).

Then the branching ratio

~ ÷ 2~ 2 ÷ + ~ +

,

should be equal, since they would be the branching ratio of all the states in The value of

bS

is

N 1/2 (as predicted by the selection rule

The first measured values of another experiment.

b~

~i)

A~ = 1/2, see 3.6).

were around i0 to 12, but a value zero appears in

The present experimental evidence is still an incompatible set

but "optimists" say it is compatible with So it is possible that

CP

b L N b S ~ 1/2.

violation is due to a still undetected inter-

action, to which no particle transition or spontaneous decay can be attributed, and which has to be superweak. CP ~£ = ~

or

violation has also been observed in

~

÷ ~± + kS + ~

(where

e) decay; there is a relative difference of 3.10 -3 in the two

conjugated rates.

But

CP

C

or

violation has not yet been observed anywhere else.

Of course physicists have proposed many theories (about thirty not yet ruled out by the meager experimental data) to explain

CP

violation.

There is no

possibility to give more details here. To conclude, let us just remark that there is no evidence against violation and there is one fact which suggests that invariance than

CP:

a small upper limit of the

CPT

K 0 - -~ K

CPT

is a "much better" mass difference is well

CP

121

known.

It is

m

< 10-14ink .

- m

K0

Such a perfect equality cannot be due to

K--0

chance and suggests an invariance in nature which contains seen that with the

C, CT, CP CPT

are ruled out, so

CPT

C.

However, we have

is the likely candidate in agreement

theorem 4.6.

Remark on Galilean Invariance

We dealt in Section 4 with relativistic P~incar4 invariance only.

Although

we sometimes spoke in Chapters 2 and 3 of Galilean invariance, such invariance was not thoroughly used in atomic and nuclear physics. characterized the irreps of

E. In~nH and E. P. Wigner

3, the covering of the Galilee group, in 1952

(N. Cim.,

9, 705). Their results did not fit with physics.

V. Bargmann

i (1954)) shewed that for central extensions of the For each irrep of parameter

m

G

(Ann. of Math., 59,

Lie algebra

8, H2(g, R) = R.

G, this yields a family of projective irreps depending on one

which corresponds to the mass of the particle.

See also 0'Raifeartaigh's

lectures where it is shown that projective

irreps of an invariance group also appear in classical mechanics.

5.

THE INTERNAL SYMMETRIES OF HADRONS

5.1.

SU(3)

5.l.a.

Symmetry

The Octets

Table 3.2 of "elementary particles" in 3.5 is reminiscent of similar tables of atomic and nuclear spectra. So, before a dozen of baryons and as many mesons were known, physicists were searching for a larger symmetry than that of which we have described in 3.5.

U2

(isospin and hypercharge)

There is no point and no time to tell here about the

ill-fated choices except to mention that of Sakata, with a U(3) mental representation was spanned by Sakata, Prog.

group whose funda-

p, n, A, the first three known baryons.

(S.

Theor. Phys., 16, 686 (1956).)

Just as Heisenberg proposed to consider neutron and proton as two states of the same spin

1/2

particle, the nucleon, by neglecting their very small mass dif-

ference (or more precisely attributing it to an electromagnetic self-mass effect), the eight known spin ½ + baryons p, n, A 0 , Z-, Z 0, Z+, Z-, ~0 could be considered as eight states of the "same" particle although the mass difference is of the order of 15 percent instead of 0.15 percent.

"122

By 1961, seven pseudoscalar mesons

( 0 - ) were known, with the same group-

ing in isospin and hypercharge y = i, t = 1/2, K+KO; y = -i, t = 1/2, K-K0; y = O, +0 t = i, ~ ~ ~ but the spread in mass was much larger. M. Gell-Mann and Y. Ne'emann independently proposed to use SU(3) as a 1+ classifying group; the eight ~ baryons and, predicting a y = 0, t = 0 pseudoscalar meson which was discovered a few month's later and called mesons form two octets = eight dimension space

E8

0,

the eight

0-

of the adjoint representation of

SU(3) ([I] in Young diagram notation). For instance the Hilbert space of states of U i+ ~] i+ one baryon is the tensor product L(m, ~ ) @ K ( h _ ; ) where L(m, ~ ) is the space of the irrep SU(3)

l+

(m, ~ )

of

P, the Poincar~ group and

K(

)

the octet space

is an exact symmetry when the baryon mass differences are neglected.

E 8.

We can

say that strong interactions will be decomposed into two parts:

a strong

SU-3

in-

variant part and a semi-strong part invariant under the subgroup

U2(T,Y)

only.

This fits the reduction*

F

of su(3)Iu(2)° dim

8

=

(01) (1½) (yt) 2

But, would it be possible to consider the

+

i

+

SU(3)

a perturbation of the very strong interaction?

3

+

51)

2

breaking semi-strong interaction as Surely, if you are an optimist.

After all 15 percent (effect in baryon mass) is small compared to i. Let us now study the mass splitting within the SU(3)

5.l.b.

multiplet.

The Mass Operator

The simple hypothesis for the mass operator

M

is that it can be decom-

posed into M = M 0 + M~(y) where

M0

algebra of

is a "scalar" tensor operator and

,

(5.2)

Mr(y)

is the image of

SU(3)) by an octet = E8-tensor operator.

irreducible representation of

SU(3).

Because

Let

SU(3)

ly has two zero roots (which are zero weights for

E

y

(of the Lie

be the space of an

is of rank two, or equivalent-

~

)##

dim Hom(E @ E8,E)SU(3) ~ 2

For

u E U(2), the black column means

(det U) -I

(5.3)

while

r-1 U means

(det u).

t+ If %1 m %2 m 0 are the number of squares in the first and second line of the Young diagram of an irrep of SU(3), one also uses the notation (%1 - %2,%2 ) for the irrep of SU(3). The contragredient of (p,q) is (q,p), so (p,p) is self-contragredient,

as in

has for contragredient

~ ~

= (i,i), while denoted

J J l]

= (3,0) of dimension I0,

i-~ by the physicists.

~23

More precisely, is

1

it is

2, except for the trivial irrep, for which it is zero, and it

for the irreps whose Young diagram has only

equal lines

X1 = %2' (i.e.,

(0,%2)):

1

line (i.e.,

for example,

~

(%1,0)), or two

and its contragredient

-

which are also denoted This is also true for ~ ] = (2,0), ~

= (0,2)

i0

= (i,0)

denoted

and

6

SU(3)

parameters

and

the

= (0,i)

denoted

3

and

U(2)

and weak interactions)

SU(3)

is an exact symmetry

the particle masses

Explicitly,

is the representation

~, the Hilbert space of hadronic

for any

p , p E K ~i ~2 operator T

and for any octet-tensor

K

depend on three

Ml(Y)).

octet-tensor

From 1.5, operators

(up to the factor states;

i) of

it satisfies

the space of an irrep of

SU(3)

in

= p + SD(x))p > ~i ~2 cos ~ + 8> sin ~, ~ = 8> cos a - i> sin a, where

SU(3)

of a 27-plet

0-.

q = 0, y = 0,

The mass formula could not apply to the known "octet"

was predicted.

figuration"

Indeed a

Cross-Sections

and Decays of Resonsances

invariance implies ratios of resonances decay rates

(measured by

the natural width and the different branching ratios) into lighter hadrons.

This

yields remarkably good predictions and explains strange facts such as the small branching ratio for the decay of

~

into

2w.

For two octet-particle reactions

A + B ÷ C + D, one can deduce that the

scattering amplitude belongs to the representation

(5.11) 8

@

8

27

8

1

symmetric which yields seven arbitrary parameters.

8

i0

i0

antisymmetric There are less in

8 ~ 8 + 8 @ i0.

to correct for the mass difference is not obvious and the predictions

The way

are not

spectacular. An anthology of original papers in

SU(3)

has been published by Gell-Mann

Eightfold Way, Benjamin, New York (1964). M. Gourdin, Unitary Symmetry, North-Holland,

and Ne'emann, The this subject by

5.2.

There is also a book on Amsterdam

(1967).

Geometry on the SU(3)-Octet#

We give here some geometrical properties of the adjoint representation of SU(3). We have defined in (1.18), product

(1.19), and (i.19 I) the

(x,y), the Lie algebra product

SU(3)

invariant scalar

x A y, and the symmetric algebra product

t Full proofs and more results are given in a preprint of L. Michel and L. Radicati, with this title. It also contains some generalizations to SU(n).

~i26

x V y

for any pair of elements

irrep of space.

SU(n).

x, y 6 En2_l, the real vector space of the adjoint

We restrict ourselves here to

Its elements can be realized as

3 × 3

n = 3

and call

E8

the octet

traceless hermitian matrices.

They

satisfy the equation x

3

- (x,x)x - ~

det x = 0

,

(5.13)

whose coefficients obey the relation

(5.14)

4(x,x) 3 ~ 27(det x) We find that 2 det x = ~ (x,x V x)

,

(5.15)

so (5.14) can also be written (x,x) 3 a 3(x,x V x) 2 Orbits of of real numbers x

SU(3)

on

E8

(x,x), (x,x V x)

is called a regular element of

(5.16)

are in a bijective correspondence with the pairs satisfying E8

(5.16).

When

(x,x) 3 > 3(x,x V x) 2,

and its isotropy group

Its Lie algebra is a Cartan subalgebra and it is generated by (x,x) 3 = 3(x,x V x) 2, x U(2).

is

x, and

U(1) × U(1). x V x.

When

is called an exceptional element and its isotropy group is

We will also call such

on only normalized vectors are the root-vectors.

Gx

:

x

a q-vector or a pseudo-root.

(x,x) = i.

Those vectors

r

We will use from now

satisfying

(r V r,r) = 0

Every pseudo-root vector is of the form q = ±/3r

V r

,

(5.17)

and also satisfies /~ q V q = ~q We call it positive or negative ( n o r m a l i z e d ) fx dx linear mappings a %- > x A a, a % > x V a.

(5.18)

q-vector.

We denote by

[fa'fb ] = fa A b, [fa'db] = da A b so for

Va, b

of a Cartan subalgebra

taneously on a basis and

da, we decompose

zk f

Cx, the

of the complexified

f(a)

rk

C . x The two eigenvalues of

Cx

and

on

Cx C

is left stable by

@ C~. x x ,6

qk = /~ rk V r k

rive unit pseudo roots of

d "a

are

the

(5.18)

+ i//3.

fa

Then (5.20)

i rk'a)zk = ~ 3 (qk'a)zk' k = 1 .... ,6

are the six unit roots of

x

can be diagonalized simul-

f'' ~ = i (rk'a)zk . k. =. i, a = 0 ' faZk .

wh ere

,

E8. m Since

= f'' • f~, d = d'' @ d a a a a a

daZk = ( r k V

fx' d

Then

(5.20 e)

are the three posi-

~27

q,

r2

-~

-r 1

r

q2

q3 r3

FIGURE 5. i.

1

- r2

Roots ± r and pseudo-roots q. = /~ r. V ri of a Cartan subalgebra. i l l The SU(3)-Weyl group 8(3) permutes the three qi"

Lemma

Every two-plane of odd function

(x,x V x)

at least a zero.

E8 x

contains at least a root. on the unit circle

Indeed, the continuous

(x,x) = i

of the two-plane has

There are linear manifolds of root vectors.

For example: Lie subalgebra of

of

SU(3)

given a pseudo-root

q, and using the same notation for a

and its vector space (subspace of

E 8 = Uq(1) @ SUq(2) @ U2(q) ~

E8)

,

where the three- and four-dimensional vectors.

SU (2) and U2(q) ~ spaces contain only rootq An octet of particles form an orthonormal basis of the complexified ES,

which diagonalizes the

fa

for all

a E C(y,q)

the Cartan algebra generated by the

hypercharge and the electric charge directions since c SU(3).

1 Q = T3 + ~ Y

among generators of

U(2) c SU(3)

are generators of

,

U (2) Y

(5.21)

is translated in the octet geometry; y, -q

unit positive pseudo-roots, Q = - 2 / ~ = F(t3).

Y, Q

The Gell-Mann-Nishijima relation

F(q), Y = 2//~

F(y),* t3

are

is a root, T 3

We give in Figure 5.2 the corresponding roots of the two lowest octets of

particles and also the weight of the lowest decuplet of baryons.

The factors 2 / ~ are found from the condition that the spectra of Q and Y are the set of integers. Equation (5.21) implies that q and y are normalized pseudo-roots of opposite sign. The choice of sign here +y, -q is conventional and corresponds to Figure 5.2.

128

AY

+

n

Ko

p

0

-

E0 A0

E-

K+

A+ ~r

0

s

A0

A+

A++

Q

•

•

-* E

)

E°*

i

-~

=-

"~0

FIGURE 5.2.

~+~ ~0.

~0

K

ROOTS OF OCTETS OF PARTICLES AND WEIGHT OF THE DECUPLET E0 = t3 = ~0A0 = Y = ~0' corresponds to the two zero roots.

5.3.

Electromagnetic and Weak Interactions in

5.3.a.

SU(3)

Electromagnetic Interaction

As shown by Equation (5.21), the electric charge operator tor of

U(2) c SU(3), so it is also a generator of 2 Q = ~F(-q)

SU(3)

Q

is a genera-

and as we have seen

,

(5.26)

where, as we have seen, q

is a pseudo-root. The SU (2) is called the U-spin group q in the literature, and we can speak of U-spin multiplets, which have the same elec~- , E- , u = i; ~, _0 trie charge u = 1/2, p+, E+ and also ~ z , 1/2 E0 + ~ / 2 A0; u = 0, ~/2

E0 _ 1/2 A 0.

electromagnetic

The electric charge is the integral of the time component of the

current Q = e[j0(x)dx

and

3/8~

course

j~(x) = 0 = Q

j0(x)

is a constant

could have any

of the non-octet part vanish. magnetic current

j~(x)

SU(3)

,

(5.27)

(more generally

P

invariant)

operator.

Of

covariance, with the condition that the integrml

The simplest hypothesis is to assume that the electro-

is the image in the direction

-q

of an octet-tensor

operator,

2 j~(x;-q)

e ~ (compare with Equation

(5.26)).

(5.28)

This allows us to draw many conclusions.

The mag-

netic moment of the particle of a multiplet is given by the expectation value of an octet-tensor operator in the direction

-q.

It thus depends on two constants only

for an octet (one for a decuplet) and the particles of the same u-spin multiplet have the same magnetic moment.

For example one predicts

~E+ = ~p+

which is well-v~ified.

Measurements of + ~

~ and ~ _ are in progress, as well as the rate of E0 ÷ A 0 A 0' ~ + E which is related (as a "magnetic dipole" transition) to the values of ~ in

this octet. example:

The ratio of rates of electromagnetic decay can be predicted.

For

129

rate n0 ~ 2y 0 rate n ÷ 2y

(t3 'q) 2 x

ratio of phase-space = 3 × ratio of phase-space

,

(5.29)

(y,q)2 +

(using (5.26) and

(Y,t3) = 0).

The observed ratio

good confirmation of the mixing angle.

~ ÷ ~+ + ~-, w ÷ ~

+ ~

is a

Finally ratios of photo production cross-

sections can also be predicted successfully. The mass differences inside a tromagnetic origin.

U (2) multiplet are thought to be of elecY They are quadratic in jia(x;-q) but to a good approximation it

seems that only the scalar and octet part are important, so to a good approximation the mass operator (5.5) can be written, when one adds electromagnetic effects, 2 M = M0 + M I ~ and inside an SU(3)-multiplet

2 F(y) + M2D(Y) + M 3 ~

F(-q) + M4D(-q)

the values of the masses are given by

m = m 0' + mlY' + m~(t(t + i) _ ~i y2) + m3qI" + m4(u(ul + i) - 71 q2)

(5.30)

which is well verified for baryons.

5.3.b.

Cabibbo generalized to

SU(3) +

the Gell-Mann Feynmann hypothesis_on the

via(x)

vector part of the weak current 2) by the assumption that

Weak Interaction

v±(x)

operator (that we shall denote

and

coupled to the leptonic current j~(x)

,~,+ia(x) (see

are images by the same octet-tensor

v (x)) of three different directions:

-q,c+.

Ex-

plicitly 2 em current = ~

ev (x;-q)

,

G weak current = ~via(x;c_+)

,

(5.31)

(where

G

is the Fermi constant).

vector parts of the weak current in the same direction

The second Cabibbo assumption is that the axial+ a-(x) are images of another octet-tensor operator,

c_+. The total weak current + + _+ h~(x;c_+) = via(x;c_+) - aia(x;c_+)

,

(5.32)

is thus also image by an octet-tensor operator. See Cabibbo's original paper (Phys. Rev. Lett., iO, 531 (1963)) in The Eightfold Way anthology (p.207) for the predictions. The

±

subscript corresponds to the electric charge of the current, i.e., [Q,h~(x)]

and using the fact that

Q

is an

this equation in the form (1.9)

SU(3)

= -+h~(x)

,

gener.ator, Q = 2 / ~ F ( - q ) ,

(5.33) we can write

~3o _ 2__ /3 [F(q),h (x,c±)] = - ~2

h (x,q A c ±) = ih (x,e±)

,

(5.34)

from (5.34) we get

/f c± q A c± = ¥~-which means that tions

e±

are eigenvectors of

(5.31, 5.34 S) imply that

root-vectors,

eI

and

as we have seen in (5.21).

= c2, q A c 2 = -e I

F(q). e2

(5.34

Writing

e± = i//2(c I i ic2)

are unit vectors

S)

Equa-

E Uq(2), so they are

Equation (5.34 t) is equivalent to

q A c1

which in turn implies ~c

I V cI = ~c

2 V c 2 = / ~ c 3 V c3 = c

,

(5.35)

where c3 = c I A c 2 This means that pseudo root

e

c, Cl, c2, c 3

span

Uc(2) ; note also that

1/2.

-

It commutes with

However, it does not commute with

violating hypercharge conservation. # - 1/2

The It is a

q, c A q = 0 = (c,q)

y; indeed, there are weak transitions

This lack of commutation is expressed by the

value of 3 (y,e) = i - ~ sin2e

where

c, e 3 E Uq(2).

is called the "weak hypercharge" or "Cabibbo hypercharge".

conserved quantity for weak interactions. =

(5.3J)

(5.36)

As we have seen in 3.6, its experimental value is ± + 15 degrees and it is rather well verified that v and a- define the same direc-

tion

e

,

c

is the Cabibbo angle.

of weak hypereharge, t

The value of this angle is empirically given by tg0 = m / m k

Cabibbo's theory not only explained the relative slower ratett transition violating the hypercharge AT = 0

nuclear B-decay were slower than the The "computation" of this angle

lems of physics. q

as function of

(by

tg2e) of the weak

y, but also explained that the super allowed

6

~ + s + ~ + ~

decay by a ratio

cos2B.

is one of the challenging present prob-

It is worth while to point out a purely algebraical relation, giving y

and

c.

Given two non-commuting

(normalized positive) pseudo-roots

y

and

e,

there is always a unique pseudo-root which commutes with both of them Xq = ~ y

1 V e + ~(y + c)

,

(5.37)

t To be more precise, the angle of cv and c with y is the same but cv and c a could be at a small angle between each o~her and this has been exploited as a possible explanation of CP violation. ttTo be accurate, it is not the rate but the probability transition = rate/phase space volume, since the phase space volumes, which should be equal in an exact SU(3)-symmetry, are in fact unequal.

where

=-(i-

(y,e))

(5.37 S )

The most commonly proposed form of non-leptonie weak interaction is HN.L

= G •

with the drawback that

E

/~=±i

~.L.

[h~(x,c )h (x,c J

~

~

)d3~

(5.38)

-~

,

is the image of a reducible tensor operator with some

component in the "27" irrep of

SU(3).

The

~T = 1/2

rule when

I&YI = 1

for those

weak transitions suggests that this 27 component is negligible compared to the octet component.

The proposal of Radieati t =

makes

~.L.

(h~(x) V h (x))(c)d3x

the component along the weak hypercharge

tensor operator.

5.4.

(5.39)

c

of an irreducible octet-

It is compatible with the known experimental data.

Critical Orbits of a G-Invariant Function on a Manifold

Given a group

G

acting on a set

M, the set of all points of

have conjugated little groups is called a stratum. orbits of the same type.

Mtt

M

which

So a stratum is the union of all

Inclusion gives a partial ordering of all subgroups, modulo

a conjugation, of a group.

It corresponds to an (inverse) ordering on the strata.

The set of fixed points form the minimal stratum (maximal isotropy group = G). the action of

G

on

M

If in

there are no fixed points, there might be several minimal

strata. For example, in 5.2 we have seen that in the action of

SU(3)

on the unit

sphere

S. of the octet space, there is the open dense general stratum l(x V x,x) I J < i//~, containing a one parameter family of six-dimensional orbits (little group

U(1) x U(1) = ± i//~.

and a minimal stratum made of two four-dimensional

orbit

(x V x,x)

In this paragraph we want to consider a) the smooth ttt action of a compact Lie group This action is given by the smooth mapping G x M ÷ M

with

b) a real smooth function

~(gl,~(g2,m)) f M ÷ R

function is constant on every

on a smooth manifold

(= manifold morphism)

= ~(glg2,m)

which is G

G

G

orbit of

,

invariant, that is, the M

t

L. Radicati in Old and New Problems in Elementary Particle Physics, Academic Press, New York (1968).

tt

This part is entirely a common work with Radicati, partly published in Coral Gables Conferences 1968, partly circulated in a preprint.

t#t We use the word smooth for infinitely differentiable.

M.

q32 g E G, m E M, f(~(g,m)) = f(m) The differential at

mI 6 M1

d~ml ; it is a linear mapping (with

m2

of a smooth mapping =

TmI(M I)

MI ÷ M2

is denoted

P(ml) ) . d~m 1 >

(5.40)

Tm2(M 2)

where

Tm.(Mi) is the tangent plane of M. at m°. So df E T S (m) the dual i l i p p vector space of T (m). We call critical point, the p E M such that df = 0. P P The stabilizer (= little group = isotropy group) G in m E M is a m closed and therefore compact subgroup of the compact group G. As is well known,* one can choose local coordinates in a neighborhood of

G

is linear.

P resentation of

V of p such that the action P E (M) be the vector space corresponding to this linear repP Vp c EP (M). Since GP is compact and M real, this linear

Let

Gp; so

action can be made orthogonal so dfp

with a vector of

is the image of denoted group

Ep(M)

g ~(P)~

E (M) is a euclidean space. We can then identify P that we shall call (grad f)p. The G-orbit of p,G(p),

~(g,p); it is a submanifo!d of

T (G(p)), is the image of d~ [p) p e transforms G(p) into itself.

G

where

e

M; its tangent plane in

is the unit of

Similarly

Tp(G(p))

G.

p,

The isotropy

is an invariant sub-

P space of

Ep(M) •

Np(G(p)) = Tp(G(p)) ~ c Ep(M)

The orthogonal subspace

variant and it is called the "slice" at

is also in-

(grad f)p E N . Indeed, by P definition, for x E Tp(M), ((grad f)p,X) = lim[(f(p + ~x) - f(p))]~-l. The bracket ~+0 is 0 when p + ex E G(p), the orbit of p, so it stays zero at the limit, when x E T (G(p)). P Note also that

(grad f)p

p.

Note that

is invariant by Gp. Let gEGp; (g • (grad f)p,X)

= ((grad f)p,g-i • x) = lim - l ( f ( p + ~g-i • x) - f(x)), and since ~+0 f(p + ~g-i • x) = f(g-i .(p + ~x)) = f(p + ~x), so = ((grad f)p;X). (grad f)p = O.

If the slice

Np(G(p))

Vx E Ep(M),

g-i • P = P,

(g - (grad f)p,X)

has no vectors invariant by

Gp, then

We can summarize this by the:

Theorem 1

Let M.

If for

G

be a compact Lie group acting smoothly on the smooth real manifold

p E M, the canonical linear representation of

not contain the trivial representation of

* Consider a Riemann metric on averaging with a G -invariant and G transform~ into each neighborhood Vp of p, take

Gp, then

G(p)

G

on the slice N does P P is a critical orbit for

M; it is transformed by the action of G_. By measure, one obtains a G -invariant Riem~nn metric other the geodesics fromPthe fixed point p. In the geodesic coordinates•

133

any real valued G-invariant smooth function on the same symbol, e.g.,

M

(where here again we denote by

SU(2), the vector space of the Lie subalgebra, and also the

group!).

Example i. a unit q-vector, and

U2(q)

We have studied the action of

of

V

on

S~ c E 8.

Let

q

be

, Nq(G(q)) = SU2(q)

acts linearly on it, without fixed vectors.

Example 2. hood

SU(3)

Gq = U2(q) , rq(M) = {q}~ C E8, rq(G(q)) = U2(q)

p

~

is an isolated fixed point in

with no other fixed points and

P

N

M.

So there is a neighbor-

= E (M) P P

has no invariant

G = G P

vector. This proves that on

p

is a critical point for every

G

invariant function

M. We shall now assume moreover,

that

M

is compact.

stratum (called generic stratum) which is open dense in closed and compact. p E C, F

c E

Let

C

be a connected component of a minimal stratum;

be the linear subspace of

P P the points of V

N F

have

so for small enough

G

fixed points.

Because

P as stabilizer so they belong to

G

P P P invariant real valued smooth function n E F

Then there is one

M; the minimal strata are

f, let

n = (grad f)p.

leI, p + sn E C.

C.

G

let is maximal,

P Given a G-

As we have seen

We can write

P (n,n) = lim - l ( f ( p e+0 so if

f

is constant on

constant on

C

C, every

p E C

+ sn) - f(p))

(5.41)

is a critical point of

f.

If

f

it has at least an orbit of maxima and an orbit of a minima.

a point of such an orbit, and

n = (grad f)p.

f(p + sn) - f(p) which means that

(n,n)

0 if ~ 6 ~ if

f f

either has the sign of

Then, in Equation is minimum is maximum ±E

at

is not Let

p

(5.41),

p

(+ at minimum, - at maximum)

which is impossible, or must be zero.

Theorem 2.t

Let fold

G

M, and let

be a compact Lie group, acting smoothly on the real compact manif

b e a real valued G-invariant, smooth function on

has at least a critical point for each connected component

C

M.

Then

f

of each minimal

stratum.

To prove this theorem, that Radicati and I conjectured, we received great help from A. Borel, C. Moore, and R. Thorn.

434

We will now be interested in a particular function on a sphere: be a compact Lie group,

E

G

the real vector space carrier of a linear representation

g ~-+ R(g), irreducible over the reals.

So

R

(up to an equivalence)

gonal representation and it is self-contragredient. variant Euclidean scalar product in

E.

dim(Hom E V E,E) G = I.

tensor product)

let

We denote by

Let us assume that (with

is an ortho-

(~,~) V

the in-

the symmetrical

As we have seen in 1.5, there is a unique

(up to a constant factor) symmetrical algebra x @ y --+xTY with

P E Hom(E V E,E) G

where

(5.42)

xTY = yT x. Since the representation is self-contragredient

and the tensor product is

associative (xTY,Z) = (x,YTZ) = {x,y,z} Hence, the invariant on

E.

Let

{x,y,z}

f({x,y,z})

Using

%

(5.43)

is a completely symmetrical G-invariant trilinear form

be a function on the unit sphere

as a Lagrange multiplier,

S = {x E E,(x,x) = i}.

critical points of

f

are given by the

equation grad(f({x,x,x}) where

ft

fs = i.

l

+ %(1 + (x,x)) = 3f XTX - 2%x = 0

is the derivative of the one variable function In other words, critical points of

f

XTX = %x i.e., the idempotents

(or nilpotents for

5.5.

f; e.g., if

f = {x,x,x},

,

% = O) of the symmetrical algebra.

SU(3) × SU(3)

Symmetry

SU(3)

for the hadronic

Of course they are coarser, but still useful as we shall see.

× SU(3) 0-

(5.44)

are given by solutions of

Physicists have considered symmetries higher than world.

,

The

SU(3)

symmetry becomes an exact symmetry of the hadronic world when the masses of

mesons are neglected.

Note that it is not much more drastic to say that those

masses are equal to zero than to say that they are equal as is already implied by SU(3).

As a matter of fact, a much milder approximation than

SU(3)

is to neglect

only the T-meson mass (only 140 MeV, and this is smaller than the 0--meson mass differences).

This corresponds to a

SU(2) × SU(2) x U(1)

subgroup of

SU(3) x SU(3).

We give in Figure 5.3, a scheme of the lattice of symmetry groups which have been considered for hadronic physics, but in this section we limit ourselves to x SU(3) metries.)

and its subgroup.

(See also O'Raifeartaigh lectures for the higher sym-

At the level of the middle line of Figure 5.3, a new feature appears; a

mixing of internal symmetry and relativity invariance. x SU(3)

SU(3)

since it concerns only the parity operator.

consider is the semi-direct product by

Z2(P)

It is very mild for

SU(3)

The total symmetry group to

135

(P0 x SU 3 x SU3) [] Z2(p ) which acts naturally on

P0

distinguish such

factors, let us denote them as

SU(3)

and exchanges the two

,

called in the physics literature the

±

chirality group.

frame for understanding the relation of interactions.

SU 3

P

is the

SU(3)

factorst int SU 3 x SU3. SU~ +) x SU~-); they are

The group (5.49) is a good

The diagonal subgroup

group of invariance of 5.1, 5.2, and 5.3.

SU(6,6)

143

SU(6) x SU(6)

70

/,,, SU(6)~

To

(parity operator) with the different

This will become clear in the following.

SU(3) d C SU +) x SU

(5.49)

~ x

SU(3)

SU(3)

35

SU(2) x SU(2) x U(1)

16

8

7

"-,7 U(2)

FIGURE 5.3.

SYMMETRY GROUPS AND THEIR DIMENSIONS Lattice of symmetry groups used in hadronic physics. + means injection as subgroup.

We will denote a vector of the 16-dimensional vector space SU(3) x SU(3)

El6

of the

Lie algebra by a direct sum of two vectors a = a+ @ a_

a_+ belongs to the

SU~+)-

,

(5.50)

octet.

The invariant Euclidean scalar product (given by the Cartan-Killing form) is, in terms of the octet scalar product ~ = (a+ @ a_,b+ @ b_) = ~(a+,b+) 1 + l(a_,b_ ) (a,b) The Lie algebra law is (we use

~

for it)

A ~ = (a+ A b+) @ (a_ A b_) and since

dim

(5.51)

I-Iom(E16V E16,E16) SU3

× SU3

,

= I, there is a unique canonical symmetri-

cal symmetrical algebra a V ~ = (a+ V b+) @ (a_ V b_)

(5.52)

136

The covariance property of the electromagnetic and weak interactions are most naturally extended to

SU(3) x SU(3)

by the following hypothesis:

the electric

" "tS)(x) a£C)(x) of the ' the axial vector part p" current j~(x), the vector part vp h(~)(x) = v ~ ) ( x ) - a~)(x) are images of (charged ~ = ±i) weak hadronic current P the sam~

El6

correspond to

tensor operator, which we will denote SU(3) d

current has pure

h (x;~); the vector currents

and the axial vector current to the anti-diagonal.

"-"

chirality.

Explicitly: 2

The weak

electromagnetic current

ehp(x;-(q @ q))

,

(5.53)

(charged) weak currents ~(h

(x;0 @

C1)

+ ih (x;0 @ c2))

(5.53')

and the Radicati form of the (non leptonic) weak hadronic interaction is ~lh The generators of

(x) V h~(x))(c)d3~ = ~-I(h G2 (x) ~ h~(x)(0 @ c)dBx SU(3) x SU(3) a

is the representation (up to

(5.54)

are the space integral of the current, i.e., F(~) = Ih0(x;a)d ~ 3÷ x

i) of the

,

(5.55)

Lie algebra on the Hilbert

SU(3) x SU(3)

space of physics [F(~),F(~)] = iF(~ A ~)

,

and for the particular case of the El6-tensor operator

(5.56) (hp(x)~)

[F(~),hp(x,~)] = ih (x,~ A ~)

(5.57)

as we saw in Equation (1.9). In the approximation where = 0

and the

F(~)

SU(3) x SU(3)

are well defined.

Since

is an exact symmetry

SU(3) x SU(3)

the usual difficulty to define the self-adjoint operator

~ph (x,a)

is a broken symmetry,

F(~)

arises.

(See

O'Raifertaigh's lectures, this Volume.) The equation f a V a = ~a for unit vectors = ±c S 0 h = +#2/3.

or

(5.58)

has two sets of solutions.

6 S15 c El6

±0 • c, where

,

c

One is the set of

i//~

is a (normalized positive) pseudo-root and

This set is made up of two minimal strata, each consisting of two pieces

of one orbit each. SU(3) x SU(3)

So each of the four orbits is a critical orbit for every smooth

invariant function in

are, up to a conjugation,

S15

SU~+)xt U(_)(2) J

C

the unit sphere of and

The stabilizers

U(+)(2) x SU~)(~ for the two strata. C

t See L. Michel and L. Radicati, preprint, Breaking of the

Hadronic Physics.

El6.

SU(3) x SU(3) Symmetry in

~37 The other type of solution is the set of vectors, ±(ql @ q2 )

'

which form two orbits of a four separated orbit stratum other orbits) whose stabilizer is of the diagonal diagonal

su(d)(3)

(±q • Sq)

(iql ~ +q2

(Uql(2) x UqI(2))~Z2.

for the two

The pseudo-roots

i(q @ q)

are on the orbits of solutions while those of the anti-

are not.

This has a bearing on parity.

It seems to us remarkable that the electromagnetic charge direction -(q @ q)

and the weak hypercharge direction

(0 @ c)

give two solutions, one of

each type, of Equation (5.58). SU(3) x SU(3)

is not only broken by electromagnetic and weak interaction,

but also by semi-strong U2-invariant interaction.

There are two different interest-

ing intermediate approximations of symmetry of strong interactions between SU(3) x SU(3); those of the fourth line of Figure 5.3,

SU(3)

U2

and

already studied, and

SU(2) × SU(2) × UI, which implies the Adler-Weissberger sum rule, and more recently emphasized by Gell-Mann, Oakes and Renner.

In both cases

H

strong

is, to a good

approximation, the sum Hstrong = H 0 + HI(~) of

H0

invariant under

SU(3) × SU(3)

SU(3) × SU(3)

and of

,

Hl(m)

(5.59) which is the image of

tensor operator for the (irreducible over reals)

resentation.

The two corresponding directions

~

tion of

SU(3) x SU(3)

model) is such that

on

~18

I refer to my

This 18-dimensional irreducible real representa-

(which is the one which naturally arises in a quark

dim Hom(El8 V EI8,EI8 )SU(3) × SU(3) = i

canonical symmetrical (real) algebra on automorphisms.

by a rep-

for these two approximations are

again idempotent or nilpotents of the canonical symmetric algebra. preprint with Radicati for details.

m

(3,5) @ (~,3)

El8

We denote this algebra law by

which has

so there is a unique

SU(3) x SU(3)

as group of

~i ~ ~2"

The equation ~ ~ = %m

,

(5.60)

has only two types of solutions (for vectors on the invariant unit sphere belonging to two minimal strata, the one for stabilizer.

The other, for

~ = 0

I%1 = 2/3

corresponds to

Theorem i shows that this latter case orbit for all 33 + 33

SU(3) x SU(3)

corresponding to

SU(3) d

S17

(unit vectors of the

(x,x ~ x)

The stratum

is also minimal; it is a nine-dimensional connected sub-

S17) made up of eight-dimensional orbits.

So from Theorem 2, each

invariant function has at least two critical orbits in this stratum. tions of

as

corresponds to a critical

irrep); this orbit is also a minimal stratum of dimension 9.

manifold (of

sud(3)

SU~+)(2)x S U y ( 2 ) x U~(1).

(% = 0)

invariant functions on

corresponds to

S17 c El8 )

these two orbits are

x ~ x = ± 2/3 x.

For all func-

138

Note Added After the Seattle Rencontres.# used in Seattle, as emphasized by Equation (5.49), SU(3) x SU(3). functions.

Then, the two orbits on

I do not understand why I have not (SU(3) x SU(3))DZ2

S17 , x ~ x = ± 2/3 x

instead of

are critical for all

Radicati and I also wonder why we have not considered before the groups

(SU(3) x SU(3))D(Z 2 x Z2)

where the discrete group

~Z2 x Z 2 = {I,P,C,PC}

ted by the parity and the charge conjugation operators.

for this group, there are four which contain only one orbit. critical ones of

S15.

Typical points

(2

is genera-

Among the strata of

S15

These orbits are the

unit vectors up to a sign) of these orbits

ar e ±q = ±(q • q)

the direction of electromagnetic interaction,

% ±c. = 0 • ±c.(i = 1,2) 1 1

the Cabibbo direction of weak coupling,

±c = 0 • ±c

the "weak hypercharge" direction proposed by Radicati,

+~ = -+(r • e r ) ,

e = +i

(root vectors)

a direction which some M. L. Good, L. Michel, Rev., 151, 1199 (1966), proposed theory of the

(5.53) (5.53')

(5.54)

authors (for instance and E. de Rafael, Phys. have used in their CP violating interaction.

Radicati and I have also included Theorem i into a more complete:

Theorem i'

Let p 6 M.

G

be a compact Lie group acting smoothly on the real manifold

M,

The three following propositions are equivalent. a) the orbit of function

f

p on

b) the orbit of Vp

of

p

is isolated in its stratum, i.e.,

5.6.

such tha~

~

6 V

~

a neighborhood

x ~ G ) = G x is not conjugate to Gp, P P c) the canonical linear representation of G on the slice N does not P P contain the trivial representation. Theorem i is simply

p

is critical (for every G-invariant real valued smooth M, dfp = 0),

and

c = a.

SU(6)~ Quarks~ Current Algebra~ Boot-Strap~ Etc.

The title of this section is a statistical sample of key words found these last years in papers on fundamental particle physics. ##

This last section is not a

After the Seattle Recontre, L. Radicati and I collected the above results to present them in a lecture on September 19, 1969 in Rome (see preprint, Geometrical Properties of the Fundamental Interaations). The following improvements were then obvious to us. tt For the last year, the passwords are Veneziano and duality. fact that there are fads in fundamental particle physics.

It is a sociological

~39 conclusion but an open-end to the description of a very rapidly changing situation; the view that physics gives us of the hadron world.

SU(6)

Symmetry.

SU(6)

1756 (1964), for mesons only). SU(3)

Symmetry was introduced independently # by GHrsey

(Phys. Rev. Lett., 13, 299 (1964)) and by B. Sakita (Phys. Rev., 136 B,

and Radicati

It was noticeable that mass-differences between

multiplets were not larger than those inside multiplets. Both groups of authors, inspired by the

SU(4) = supermultiplet Wigner

theory for nuclei (3.3) extended it to fundamental particles by enlarging the isospin to

SU(3).

SU(2)

So in the non-relativistic version, the space of the one particle

hadron states is the tensor product ~C(I) = L2(R3,t) @ K here

K

and

the action of ~(i)

Kh

@ K%

;

are respectively, two- and three-dimensional Hilbert spaces and

G, the central extension of the Galilee group, and of

are respectively, (with

G ~ SU(2)

SU(6)

on

also (2.9) and Equations (2.57) and (2.5~ff

~(i) = L2(R3,t) ~

Ko @ Kh

g E G ÷ ~(g) @ ~(g) 8 1

(5.61)

u E SU(6) ÷ I @ u The lowest two multiplets of baryon, it belongs to the irrep

~

SU(6)

are given in Figure 5.4.

of dimension 56;

For the

for the meson, to the

~, I.I

the 35-dimensional adjoint irrep of singlet.

SU(6).

The

X0

(not discovered in 1964!) is a

We give here the decomposition of these irrep into

SU 2 × SU 3

irrep

m

rrm

SU2 × SU 3

= [] × [3-'@ ,,,, x 2 × 8 + 4 × i0

=

56

SU 2 × SU 3 (1 × 8) + (3 × 8) + (3 × Z)

The mass formula for each

SU(6)

=

35

multiplet becomes

i i m = m 0 + mlY + m2(t(t + i) - ~ y2) + m3 j(j + i) + m4q + m5(u(u + i) - ~ q2)

# In fact, Gell-Mann in, Physics, !, 63 (1964), page 74 (reproduced in, The Eightfold Way, anthology, p. 203), was the first to introduce SU(6) in the physics of elementary particles but, for once, he did not work out its physical applications. t? For more details, see L. Michel, "The Problem of Group Extensions of the Poincar& Group and SU(6) Symmetry', p. 331; 2nd Coral Gables Conferences, Syn~etry Principles at High Energy, Freeman and Co., San Francisco (1965).

140

-2 BeV

Baryons

Mesons

E* ~-~0 E- X0 X+

i BeY A0

. . ° o

~*

W

p- p

K-K--0

-

-

0

+

P

Ko K+

D

o

FIGURE 5.4.

K*

X

np

+

THE (8 x 2) + (i0 x 4) = 56-PLET OF BARYONS AND THE (8 x I) + (9 x 3) 35-PLET OF MESONS IN THE SU-6 CLASSIFICATION OF HADRONS.

=

Neglecting the electromagnetic mass difference

(m 4 = m 5 = O)

formula predicts well the masses of the eight lowest

U(2 )_

The magnetic moment of baryons depends on only one parameter

the four-parameter multiplets of baryons. ~

so we have the P

relation 2 ~n = - ~ ~p

'

(5.62)

which is within 3 percent of reality (this is too good!). How should one apply

SU(6>

invariance to particle reactions?

physics and empirical rules (e.g., so called

Some

SU(6)w ) have to be injected, and the

symmetry is still useful. However, the drawback is the difficulty in reconciling

SU(6)

with rela-

tivistic invariance. *

Quarks.

It is a natural tendency in science to try to explain the uni-

verse by the smallest number of different types of building blocks, such as the four elements of the Greeks, which at the end of the XIXth century had reached nearly ninety chemical elements.

From 1910 to 1929 (measurement of the spin and statistics

This will be dealt with by O'Raifeartaigh, when he studies the two upper lines of the diagram of Figure 5.3.

141

of the

N 14

+ p , e-, y, were known and

nucleus, see 2.10) only three particles

needed to build the universe again.

But one had to add

~

in 1931, n

and

e+

in

1932, etc., so now we have the Table 3.2 of 3.5 = spectroscopy of hadrons. Is it possible to return to "simplicity"?

The hoped for building blocks

have been called quarks by Gell-Mann: the 3 spin 1/2 quarks for the multiplet 3 (= fundamental irrep

D) of

SU(3)

and

6 (= irrep

~)

of

SU(6).

There are also

3 antiquarks belonging to the contragredient irrep

= ~

of

SU(3)

or

6 =~

of

su(6)

Mesons of Table 3.2 are formed of one quark and one antiquark bound states of

q + q

yield all expected meson states.

are made of 3 quarks, which are, for the lowest state, in the so they must have a space symmetry

~

~.

Lowest

Baryons of the same table SU(6)

state

K~D,

to obey Fermi statistics; this from our ex-

perience acquired in Chapter 2 and 3 does not seem compatible with attractive forces. And how to explain the saturation by 3 ; why should 2-quark or 4- or 5-quark states not also be stable? # Forgetting these difficulties one can search for quarks. (They should be very heavy, stable, have fractionallquantum numbers

b = 1/3, q = 2/3

or

- 1/3)

and compute with them (good prediction of the "quarks model", e.g., by Dalitz, Lipkin.)

They have not been found experimentally, and quarks can simply be looked

at as the physicists' name for an orthonormal basis of the fundamental

~

irrep of

SU(6), used in their computations!

Current Alsebra.

Let

a ~+ D(~)

be the SU(3) x SU(3)

Lie algebra ad÷

joint irrep

El6.

Any

E-tensor oFerator function on space time

f(y,m)

will

satisfy Equation (1.9) at any fixed time [F(~),f(y,m)] = if(y,D(a)~) where

m E E.

Equation (5.57) is a particular case for

Replace and

F(~)

by its expression (5.55).

, Equation (5.63) reads (use

(i =

,

(5.63)

f(x,~) = h (x,b).

After commuting the symbols

6 ( ~ - y)d x)),

÷ + + (%)~) d3~[h 0(~,~) ,f (~,m)] = i d 3÷ x~(x - y)f(y,D

I

for any tensor operator function of

÷ x.

It is very suggestive to write the equality

for the integrands 0

÷

tU

÷

÷

÷

[h (x,a) ,f (y,m)] = i6(x - y) f (x,D(a)m)

(5.64)

# There are several ways out of these difficulties, but the most efficient seems to me that of O. W. Greenberg and collaborators who have introduced three types of 3(q and q). They obtain a remarkable hadronic spectrum.

qzl-2

Equation (5.56) is written in this local form 0÷~ ~ ÷ ~ ÷ ÷'~ [h (x,a),h (y,b)] = i 6 ( ~ - y)h~(x,a A 6) This is called current algebra in the literature. one speaks of the current algebra of charges. troduce in the second member a distribution

(5.65)

For the time component

~ = 0,

For a space-component one has to in-

(usually called Schwinger terms, see

O'Raifeartaigh lectures). Very few physical results require the local form of current algebra and cannot be deduced from the form (5.63). current algebra as an hypothesis.

However, physicists prefer to consider

They like the analogy with quantum mechanics

which is expressed by the algebra (= Lie algebra of the Heisenberg group) of and

q's

at a given time.

p's

Let us note also that in this frame B. W. Lee (Phys.

Rev. Lett., 17, 145 (1965)) has given a meaning to

SU(6)

symmetry.

There is an

anthology on "current algebra" physics (see below).

Boot-Strap.

When there are so many particles, one hesitates to distingu~h

which ones are elementary.

Boot-strap is a physical concept # which deals with

particles on a more democratic basis.

Boot-strap is expressed by non-linear

quadratic) equations, invariant under the hadronic symmetry group group than

SU(3)

symmetry of

G.

has been used).

G

(simply

(no larger

Such equations yield solutions which break the

Indeed, from the abstract point of view of group invariance,

these

equations are of the form aVa=

~a

and we have already shown how this yields the directions in nature which break the SU(3) × SU(3)

symmetry.

For the readers who wish to read the physics literature, we recall the existence of the anthologies

(with commentaries)

of original papers, that we have

already mentioned.

•

~wi~

Theory of Angular Momentum, Biedenharn, L. C., and van Dam, H.,

•

Symmetry Group in Nuclear and Particle Physics, Dyson, F. J., Benjamin,

•

The Eightfold Way, Gell-Mann, M., and Ne'aman, Y., Benjamin, New York

•

Current Algebras, Adler ~, S. L., and Dashen, R. F., Benjamin, New York

Academic Press, New York (1965).

New York (1966)

(which also contains three lectures by Dyson).

(1964).

(1968).

# Although its father, G. F. Chew has written recently a paper entitled "Boot-strap, a scientific concept?", and given an ambiguous answer!

143

ACKNOWLEDGMENTS

It was very exciting to prepare these lectures, and discuss some points with my colleagues in or near Bures (Deligne, Fotiadi, Lascoux, Radicati, Stora, Thom, et al.).

For the preparation of these notes, I benefited from many discussiorm

with the participants of the Rencontres, among them B. Kostant, G. Mackey, C. Moore, L. 0'Rafeartaigh,

and more especially the Rencontres Director, V. Bargmann.

friendly advice helped me to improve many points of the original draft.

His

By their

careful readings, Dr. Abellanas and Professor Bargmann suppressed most of the misprints of the original draft. Unhappily, lectures.

these notes do not convey the lively interruptions during the

They are incomplete

(no time to deal with molecular and solid state

physics!) and written much too hastily.

I apologize to the reader, asking him to

remember that he is not reading a book, but perishable lecture notes.

I still hope

they will incite some readers to better learn this fascinating part of physics. I acknowledge the wonderful hospitality offered by the Battelle Memorial Institute, to the participants of this fruitful Rencontres.

(and their families!), and the perfect organization The only sad point was the absence of E. P. Wigner, the

most, and yet not enough, quoted scientist in these notes.

UNITARY REPRESENTATIONS OF LIE GROUPS IN QUANTUM MECHANICS by L. O'Raifeartaigh*

1.

Let a mapping

S(3)

NON-RELATIVISTIC CLASSICAL MECHANICS AND THE GALILEAN GROUP**

denote Euclidean 3-space.

s E S(3) ~ x E (XlX2X3) E R 3

A Cartesian observer of

for which the metric

p(s,s t)

S(3) of

is

S(3)

may be written as 1 r 2 + (x 3 - x~)2} 2 p(s,s') = {(x I _ x~) 2 + (x 2 _ x 2)

(i.i)

The group of transformations between Cartesian observers is the Euclidean group E(3) Xta = RabXb + C a where

Rab

,

a = 1,2,3

is any real orthogonal matrix and

Ca

,

(1.2)

any real vector

(independent of

x). Let

t

denote Newtonian time, which is simply a parameter assumed to be

the same, up to a change of origin

t ~ t p = t + to, for all Cartesian observers.

Note that in general

are functions of

Rab

and

Ca

t, i.e., Cartesian observers

may be accelerating relative to each other. Newtonian physics assumes that physical objects occupy volumes in

S(3)

and vary their positions continuously with time, the variation of any body being determined by the others.

The business of physics is to determine the laws of

variation. We shall be concerned mainly with a simplifying limiting case of physical objects, namely, Newtonian particles.

A Newtonian particle is a physical

object to which is attached an intrinsic label called its mass

*

m

(which will be

School of Theoretical Physics, Dublin Institute for Advanced Studies, 64-65 Merrion Square, Dublin 2, Ireland.

** Throughout this paper an asterisk (*) used in a mathematical expression denotes complex conjugation and a dagger (t) passing to the adjoint operator.

~45

discussed in more detail in a moment) its distance from other particles) and shrunk to a point in any time

t

S(3).

by a point in

and whose volume is so small (relative to

that for practical purposes it can be neglected

Thus, a Newtonian particle is characterized at

S(3)

and its mass.

In view of the importance of the mass of a particle for our later discussion, we consider in a little detail how it enters in Newtonian theory.

Its

existence is, of course, empirical and may, in principle at least, be established as follows:

If any ~ particles interact in isolation

far from other objects),

(in practice, sufficiently

then there exists a set of Cartesian observers such that

the quantity

ml 2

= _ d2x~l) / d 2 x ~ dt 2 / ~

(the ratio of the acceleration of the particles) a, t, x (I), x (2)

2) (1.3)

is positive and is independent of

and the nature of the interaction.

intrinsic property of the pair of particles 1 and 2.

In other words, m12 Furthermore,

if

is an

o, B, y

are

any ~ particles then (again empirically) mob = moy • myB Equation

(1.4), however,

(1.4)

implies the existence of a set of intrinsic masses

mo,

one for each particle, and unique up to a common scale factor, such that mo~ = mB/m ° As the masses

m

o

are relatively positive,

(1.5)

they are chosen by convention to be

positive. The result that

mob

is constant already lays the foundations for the

law of variation of the positions of the particles with respect to time. general law (Newton's law) is a linear generalization, isolated particles

(m , x °, o = l,...,n), o

The

namely, given a set of

n

there exists a set of Cartesian

observers such that n

m ° d2x° = 0 o=l

(1.6)

dt 2

This law, in turn, brings out the importance of the force, defined by

F

as a basic physical concept.

d2x °

= m o

o

dt 2

'

(1.7)

Forces are additive, from (1.6), and have additional

good properties, which might be described as follows: What we are looking for is a description of the interaction of particles which is as simple and as universal as possible.

Now a description would be

~46

provided by simply stating what each of particles,

x

is as a function of

t

for each ensemble

(this is what Kepler actually did for the planets), but such a des-

cription would be neither simple nor universal

(as Kepler found to his cost).

What

Newton discovered is that there exists a quantity that is simple and universal, namely, F .

The classic example of a simple universal

theory of gravitation,

F

is in the Newtonian

for which the simple inverse square law

sufficient to explain all (non-relativistic)

effects.

F = mlm2/r2

is

(Of course, one can reverse

the logic and define gravitational effects to be those for which

F = mlm2/r2.

However, the point is that gravitational effects so defined cover a huge class of observed phenomena--falling bodies, projectiles, planetary motion, etc.) From the group theoretical point of view, the interesting aspect of Newton's Equation (1.6) is its invariance group.

Equation

all Cartesian observers, but only for a subclass. Galilean observers.

(1.6) does not hold for

Let us call the subclass

By noting that any Cartesian observer is related to a

Galilean observer by a transformation of the form Xra = Rab(t)Xb + Ca(t)

"

tt = t + ~

,

(1.8)

and inserting this result in (1.6), we see that the Galilean observers are those, and only those, for whom Rab(t) = Rab where

Rab , Ca, and

Va

,

Ca(t) = Ca + Vat

are independent of

t.

which (1.9) holds is called the Galilean group

,

The subgroup

(1.9) G

of (1.8) for

G.

The geometrical significance of the Galilean group becomes clear if we note that it is formed exhaustively from the four subgroups: i)

Time-translations

t r = t + tO

2)

Space-translations

x ar = x a + C a

3)

Rotations

x ar = RabXb

4)

Accelerations

x ar = x a + V a t

(i.io)

The invariance of (1.6) under (i.i0), i) to 3), means that (1.6) does not prefer any origin in space or time or any direction in space, which is understandable. The invariance under 4) means that observers with different but constant velocities are equivalent.

This is far less obvious,

and was first discovered by Galileo.

The invariance under 4) does have, however, a geometrical significance, namely, in the 4-space spanned by

S(3)

and

t, (1.6) does not prefer any slope for the

t-axis.*

* I am indebted to Henri Bacry for this remark.

~47 The force defined by Equation (1.7) is clearly Galilean invariant, provided that the Galilean transformation is universal, i.e., it is a transformation of the coordinates of all the particles.

Thus, in guessing the forces for any

problem, one can restrict oneself to those that are Galilean invariant. Let us now consider the Galilean group by itself.

By definition,

it is

a 10-parameter Lie group, which is the semi-direct product of its connected part (det Rab = +I) and the 2-element space reflexion (parity) group. dG

Its Lie algebra

has the basis: i)

Time-translations

E

2)

Rotations

L

3)

Space-translations

P

4)

Accelerations

K

a a a

with commutation relations

[E,M a] = 0

[E,P a] = 0

[E,Ka] = Pa

[Ma,~]

[Ma,P b] = SabcP c

[Ma,Kb] = gabcKc

= gabeMc

(i.ll) [Pa,eb] = 0 [Ka, ~ ]

where

[ea,~]

= 0

= 0

a,b,c = 1,2,3

and

~abc

is the Levi-Civita symbol.

semi-direct sum of the rotation algebra

L

K

onto

P

and commutes with

dG

is the

and a 7-dimensional solvable algebra

made up of the two abelian commuting vectors projects

In words,

P

and

K

and a scalar

E

which

P.

One of the most important properties of Galilean transformations is that they are a special case of contact transformations x ÷ xr(x,p),

p ÷ pr(x,p)

which leave the symplectic form {A,B} = ~

where

pa = m

[I], namelF, transformations

~(~pA~x~B

~x~A ~ I

'

(1.12)

dx~ d-~ ' invariant.

Now a property of the group of contact transformations[2]

is that if

a

is the parameter of any 1-parameter simply connected Lie subgroup, then there exists a function

G(p,q)

such that ~F 6a - {G,F}

where

F

is any regular function of

with respect to the group parameter

p a.

and

, q, and

The function

(1.13) 6F ~ G

is its rate of variation is called the generator

function for the i-parameter subgroup. Furthermore, with parameters

for an n-parameter Lie subgroup of contact transformations

a, B,...

'14-8

(f~ ~ ~B where

CY eB

6 6 ) cy 6F 68 ~e F = ~6 ~ Y

'

are the structure constants of the group.

(1.14) and using the Jacobi relation for

(1.14)

Hence, inserting

(1.13) into

{A,B}, we obtain

{{Ge,GB}F 1 = C~B{G ,F}

,

(i.15)

whence, {Gy'GB} = C~6Gy + %~B where the

%~B

have zero bracket with all

under the bracket operation, tact transformations algebra. form

F

'

and hence are constants.

the generator functions

form a representation

The number of constants

Ge + Ge + X~, where the

%eB

%~

(1.16)

Ga

Thus,

of a Lie group of con-

(up to the constants

%e6) of the Lie

can be minimized by transformations of the

are constants, but whether the

Xe6

can be

eliminated entirely depends on the group structure. The above results hold for any Lie group of contact transformations. Let us now return to the connected Galilean group

G.

For

G, the generator

functions corresponding to the generators in (i.ii) can be seen to be L=

Ix e

P=[

x p

e

P~ e (1.17)

K = ~ mcxc~ - P t

E=

where

dx~ Pe = ms ~ ' and

~

2-~--- pa +qo e

is the potential from which the

F

can be derived,

i.e., =

F

_

~

~x e

If we compute the brackets

{L,E}, etc., for the generator functions

obtain, as expected, the Lie algebra (i.ii) up to constants. only one constant; namely, the relations

(I.Ii) hold as they stand except that

[Pa,Kb] = 0 ÷ {Pa,~} where

is such that later).

is the total mass.

M = ~ M M

cannot be eliminated

(1.17), we

In fact, there is

= 6abM

,

(1.18)

Further, the structure of the Galilean group (we shall be discussing this question again

149 Note that the generator for the time translations is just the Hamiltonian H

for the system.

Note further that

[H,K] ~ 0, although

[H[H,K]] = 0.

Thus,

although the Galilean group is an invariance group of Newton's Equations (1.6), it i~ not quite an invariance group of the Hamiltonian, or of Hamilton's equations of motion, dpa dt

~H ~x

dx dt

~H ~p

(1.19)

This is understandable since a choice of Hamiltonian forces a choice of direction for the t-axis in tally, the term definition of

S(3) @ R

and thus destroys the Galilean invariance.

Inciden-

-Pt, which is explicitly time-dependent, is inserted in the

K, so that in spite of the fact that

[H,K] ~ 0, K

can be a con-

stant of the motion, i.e., so that dK

~K

d-~ = ~-~ + {H,K}

2.

= -P + P = 0

(1.20)

NON-RELATIVISTIC QUANTUM MECHANICS

As is well known, Newton's laws, or the more general and sophisticated versions of them, such as Hamilton's, sufficed to explain all physical phenomena until the end of the last century.

But after the turn of the century, the New-

tonian framework was shattered both by the theory of relativity and by the quantum theory.

In this lecture, we shall be concerned only with quantum theory. As is also

well-known, the crux of the quantum theory is to replace the functions x and p dx = m~ needed to describe particles, by linear operators X and P on a Hilbert space, satisfying the relation [X,P]

= i~

(2.1)

(This relation will be made mathematically more precise later.)

For the moment, we

shall only emphasize that the assumption (2.1) is the only new assumption made in the quantum theory.

The old equations of motion dX dt

are retained with

~ ~P

'

dP dt

~H ~X

x ÷ X, p ÷ P (which is unambiguous since

H = p2 + ~(x)). 2m

There

are four questions which we wish to discuss briefly: i)

How one arrives at the particular Ansatz (2.1)

2)

How to make it mathematically precise

3)

How to relate it to experiment

4)

How the group structure of Newtonian theory is affected.

Let us begin with i).

The decision to replace

x

and

p

by operators

150

was based on a large number of empirical observations and on partial theories formed from these observations [i].

Since we could not even begin to describe the

general picture in a part of one lecture, let us concentrate on one experimental result, namely, the discrete frequency of the light emitted from atoms, and try to sketch the motivation from that result.

It was known at the time the quantum

theory was founded that the atom consisted of a positively charged kernel of very small radius with negatively charged electrons circling it, about 10 -8 cms out. For such a system Newton's laws (extended to include Maxwell's) would predict a continuous emission of radiation from the circling (and therefore accelerating) electrons, leading to a continuous loss of energy on the part of the electrons (so that the atom would finally run down) and a continuous change in the frequency of the emitted radiation. the opposite. not exist).

The experimental situation, however, was quite

First, the atoms were quite stable (otherwise, our universe would Second, from spectroscopy it was known that the frequency of the

radiation emitted from atoms, far from being continuous, could only have special sharp values (spectral lines) characteristic of the atom (yellow for sodium, green for copper, and so on). the atomic level.

Hence, Newton's laws were incompatible with experiment on

The question was:

One worked backwards. i)

how to change them?

If one assumes

Einstein's empirical law

E = h~, where

h

the frequency of the emitted light, and 2)

is Planck's constant, E

its energy, and

conservation of energy, i.e., energy lost by electron in the atom energy of emitted radiation,

it follows from the discreteness of the frequency of the emitted radiation that the energy levels of the electron in the atom must be discrete.

It follows that the

Hamiltonian

1

p2

(2.2)

Ze2 r

H=~m

'

for an electron in an atom with nucleus of charge values.

Ze, cannot take continuous

This leaves one with three options: i)

Abandon the Hamiltonian (2.2)

2)

Impose some conditions on it from outside

3)

Change it so that it can naturally take only discrete values.

i) has the difficulty that it is almost impossible to think of a classical Hamilionian which would take discrete values.

2) is what was done in the so-called "old

quantum theory" (1900-25), and is very ad hoc. Schr6"dinger and Heisenberg.

3) is the option chosen by

The choice they made was to interpret

operator, since

H

could then take discrete values naturally.

preting

p

as linear operators

operators

x

and P

and

X

X

and

P°

should be, one must do more.

H

as a linear

This means inter-

To determine the kind of Heisenberg analyzed the atomic

spectra in detail and concluded that

P

and

X

should be the matrices u

X

P = i----~

where

-¢~

h ~ = ~.

de Broglie.

=

i

0

~f

~

0

0

- -

0

¢~

0

¢T

0

£f

0

-/f

0

/f

0

~f

0

¢T

0

-/f

0

0

0

/f

0

Jo

°

Schrodlnger,

(2.3)

on the other hand, built on a partial theory due to

According to de Broglie, free particles should diffract like light

from sufficiently small gratings and should therefore satisfy, in the relativistic case, a wave equation of the form

[

~2 ~t 2

V 2 - m2| ~(x) = 0

(2.4)

J

Comparing this with the classical energy moment relation, ~2 _ p2 _ m 2 = 0 Schr~dinger concluded that

P

,

(2.5)

should be the operator

i ~x

'

(2.6)

on

L3(-~,~), and went on to postulate that this identification should persist in 2 the non-relativistic limit and in the presence of a potential. One sees that the Schrod~nger and Heisenberg Ansatz are equivalent by

noting that they are special realizations that the Ansatz

of the Ansatz

is therefore a kinematical Ansatz.

Note, incidentally,

Newton's laws then guarantee it for all times.

It might be wondered if the Ansatz obtain agreement with experiment, Wigner

(2.1).

(2.1) need only be made at a single (initial) instant of time and it

(2.1) is absolutely necessary to

or whether one could get away with less.

[2], for example, has proposed that

(2.1) might be replaced by the weaker

commutation relations ~H

[H,P] = i ~--~ where

~

,

~H

[H,X] = -i ~-~

,

(2.7)

is the Hamiltonian, which would seem to be necessary from Heisenberg's

analyses of the spectral lines.

However,

except in the case (2.1), the Ansatz

(2.7) would make the commutation relations depend on

H, i.e., on the dynamics.

Let us now turn to question 2), namely the question of putting the Ansatz

IX,P] = ih

on a better mathematical footing.

For this we proceed as

follows: Let

~

be a Hilbert space, and let

X

and

P

that there exists for them a common invariant dense domain

be operators on it such ~

on which

152 a)

X

b)

X 2 + p2

c)

IX,P] = i~

d)

and

P

are symmetric is essentially self-adjoint

the only bounded operator which commutes with

X

and

P

is a

~

up to a unitary

multiple of the unit operator. Then

X

and

P

transformation on

~.

are uniquely and rigorously defined (which may depend on the time).

A realization of

X

and

[3] on

They are essentially self-adjoint

P, is the Schr~dinger realization

x

and

ha

on L3(-=,~), where the domain ~ could be, for example, the space K of i ~x 2 all infinitely differentiable functions of compact support, or the space S of all infinitely differentiable functions of fast decrease (i.et, which decrease faster than any inverse power of

x

as

Ixl ÷ ~).

We shall see later (from Nelson's

theorem) that conditions a) to d) are precisely the necessary and sufficient conditions, that

X

and

P

can be exponentiated

representation of the Weyl-Heisenberg group e

to form a unique unitary irreducible

W, i.e., that

i~X iTP iTP laX i~T~ e = e e e

Thus, an alternative definition of

X

and

P

, on

~

(2.8)

is that they satisfy (2.8), i.e.,

that they are the generators of the unitary irreducible representation W, [4].

In fact, this definition of

X

and

P

(UIR) of

was the starting point for

von Neumann's celebrated proof [5] of the uniqueness of

X

and

P

up to a unitary

transformation. Having disposed of these mathematical points, we come to the experimental numbers. operators

A

To extract the experimental numbers, we first put the self-adjoint

on

~

into a i-i correspondence with the measurable quantities

(observables) which we shall then also denote by adjoint operators for which it is meaningful, A = f(P,X) =

where

f(p,x)

A.

In practice, for the self-

the correspondence is [4]

i I e~(P~+X~)d~d~ f e-i(P~+X~)f(p~x)dpdx (27) 2

are the corresponding classical functions.

,

(2.9)

(The bounded subset of

the operators for which (2.9) is meaningful form a dense set in the ring of bounded self-adjoint operators.) Now let

P%(A)

longing to the eigenvalue

be the projection operator on the eigenspace of %, where for the moment we assume

the eigenspace finite dimensional.

in the case that

that both eigenspaces are infinite dimensional. is that they are probabilities;

be-

to be discrete and

The numbers to be extracted are then

trace (P~(A)P (B)) with appropriate modifications

%

A

, X

and

(2.10) ~

are not discrete and

!The meaning of the numbers

namely, trace (P%(A)P (B))

(2.10)

is the probability of

~153

finding

the value

from a measurement probabilities

~

from a measurement

of

P%(A)

and

B, having

P (B)

numbers

case that the eigenvalues

project

onto 1-dimensional

fx(A)

and

f (B)

are any unit vectors

is the case which will be of most interest

%

~i~f~(A),

a system are in 1-to-i correspondence

quantum mechanics,

D

above.

Hence, by Stone's

equations

U(t)

Such a set Thus,

4), the group theoretical

H

the states

of

properties

of non-

the Hamiltonian

,

theorem

is essentially

[6], there exists

transformations dU(t) dt

We now show that

is defined

(2.12)

potential.

In most cases of interest,

meter group of unitary

we shall

with the rays.)

and first consider

in an external

is simple,

eigenspaee.

i p2 H = 2~m + ~0(X) for a single particle

~

This

The state of a

is often called a ray.

Let us turn now to question relativistic

state.

%, where

in the 1-dimensional

0 ~ ~ < 2~

to

subspaces.

(For future reference,

A

of unit vectors

i.e.,

(2.11)

system after a measurement

with result

are simple,

(2.10) reduces

in the respective

to us.

the

can predict.

,

of a quantum mechanical

to be the set of unit vectors

~

subspaces,

need for this case the concept of

and

found

operators,

that quantum mechanics

I(fx(A) , f (B)) I where

just previously

A, except points in spectra of self-adjoint

are the only experimental

In the particular that

of

U(t) HU(t)

on on

self-adjoint a unique

continuous

1-para-

~, such that ~

(2.13)

is the group of time translations.

of motion are the same in classical

on the domain

and quantum

Since the Newtonian theory, we have in both

cases HX = i e dt m In the quantum mechanical

'

case, however,

HP = _ ~(X) dt ~X

(2.14)

we have the extra condition

IX,P] = i~ Inserting

this equation

into

(2.14)

and (2.12), we see that in the quantum mechan-

ical case we have dX i d-~ = ~ [H,X] If we assume

that the domain

D

'

dP i --=[H,P] dt

on

D

is invariant with respect

(2.15) to

U(t),

it follows

at

once that X(t) = U(t)X(0)U-I(t)

,

P(t) = U(t)P(0)U-I(t)

on

D

,

(2.16)

154

and, in general, for suitably defined

F(P,X)

in (2.9)

F(P(t)X(t)) = U(t)F(P(0)X(0))U-I(t) Thus, U(t)

is the group of time translations.

Hamiltonian

H, like

P

and

(2.17)

In quantum mechanics,

X, plays a dual role.

therefore,the

It is a physical observable

(energy) and it generates the group of time translations. It may happen that

H

is not essentially self-adjoint on

9.

In this

case, there is usually a good physical reason, and the corresponding classical Hamiltonian also has bad properties, e.g., sends the particle off to infinity in a finite time [7]. Turning now to the Galilean group for a system of interacting particles, we find that, in analogy to

P, X, and

H, if we replace the classical generator

functions of the Galilean group by their quantum mechanical counterparts

E=H

= ~ 1

L=~X

to obtain

p2+~

xP (2.18)

p = ~ P~ K = ~ m X

then, in analogy to

P, X

and

- Pt

,

H, these ten operators (2.18) play a dual role.

They are physical observables and at the same time they are the generators of unitary representations ~, i.e., if

a

of the 1-parameter subgroups of the Galilean group

G

on

is a parameter, dF i d-~ = ~ [Ga'F]

This is the quantum-mechanical

,

(2.19)

o = i...i0

analogue of the classical Poisson bracket relation dZF = {~a'F} da

(2.20)

Using the quantum mechanical relation the commutators of the operators

[X,P] = i~, we can easily compute

(2.18) amongst themselves.

We obtain

[Pa,Pb] = 0

[Ka,~]

: 0

[Ma,P b] = ieabcP c

[Pa,~]

= i6abM

[Ka,H]

= 0

[Ma,~]

= ieabcK c

[Pa,H]

= 0

[Ma,H]

= 0

[Ma, ~ ]

= ieabcM c

(2.21) ,

455 These relations are the analogue of the classical Poisson bracket relations for the generator functions amongst themselves. term

~abM

Nete that (2.21) even contains the

which occurs in the classical Poisson bracket relations, but not in

the Lie algebra of

G.

Apart from the term if the term

~ab M

~ab M, (2.21) is just the Lie algebra of

were absent, the 1-parameter subgroups of

G , would mesh together to form a unitary representation of

G.

Hence,

G, generated by the G

on

X

(modulo some

domain restrictions which will be discussed later and which are normally satisfied). Thus, in quantum mechanics the generators and generators (modulo

G

play the dual role of observables

~ab M) of a unitary representation [8] of

G

on

~.

This

is true, of course, in classical mechanics also, where the generator functions are observables and generators of group transformations in the sense of Poisson brackets.

But the relationship in quantum mechanics is more direct.

In particular,

the operation of commutation is simpler and more direct than the operation of forming Poisson brackets.

In this sense, group theory, which plays a background

role in classical theory, may be said to come into its own and play a central role in quantum mechanics. Let us now consider the term

6abM.

Since it commutes with all the

it cannot make a big difference to the representation of

G

on

~.

checked that the difference it makes is that the 1-parameter subgroups of instead of meshing together to form a true unitary representation of mesh together to form a unitary ray representation of sentation by unitary operators

U(g)

and

g,gl 6 G

~

is real.

exp im(g,g r)

that the factor

on

0 ~ ~ < 2~).

G, on

~,

X, i.e., a repre-

,

(2.22)

The reason for the name ray representation is

is irrelevant for rays, (where rays are defined as

above to be sets of unit vectors related to a given unit vector where

G

satisfying

U(g)U(g r) = U(ggr)e i~(g'gr) where

G

G ,

It is easily

f

by

exp(i~)f,

If we now recall that the experimental numbers which can be

extracted from quantum theory are l(f,g) l where

f

and

g

,

(2.23)

are unit vectors, we see at once that they do not distinguish

between vectors in the same ray.

Thus, the experimental numbers do not distinguish

between unitary ray representations and true unitary representations.

We shall be

returning in more detail to this point later, but for the moment we merely note that the failure of the experimental numbers to distinguish between true and ray representations means that the appearance of ray representations and hence, in particular, of the term quantum mechanics.

~ab M

in the Lie algebra (2.21), is quite natural in

~56

In the case of a single free particle,

the generators reduce to

i M a = ~ SabcPaXc p

a

= p

a (2.24)

K

a

m

a

i =Tm

E where

=mX

-Pt

a

p2

is now the mass of the particle and

E

is both a generator of the

Galilean group and the generator of time translations. "carries" a unitary ray representation of

0.

Thus, a free particle

Furthermore,

if the quantum mechani-

cal commutation relation [X,P] = i~ is irreducible on

,

~, then so is the representation

(2.21) of

G.

A non-relativ-

istic free particle may, therefore, be said to carry an irreducible unitary ray representation of

G.

An interesting question is what would happen if we reversed our line of approach and demanded that a free non-relativistic particle carry a true unitary representation of

0.

This question has been investigated by InSn~ and Wigner

They showed that in a true irreducible unitary representation of

G

[9].

the quantum

mechanical relation IX,P] = i~

,

cannot be realized, which has the unpleasant physical consequence that be localized.

The crucial point is that

in any unitary irreducible representation, form

~CX)

of any

f(P)

p2

is a Casimir operator for

cannot G.

Hence,

it is a number, and the Fourier trans-

must therefore have a spread in

In a ray representation,

X

X.

the situation is saved by the ray relation

i[Ka,Pb] = 6abm

,

(2.25)

or (2.26)

i[Ka,P2 ] = 2mP a The latter relation implies that

p2

assumes all values in the range

0 ~ p2 < ~,

which together with [Ma,Pb] = iSabcPc implies that is localizable.

takes all values in

,

(2.27)

R 3, in which case the Fourier transform

15V

In conclusion quantum mechanics, pendent.

I(f,g) l2 = probability,

The second

can be deduced

other general arguments

3.

it might be worth remarking

GROUPS

n

the Galilean

invariance

by introducing

i X = ~ (mlXl+ m2x 2)

,

of motion

a 2-particle

6).

,

G

was the

of an isolated

system and "factor-off"

center of mass and relative

P = Pl + P2

indeand

QUANTUM MECHANICS

equations

Let us now consider

of

(see ref. 4, lecture

we saw that the Galilean group

of the non-relativistic

particles.

are not entirely

nature

IN NON-RELATIVISTIC

In the last two sections,

system of

[X,P] = i~

from the first, using group theoretical

of a more or less plausible

INVARIANCE

group of invariance

and

that the twin postulates

coordinates.

M = mI + m2

,

and 1 ~ = ~ tm2Pl - mlP2) f

y = xI - x2 , respectively. H =

Because

of Galilean

invariance

(3. i)

the Hamiltonian

splits

into

HCM+ Hr, where p2 HCM

=

2--M '

IX,P] = in

,

and ~2 Hr = ~+ where

~ = mlm2/M Clearly

relative motion

HCM

describes

the motion

,

(3.2)

of the centre of mass and

of motion derived

from the "relative"

retain any of the original Galilean

(i.e., for particular

ance under a subgroup happen

[y,~] = i~

potentials

of the Galilean group

~(y))

(e.g.,

Hamiltonian

invariance.

In this lecture we wish to consider

define an invariance Definition:

(3.2)

However,

in

such cases.

group)

or they may

to do with Gslilean For this purpose,

we

group.

An invariance

group is defined

~£,

the Hilbert

a)

the Hamiltonian

b)

the absolute values

We first discuss

the

they may retain invari-

the rotation

to be invariant under special groups which have nothing

invariance.

Hr

of the particles.

The equations

cases

,

is called the reduced mass.

will not, in general, particular

~(y)

space of

the motivation

to be any group of transformations

y, ~, which

leaves

on

invariant

H of the inner products

for this definition.

l(f,g) l.

That the group should

leave

158

the Hamiltonian invariant is practically self-explanatory since this is true of an invariance group even in classical mechanics. and quantum mechanics)

the invariance of

H

We only note that (in both classical is slightly stronger than the demand

that the group leave the equations of motion invariant.

(For example, as we saw for

an isolated system, the Galilean group left the equations of motion invariant but not the Hamiltonian.)

However, for invariance groups of the relative Hamiltonian,

the distinction between the invariance of

H

H

and the equations of motion usually does not arise, and

is used as the simplest and most compact was of defining

invariance. The more interesting question concerns b), namely the invariance of the inner products

l(f,g) I which are peculiar to quantum mechanics.

The question is

whether this demand is necessary, or at least reasonable. For a group of transformations which have a passive interpretation, the case for the Galilean group

G, the answer is yes.

as is

For if we change the

observer of a system, without changing the system itself, the probability of the system making any particular transition

g ÷ f

cannot change (since the system

"does not know who is looking at it") and this is just another way of saying that I(f,g)I

is invariant. For transformations which do not have a passive interpretation,

i.e., for

which we must change the system itself to implement them (these are usually transformations which have no geometrical interpretation), to establish.

the argument is not so easy

However, it is usual to demand the invariance of the probabilities in

this case also, if only for simplicity and to preserve the analogy with the active case. Demanding that the probabilities a second question:

I(f,g)I 2

Are unitary ray representations

sentations which leave the probabilities

remain invariant, we come to the most general group repre-

invariant?

To answer this, one first concentrates on a sin$1e transformation and asks:

What is the most general

T

such that

I(Tg,Tf) l = ](g,f) l If

T

,

is linear, then the answer is simple:

however, there is no need for

T

T

T

to be linear.

following remarkable theorem due to Wigner

g,f E ~

(3.3)

must be unitary.

In general,

In that case, we fall back on the

[i].

Theorem

Let

T

be a transformation satisfying

unitary or anti-unitary transformation

U

(3.3).

Then there exists a

such that for all

(U-IT)f = ei~(f)f

f E (3.4)

159

Note that f.

U

is then unique up to a phase-factor,

exp(i~), which is independent of

[An anti-unitary transformation is defined to be a transformation such that (Uf,Ug) = (g,f) = (f,g)*

This theorem means that, for rays, T

]

(3.5)

is equivalent to, and may be replaced by, a

unitary or anti-unitary transformation. This theorem was first stated by Wigner in his book on group theory in 1931.

[i]

However, the proof given in the book is not complete,

many papers

and since then

[2] have been devoted to completing, simplifying and generalizing the

proof. The most definitive proof is that given by Bargmann

[3] in 1964.

This

proof has the advantage of being basis-free and hence valid for non-separable as well as separable Hilbert spaces. Wigner's

theorem applies to any fixed transformation

group of transformations

T(g).

For each fixed

unitary or anti-unitary transformation exp i6(g).

g, T(g)

Equation (3.4), and the unitarity

can be replaced by a

U(g), unique up to a phase-factor

m(g,gr)

is a real number.

U(g)

=

of

U(gg r )e i~(g'g~)

U(g), one sees that ,

(3.7)

(3.2) is equivalent to a set of unitary or anti-

forming a ray representation of the group.

sense, unitary of anti,unitary ray representations representations

(3.6)

It follows that any group of transformations

preserving the probabilities

unitary transformations

,

(or anti-unitarity)

U(g)U(g r)

T(g)

Consider now a

Using the group relation T(g)T(g t) = T(gg r)

where

T,

In this

are the most general group

preserving the probabilities.

In practice, only one anti-unitary transformation is used in physics. This is the time-reversal transformation.

To keep the quantum mechanical equations

of motion d F = ! [H,F] dt h invariant under time-reversal, when

t ÷ -t • H ÷ -H

(3.8)

it is necessary to let either

is ruled out because

H m 0.

H ÷ -H

or

i ÷ -i

Hence, i ÷ -i, and this leads

to an anti-unitary transformation. We turn now to some examples of invariance groups in quantum mechanics. For this purpose, it is usual to consider the relative motion Hamiltonian ~2 H = ~+ The problem is, given commute with this

~(y),

~(y)

(3.9)

to find unitary groups of operators which

H, and have a direct physical meaning.

Indeed, in practice, it

160

is usually the physical meaning that enables us to find the groups.

The advantages

of finding such groups are: l)

Since for the group generators

G,

[H,G] = 0 the group provides in the

,

G's

(3.10)

at least some of the constants of the

motion.

2)

At the same time, the

3)

The group can be used to reduce enormously the labor involved in

simultaneously with

G's

are natural operators to diagonalize

H.

making a calculation with the Hamiltonian, energy level, an emission probability, Note that Equation group generated by of motion).

G

e.g., calculating an

or a scattering amplitude.

(3.10) can be looked at from two points of view:

leaves

H

invariant

The

(is an invariance group of the equations

Conversely, the group generated by

H

leaves

G

invariant

(G

is

conserved). Let us illustrate points i), 2), and 3) above with the most important special case of an invariance group; namely, the case when central, i.e., depends only on

~(y)

in (3.9) is

r 2 = Y l2 + Y22 + Y3" 2 In this case, H c o m mutes with the rotation group generated by the three operators L = y x ~, with Lie algebra

r

where

[L,L] = iL, and which are at the same time identified with the relative

angular momenta of the particles in the i, 2, 3 directions.

[The transition from

the group to the algebra and back will be justified in the next section.] respect to I) above it is clear that

LI, L2, and

L3

are conserved.

Now with

With respect

to 2) it is not difficult to show that the so-called total relative angular momentum L 2 = L 2 + L 2 + L 2 and any one of L L L 3 (usually L3) can be added to H to i 2 3 I ~ 2' form a complete set on ~ ~ being assumed irreducible with respect to [y,z] =i~). Thus, a convenient and physically relevant basis in

~

is

f(e,Z,m)

where

Hf(s~m) = sf(s~m) L2f(e%m) = Z(Z + l)f(eZm)

,

(3.11)

L3f(E~m) = mf(eim) where, because the rotation group is compact, Z

is a non-negative integer and

-~m~Z. With respect to 3), we see at once that in calculating the eigenvalues of H, which are the eigenvalues of the differential operator

~2 V 2 + V(r) 2m on

L2, the use of (3.11) reduces the partial differential operator

simple differential operator

(3.12) (3.12) to the

and s o s i m p l i f i e s t h e c a l c u l a t i o n . But t h e group does much more f o r us t h a n t h a t . t o c a l c u l a t e t h e p r o b a b i l i t y of a p a r t i c l e i n t h e s t a t e

-+

photon w i t h momentum k

f(E,k,m)

emitting a

f ( E ' , Q ' , m r ) , t h e n , t o lowest

and ending up i n a s t a t e

o r d e r i n t h e EM c o u p l i n g c o n s t a n t

For example, i f we wish

e , and provided t h e wavelength of t h e e m i t t e d

photon i s l a r g e compared w i t h t h e s i z e of t h e atom [ 1 , 4 ] , t h e r e l e v a n t i n n e r prod u c t s t o compute a r e t h e m u l t i p o l e moments of t h e p a r t i c l e .

A t y p i c a l one of t h e s e

i s t h e d i p o l e moment,

Now f o r even q u i t e low v a l u e of

R

and

R',

computed i s q u i t e l a r g e , s i n c e

-Rr

5 m'

5 R',

p r o p e r t i e s of

t h e number of q u a n t i t i e s (3.14) t o be 5 m 5 R.

-R

But thanks t o t h e group

( i t i s a p o l a r v e c t o r w i t h r e s p e c t t o r o t a t i o n s and space r e f l e x -

y

i o n s ) , we can

lation.

where

a)

show t h a t t h e

b)

for

+

R' = R

Ea

vanish unless

R ' = R i 1, m' = m, m i- 1,

1, reduce t h e c a l c u l a t i o n s i n each c a s e t o

one calcu-

I n f a c t , t h e group i n v a r i a n c e i m p l i e s t h a t

-

m'

m = 0, + 1 f o r

a = 3, 1 i i 2

r e s p e c t i v e l y and t h e

f u n c t i o n s of t h e simple d i f f e r e n t i a l o p e r a t o r (3.13). is that the

m'

and

m

FEQ a r e t h e eigen-

The c r u c i a l p o i n t about (3.15)

d=<errdence appears only i n t h e c o e f f i c i e n t s (Clebsch-

Gordon c o e f f i c i e n t s ) which a r e independent of

V(r).

Thus, t h e s e c o e f f i c i e n t s need

o n l y be c a l c u l a t e d once and f o r a l l (Figure 3 . 1 ) , and then they can be used f o r any central potential.

,

+

lI

/(a

(R

+m')(R + m r + 1) 2) (29. + 1 ) (2R

+

-

11

1/

-

(R

m' (2R

+ +

1 ) (R 1 ) (R

+ m + 1) + 1)

(a

- m') (a - m ' + 1 ) (2R + 1 ) (2R + 2)

+ m') (R + m' + 1 ) 2R(R

R

(The f u n c t i o n s i n t h e i n t e g r a l w i l l , of c o u r s e , depend on V(r).)

/ (L -

+

mr)(R 2R(2R

1)

+- m'1 ) + 1 )

I

FIGURE 3.1.

I/

/ VALUES OF

@(R'mr,Rm)

(Q

+ m' + 2R(2R

+ m')

1)(R + 1)

~62

The labor saved by using one group to obtain the results a) and b) in this example is obviously immense.

Furthermore, the use of the group gives a much

deeper insight into what is going on.

It isolates the group properties of a

central potential (independence of the potential of the angular variables from the dynamical properties (form of the dependence of

V(r)

on r).

6, 9)

The results

a) and b) for this example are, of course, a special case of the Wigner-Eckart theorem, which has already been mentioned by Louis Michel and will be formulated for completeness in the next chapter. We conclude by considering two Hamiltonians which have special invariance groups.

The first is the harmonic oscillator Hamiltonian i

H = ~m

where

~

is a constant.

72

i

y2

+ ~

,

(3.16)

This is centrally symmetric and has the angular momentum

invariance group generated by

L

discussed above.

But, in addition, H

commutes

with the six operators Mab = X a ~

where

+ PaPb

,

(3.17)

X = ~y, P = ~-i~, ~4 = m 0

,

(4.8)

is the unique self-adjoint extension of the operator

A = i - G 2 - G 2 - ... - G 2 I 2 r

on the GSrding domain

Dg

have Gaussian decrease for

(4.9)

'

for the regular representation.

The functions

a(g)

t > 0.

It is interesting to note that the above results concerning the existence of a GSrding and analytic dense domain are not confined to unitary representations. They hold for any continuous representation by bounded operators.

This is clear

for the G~rding domain and follows for the analytic domain because, for a continuous representation, tial. D

the growth of

Even the result that the

U(g), like the Haar measure, is at most exponenG

are skew-adjoint on any group invariant domain

generalizes; namely, if superscript

c

denotes contragredient quantities, we

have (G~IDc)t = - (G ID)

167

So far, we have given the group representation about the Lie algebra.

Now we ask the converse question:

sufficient condition that a Lie algebra on

~?

U(g)

G

and asked questions

What is a necessary and

generate a unique unitary group

U(g)

An answer was given by Nelson [7] in 1959, who established the following

theorem:

Theorem

A necessary and sufficient condition that a Lie algebra of symmetric operators

iG

be the Lie algebra of a unique unitary Lie group

that there exist in the

iG

X

a common dense invariant domain

~

U(g)

for the

G

on

}[

is

on which

are symmetric, and the operator r =-

G2+I

,

~=1 is essentially self-adjoint. In the course of the proof, Nelson has shown that the analytic domain for the self adjoint extension

~

of the operator

A

is a common analytic domain for

the Lie Algebra, and thus furnished an alternative proof of the existence of a dense analytic domain for the unitary representations.

The essence of Nelson's proof is

to obtain, from the general form of the commutation relations and the obvious bounds

IIGII < II~I fIG211 < II~I '

(t~) n ~ < ~

for all

llGnll < Cnli~nll, where

C

G.

~ n!.

Then, if

n

t, ~ ~tg)n n!

general, the entire vectors for entire vectors for

a bound

'

< ~

for

t < to, where

(t~) n ~, i.e., ~ - - ~ ,
0.

Note that, in

t, are not necessarily

Indeed, in general there do not exist any entire vectors

for the Lie algebra of a unitary Lie group. already provide a counter-example.

The unitary representations of

SL(2,C)

Recently it has been shown by R. Goodman [i0]

that the analytic domain for the Lie algebra is exactly the analytic domain for the operator vectors

A I/2.

Goodman has also discussed the question of the existence of entire

[ii]. From the above results, namely the existence of an analytic domain for

any continuous representation, and the existence of a unique continuous unitary representation when

&

is essentially self-adjoint, it is evident that for con-

tinuous Lie groups the relationship between Lie algebra and Lie group representations is all that could be required.

We can operate relatively freely with the

algebra in spite of the unbounded nature of the operators, a circumstance we had anticipated earlier.

We close with a few incidental remarks:

First, in the case of U!R's of semisimple Lie groups, there are some stronger results due to Harish-Chandra.[9]

For example, the vectors in the (neces-

sarily finite dimensional) subspaces, which are invariant with respect to the

168

maximal compact subgroup of the group, are analytic vectors for the whole group. Furthermore,

the linear span of such vectors, which is dense in

~, can be gener-

ated from any one such vector using the enveloping algebra of the Lie algebra. Second, there are still some outstanding problems. analogue of Nelson's results representations.

(A

One is to find an

for non-unitary

Another is to ask for statements concerning the analytic

continuation of the functions parameters.

essentially self-adjoint)

(h,U(g)a)

to complex values of the group

How close are the singularities?

Are they poles or cuts?

And so

on.

We next consider briefly the domain question for tensor operators. a set of operators

T a, a = l...s

U(g), we need only a dense domain i)

the

2)

D

T~

to transform as a tensor under a unitary group D

with

essentially self-adjoint on

D,

stable with respect to U(g), A U(g)T u-l(g) = Db~(g)T b on D, where

3)

For

DA(g)

is a representation of

U(g). DA(g)

is usually finite-dimensional

(r < ~), but the definition can be extended to

cover infinite dimensional representations If the group

U(g)

as well.

is compact, one is usually interested not in the

full (generally unbounded) tensor components P 'TaP , where

P,P~

Ta, but only in the restrictions

are the projections onto finite dimensional subspaces of

which are invariant with respect to

U(g).

For the restrictions

P'TaP

to

exist, one needs only the weaker condition that there exist a dense domain

D

for the

Ta

extension of

such that T .

P~ c D(~a) , where

~a

is the unique self-adjoint

The physical conditions are usually enough to guarantee

this. For example, in the dipole radiation example of the last section, the relevant matrix elements were of the restrictions of

Ya

(fEr~rm,,YafEEm),

i.e., they were the matrix elements

to the finite spaces

fs%m"

One can see that these

restrictions must exist from the physical point of view as follows:

The dipole

radiation is actually just the first coefficient in the expansion of (f t%rmr,eiYa/%f %m) radiation.

in powers of 11% , where

Now the restriction

P~ exp iYa/%P

%

is the wavelength of the emitted certainly exists since

exp iYa/%

is a bounded operator, so the only question is the validity of the subsequent expansion in powers of

ii%.

that the wavelength

can be (and in practice usually is) large compared with the

mean value of

%

This expansion is justified on the physical grounds

IYl for the w a v e f unction

fg%m' i.e., compared with the "size" of

the atom. Finally, we consider the WE theorem. tation of DA(g).

Let

G

on

~

~i~ 2

and

Ta

Let

U(g)

be a unitary represen-

a tensor component belonging to the representation

be irreducible subspaces of

~

with respect to

U(g), let

169

]CA be the Hilbert space for

A D (g), and let the product space

]CA ~ ~2

decompose

into ]CA @ ~2 = ~ @]C%

with respect to

U(g).

where the sum is taken over all (U(g)/~l)

are equivalent and

%

~ (f%,f~f2)~0~iT~2) % %

fA,f~

(4.11)

D%(g)

are vectors in the directions

In other words, the T-dependent tensor

In particular, if

,

such that the representations

linearly in terms of the T-independent tensors ~IT~)%.

(4.10)

The WE theorem states that (fl,T&f2) =

respectively.

'

U(g)~ I

(fl,T f2)

(f%,f f2)

fl

and and

Ta,

can be expanded

with scalar coefficients

occurs only once in the decomposition

(4.10), then (fl,Tf2) = (fl,f f 2 ) ~ 2 ) i.e., (fl,T~f2)

is parallel to

The coefficients

Note that the

(fA,f f2)

are usually called reduced matrix elements,

(fl,f f2 )

are called Clebsch-Gordon coefficients.

are just the matrix elements of the unitary (intertwining)

operator which transforms the direct product basis in ]CA @ ~ 2 which

U(g)

(4.12)

(fl,faf2).

~ ) A

and the T-independent tensors

,

into the basis in

is diagonal.

5. SURVEY OF EXPERIMENTAL AND THEORETICAL BACKGROUND TO ELEMENTARY PARTICLE PHYSICS

The rest of these chapters will be devoted to th~ group theory of elementary particle physics.

But before going on to the group theory proper, it might

be worthwhile to fill in a little of the experimental and theoretical background. This we shall do in the present chapter. First we consider the experimental background [i]. The non-relativistic quantum mechanics discussed up to now suffices to describe completely the greater part of modern physics--atomic, molecular, plasma, solid state, low temperature, etc., physics. Newton's laws and

IX,P] = i~.

It is built on the twin postulates of

The basic constituents of matter for all these

branches of physics are the protons, neutrons, and electrons which form the atoms9 and the photons, which carry the EM (electromagnetic) field. matter, or particles, are regarded as elementary.

These constituents of

In particular, the protons,

neutrons, and electrons are regarded as indestructible.

~7o

As soon, however, as one wishes to inquire into the finer features of atomic phenomena or wishes to investigate the structure of the atomic nucleus or the structure of the protons, neutrons, and electrons themselves, then the situation changes drastically. relativistic.

First, the energies necessary for the investigation are

Second, the electrons, protons, and neutrons are found to be far

from indestructible.

They can be destroyed and created almost at will.

Third, not

only can these particles be destroyed and created, but new particles are created and destroyed along with them.

The new particles include the anti-particles of the

proton, neutron, and electron, the q-meson which keeps the protons and neutrons bound in the nucleus, and many other particles (along with their anti-particles). To date, the number of new particles which have been produced is of the order of i00. It should, perhaps, be emphasized that the particles referred to here differ in some fundamental ways from the Newtonian particles defined in the first lecture; namely, a)

they can be created and destroyed.

b)

Although they can be created and destroyed, their masses are not arbitrary but are fixed by nature to have definite values outside our control.

c)

For example, the electron has a mass 9.11 × 10 -28 grams.

As well as an intrinsic mass, the particles have an intrinsic angular momentum.

The Casimir operator of the intrinsic angular

momentum group takes the values

J(J + i), where

J

(the spin of

the particle) is half-integer. Thus, the particles appear to be particles in the sense of Democritus (fixed, ultimate constituents of matter) rather than of Newton (fictitious limits of small bodies).

For this reason they are called elementary particles.

Of course,

it is difficult to believe that I00 particles can be elementary, but until something more elementary is discovered, they are regarded as such.

(An analogy is

provided by the chemical elements, all 92 of which were regarded as elementary until the advent of atomic theory.) In Figure 5.1, a list of the particles is presented.

They are grouped

together into multiplets (so-called isospin multiplets) of particles with approximately the same mass and spin.

Even so, the number of multiplets is very large and

it might help to clarify the situation a little if we briefly classify them by word. The broadest classification of the particles is in terms of their interactions.

Apart from the gravitational interactions, in which all the particles

participate, but which are so weak as to be negligible, the particles can interact in only three ways: a)

By electromagnetic interactions, with coupling constant

b)

By weak interactions, with coupling constant

c)

By strong interactions, with coupling constant

e2/~c~i/137

g2 > e2/~c.

Name

IG($)cn

Isospin 0

SU(3) m u l t i p l e t s

estab. ?=guess

x+(140)

I-(0-)+

x0(135)

~(549)

o+(o-)+

-

n(550)

0

~(780)

1-

n*(960)

0- (?)

$(1020)

1-

n(1070)

0'

f (1260)

2'

D(1285)

P=(-1)

E(1420)

0-

f*(1515)

2'

n(700)

0'

J+1

(?)

I s o s p i n 112

-

K(490)

0

K*(890)

1

K(1320)

1'

K**(1420)

2'

K (1780)

P= (-1)

J+1

Isospin 1 ~(140)

0

-

~(760)

1

~(1016)

0'

A1(1070)

1' (?)

A2 (1270)

P=(-1)

A2(1315)

(?) J+1 P= (-1)

T

(1640)

J

2'

p (1650)

P= (-1)

B(1235)

1' ( ? )

J

FIGURE 5 . l b

MESONS

The f o l l o w i n g bumps have a l s o been o b s e r v e d , b u t t h e i r s p i n s and p a r i t i e s a r e n o t y e t known; ~ ( 4 1 0 ) ;

nV (1080) ; Al.5 (1170) ; A22 (1320) ; p p (1410) ; KSKS(1440) ; $ (1650), R(1750) ; n o r p (1830) + 4 ~ ; H(99O) ;

$ or

IT

(1830)

p (2275)

+

m a ; S (1930) ; p (2100) ; T(2200) ;

#1=0(2380)

; ~ ( 7 2 5 ;) \(1080-1260);

: KA(I=3/2) (1265) ; KN(I=1/2) (1660) ; K ~ ( ~ = 3 /(1175) 2)

K*(2240) +

YN;

X-(2500)

: X-(2620; X-(2880).

172 Particle or resonance 2

p

1 "'(jP)

A0815)

1/2(1/2 +)

A(183o)

n

N" (1470)

SU(3) multiplets

n

A(23.50)

o(~)

A

Z

N" (1 520)

1/2(3/2-)

N'(1535)

1/2(1/2-)

~,

1(1/2+)_

N(1670)

1/2(5/2-)

$(138,5)

1(3/2 +)

N(1688)

1/2(5/2 + )

3+ -2 "

A(1236) Z(1385)

1/2(1/2-)

~

N'" (1780)

1/2(I/2 +)

Z (1765)

1(5/2-) D,5

N(1860)

1/2(/2+)

z (1915)

±(5/2*) ~,5

N(1990)

V2 0 to

p2 < 0

states, and simi-

larly for any other interactions which are local in the Fourier transformed space. Now, of course, one might do better with some more complicated, non-local interaction.

But since the purpose of the manifestly covariant wavefunctions is to pro-

vide a framework for introducing simple, local commutation relations and interaction~ this would defeat the purpose. ficulty in Majorana's

For this reason, the

p2 < 0

The second model we consider is a wavefunction @

orbits are a real dif-

theory. ~(p)

carrying a Dirac

unitary spin representation and satisfying the subsidiary condition (y • p + M)~(p) = 0

where

M

M = m 0 + ml~pvEp~ where

,

(8.7)

is a spin invariant, e.g.,

m0, m I

are constants and

~

and

Epv

, are the generators of the Dirac

and unitary representations,

respectively.

This equation was first studied by

Abers Grodsky and Norton E3] (AGN) in 1965 and has since been used in current algebra theory.

An analysis of the equation, similar to that described above for

the Majorana equation,

for the case in which the unitary representation is

(J0' C = 0), shows that for the

p2 > 0

orbits there is a mass-spin relationship

i ±{(m 0 _ m I )2 + m~[J(J + i) - j0(j 0 + i) - ¼]} ±m = m l(J + 7) which can be drawn graphically as in Figure 8.1. well with the observed particles cribing later). pretation.

However,

(The

m < 0

The rising curve for

,

(8.8)

m > 0

fits

(and with Regge theory, which we shall be des-

the falling curve for

m > 0

has no satisfactory inter-

curves can be identified with anti-particles.)

FIGURE 8.1.

P~ASS-SPIN RELATIONSHIP FOR AGN EQUATION

Leaving aside the interpretation of the falling curve, we ask again whether

~(p)

carries unphysical

p2 < 0

orbits.

The answer is yes.

The proof

is perhaps worth giving. Proof.

Write the subsidiary condition

(~ where Now and

~

and

B

(8.7) in the form

+

• p + BM)~(p) = p0~(p)

are the self-adjoint Dirac matrices

,

(8.9)

y0 ~

and

Y0" respectively. ÷ p = 0,

BM

must be self-adjoint to provide a mass spectrum in the rest frame

÷

÷

~ • p

÷

is self-adjoint and bounded.

Hence, for each

÷

p, ~ • p + BM

is self-

198

adjoint.

Hence,

(8.9) may be regarded as an eigenvalue equation for the self-

adjoint operator

~ • p + SM, i.e., P0

is any point in the spectrum of

~ • p

+ ~M. The condition that there be no since

is

P0

any point

in

the

spectrum

p2 < 0 of

orbits is that

a • p + BM, t h i s

• p + ~M)2 ~ p2

or, since

P02 ~ p2.

implies

,

But

that (8.10)

~2 = i, ->

->

(8.11)

p " [~,BM]+ + (BM) 2 > 0 But since

p

varies over the whole Euclidean 3-space, this is possible if and only

if

[~,~M]+ = o which on account of the anti-commutativity of

(8.12)

, ~

and

B, reduces to (8.13)

It,M] = 0 But since (INk)

this means that

p2 ~ 0

=

~,

[o,M] i.e., if and only if

x O~

~

M

= 0

,

(8.14)

is a Dirac invariant, in which case equation (8.7) can be

reduced to a direct sum of Dirac equations with Thus, the AGN equation, unphysical orbits

~,

is possible if and only if

#~ < 0

M = constant.

like the Majorana, is either trivial or contains

and, once again, it can be checked that local inter-

actions couple the physical orbits to unphysical ones. Note that the tion

~(p)

p2 < 0

is quantized.

difficulties ite component

difficulties arise whether or not the wavefunc-

If the field is quantized,

(b) concerning locality. wavefunction

~(p)

then there are the further

To illustrate the point, consider an infin-

which has not yet been quantized, introduce a

set of creation and destruction operators for particles satisfying Bose-Einstein or Fermi-Dirac statistics on a Hilbert space

~, i.e., satisfying

[a(p),a+(p r)]± = d(p _ P') etc., and construct from

~(p)

and

a(p)

,

(8.15)

a quantized field in the standard way,

namely, ~(x) = ~ d~(p){eip'Xa(p)~(p) where into

~(p)

transforms like

~(p).

+ e-ip'xbt(p)~(p)}

The locality difficulties

,

(8.16)

can be subdivided

~199

a)

locality proper

b)

spin-statistics

c)

CPT-invariance

d)

analyticity.

Locality proper is the question whether the commutator

[~(x),~(x')] vanishes for

(x - xP)2 < 0.

,

(8.17)

In the finite-dimensional case, the commutator does

vanish for suitable choice of

±

in (8.16).

In the infinite dimensional case,

however, in general no choice of sign in (8.15) and no simple modification will make (8.17) vanish.

The possibilities for evading this difficulty have been inves-

tigated in some detail in the recent literature E4], but with no particularly attractive solution. The spin-statistics difficulty is an extension of the problem: finite-dimensional cases, (8.17) vanishes for is not arbitrary.

It must be

half-odd-integer spin and

(-)

(+)

±

In the

in (8.15), but the choice of

±

(Fermi-Dirac statistics) if the field carries

(Bose-Einstein statistics) if the field carries

integer spin, a correlation which is verified experimentally and is regarded as one of the most fundamental results of quantum field theory.

But in the infinite

dimensional case, since (8.17) does not vanish for either choice of sign, the spinstatistics correlation gets lost.

(In the cases that (8.17) can be made to vanish,

it can be made to vanish for either choice of sign, so the correlation becomes, at best, arbitrary.) The other two difficulties, CPT invariance and analyticity, are special cases of the general result that for finite-dimensional spin groups, the Lorentz transformations can be continued to any complex values of the parameters whereas for infinite dimensional spin representations, this is not the case. dimensional representations of no entire vectors.) S

SL(2,C)

(Infinite-

have dense sets of analytic vectors, but

As a result, the EM form factors and the scattering matrix

have different analytic properties (as functions of the inner products of the

momenta) in the finite and infinite-dimensional cases, and the analytic properties in the infinite-dimensional case do not seem to be the most desirable. All models so far constructed using infinite-dimensional representations of

SL(2,C)

ways.

have been found to be unsatisfactory in at least some of the above

This suggests that it might be possible to rule out infinite component

fields on quite general grounds and, thus, restrict oneself to the finitedimensional spin representations without any real loss in generality. One such general set of conditions was found recently by Streater and Grodsky [5].

Their argument is as follows:

200

Let

~(o,x)

Hilbert space

X

be an infinite component field operating on a physical

with vacuum state

h, and carrying a continuous bounded irreduc-

ible infinite dimensional spin group, S X . ~(o,x)

is quantized,

Rather than specify precisely how

they assume only that it has been quantized in such a way

that the vacuum expectation value

F(~,~',x,x I) = (0,~¢(o,x)~(~1,xg0) with unique vacuum state

,

(8.18)

0), has the following properties:

a)

Translational invariance:

b)

Reasonable spectrum:

c)

Causality (locality):

d)

Temperedness:

F.(o,or,x,x t) = F(~,~r,x - x r)

~(~,o',p)

= 0

for

p2 < 0, where

F(o,orx) = 0

for

x2 < 0

~

denotes

x

for all

Fourier transform

0,0

e)

F(o,o~,x)

is a tempered distribution in

w

Finite

degeneracy o f t h e l o w e s t i s o l a t e d

These a r e a l l

mass-hyperboloid.

assumptions t h a t a r e made n o r m a l l y i n quantum f i e l d

theory.

The temperedness assumption is a strengthening of locality (it implies that f(o,o',x)

is not too singular on the light cone) and, although this assumption

can be relaxed, it cannot be relaxed very much if the correct analyticity properties are to be obtained for the S-matrix. Grodsky and Streater now claim that these assumptions are incompatible. To prove this, they make use of a theorem due to Bogoliubov and Vladimirov [6] which states that if

f(x)

and the Fourier transform i.e., ~(p)

is a tempered distribution with ~(p) = 0

for

p2 < 0, then

~(p)

for

x2 < 0

is a finite covarian~

has the representation n n ~(P) =[!]C[n]P00...p33~

dm2P[n](m2)~(P2 - m 2)

where the sum is finite, [n] = [nln2n3n0] theorem to

f(x) = 0

and

Pin]

,

is tempered.

(8.19)

Applying this

F(o,or,x), which obviously satisfies the conditions, and smearing with

a test function

f(x)~(p)

with support only in the neighborhood of the lowest

mass-hyperboloid in p-space, one obtains

(0,~#(~,f)~(or,f)0)

= Const. ~ ( p r

n n _ p) ~ C[n](~,~,)p00...p33

In] But since the spin-representation

.

is assumed to be continuous,

C[n](G,or ) L

tinuous in

o

and

or.

Hence, C[nq(O,or )

linear operator in spin space garded as a vector in

V

V.

is con-

J

is the matrix element of a bounded

Hence, for fixed

~r, C[,](o,or )

and since there are only a finite number of

may be reC[n], the

linear span

~]C[n](~,~ ,)p~0 [

•

"'P3n3

'

(8.20)

20q

for all

p0...p3

and fixed

o I, is finite dimensional.

It follows that the expres-

sion (8.20) vanishes for an infinite number of values of

o.

Referring back to

(8.18), we see that there are, therefore, an infinite number of states in

~, orthogonal to the state

~(~rf)O)

the spin group is irreducible, ~(o,f)0) ishes.

for all

p

and

~t.

Furthermore, since

vanishes if and only if

It follows that the orthogonal states are not zero.

hyperboloid is infinitely degenerate.

~(o,f)0)

~(ot,f)0)

van-

Thus, the lowest mass-

This is the result of Grodsky and Streater.

A corollary to their result, which has been pointed out by Grodsky and Streater, is that since any field usual manner (8.16), a wavefunction SL(2,C)-space projection on

~(x)

which is obtained by quantizing in the

~ (p)

P0 > 0

whose support is in

p2 > 0

is polynomially bounded in

and whose

p, will be auto-

matically tempered and causal, it must belong to a finite dimensional representation of

SL(2,C). What does this result mean physically?

dimensional representations of SL(2,C)

It means that if we use infinite-

one of two things must happen.

Either the

subsidiary conditions imposed on the wavefunctions are too weak, in which case there is an infinite number of spin states on each mass-hyperboloid (in gross contradiction to experiment), or else the subsidiary conditions are too strong (as in the Majorana and AGN cases discussed above).

In that case, there is no spin

degeneracy but the wavefunction cannot be quantized so as to describe a tempered local field with

p2 > 0.

Note that the temperedness of the distribution plays a critical role in the above arguments.

It leads directly to the finiteness of the expansion (8.19),

which leads in turn to the finiteness of the linear span (8.20) and hence to the infiniteness of the orthogonal complement.

(Note added in proof:

a generalization

of the GS theorem which allows more general distributions, including Jaffe distributions, is now available [7].) Perhaps the best way to summarize the results of this chapter is to say that while there are no group-theoretical reasons for excluding infinite spin groups, there appear to be other reasons to exclude them, namely, mass-spectrum, locality, and finite-spin degeneracy considerations,

Thus, one can return, (with

some relief!) to the finite dimensional spin representations.

9.

LITTLE GROUP DECOMPOSITION OF THE SCATTERING AMPLITUDE

In the last couple of chapters we saw how the Poincar~ group little group for

p2 ~ 0

this chapter I should like to mention briefly how used to analyze scattering processes. in spite of the spectral condition, the p2 < 0

P++

and its

could be used to characterize relativistic particles.

will also be relevant.

P++

In

and its little group can be

One of the interesting features will be that, SU(I,I)

little group for the orbits

202

To put the role of the little groups into perspective, scattering amplitude necessarily

(Figure 9.1) for 2-particles

the same), e.g.

~N ÷ EK.

we consider the

scattering into 2 particles

The probability

of the particles

(not

1 and 2

K

P2

P4

FIGURE 9.1.

with momenta P4

Pl

and

P2

N

SCATTERING

IN S-CHANNEL

scattering into particles

3 and 4 with momenta

P3

and

is given by P(plP2 + p3p4 ) = l(p3P4 , T plP2)l 2

where

T

amplitude

is the scattering matrix. (pSp4 , T plP2 )

,

(9.1)

Because of Poincar4 invariance,

is (apart from some kinematical

a function of two invariant variables,

s

and

the scattering

factors, which we omit)

t ,

(9.2)

t = (Pl - P3 )2

(9.3)

(p3P4, T plP2 ) = F(s,t) where

s = (Pl + P2 )2 For symmetry we can also define variable.

In fact

u + s + t =

'

u = (Pl - P4 )2' but u is not an independent 4 ~ m~, where m are the masses. (In general,

the

~=i

scattering amplitude

for 2 particles

iant variables,

3n

involved, P:.)

the

into

n-2

variables being the

the ten constraints

If the four particles

particles n

depends on

3-momenta of the

coming from the conservation

n

3n - i0 invarparticles

of the ten generators

involved in the scattering of Figure 9.1 are spinless

(as we shall assume for simplicity)

then

F

is a scalar function.

of

203

Now consider the process of Figure 9.2, namely the scattering

of particles

3

/ p2

Z

\p4

FIGURE 9.2.

i and 3 with momenta (e.g.

~K ÷ NE).

Pl

and

SCATTERING IN S-CHANNEL

P3

The probability

into particles

2 and 4 with momenta

for this scattering

P2

and

P4

is given by

P(plP3 + p2p4) = I (p2p4,T plP3)l 2

,

(9.4)

where (9.5)

(pZp4,T plP3 ) = F'(s',t') and s' = (Pl - P2 )2

'

t' = (Pl + P3 )2

(9.6)

One of the most basic and fruitful ideas to emerge in particle physics during the fifties was that the two scattering

amplitudes

related, but are in fact the same analytic considered and

t'

s'

to be the analytic

the analytic

continuation

F(s,t)

function [i].

continuation of

and

of

F'(s',t')

That is to say, if one

s = (Pl + P2 )2

t = (Pl - P3 )2

to

are not only

to

P2 ÷ -P2

P3 ÷ -P3' then

F(s,t) = F'(s,t) The process of Figure 9.1, for which Figure 9.2, for which analysis

condition,

and of axiomatic

The hypothesis

(9.7) is based upon an

field theory [2].

causality and the temperedness

Returning to the s-channel, are

s > O, is called the s-channel and that of

t > 0, the t-channel.

of Feynman diagrams

the spectral

(9.7)

It is related to

of the field-distributions.

an alternative pair of variables

to

(s,t)

(s, cos 0), F(s,t) = f(s, cos 6)

,

(9.8)

204-

where

e

is the angle between the three-momenta

mass frame of

Pl

and

P2 (Figure 9.3).

and

P2

The relationship

in the center of

between

t

and

cos 9

is

S(t - u) + (m~ _ m4)(m122 _ ml ) cos e =

This looks complicated unless the masses are equal. is that

(9.9)

1 {Is - (m I - m2)2][s - (m I + m2)2][s - (m 3 - m4)2][s - (m 3 + m4)2]}~

cos O

is linear in

t.

to make a "partial wave decomposition" in terms of Legendre

However,

of

data it is usual

f(s, cos 0) i.e., to expand

[3]

f(s, cos 0)

functions f(s, cos 8) = [ (2£ + l)a£(s)Pz(cos

Pl

FIGURE 9.3.

8)

(9.10)

P3

SCATTERING IN CM SYSTEM IN S-CHANNEL IN 3-SPACE

This is done for two reasons. total probability

the important point

In the analyses of scattering

(a)

for scattering

The unitary condition,

which says that the

is unity, is diagonal in the

P%

basis.

In fact,

it reads a~(s) = sin ~ ( s ) where

~(s)

"phase-shifts" ~(~ = 0,1,2)

exp i6~(s)

is real, and a scattering analysis ~%(s).

(b)

dominate.

For low-energies,

is normally an analysis of the

s ~

(m I + m2)2,

the low values of

(One can see this intuitively by noting that for low

energy we have low relative angular momentum of the two particles, see later, ~

and as we shall

is the relative angular momentum.)

Regge Theory

One of the problems city (9.7) with the expansion and

of scattering (9.10).

As we go from the

I c o s e I ~ i, to the t-channel, where

expansion

(9.10) diverges.

theory was how to combine the analyti-

t > (m I + m2)2

To overcome this difficulty,

s

channel, where and

t < 0

Icos 9 1 ~ i, the

Regge [4] showed that, at

2o5

least for a class of non-relativistic tinue

cos 0

potential scattering

was to express the expansion

theories,

the way to con-

(9.10) in integral form.

First, one

writes i I C (2~sin+ w£ l)d£ aZ(s)P£(cos f(s, cos 8) = 2--~i where

C

G)

,

(9. ii)

is the contour of Figure 9.4, then divides the integrand into

+

and

-

signature parts + f-(s, cos 0) = 2 -I~ [J C (2£sin+ ~l)dZ ai(s)[p

which have independent

physical properties,

rately on the circle at infinity,

(cos e) ± Pi(-cos

(9.12)

and then, because each converges

opens up the contour to

iA

0)]

sepa-

L, which is the furthest

I

L

C


Z

>

I| FIGURE 9.4.

THE CONTOURS OF

C

AND

L

line to the left allowed by the Pz(cos O). On the way, one picks up the poles of + a~(s), which for the class of potentials considered is a meromorphic function of +

to the right of

L, and obtains

(simplifying

has only one pole to the right of

i

I

B±(s)+ sin ~ - ( s )

+ a-(s)

the pole.

e)]

e)

,

(9.13) ±

is the position of the pole, and

The expression

and indeed to

[P + (cos 8) _+ P (-cos 8)] a-(s) e±(s)

(2~ + l)d£ ± sin ~ a£(s)[Pz(cos

L

± P~(-cos where

a~(s)

L)

f-(s, cos 8) = (2~±(s) + i) +

+ ~

for clarity to the case when

~i(s)

the residue of

(9.13) can now be continued in

cos 8

a~(s)

at

into the t-channel,

t N cos 8 ÷ ~.

~fhat is the relevance of all this to relativistic is that one now makes the hypothesis be quite different ture of it, namely,

[5] that although relativistic

from non-relativistic the fact that

scattering?

aZ(s)

scattering,

The point

scattering may

it retains at least one fea-

is meromorphic

to the right of

L.

206

This is quite an assumption,

and indeed, has had to be modified.

But it

is at least within the general philosophy that nature is simple if looked at the right way--and here the postulate is that the right way to look at

f(s, cos 0)

from the point of view of its properties in the %-plane to the right of case, let us investigate

L!

is

In any

[6] the physical implications of (9.13).

The physical implications of (9.13) are best seen by noting that the pole ~(s)

is not fixed, but varies with

function of

s

(Figure 9.5).

s, and drawing the path of its real part as a

There is good reason to believe, as we shall see in

5

4¸

R~ ~ ( s ~ i

FIGURE 9.5.

REGGE TRAJECTORIES

a moment, that its path is as in this figure.

The physical implications are then

two-fold: (i)

for

t ~ cos e ÷ ~, s < 0, we have from (9.13)

+ + 1 Bi(s)(cos 0 )~±(s) f-(s, cos O) + 2a-(s) + sin ~ - (s) +

This means that in the t-channel, as

(9.14)

t ÷ ~,

+ ± f±(s,t) ÷ A-(s)t ~ (s)

,

(9.15)

i.e., we have an explicit statement about the behavior of the scattering amplitude as a function of the energy energy.

(t)

for high

This is a result which could not be obtained experimentally

and was not obtained theoretically before the advent of Regge theory.

What was known theoretically before was that, because of

the unitary condition for

T, f(s,t)

decreased, as a function of is assumed to be less than

t 1

for for

was bounded, and probably t ÷ ~. s < 0

This is why in Figure 9.5.

R1 ~-+(s) But the

explicit t-dependence was first obtained in Regge theory, and is clearly controlled by the Regge-pole at

~ = ~+(s).

20?

(ii)

+ Im a-(s)

If

is small, then when

+ + RI a-(s) = integer, i/sin ~ - ( s )

is large.

Hence, remembering the factor Pa-(s) ÷ (cos 6) i Pei(s) + (-cos 0), which is small for R1 a-(s) = even/odd integer, we see + + that f-(s, cos 0) is large for RI a-(s) = even/odd integer.

Returning to the s-channel, s > (m I + m2)2 , we see that the schannel amplitude therefore becomes large, or resonates, whenever + RI a-(s) = even/odd integer. Furthermore, a simple analysis of how the amplitude resonates near

~±(s) = even/odd integer, shows that

it behaves as if it were the contribution to the s-channel scattering of an unstable bound state particle or resonance of mass = ~ss, spin = R1 ~-+(s), and life time

=[Im ~+-(s)]-I, Figure 9.5. This + result clearly suggests that the R1 ~-(s) = even/odd integer points on the Regge-trajectory of Figure 9.5 should be interpreted as unstable particles of increasing mass and spin.

And indeed, if one

examines Figure 5.1a, one sees that the baryons for which it can be checked do indeed lie on Regge trajectories.

The mesons do not have

sufficiently well-determined spins and parities for a direct check but other considerations support the conjecture that they also lie on Regge trajectories.

A typical conjecture [ii] is shown in Figure

9.6.

67_-

U

(

2

T(2200)~ s (193

~

.

a3

1 1 .-I b L

1uJ~

4

--~

J

R 1 (1660)

2--

i-

A (1286) ~2 ~ 1

1.0v

~

2.0

I " I" ~

~" " ( 1

7

4

3.0 t(GeV) 2

FIGURE 9.6.

~ 0

4.0

)

0

~

)

5 0

6.0

7

208

The most beautiful part of the results tion.

(i) and (ii) lies in their combina-

By combining them we see that the resonant stateS, or unstable particles,

which are produced in the s-channel, channel

dictate the high-energy behavior in the t-

(and, of course, conversely).

This unexpected relationship

between these

hitherto unconnected phenomena is a result that is almost certain to survive, no matter how the details of the Regge theory may have been modified. A further beauty of the result is that it simultaneously standing puzzle in scattering tribution to

F(s,t)

theory, namely,

of a particle with a fixed high spin

cos 8, therefore high powers of

solves a long-

that if one were to continue the con(therefore high powers of

t) from the s-channel to the t-channel,

this con-

tribution alone would violate the unitary condition for large t. The Regge result + solves the problem by showing that the spin is really R1 a-(s), and hence is not fixed, but varies with

s

and becomes

less than 1 for

s < 0

in the t-channel.

After the above rather lengthy description of the background,

let us turn

at last to the little groups. Consider first the two-particle state can equally well be described by Since

Pl2 = m~, P22 = m 22' if we consider

two constraints

on

q.

q0

longing to the little group

(@2~2)

SO(3)

P' = P3 + P4, R(62~2)

of

in the s-channel.

as 4 independent

P.

IP,q>

(The angle

can be written as

variables

as

(@i~i)

is a rotation beis the angle between

diagram of Figure 9.3.)

Is, = p,2, ~,, R(02¢2)q0> '

is an element of the little group

is the angle between the fixed z-axis and

there are

IP,R(OI¢I)q0 > = Is

R(OI¢ I)

line in the 3-dimensional

Ip3p4 >

This

P = Pl + P2' q = Pl - P2"

is a fixed vector and

Pl - P2

In a similar way the state where

P

IplP2 > where

As a result, we can write

= p2, p, R(81¢l)q0> ' where

a fixed z-axis and the

state IP,q>

q~ = P3 - P4"

S0(3)

of

However,

P', and from

energy momentum conservation we have S = S

Hence P = P'.

R(81¢I)

and

R(@2¢ 2)

'

~

=

(9.16)

P'

are elements of the same little group, namely that of

For the scattering amplitude,

ally invariant,

,

which is Pofncar~,

and therefore

rotation-

we then have



: <s,P,R(O2¢2)q01TIs,P,R(81*I)q0>

= <s,~,q01rls,P,R-l(e2¢2)R(61¢l)q0 >

: <s,P,q01Tls,P,R(o3,~3)q£

where

(83,¢3) = 8

Figure 9.3.

is the angle between the lines

Pl" - P2

,

(9.17)

and

P2 - P4

in

Hence f(s, cos 8) = = <s,~,q01rls,P,R(8)q0>

(9.18)

209

But since

R(e)

is an element of the little group

as a function of

0, f(s, cos e)

S0(3)

of

P, this means that

is a function over the $ittle group

S0(3).

Hence the expansion f(s, cos 0) = ~ (2~ + l)a~(s)p~(cos

6)

,

(9.19)

emerges as nothing but the expansion of the scattering amplitude as a function over the little group

SO(3)

in terms of the irreducible representations

Once it is realized that the partial wave decomposition restricted to the part of the

s - t

of

SO(d)

(9.19) which is

plane which belongs to the s-channel,

is nothing but an expansion over the little group

S0(3)

of

[7].

s > 0,

P = Pl + P2' a method

for extending the expansion to other channels immediately suggests itself, namely, to make a little group expansion in the same variable in the other channel also. In the other channel, s = p2 < 0, since little group is

However,

there is a snag.

unitary condition guarantees that little group

P = Pl + P2

with

p~ < 0.

Hence the

S0(2,1). The snag is that whereas in the s-channel the f(s, cos 0)

will be square integrable over the

SO(3), in the other channels there is no guarantee that it will be

square-integrable

over

S0(2,1), and in gemeral it is not.

At this point, however, one can return to Regge theory. Regge expansion

Looking at the

(9.13) one sees at once that the background integral is nothing but

the expansion of the scattering amplitude in terms of the principal series of S0(2,1).

Thus what Regge theory says is that, although the full scattering ampli-

tude is not square-integrable

over

S0(2,1), when one removes the contribution of

the Regge poles, the remainder i__sssquare-integrable. looked at from two points of view.

Thus equation

(9.13) can be

From the Regge, or physical, point of view, the

pole term is the important term and the integral just an incidental background term. From the group theoretical point of view, the integral is the interesting group expansion term, and the pole term just an incidental subtraction term to make the integral converge. One may ask why only the principal series appears in the Regge formula. First there is a theorem due to Bargmann ble function over

SO(2,1)

[8] which states that any square-integra-

can be expanded in terms of the principal series and 1 g > - ~. This theorem explains why the supplementary,

the discrete series with 1 trivial and D±(-- ~) representations do not appear. Secondly, the discrete series 1 with g > - ~ do not appear in our case because we have left out the spin of the particles

[9]. In conclusion, I should mention that the further generalization to an ex-

pansion of for

f(s, cos 6)

t N cos es, and

in terms of the Lorentz group

SO(3)

for

S0(3,1)

s) has been considered.

The

becomes particularly interesting and illuminating at the point

(to include S0(3,1) s = 0

SO(2,1)

expansion in the

2~0

continuation of

s

from the

s

to the t-channel, because we can choose

our reference frame, and then, for the little group.

s = 0, P = 0, in which case

The fact that at

s = 0

S0(3,1)

the little group expands to

~ = 0

as

is itself S0(3,1)

has physical consequences, notably that to every Regge trajectory crossing the line s = 0

in Figure 9.5 there is a family of trajectories with values

~(0) - 2 . . . .

corresponding to the representations

occurring in an irreducible representation of

i0.

~(0), e(0) - i,

J = J0' J0 + i, ...

S0(3,1)

of

S0(3)

[i0].

INTERNAL SYMMETRIES

In the previous three chapters, we have considered the space time properties of relativistic Hilbert space in general, and of 1-particle states in particular.

In the present lecture, we should like to consider some properties of

the particles which are independent of space-time. space-time,

these properties,

or symmetries,

Because they are independent of

are called internal symmetries

[i].

The first internal symmetries came to light when the structure of the atomic nucleus began to be investigated in the early thirties. found to consist of protons and neutrons

The nucleus was

(each about 2,000 times the mass of the

electron, i.e., each about 10 -25 grs.) and investigation of the forces that held them together i)

(the nuclear forces) showed that they were much stronger than the electromagnetic

forces (they are

the strong forces mentioned in Section 5) and 2)

they were charge-independent.

That is to say, apart from statistics,

they did not distinguish between protons and neutrons--the

force

between two protons was the same as the force between two neutrons or the force between a proton and neutron. to the electromagnetic

(This is in marked contrast

forces which distinguish clearly between the

proton and neutron, since the proton is charged and the neutron is not.) To formulate charge-independence,

it was convenient to introduce on the

physical Hilbert space, an abstract invariance group. SU(2)

The group used was the

group (isotopic spin group) and the idea was to assign the proton and the

neutron,

respectively,

to two orthogonal vectors

sional representation of Hamiltonian

HN

under the

SU(2)

IP)

and

In)

in the 2-dimen-

and then to demand the invariance of the nuclear

SU(2)

group

[SU(2),H N] = 0

(i0.i)

211

The generator

I

of

SU(2)

for which 1 I [ p ) = ~ IP)

is called by convention

13

,

-i I[n) = --~ In)

,

(10.2)

or the third component of isospin.

For many-nucleon

states (nucleon = proton, neutron), one uses tensor products of the states In)

[p),

in conjunction with (10.1). The mathematics of the isospin group is the same as for the ordinary

electron spin group

SU(2), but the physics is quite different.

First, every vec-

tor in the ordinary 2-dimensional spin space is realizable in nature, but only and

In)

in isotopic spin space are realizable.

[p)

(Nobody has ever succeeded in

constructing a state which is a linear superposition

alp) + b[n)ab # 0.)

Secondly,

the operators in spin space transform non-trivially under space rotations, whereas the operators in isospace are independent of space-time. Later in the investigation of nuclear structure, it was found that the nuclear forces between protons and neutrons were mediated by there are three, T 0 that the

T's

and

T-mesons, of which

T ±, the index referring to the charge.

It turned out

could be incorporated into the isospin scheme by assigning them to

the 3-dimensional representation of

SU(2)

13 IT±) = ±

[~±)

with I3[T0) = 0

(10.3)

Once higher (relativistic) energies became available, and the bombardment of nuclei with protons and neutrons of high energy resulted in the production of i

new particles, it was found that the forces producing the new particles were a)

of the same order of strength as the nuclear forces

b)

still charge independent.

These forces are, therefore,

called generically "strong" forces, and the associated

Hamilton±an the strong Hamilton±an

H . s From the charge independence of the strong interactions and the assign-

ment of the creating particles

(p,n)

and

tively, to irreducible representations of

(~±0)

denoted by

N

and

~-

of

SU(2)

In fact, the metastable hadrons

are found to belong to the

2,2,1,1,3,2

and, hence, are denoted by

SU(2). +

and 1-dimensional representations

K, K, n, A, E, E, and

of

SU(2)

[SU(2),Hs] = 0

And this

K 0, ~0, q, A, E±0, E 0,

g, respectively.

For the unstable hadrons, the same results are found. assigned to irreducible representations

~, respec-

SU(2), it follows that the created

particles should also belong to irreducible representations of + turns out to be the case.

and

All can be

with ,

(10.4)

and, indeed, the number of unstable hadrons is now so large that one no longer refers to them individually but refers only to the isospin multiplets to which they belong.

This method of referrin~ to them has been anticipated in Table 5, in which

212

i

(the total isospin) labels the representation of

belong (dimension = 2I + i).

SU(2)

to which the particles

Note that all particles in the same

SU(2)

multiplet

have the same mass (up to electromagnetic and weak interaction corrections) and the same spin, indeed have the same space-time properties in general. the internal invariance group

SU(2)

This is because

and the space-time Poincar~ group are inde-

pendent, i.e.,

(10.5)

[SU(2),P:] = 0 In the production of new particles from

N,N,~, a new invariance law

became evident, namely, that when new particles are produced from they come only in certain combinations.

multiplets to integer eigenvalues of

Y

[Y,Hs ] = 0 i.e., t h a t

Y

and

The simplest description of the allowed

combinations is obtained by introducing a new operator SU(2)

N,N

Y

on

~, assigning the

and demanding that ,

(10.6)

be conserved in the strong interactions.

Because of its analogy to

the electric charge, Q, which takes integer values on the particle states and is conserved in interactions, Y whereas

Q

is conserved in

in the strong ones. and

is called the hypercharge. all interactions, Y (like

The hypercharges of

(1,0,0,-1,-2), respectively.

~,K,K,~

and

Note, however, that SU(2)) is conserved only N,A,~,H,~

are

(0,I,-i,0)

In general, we have also the relation

1 Q = 13 + ~ Y

,

(10.7)

which was first discovered by Gell-Mann and Nishijima. The strong interactions are found to be invariant, therefore, under SU(2) by

and

Y.

Y, and hence under the group

SU(2) x U(1)

ger, means that only those representations of Y = D (modulo 2), where

D

such representations of

SU(2) x U(1)

of

where

But, as Louis Michel has pointed out, the fact that SU(2) x U(1)

is the dimension of the

SU(2)

is generated

in (10.7) is inte-

occur, for which representation.

Since

are exactly those which are representations

U(2), it follows that we can replace the invariance under

invariance under

U(1) Q

SU(2) x U(1)

by

U(2).

A glance at Figure 5.1 will show that

U(2)

is the maximal group with the

i

property

[U(2),P:] = 0, up to electromagnetic and weak corrections, since any

other transformations among the particles will not commute with the mass and spin. In spite of this, Figure 5.1 does suggest that one could go beyond The reason is that the different

U(2)

U(2).

multiplets seem to fall into sets which,

while they do not have the same mass, do have the same spin and have approximately the same mass.

Examples of such sets are

(~, K, K, n)

and

(N, A, E, Z).

For

2~3

this reason, it has been proposed that original choice i)

SU(3)

SU(3)

U(2)

be enlarged to a group

SU(3).

The

rested primarily on two factors:

allows two and only two commuting generators, and two and only

two additive operators

(13

and

Y)

are necessary to label the

particles. 2)

SU(3)

has an irreducible representation

(the 8-dimensional adjoint

representation) which can accommodate the two sets of eight particles (~, K, K, ~) values. (Actually, the

n

SU(3).)

Having adopted

SU(3)

multiplet

sional representation 2+

with the correct

(I(I + l) = 121 + 122 + I~)

unstable, particles to JP = ~3+

(N, A, E, E)

multiplets.

SU(3)

SU(3)

was proposed

SU(3), one tries to assign the othe$

which is assigned to the 10-dimen-

(Figure 5.1) and the

JP = ~3+ ,~ 5+ ,i- , and

multiplets which are assigned to the 8-dimensional representation

higher like

JP

(spin-parity) multiplets look equally promising.

~, was actually predicted by the

SU(3)

Thus, for particle assisnmentsj the exact invariance group

U(2)

Y

Success has already been achieved with

(N*, E* ~ Y*, ~*, ~)

(decimet) of

I, 13, and

See Figure 5.1.

was not known experimentally at the time

and was predicted by

the

and

SU(3)

(octet).

The existence of

The ~,

assignment. turns out to be as successful as

had been before it.

Unlike

U(2), however, SU(3)

is not an invariance group, i.e.,

[x,H s] ~ 0 for those generators U(2).

X

of

SU(3)

,

(10.8)

which do not generate transformations within

Indeed, if (10.8) were zero, SU(3)

transformations would commute with the

mass-operator and all the particles assigned to an

SU(3)

multiplet would have to

have the same mass, which is manifestly not the case. One might then ask: predicting particles?

Of what use is

SU(3)

The answer is that although

apart from classifying and SU(3)

is not an invariance

group, it is approximately an invariance group, and by stating how certain operators transform with respect to it, one can obtain physical predictions, within the approximation

correct to

(N20%).

The operators whose transformation properties are of most interest are i)

the Hamiltonian

H

2)

the mass operator

3)

the electric current

4)

the weak current of the hadrons

s M j~(x) 3~(x)

The transformation properties of these operators are assigned on physical grounds and amount to a statement about the tensor character of the operators. cal predictions are then extracted by using the WE theorem. operators i) to 4) briefly:

The physi-

Let us consider the

2"14

i)

The Hamiltonians:

one usually assumes that H

= H (0) + H (1)

S

where

H (0)

H( 1 ) .

Hence,

S

ant.

is

S

SU(3)

S

,

S

invariant and about five times as large as

to within

20%, H

can be r e g a r d e d

S

as

SU(3)

invarJ-

This allows us to obtain 20% estimates for the relations

between strong decay processes such as N* ÷ Nw

÷ AT ~ ÷

'IT

for example, N* ÷ N~ ,. p_ (N*'HsN~) -- O (N*'H~ 0)N~) ~* + AT

o (Z*,HsA~)

o (E,,H(0)N~) S

p

N*

~

Cg,

~

,

~ 0

,H(0) l- ? s 2 0J

1088 p CN* N ~ (10.9) d CI 0 8 8 E*A~ where

0, a

the masses

are kinematical phase-space factors, depending only on (see Equation

coefficients,

(10.15) below),

13- .(0)l

and

L~ '~s ~ vj

the

C's

are Clebsch-Gordan

is a WE reduced matrix element.

In a

similar way, one can relate scattering processes such as ~N ~ ~N ÷ KZ ÷KA [N÷~Z m~

etc., for particles

(~,K,K) and

(N,I,A,E)

in the same

SU(3)

multiplets. 2)

The Mass Operator

M, like

H , is assumed to be of the form S

M = M (°) + M (I)

where

M (0)

is

SU(3)-invariant and

,

M (I)

is not.

However,

in the

2~5

case of the mass,

one goes further and also makes a positive

ment about

namely,

M (I)

eight-dimensional Using then possible

M (I)

transforms

representation

of

as the Y-component

the WE theorem or general

of

tensor techniques,

and

SU(3),

I(I + i) = I~ + I~ + I~ ~

are

SU(3)

the mass-formula

it is

on the space of an irreducible

1 M = ~ + BY + y[I(I + i) - ~ y2] where

of the

SU(3).

to show that operating

representation

state-

scalars.

,

(i0.i0)

is the total isotopic Thus,

(i0.i0), where

for any

~

SU(3)

spin and

~, B

multiplet,

is the mean value of

M

we have

for the

multiplet. This formula agrees with experiment metastable

baryon octet,

~ o r which it makes the

~

to within 4% for the 30.5% for the ~ baryon decimet

to within

two predictions,

one of which is the prediction

at exactly the right mass value)

(with some "representation SU(3)

and it agrees reasonably well

mixing" modifications)

operator

to illustrate

in finite-dimensional dimensional ~

the following spaces.

on

A.

Then

Hilbert

on

~,

L(A)

G.

Hence any given operator

G.

But this means

Thus,

a "tensor"

M

respect

A E L(A)

that amy . operator

dominate.

SU(3).

subspace

Hence,

operators

M

in terms of in the tensor oper-

that an operator

one (or a few) irreducible

A

is completely

the direction determined

of

space

1,8,8,27,10,10

is limited

M

ten-

must be one of the 64 on this space.

With

(L(A))

to be (i0.ii)

within each irreducible

by the condition

splits

with respect

M = M I + M 8 + M 8 + M I0 + M I0 + M 27 In addition,

is

in the case of the mass oper-

this 64-dimensional

of dimension

a p~o~,

space for

of irreducible

to the octet space, M

SU(3), however,

into irreducible to

the space of linear

is a tensor operator

For example,

linear independent

respects.)

a group implemented

can be expanded

as a series

that in its expansion,

restricted

to

L(A)

the real content of the statement

is

sor components

possible

and

G

mass

to infinite

in several

space,

SU(3)

tensor operators

is itself a representation

sense that it can be expanded ators.

point concerning

spaces but it is more complicated

transformations

operators

and use the

(There is a generalization

be a finite-dimensional

by unitary

ator

for the remaining

multiplets. At this point, we make a digression

Let

of

that

M

be a

subspace U(2)

invar-

iant [U(2),M]

= 0

,

(10.12)

2~16

and, in fact, this condition kills

i0

and

i0, and constrains

M

to be

M where

Y

M1 + 0

+

+

(10.13)

is the direction Qf the hypercharge.

Thus, the only

assumption that goes into the mass formula (i0.i0) which is not already implied by general considerations is that in (10.13) the component of

M

in the

27

is suppressed.

This assumption is made

for other tensor operators also (see below), in which case it is called explicitly "octet dominance".

Incidentally, it might be worth remarking that the analogue of (10.13) for the decimet is M = M 01 + ~

+ ~?

+ ~,

so that the octet dominance assumption kills two

parameters (27 and 64) in this case.

This is why the mass-formula

yields two predictions for the decimet. The strong decay rates and the mass formulae provide stringent cross-checks on the

SU(3)

particle assignments.

In fact,

since the mass of a particle is usually known experimentally long before its spin-parity, in practice one assigns according to the mass formula first and checks with spin-parity afterwards. An interesting feature emerges in the case of the mesons 0-, i-, and

2+ .

These come not in octets but in nonets and, to fit

the mass formula, one must assume that the two particles

(~,n*),(~,~)

and

nonets, do not have definite

U(2)

scalar

(f,f*), respectivel~ which occur in the SU(3)

properties; rather, the linear

combinations q8 = cos 8 n + sin 8 ~*

(10.14) q0 = cos @ ~* - sin @ n etc., belong to the pectively.

SU(3)

octet and scalar representations, res-

This phenomenon is called "representation mixing" and

it robs the mass formula of its direct predictions for these particles by adding the new unknown parameter formula can do is determine the various 80- ~ i00' 61- ~ 600, 02+ ~ 300")

0.

The best that the mass-

@'s.

(They turn out to be

The interesting point, however, is

that in spite of this, the mass formula is not empty.

One can get

indirect predictions, and indeed one of the most remarkable features of the decay rate analyses is that in two cases in which the experimental decay rates are in complete contradiction with phase space and SU(3)

without mixing, the use of mixing accounts for the discrep-

ancy.

For example, experimentally

2~ 7

f ÷ 2-------=-!~ 50 f ÷ KK

,

f* ÷ 2__~~ i f* ÷ KK ~

'

(10.15)

whereas, on account of the much smaller mass of the pion, phase space would predict the ratio ~

1

in both cases.

With mixing, SU(3)

predicts, using the WE theorem, [c°s O

8>

+ sin 0 / @ I S 0 8 ~

f "->" 2"rr _- p

2,kfoI

o

0

8

S0 8

8

0

0

8~12

I~/1~I|

so 8

8

88ol2 888s + a sin 0 2 C 0 | __c°s 0 2 C 8

~

I

o

888s COS @2CKK8 + a sin @ C880l 2 KKOJ

L

3(2a sin @ 2 + cos @ )2 =

p__

o where

SO

2

,

(i0.16)

4(a sin @ 2 - cos @2 )2

is the scalar approximation

to the S-matrix

S,



,

(12.22)

and such quantities have the property that their Fourier transforms with respect to

x

ment.

are the form-factors Unfortunately,

culties of principle.

for the particles and so are within reach of experi-

the saturation with 1-particle states raises some diffiOne can show, for example,

that unless the current

j~(x)

is trivial, the current x current algebra (12.21) cannot be even approximately saturated with 1-particle states

(even if an infinite number of 1-particle states

are used) unless the masses are degenerate. that in the limit that

However, it has been conjectured

[7]

Pz' the third component of the total momentum of all the

states, becomes infinite, the saturation with 1-particle states may become exact and lead to predictions for the mass-spectrum and the form factors, or at least to correlations between the two.

This conjecture, which is based on experience with

the free-Dirac equation and the charge x current algebra, is still open. inary investigations,

using, for simplicity,

j0(x)

[~,~6]

=

Prelim-

the special case of a factored current

~ j0(x)

= is By~ Y

,

,

j0(x)j0(y) = j0(x)6(x - y)

(12.23) ,

show that in the factored case the solutions can be written as infinite component wave equations.

This result furnishes another link between conventional physics and

infinite component wave equations, but since, as we have seen in Lecture 8, infinite component equations have some undesirable physical properties, be an indication that the factorization hypothesis

the result may only

(12.23) is too strong.

232

13.

REFERENCES

Chapter 1

[1]

E. %qhittaker, Analytical Dynamics, Cambridge (1937). Pri~aiples of Mechanics, Toronto Univ. Press (1962).

C. Lanczos, Variational

[2]

P. Dirac, ReV. Mod. Phys., 21, 392 (1949). T. Jordan, E. C. G. Sudarshan, ibid., 33, 515 (1961). M. Pauri, G. Prosperi, J. Math. Phys., ~, 366 (1966). M. Hamermesh, Group Theory, Addison-Wesley, Cambridge, Mass. (1962).

Chapter 2

[i]

N. Bohr, Constitution of Atoms and Molecules, Munksgaard, Copenhagen and Benjamin, New York (1963).

[2]

E. Wigner, Phys. Rev., 77, 711 (1950).

[3]

F. Rellich, Nachr. Akad. Wiss. G~ttingen, IIA, 107 (1946). Math., 13, 263 (1958).

[4]

H. Weyl, Theory of Group Representations and Quantum Mechanics, Methuen, London (1931).

[5]

J. von Neumann, Math. Ann., 104, 570 (1931).

[6]

F. Riesz, B. Nagy, Functional Analysis, Blackie, London (1956).

[7]

A. Wightman, Proceedings Fifth Eastern Theoretical Conference, edited by D. Feldman, Benjamin, New York (1967).

[8]

M. Hamermesh, Ann. Phys., 9, 518 (1960). J. Levy-Leblond, J. Math. Phys., ~, 776 (1963). J. Voisin, ibid., ~, 1822 (1965).

[9]

E. InSnd, E. Wigner, Nuovo Cimento, ~, 705 (1952).

J. Dixmier, Comp.

Chapter 3

[i]

E. Wigner, Group Theory, Academic Press, New York (1959), p. 233.

[2]

A list of papers is given in [3] below. See also J. Jauch, in Group Theory and its Applications, edited by E. Loebl, Academic Press, New York (1968), G. Ludwig, Grundlagen der Quanten Mechanik, Springer, GSttingen (1954), A. Messiah, Quantum Mechanics, Vol. II, North-Holland, Amsterdam (1962), L. O'Raifeartaigh and G. Rasche, Ann. Phys., 25, 155 (1963).

[3]

V. Bargmann, J. Math. Phys., ~, 862 (1964).

[4]

E. Condon, C. Shortley, Theory of Atomic Spectra, Cambridge (1935).

[5]

P. Xramer and M. Moshinsky, in Group Theory and its Applications, edited by E. Loebl, Academic Press, New York (1968).

[6]

The S0(4) symmetry of the H-atom was first analyzed by W. Pauli, Z. Physik, 36 (1926), V. Fock, ibid., 98, 145 (1935), V. Bargmann, ibid., 99, 576 (1936).

233

[7]

C. Fronsdal, Phys. Rev., 156, 1665 (1967). 1149 (1967).

A. Barut, H. Kleinert, ibid., 160,

Chapter 4

[i]

C. Eckart, Rev. Mod. Phys., ~, 302 (1930). Press, New York (1959).

E. Wigner, Group Theory, Academic

[2]

N. Akhiezer, I. Glszman, Theory of Linear Operators in Hilbert Space II, Ungar, New York (1963).

[3]

L. GSrding, Proc. Nat. Acad. Sc/. U. S., 33, 331 (1947).

[4]

I. Segal, Duke Math. J., 18, 221 (1951).

[5]

E. Nelson, Lecture Notes, ETH, Z~rich (1963).

[6]

P. Cartier, J. Dixmier, Amer. J. Math., 80, 131 (1958).

[7]

E. Nelson, Ann. of Math., 70, 572 (1959).

[8]

L. GSrding, Bull. Soc. Math. France, 88, 73 (1960).

[9]

Harish-Chandra, Proc. Nat. Acad. Sci. U. S., 37, 170 (1951).

[i0]

R. Goodman, J. Functional Analysis, ~, 246 (1969).

[ii]

R. Goodman, Trans. Amer. Math. Soc., (to appear).

Chapter 5

[1]

For more detailed information, see for example, E. Segr&, Nuclei and

Particles, Benjamin, New York (1964). [2]

P. Dirac, Proc. Roy. Soc., II4A, 243 (1927). Z. Physik, 56, i (1929), 59, 160 (1930).

[3]

For precise mathematical formulation, see R. Streater and A. Wightman, PCT, Statistics and All That, Benjamin, New York (1964) and R. Jost, The General Theory of Quantized Fields, Amer. Math. Soc., Providence, R. I. (1965).

W. lieisenberg, W. Pauli,

Chapter 6

[i]

The lifting problem has been analyzed for general topological groups by V. Bargmann, Ann. of Math., 59, i (1954), D. Simms, Lecture Notes in Mathematics, Springer, Heidelberg (1968), K. Parthasarathy, ibid. (1969).

[2]

H. Bacry, J. Levy-Leblond, J. Math. Phys., ~, 1605 (1968).

[3]

E. Wigner, Ann. of Math., (1958), H. Joos, Fortschr. Phys., ~, 949 (1966). J. ibid., ~, 532 (1968). H.

[4]

G. Mackey, Induced Representations of Groups and Quantum Mechanics, Benjamin, New York (1968).

40, 39 (1939). See also Y. Shirokov, JETP, 6, 919 Physik, iO, 65 (1962), A. Chakrabarti, J. M~th. Lomont, H. Moses, ibid., 3, 405 (1962). N. Mukunda, Moses, ibid., ~, 16 (19687 and references therein.

234

[5]

R. Newton, E. Wigner, ReV. Mod. Phys., 21, 400 (1949). 34, 845 (1962).

[6]

B. Schroer, Fortschr. Physik, 2_w i (1963). U. S. A., 57, 194 (1967).

A. Wightman, ibid.,

I. Segal, Proc. Nat. Acad. Sci.

Chapter 7

[I]

The ideas of the present chapter are implicit in the work of Wigner (Ann. of Math. (1939)), V. Bargmann and E. Wigner, Proc. Nat. Acad. Sci. U. S. A., 34 211 (1946), and are treated explicitly by Pursey, Ann. Phys., 32, 157 (1965), to which the reader is referred for many other references.

[2]

L. Foldy, Phys. Rev., 102, 568 (1956).

[3]

M. Jacob, G. Wick, Ann. Phys., ~, 404 (1959).

[4]

A. Wightman, Symmetry Principles at High Energy, edited by A. Perlmutter et al. Benjamin, New York (1968). S. Weinberg, Phys. Rev., 133B, 1318 (1964).

[5]

K. Johnson, E. Sudarshan, Ann. Phys., 13, 126 (1961).

[6]

I. Gelfand, R. Minlos, Z. Shapiro, Representations of the Rotation and Lorentz

Groups, Pergamon, New York (1963). [7]

M. Fierz, Helv. Phys. Acta, 12, 3 (1939).

[8]

W. Rarita, J. Schwinger, Phys. Rev., 60, 61 (1941).

[9]

V. Bargmann, E. Wigner, Proc. Nat. Acad. Sci. U. S. A., 3-4, 211 (1946).

Chapter 8

[1]

E. Majorana, N~ovo Cimento, 9, 335 (1932).

~2 ] See reference [6] of Chapter 7. [3]

E. Abers, I. Grodsky, R. Norton, Phys. Rev., 159, 1222 (1967).

[4]

G. Feldman, P. Mathews, Phys. Rev., 154, 1241 (1967). C. Fronsdal, ibid., 156, 1665 (1967). Further references can be found in L. O'Raifeartaigh, Symmetry Principles at High Energy, edited by A. Perlmutter et al., Benjamin, New York (1968) and I. Todorov, Proceedings of Theoretical High Energy Conference, Rochester (1967).

[5] I. Grodsky, R. Streater, Phys. Rev. Lett., 20, 695 (1968). [6]

N. Bogoliubov, V. Vladimirov, Nauch. Dokl. Vysshei, Shkoly (1958), 3, p. 26. R. F. Streater, Ph.D. Thesis, London (1959), p. 41. J. Bros. H. Epstein, V. Glaser, Comm. Math. Phys., ~, 77 (1967).

[7] A. Oksak, I. Todorov, Degeneracy of the Mass-Spectrum for Infinite-Component Fields, Princeton Institute for Advanced Study, Preprint (1970).

255

Chapter 9

[i] J. Hamilton, Theory of Elementary Particles, Oxford (1959). G. Chew, S-matrix Theory of Strong Interactions, Benjamin (1961). G. Chew, M. Jacob, Strong Interaction Physics, Benjamin (1964). [2]

R. Eden, P. Landshoff, D. Olive, J. Polkinghorne, The Analytic S-matrix, Cambridge (1966). R. Eden, High Energy Collisions of Elementary Particles, Cambridge (1967). G. K~llen, Elementary Particle Physics, Addison-Wesley, New York (1964). A. Wightman, Dispersion Relations and Elementary Particles, edited by C. de Witt & R. Omnes, Wiley, New York (1960).

[3]

L. Schiff, Quantum Mechanics, McGraw-Hill, New York (1949).

[4]

T. Regge, Nuovo Cimento, 18, 947 (1960).

[5]

G. Chew, S. Frautschi, Phys. ReV. Lett., 8, 41 (1962). M. Gell-Mann, S. Frautschi, F. Zaehariasen, Phys. Rev., 126, 2204 (1962).

[62

E. J. Squires, Complex Angular Momentum and Particle Physics~ Benjamin, New York (1963). M. Froissart, R. Omnes, Mandelstam Theory and Regge Poles, Benjamin, New York (1963). R. Newton, The Complex J-plane, Benjamin, New York (1964). S. Frautschi, Regge-Poles and S-matrix Theory, Benjamin, New York (1963).

[7]

M. Toiler, Nuovo Cimento, 37, 631 (1965). H. Joos, in Lectures in Theoretical Physics, University of Colorado, Boulder (1964), Fortschn Physik, lO, 65 (1962).

[8]

V. Bargmann, Ann. of Math., 48, 586 (1947).

[9]

J. Boyce, J. Math. Phys., 8, 675 (1967).

[i0]

D. Freedman, J. Wang, Phys. Rev., 153, 1596 (1967). G. Domokos, G. Tindle, ibid., 165, 1906 (1968). M. Toiler, Nuovo Cimento, 54, 295 (1968).

[ii]

A. Ahmadzadeh, R. Jacob, Phys. Rev., 176, 1719 (1968).

Chapter i0

[1] A few references for isotopic spin are: J. Blatt, V. Weisskopf, Theoretical Nuclear Physics, Wiley, New York (1952); P. Roman, Theory of Elementary Particles, North-Holland, Amsterdam (1960); S. Schweber, Relativistic Quantum Field Theory, Row-Peterson, New York (1961). Some references for SU(3) are: M. Gell-Mann, Y. Ne'eman, The Eightfold Way, Benjamin, New York (1964); M. Gourdin, Unitary Symmetries, North-Holland, Amsterdam (1967); P. Carruthers, Introduction to Unitary Symmetry, Wiley, New York (1966); E. Loebl, Group Theory and its Applications, Academic Press, New York (1968). [2]

See also: F. Lurgat, L. Michel, Nuovo Cimento, 21, 575 (1961); L. Michel i_~n Group Theoretical Concepts and Methods in Elementary Particle Physics Istanbul Summer School 1962, edited by F. G~rsey, Gordon & Breach, New York (1964).

Chapter ii [i]

M. Gell-Mann, Phys. Lett., 8, 214 (1964). TH. 401 and 6419/TH. 412 (1964).

G. Zweig, CERN Reports nos 8/82/

236 [2]

Although it is convenient to describe SU(6) in terms of quarks, they were not used explicitly in the original introduction, F. G~rsey, Phys. Rev. Lett., 13, 173 (1964), A. Pals, L. Radicati, ibid., 13, 175 (1964), F. GNrsey, A. Pals, L. Radicati, ibid., 13, 299 (1964), B. Sakita, Phys. Rev., 136, B1756 (1964). For a review article on SU(6), containing an extensive list of references, see A. Pals, Rev. Mod. Phys., 38, 215 (1966).

[3]

K. Bitar, F. GHrsey, Phys. Rev., 16___~4,1805 (1964).

[4]

B. Sakita, K. Wall, Phys. Rev., 13__~9,B1355 (1965). A. Salam, R. Delbourgo, J. Strathdee, Proc. Roy. Sot., 284A, 146 (1965). M. Beg, A. Pals, Phys. Rev. Lett., 14, 267 (1965).

[5]

M. Beg, A. Pals, Phys. Rev. Lett., 14, 509 (1965).

[6]

S. Coleman, Phys. Reu., 138, B1262 (1965). Rev., 159, 1251 (1967).

[7]

For a review of the mathematical aspects of the mass-spectrum problem and other mathematical aspects of the difficulty of combining SU(3) and + P+ in G see G. Hegerfeldt, J. Henning, Fortschr. Physik, 16, 491 (1968),

S. Coleman, J. Mandula, Phys.

17, 463 (1969).

[8]

L. Michel, Phys. Rev., 137, B405 (1965). H. Lipkin, in Symmetry Principles at High Energy, edited by A. Perlmutter et al., Benjamin, New York (1968). See also: W. McGlinn, Phys. Rev. Lett., 12, 467 (1964), E. C. G. Sudarshan, J. Math. Phys., 6, 1329 (1965) and reference [6].

Chapter 12

[1]

Current Algebra was originally proposed by M. Gell-Mann, Physics, ~, 63 (1964); Phys. Rev., 125, 1067 (1962). The two standard books on current algebra are: S. Adler and R. Dashen, Current Algebras, Benjamin, New York, (1968); B. Renner, Current Algebras and their Applications, Permagon Press, Oxford (1968). See also A. V~ikel, U. V~ikel, NuoVo Cimento, 634, 203 (1969).

[2]

J. Schwinger, Phys. Rev. Lett. !, 296 (1959).

[3]

M. Gell-Mann, Proceedings Conference High Energy Physics held in Rochester, 1960, p. 508 i__n_nThe Eightfold Way, Benjamin, New York (1964).

[4]

S. Coleman, J. Math. Phys., ~, 787 (1966).

[5]

S. Adler, Phys. Hey. Lett., 25, 1051 (1965). (1965).

[6]

B. Lee, Phys. Rev. Lett., 14, 676 (1965).

[7]

R. Dashen, M. Gell-Mann, Phys. Rev. Lett., 17, 340 (1966). S. Fubini, Proceedings Fourth Coral Gables Conference 1967, W. H. Freeman & Co., San

W. Weisberger, ibid., 25, 1047

Francisco (1967).

[8]

S.-J. Chang, R. Dashen, L. O'Raifeartaigh, Phys. Rev. Lett., 21, 1026 (1968). B. Hamprecht, H. Kleinert, Phys. Rev., 180, 1410 (1969). M. Gell-Mann, D. Horn~ J. Weyer, Proceedings Heidelberg International Conference, North-Holland, Amsterdam (1968). H. Leutwyler, Phys. Rev. Lett., 20, 561 (1968). H. Bebi&, F. Ghielmetti, V. Garg&, H. Leutwyler, Phys. Rev., 177, 2196 (1969).

ON CERTAIN UNITARY REPRESENTATIONS WHICH ARISE FROM A QUANTIZATION THEORY by Bertram Kostant*

In this paper we are concerned with certain explicit constructions of unitary representations which arise from a general theory relating quantization and unitary representations.

We shall not go into the general theory here but we

can refer the reader to a forthcoming publication entitled "Quantization and Unitary Representations,

Part I - Prequantization" which will appear as part of the

series "Lectures in Modern Analysis and Applications" edited by C. T. Taem, in

Lecture Notes in Mathematics

published by Springer-Verlag.

Those considerations

here for solvable groups are part of a joint work of L. Auslander and myself.

i.

Let

G

THE REPRESENTATION

indG(ng,h)

be a Lie group, not necessarily connected, and let

g

be its Lie

algebra. Now let

g E g'

be a linear functional on

algebra of the isotropy subgroup tion of

G

on

g'.

Thus if

Bg(x,y) =

B

gg

is the radical of

(i)

g

g

~ G with respect to the coadjoint representag is the alternating bilinear form on g given by

g

the radical of

for all

and

g~/h

y E g}

Bg.

is a complex subalgebra

gg ~ h

dim~

be the Lie

as a complex valued linear functional on

gg

necessarily connected even if (2)

gg

then

We may regard A polarization at

and let

G

gg = {x E glBg(x,y) = 0 That is

g

h ~ g~

is stable under G

= 1/2 d i ~

Ad Gg

g~ = g + ig.

such that (note that

Gg

is not

is connected) g/gg

(recall

di~R g/gg

is even since

gg

Bg)

Department of Mathematics, Massachusetts Institute of Technology, Cambridge, Massachusetts. Currently at Tata Institute, Department of Mathematics, Bombay, India.

is

238

(3)

gi[h,h] = 0,

(4)

h + h

Now let

i.e.,

gI h

is a homomorphism

is a Lie algebra of

d = h N g

g~.

so that if

d~ = d + id

one has

d~=hnK Also let

e = (h + ~) N g

so that if

e¢ = e + ie

one has

e~= h + ~ Now clearly extension of space to

e

B

h

to

g relative to

under the quotient map form

B

on

g

e/d

is equal to its own orthogonal subspace relative to the

g~.

It follows easily then that B

and hence if

g

e ÷ e/d

e/d

x 6

d

is the orthogonal sub-

denotes the image of

x E e

one defines a non-singular alternating bilinear

by the relation (~,y) =

for

x,y E e.

(e/d)~

Next note that we may identify

with

e~/d~

so that

(e/d)~ = h/d~ ~ / d ~ is a linear direct sum.

~/d~

Since

= (~)

relative to conjugation over the real

form e/d of (e/d)~ one defines a non-singular operator j E End e/d where .2 3 = -I and (upon complexification) j = -i on h/d~ and j = i on ~/d~.

Remark i.

Note that if

h/d~

u + iju E Let

S

g

u E

be the bilinear form on

e/d

and

e/d

one has u - iju E

~/d~

given by

{u,v} = (ju,v)

Proposition i

Sg is a non-singular symmetric bilinear form on e/d. orthogonal relative to both

Sg and Bg

{ju,jv} = {u,v}

Proof.

e/d

B

g

to

u,v E

j

is

(ju,jv) = (u,v)

It is clear that by definition

relative to the extension of u,v E

and

That is, if

Moreover, e/d one has

(e/d)~.

h/d~

is orthogonal to itself

Thus by Remark i, one has for

0 = (u + iju,v + ijv) = [(u,v) - (ju,jv)] + i[(ju,v) + (u,jv)] Since the imaginary part is zero this implies that (ju,v) = - ( u , j v )

= (jr,u)

(i.i)

239

That is since

{u,v} = {v,u} j

and hence

is symmetric. It is clearly non-singular g .2 The relation (i.i) together with ] = -I clearly

is non-singular.

implies

j

S

is orthogonal relative to both

S

We will say that the polarization positive definite bilinear form. where

and

B . g is positive in case

g h

S

is a

g

e/d = O,

(This includes the case where

that is

h = ~.)

A simple criterion for the positivity of the polarization

Remark 2.

e/d

without going to the quotient

is as follows:

We assert that

h

is a positive

h

polarization if and only if

-i(z,7) m 0 for all

z 6 h.

Indeed if

write

z 6 h

z = x + iy

where

x,y 6 e.

y = ~j~

Thus

J

and hence

relation then follows since the correspondence Now let in

d

b = {x 6 dl} = 0.

if and only if

Remark 3. gI d # 0

d = h N g

that

If

DO

and

groups of

DO

and E0

z ~ x

maps

h

It follows that

e/d.

onto

b

has codimension

i

gI d # 0.

g

if and only if Now let

to

The

-i(z,7) = -i(x + iy,x - iy) = 2(y,x) = 2(y,x) = 2(jx,x) = 2{x,x}.

is nilpotent one knows that

glgg # 0

and hence

g # 0. and

E0

be the connected Lie subgroups of

e = (h + ~) N g.

Since

are normalized by

Gg

h

and

is stable under D = GgD 0

and

G

corresponding

Ad G

it follows

g E = GgE 0

are sub-

G.

Proposition 2

The groups component of

D

Proof. to

Bg,

x E d.

D

and

so that

Since

d

d

one has that if

are closed in

DO

G.

is the Lie algebra of

and

e

Also

DO

is the identity

D.

are each other's orthogonal subspaces relative

x 6 g. then

<x • g,y> = 0

for all

y E e

if and only if

Thus = 0

for all T0

a 6 DO

and hence for all

then clearly

<x • g,y> = 0

a 6 T 0.

for all

have the same Lie algebras and hence Now let has

D1

= 0

algebra of

DI

one has

But if

y 6 e

d I ~ d.

lies in the Lie algebra of

so that

x 6 d.

Thus

DO

and

T0

D O = T 0.

be the identity component of for all

x

a 6 D I,

and

But of course

D = DoGg.

y 6 e. d = dI

Then if

Then if dI

a 6 D1

is the Lie

since D O ~ D I.

Thus

one

24-0

d = dI D

is

so that

DO = DI

also closed and

is the identity

DO

component

is the identity

of

component

D.

of

But

D O = D = D.

Hence

D. QED

Now consider its orthogonal

Proposition

DO

•

in

D ° g = g'.

a ~ g

let

a

be

is an open set of the affine plane

g + e in

g'.

Also

D • g

g.

Proof. Indeed since since

e

We first observe is

D = DoGg

stable

under

one has

that

g + ~

Ad D

for some under

is

But now clearly morphism

d/gg.

d • g ~

d • g

is

is open in

the

stable

d • g ~ ~. But then

tangent

e

is

under

b E D

a E DO •

and

But then

D.

However,

f E e

one has

b • (g + f) - g ~

On the other hand one has a natural

g

D.

D.

d/gg = dim e.

dim d • g = dim

space at

stable

and hence if

(as above) so that

g + ~

is stable under the action of

clearly

D • g = DO • g

b • (g + f) - g = a • g - g + b • f

But

For any subspace

g'.

3

D • g =

the D-orbit

subspace

to the orbit

Hence

DO • g ~ g + ~.

iso-

d • g = ~. Thus

D • g

g + e.

We w i l l (see [4]) if

QED say that

E • g

the polarization

is closed;

h

satisfies

in which case

E

the Pukansky condition

is closed and

D • g = g + e

(1.2)

Lemma i

If

h

satisfies the Pukansky condition then

identity component of group to

and

DO

Gg.

Furthermore, if

T: D I ÷ D O

DI

D O N Gg = (Gg)0,

the

is the simply connected covering

is the covering map then

T-I((Gg)0 ) = (Gg) I

is

connected. Proof. But since

As a

(Gg)0 = D O

DO

homogeneous

one has that

However by (1.2) one has that connected.

Thus

of

DO • g

(Gg) 0

DO • g

D O N Gg = (Gg) 0.

simple connectivity

space one has

D • g = D O ° g ~ D0/D 0 N Gg.

is the identity

is simply connected

component so that

of

D O N Gg.

D O N Gg

is

But now also since

implies

that

D~/(G )4 ~ D0/(Gg) 0 the ii g ± (Gg)l = t-~ ((Gg)0) is also connected. QED

Now

g

[gg,gg]

or

that

is integral

g

glgg

vanishes

on

[gg,g]

is a homomorphism if there exists

so that in particular gg ÷ I R

of Lie algebras.

a character

ng: Gg ÷ ~

g

vanishes

on

We will say whose differential

is

241

2~iglgg.

That i~ if for all

x Egg

d__dt~g(eXp tx) t = 0 = 2~i(g,x> When this is satisfied we will say that

Remark 5. existence of

~g

If

G

~g

corresponds to

g.

is connected and simply connected one knows that the

is equivalent to the integrality of the de Rham class of the

canonical symplectic 2-form on the orbit and Unitary Representations, Now since morphism

.

G ° g ~ g'

(see Kostant, Quantization

Part I).

= 0

then

gld

also defines a Lie algebra homo-

d ÷IR. Until otherwise stated we will assume

character on

G

corresponding to

g

g

is integral and

~

g

is a

g.

Proposition 4

If the Pukansky condition is satisfied then

n

extends to a unique

g

character Xg: D + ~

whose differential is

Proof.

2~igld.

Now let the notation be as in Lemma i so that

connected covering group to character

X~: D 1 ÷ ~

is satisfied, = ngl(Gg)0 has

o T.

But then if

0 Xg: D O ÷ ~

Now

Gg

0 However, Xg

G • g[d = gld g differential). then and

(~g,X~) ~g = X~

surjection Xg o y tion.

(Gg) 1 Z

normalizes

DO

Gg

2~ig[d.

DI

Z

is trivial.

1 0 Xg = ×g Q -r.

and hence

Now if the Pukansky condition X~](Gg) 1

g

T: D I + D O

one

Hence there exists a unique

2~ig]d

Clearly

G

is the simply-

there exists a unique

is connected and clearly

is

the differential

operates on the character group of

is invariant under this action since

G

g

• g = g

and hence

(of course a character on a connected Lie group is determined by its It follows then that if we form the semi-direct product defines a character onthis group. on

(Gg)0

so that

Xg

and

(~g,X0g)

given by

is a character on

As such it is unique since

mined on

= 0

is the kernel of the covering map

and

such that

o: Gg x D O , D

where

Now since

whose differential is

then by Lemma 1

Z ~ (Gg) 1 = T-I((Gg)0)

character 0 of Xg.

DO .

DO .

DO .

is trivial on the kernel

(a,b) + ab. D

However by Lemma 1

Thus

(~g,X~)

Gg × D O

Gg N D O = (%)0 K

of the

is of the form

satisfying the conditions of the proposi-

D = DoGg

and

Xg

is obviously uniquely deterQED

2#2

Assume that Now let

h

is a polarization satisfying the Pukansky condition.

X = E/D.

the other hand since

B

Since

g is invariant under the action of variant under the action of

such that

~(ab) = Xg(b)-l~(a)

D

M(E,Xg ) for all

Then if

tion for

~C(E,Xg )

indEX.

Since

~X

Now recall

is a right

C~(E) e~

one puts

a 6 E, b E D.

has a measure

Then

such that

~X

which in-

M(E,Xg)

~

on

E

is an

is given by

II~II2 = [I~I2d~x

is an E-invariant measure one has

is

((indEX)(a))~ = a •

(conforming to the usual abuse of language).

h N ~ = d~

and

h + ~ = e~.

is the space of all

module where if

(a-

o E X = E/D

may be identified with

C~

functions on

z = x + iy E g~ and if

E C~(E), a E E, z 6 e~

Now if

On

e/d

is the space of equivalence classes (de-

d (~ • x)(a) = ~

o

X

a • ~ E M(E,Xg)

~ 6 M(E,Xg)

~ • z = ~ • x + i~ - y

Clearly if

is connected.

is the Hilbert space associated with the unitary representa-

a E E, ~ 6 ~(E,Xg)

If

X

of all measurable functions

then

~C(E,Xg)

fined by sets of measure zero) of finite then

it is clear that

a E E, ~ E M(E,Xg)

(a • ~)(b) = ~(a-~).

it ~s clear that

E.

Now consider the space

E-module where if

E0D = E

is a non-singular alternating bilinear form on

~)

we note that

x,y E e

then if

C~(E)

# E C=(E)

a E E

~(a exp - tx) t = 0 then

• z = a-

(~ •

is the coset

e/d.

with

E

D

(1.3)

z)

then the tangent space

To(X )

at

Hence upon complexification

(To(X)) ~ =

e~/d¢ = h/dl • ~/dl

Proposition 5

There is an E-invariant complex structure on space of anti-holomorphic vectors at

Proof. p E X

such that

h/d~

We define a complex distribution

F

on

X

such that for any

one has P

by p u t t in g

Fp

a E E

a,(h/d~) w h e r e a • o = p , a since h/d~ is invariant under

=

to prove that F P p, we have only to prove that

P

6 E.

Ad D.

T h i s d e p e n d s o n l y on

Clearly

F

By Nirenberg-Newlander,

is the space of anti-holomorphic F

are two complex vector fields on for all

p E X

where

X

such that

~ = [~,q].

is involutory. ~p,~p 6 Fp

That is, if for all

p

and

is E-invariant.

vectors at

~p E Fp

is the

o.

(Tp(X))¢ = F • g

not on

X

X

tangent $,~

then

But this condition is purely local.

If

243

p E X

let

U ~ X

be a neighborhood of

p

with the property that

~: U ÷ E is a smooth section of the projection neighborhood

V

of the identity on

~: E ÷ E/D = X. D

Then there exists an open

such that the map

o: U × V ÷ W E E is a diffeomorphism onto an open set be the complex vector fields on Clearly

~,$ = $,~,n = n.

bution on

E

h = ~, l(h/d~) .

Then

is ~-related to Hence

F

~, then

Fh

However,

n

Ph

o(a,b) = o(a)b.

But let

$,n

~ = (~),(~,0), ~ = (o),(n~,0).

is the left invariant complex distri-

is involutory since

~a,na E (F~) a

[~'~]a ~ E (F~) a ~, and

where

defined by

But then if

defined by

are in the group case).

W

W = E

for any

a E W.

is n-related to

h

for any

n.

is a subalgebra

a E W

~ ~ = ~,[~,B]

However,

Thus

(we

since

(~)p E Fp

for all

since p E U.

is involutory.

QED

We can now speak of holomorphic functions on any open set

V ~ X = E/D.

In fact if ~: E ÷ X is the quotient map then these are just the elements of all

(4" in

6 C=(V)

such that, for

z E h, ~) " z = 0

(1.4)

- i (V). Now let

C(E,Xg,h)

be the set of all

C~

functions

~

in

M(E,Xg )

such

that • z = 2~i(g,z>~ for all E

z E h.

By (1.3) it is clear that

C(E,Xg,h)

is stable under the action of

and hence if ~(E,~g,h)

(abuse of language)

then

Remark 6. notation rather than

~(E,Ng,h)

Since

Xg

= C(E,Xg,h) N ~C(E,Xg) is stable under

is determined by

ng

ind E Xg.

and

h

we use

ng

in the

Xg.

Proposition 6

~(E,ng,h)

Proof. ~C(E,~g,h) # 0 such that

is a closed subspace of the Hilbert space ~(E,Xg).

We may assume

~(E,ng,h)

~ 0.

Let

a E E

there exists (by translation if necessary)

@(a) # 0.

Let

U

be an open neighborhood of

and

p = ~a E X.

an element a

Since

~ E ~C(E,~g,h)

with compact closure

24-4-

such that

A > I~I > E > 0 Now if

in

U.

B E M(E,Xg)

B E ~(E,ng,h)

V.

Also

so that for

= ((~ o ~) • z)~ + (~ o ~)(~ • z). ((~ o ~) • z)~ = 0

which implies

holomorphie and hence

V = ~(U) = X.

then clearly one has that

is a measurable function on But now

Let

B ~ ~

~ E C~(V)

z E ~

one has

But also

B = (~ o ~)~

if and only if

U

where

2~iB = B " z

~ • z = 2~i~

(# • ~) • z = 0.

in

BI U E C~(U).

so that one has

Thus by (1.4) one has

~

is

defines a map ~(E,ng,~) ÷ Bo(V)

where

(Bo(V))

is the space of all bounded holomorphic functions in

On the other han@ (taking morphic coordinates in

V.

small enough) if z I ,.-.,zm are the holo2 then the measure im dZlA'''AdZmAdZl ^'''Adam is abso-

V

U

lutely continuous with bounded (from above and below) Radon-Nikodyn derivative with respect to U

where

~xIV.

But now if

~n E B0(V)

notation of

Bn

is Cauchy in

then clearly

(Weil, [5], p. 59).

~ndzlA...Adz m Since

B(V)

it follows that

~ndzlA..'Adz TM ÷ pdzlA.-.Adz n

in

converges to

V.

But

~n

5 in Weil. for = p for

~

p

On the other hand if

a measurable function on almost everywhere.

z ~ ~.

~(E,qg,~)

Now since representation

and

~n = (~n ° ~)~

is Cauchy in

B(V)

Bn ÷ ~

in

B(V)

V

in

~(E,Xg)

one has

~n ÷ #

But clearly

in

using the

is complete (see again Weil, p. 59) where

p

uniformly on compact subsets of where

is holomorphic V

by Proposition

~ = (~ o ~)~

almost everywhere.

in

~

contains an element in

U

Thus

((p o ~)~) . z = 2~i(g,z>(p ~ ~)~

Thus the equivalence class of

proving that

~C(E,ng,~)

on

U

~C(E,~g,~)

is complete.

~(E,~g,~)

indE(~g,~)

QED

is stable under

of

ind E Xg.

indE Xg

it defines a sub-

But since

indG(ind E Xg) = ind G Xg it follows that if indG(~g,~) = ind G indE(~g,~) then

indG(qg,~)

Hilbert space by

is a subrepresentation of

then

We denote the corresponding

~(G,~g,~).

Remark 7. G/E

ind E Xg.

~(G,~g,h)

It is clear that if

~Z

is a

G-quasi invariant measure on

can be taken to be the set of all equivalence classes of on

measurable functions

G

such that

~a E ~ ( E , q g , h )

for all

a E G, and such

that IZll~all2d~z (7) < where

#a(b) = #(ab)

for

b E E

and

~ E Z

is the image of

a

in

Z.

245

Remark 8. g E g'

We recall for emphasis that

is integral and (2)

h

indG(ng,h )

is defined when (i)

is a polarization satisfying the Pukansky condition.

However it may reduce to the zero representation if

~(E,~g,h)

From the point of view of the general quantization theory

reduces to zero.

indG(~g,h )

is a "zero

cohomology" representation.

2.

THE SOLVABLE CASE~ EXISTENCE OF ADMISSIBLE POLARIZATIONS

Although one is forced into considering higher cohomology representations in the case where resentations

G

of the form

sufficient to give resentations

is semi-simple, L. Auslander and I have shown that the rep-

of

indG(~g,h)

G

G

for one thing we have shown that integrable and (2) all orbits

G

is of type

G • g = O ~ g'

at

f = gln E n'.

f.

Since

Obviously

n

Gg ~ Gf

A polarization (i.e.

'

the bilinear form

stable under

are

I

if and only if (i)

Gf

h

at

Sg

is stable under

Ad G

G

Gf

and g

on

on

n'.

Let

gg ~ gf

Then

g E g'

are

are the intersections of a closed

Furthermore in such a case we may explicitly give

grediently the representation of G

I

is a solvable simply connected Lie group.

G.

To do this consider first the maximal nilpotent ideal and let

of type

G.

More precisely assume

and open set.

for a solvable Lie group

G, the set of equivalence classes of irreducible unitary rep-

where

n ~ g.

Let

g E g

one may consider contrabe the isotropy subgroup of

gf

is the Lie algebra of

Gf.

is called admissible in case (i) it is positive

e/d

is positive definite) and (2)

and is a polarization at

h A n~

is

f.

Then the following is proved in [i].

Theorem i

For any

g E g'

sible polarization at

g.

whether or not

Pakansky condition so that if more, assuming polarizations

g h

g

is of type

is integrable,

is integrable then and if

G

I

there exists an admis-

Moreover, any admissible polarization indG(ng,h)

G is of type

I

then

indG(ng,h)

h

satisfies the

is defined.

Further-

is independent of the choice of indG(ng,h)

is irreducible and

every irreducible unitary representation is equivalent to a representation of this form.

Finally if

G

lent if and only if of an element

a E G

is type

I

then

G • g = G • gl

such that

and

indG(ng,h) ng

and

indG(ngl,hl)

corresponds to

~I

are equiva-

under the action i

a • g = gl"

We cannot go into the proof of this theorem here but we will prove two relevant facts which are needed in the proof.

The first of these asserts the in-

dependence of the polarization in the nilpotent-case.

This generalizes a result of

2#6

Kirillov who proved where

h = ~

or

e = d.

fact to be proved.

Theorem

a similar

One is forced

Let

g

is nilpotent,

b = Ker

In particular Proof. ad x.

If

Since



= 0

d/b

d

by the second

implies

let

skew-symmetric d



Hence e/b.

then for

x E

one has

= 0

d/b

induced

it follows

obviously

that

commutes with

[d,e] a b e.

j

e/d

d/b

y

d/b

since

Indeed

for all

so that

d/b

as center,

e/b. But for

of

e-d. But from the # 0.

the center

that to prove the theorem it suffices

is

if this were the case

y E



is exactly

d/b

g # 0 (see Remark 3).

is the center

is abelian.

[x + b,y + b] ~

so that in

Furthermore

Lie algebra with

d/b

is abelian and

of B we can choose g [x + b,y + b] = d/b. Hence We assert

e/d

on

is an algebra, ~(x)

is 1-dimensional

non-singularity implies

g

is commutative.

then implies

is a Heisenberg

only to show that

e-d

e

is also an ideal in

e/b

e/d

to show that

this it suffices

b

Also

Now to prove that it suffices

at

e.

[b,e] ~ b. in

h is a

S . Thus n(x) is both nilpotent and g to a positive definite bilinear form. Hence ~(x) = 0 so

relative

central

e/b

On the other hand the relation

since

B . g relative to

But the relation

obviously

e and

be the operator

~(x).

However,

to

e/d

e/d

E End

so is

(g,[d[e,e]]> = 0

is an ideal in

particular

~(x)

e and

relative

so that it is skew-symmetric

and the polarization

is an ideal in

as the 1-dimensional center.

is nilpotent

is skew-symmetric

b

is an ideal in

x E d

ad x

0 # g E g'

Then

(gld).

Heisenberg Lie algebra with

that

polarizations

To begin with we need

Assume that

~(x)

into non-real

i.e.,

2

is positive.

by

theorem for the case of real polarizations,

This however

e/b.

of

only to prove

Lemma 2

The center of Indeed is non-singular under

j

e/d

it follows

a

since that

B

gonal complement

to

a

algebra.

if

y,z E V

S g

in

since

(x,[y,z]) a

[d,e] ~ b.

= (g,[x[y,z]]>

is central

in

This proves

a

g is also non-singular

e/d and

be the center

is positive definite.

relative x E a

(2.1) so that

Let

v

S

g

is stable

be the ortho-

: 0

(2.1)

+ {g,[y,[x,z]]>.

[[x,y]z] v

a.

a

B . We assert that v is a subg where x,y,z E e we must show

=

e/d. But then

on

e/d. Now

of

But since

to

(:~,[~,f~]) But

j.

assume Lemma 2 is true and let on

Indeed

is stable under

But

and [y,[x,z]]

is a subalgebra.

[x,y],[x,z] lie in

b

E d

since

But it is obviously

247

nilpotent so that if

e/d = a

u = cent abelian.

v # 0

then center

which is a contradiction.

However,

Thus

v = 0

Let

v 6

e/d.

since

so that

a = e/d

is

Let

£/d.

u 6 center

We must prove

j u

ju

is central in

We first observe that j[ju,v] = [ju,jv]

That is

clearly center

We proceed now to the

Proof of Lemma 2.

e/d.

v # O.

commutes with

ad ju.

Indeed

•

u + iju

(2.2)

and

v + ijv

h/d~

lie in

and

is central [u + iju,v + ijv] = -[ju,jv] + i[ju,v]

h/d~

However since

is an algebra it follows that

to both sides yields B = 0. S . g

Let

Hence

(2.2).

A = B + Bt A = At

Now let

B = ad ju

where superscript

t

is a symmetric operator.

[ju,v] = -j[ju,jv].

Applying

so the problem is to show that denotes the transpose relative to

We next establish the relation

{Av,w} = {[jw,v],u} for any

v,w 6

e/d.

j

(2.3)

Indeed we first observe that for any

z. 6 i

e/d,

i = 1,2,3

one

has ([Zl,Z 2],z3) + ([z2,z3],Zl) + ([z3,zl],Z2) - 0 This of course follows from the relation Yi 6 e

and

{Bv,w} = {[ju,v],w} = (j [ju,v] ,w) = -([ju,v],jw)

{Btv,w} = {v,Bw} = (jv,[ju,w]) = - ( v , j [ j u , w ] )

j[ju,w] = [ju,jw] nating.

([Zl,Z2],z3) =

by (2.2) so that

{Btv,w} = -([jw,ju],v)

by (i.i).

again by (i.i). since

Bf

On the But

is alter-

Thus {Av,w} = - ( ( [ j u , v ] , j w )

Hence

where

Yi = z i •

Now other hand

(2.4)

{Av,w} = ([v,jw],ju)

by (i.i) establishing

by (2.4).

+ (([jw,ju],v))

But then

{Av,w} = (j [jw,v] ,u) = {[jw,v],u}

(2.3).

As a consequence of (2.3) note that

Au = 0

and since

A

is symmetric

one t~erefore has, by (2.3), 0 = (Av,u) = {[ju,v],u} for all

v 6

£/d.

That is since

A

We now assert that

AB

(2.5)

is skew-symmetric or that

AB + (AB) t -- 0.

is symmetric w e assert {ABv,w} + {Av,Bw} = 0

for all

v,w 6

e/d.

Indeed replaces where

v.

[ju,w]

(2.6)

{ABv,w} = {A[ju,v],w} = {[jw,[ju,v]],u}

On the other hand replaces

w.

But

by (2.3) where

{Av,Bw} = {Av,[ju,w]} = { [j [ju,w] ,v] ,u} j[ju,w] = [ju,jw]

by (2.2) so that

[ju,v] by (2.3)

248

{(AB + (AB)t)v,w} = {([jw,[ju,v]] + [[ju,jw],v]),u} = { [ju, [jw,v] ] ,u} by Jacobi. proves

However,

AB

(2.7) vanishes by (2.5) where

AB = (B + Bt)B = B 2 + BtB.

B 2 + BtB = -(Bt) 2 - B t B

= B 2 + (Bt) 2 + BB t + B t B However, since implies

[jw,v]

replaces

v.

This

is skew-symmetric. Now

Thus

(2.7)

A

A = 0.

or

= BB t - BtB.

is symmetric Thus

B

But

AB = -(AB) t = -BtA = -((Bt) 2 +BtB).

B 2 + (Bt) 2 = -2BtB.

A2

But then

Therefore,

tr A 2 = 0

A 2 = (B + Bt) 2

since

tr BB t = tr BtB.

is positive semi-definite so that

is skew-symmetric.

But

B

tr A 2 = 0

is clearly nilpotent.

Hence

B = 0.

QED One now deduces the following generalization of a result of Kirillov.

(See [3]).

Theorem 3

Let

G

Lie algebra.

be any simply connected nilpotent Lie group and let

Let

g 6 n'

and let

h

~e any positive polarization at

is irreducible and up to equivalence is independent of

indG(~g,h)

Proof.

(Sketched).

the Bargmann-Segal

It follows from Theorem 2 that

ind E ~g

b = Ker gld.)

where

K.

But then

that

k

D

and

(B = E

indE(ng,h )

Then

is just

is the subgroup corres-

indE(ng,h)

2~iglk.

is equivalent to E

Here

ind G Bg.

is equivalent to

g.

k

E/B

gg

and

Bg

is the Lie algebra

However, since

it is also "half-way" between

defines a real polarization at

ind G Bg

g.

h.

is a maximal commutative subgroup of

whose differential is

indG(~g,h)

"half-way" between has that

K

E/B.

One knows therefore that

B ~ K = E, K/B

is the character on of

be its

(see e.g., [2]) holomorphic construction of an irreducible uni-

tary representation of the Heisenberg group ponding to

g

and

g.

K

is One thus

By Kirillov's result one knows

is irreducible and that any real polarization gives rise to an

equivalent representation.

QED

Now returning to previous notation where

g

is solvable one is forced

into considering complex polarizations of the nil-radical general, there exists no real polarization at

f = gln

n

of

g

since, in

which is stable under

Gf.

However, by the next lemma there exists complex polarizations and in fact positive polarizations stable under corresponds to

n

Gf.

Since the commutator group

it follows that

lemma is satisfied where

F = Gf.

Gf' ~ N

G' = N

where

N ~ G

so that the hypothesis of the following

249

Lemma 3

algebra.

Let

N

be a simply connected nilpotent Lie group and let

Let

Aut n

Regard f ~ n'. on

n

(2)

F

as operating by contragredience on the dual

is a group and a homomorphism

F ÷ Aut n

n') such that (i) the commutator subgroup

Let

n/m

m

= Ker flcenter n.

is an ideal in

F 0.

m.

Moreover,

F 0.

stable under

F, where

~-id 0

and

dim m = 0 where

But then

F I ÷ Ad N/M

f0 E (n/m)'

~: n ÷n/m

Since

Aut n

action of this group. Thus the abelian group

f

dim n.

F.

Thus

if

M

F

f

then

is fixed

is clearly a positive polarization at (indeed

f0 f

e = q-le0, d

is one-dimensional,

is fixed by

F

spanned by an element

clearly

z

z

is also fixed under the

k = center n/(z)

However F/F'

we may write and

kI

so that

n/(z)

Ad N

and

so that

k I = k 0 @]Rz

is stable under

F

k.

Let

dim p

where

where

p ~ k

k

since

k 0 = Ker flk I.

it follows that

k0

is the quotient map then

Case i.

Assume

dim P0 = i

along the lines used by Kirillov. defined by the relation be central in

n

such that

contradicting [x,w] = z

f

is fixed F

and

induces an F-isomorphism to

p ~ k.

Note

k0.

so that

P0 =l~w.

That is, let

[y,w] = (g,y>z.

Now since

Since

be the F-irreducible subspace corresponding

must operate trivially on

[n,kl] ~IRz.

is stable under

7: ~ ÷ n/(z)

F'

~

is an

be an irreducible sub-

is either i or 2.

Let

P0 ~ ~0

kI ~ n

is clearly stable under the

operates trivially on

operates on

F/F'

k = kl/(Z) k

k0 ÷ k.

x E g

f0

h 0 = (n/m)~, a positive polarization at

that if

then that

operates on

is the subgroup cor-

is induced by

is the quotient map

operates on

space under the action of

F

which is

F.

Clearly

= i

f

and

Thus we are done in this case so that we may assume

center n

Now consider ideal.

under

where

if

~-lh 0 = h

e/d ~ eo/do).

and hence

(f,z> = i.

action of

at

operates

Assume this space has positive dimension.

which is stable under

Now by induction there exists

stable under

=

n

F ÷ Aut n/m

inducing a map

responding to by

hI

F Ad N

We assume inductively that the result is true for all simply con-

nected nilpotent Lie groups of dimension smaller than

m

(so that

Let

F.

Proof.

Clearly

so that

n'.

maps into

F'

Then there exists a positive polarization

F • f = f.

stable under

n

Autn.

Aut n

Assume and

be its Lie

be the group of all Lie algebra automorphisms of

is a subgroup of

Ad N

n

g E n'

One has

the fact that

since otherwise

center n =]Rz.

and hence n = ~ n

g # 0

In this case we proceed be the linear functional

0

w

would

Thus there exists

250

where

n O = Ker g.

subalgebra

But then

stable under

is nilpotent,

no

nO

F.

However,

is an ideal in

are the subgroups

corresponding

Now the action of where on

F' ÷ F 0.'

~w

However,

since clearly

w E P0

as observed

operates

is the centralizer

trivially

n. IRx

on

n0

and

w

and hence

N = XN 0

But

i in

where

n X

is a and

n

and

NO

[x,w] = z

F ÷ F 0 = Aut n o

AdnN 0

On the other hand

so we must have

nO

R0"

induces an epimorphism

But since

f0 = fln0

w

has codimension

In particular

to

w E center n o .

Now clearly

no

F' ÷ AdnN = AdnXAdnN 0.

above. on

F

since

of

operates

F'

operates

no non-trivial

F' ÷ AdnN 0

trivially trivially on

element of

AdnX

F~ = Adn NO. 0 Furthermore, we assert

which implies

is invariant under

F 0.

that

(2.8)

(~0)f0 = nf OIRw Indeed

w 6 (n0)f0

to observe that such that

since

w 0 6 center n o .

nf ~ n o .

[y,w] = z.

To see that

~f = (nO)f0

But this is clear since otherwise

we have only

there exists

y E nf

But then i = (f,[y,w]> = -

contradicting

the fact that

<w • f,x) = = .

But

y • f = 0.

= (f,z> = i.



= c.

(y - cw) • fln 0 = (y - cw) • f0 = 0 so that

y E nf +IRw.

since

This establishes

Now by induction which is stable under

Also one has

Finally if

nf N IRw = 0

y E (n0)f0

Thus

let

= 0.

w E (n0)f0.

But then

But

y - cw = Yl 6 nf

(2.8).

there exists a positive polarization

F 0.

since

c = (y • f,x>

h 0 ~ (no) ~

at

f0

Clearly then one has

(nf)~ ~ ((n0)f0) ¢ ~ h 0 = (no) ~ ~ n~ But since between

h0

is "half-way" between

(nf)~

and

codimension i in zation at

f

n~

n.

because

Thus,

if

((n0)f0) ~

and

nf

has codimension

h = h0

it follows

which is stable under the action of

Now if gj E R', j = 1,2

dim PO = 2

linearly independent P0 N center n = 0

we may write

by the relation

that

h

it is also "half-way" (n0)f0

P0 N center

and

no

has

is a positive polari-

F.

P0 =IRWl @IRw2"

[y,wj] = z

since otherwise

since

(n0) ~ i in

then n ¢ 0.

If we define gl

and

g2

are

But of course

center n =]Rz.

But then we may find elements

Xl, x 2 E n

such that

[xi,w j] = Bijz

(2.9)

Clearly then n =IRx I @]Rx 2 • n O where

n o = Ker gl N Ker g2

is the centralizer

of the subspace

(2.10) P0"

Since

P0

is

stable under

F

since

annihilates

[n,n]

it follows

that

nO

is a subalgebra

k I m k 0 m P0' it follows

stable under

F.

In fact

that (2.11)

In,n] ~ n o

and hence

nO

is an ideal in

F * F 0 ~ Aut n O jective where to

n

where

F'

n.

The action of

maps into

(al,a2,b) + ala2b

F 0.

F

on

no

But the map

and where

NO ~ N

induces an epimorphism

X1 × X2 × NO + N

is the subgroup

is bi-

corresponding

operates

and

X. is the subgroup corresponding to ]Rxj, j = 1,2. But now N O J trivially on P0 ~ k0" But since no non-trivial element of XIX 2

operates

trivially on

hence

by the relations

(2.9) it follows

that

F' ÷ AdnN 0

and

F$ ~ Adn0N 0. Now let

h0

P0

at

f0

f0 = flno"

By induction there exists a positive polarization

which is stable under the action of As in the case where

dim P0 = i

F 0.

one has

[nf'P0] = 0

so that

nf ~ n o

and hence nf ~ (n0)f0

(2.12)

Next observe that nf + P0 = nf @ P0

(n0)f 0 Indeed if then

g = (y- ClW I - c2w2) • f

However,

clearly

c2w 2 E nf

-

y E (n0)f0

g

and

is orthogonal

is orthogonal

and hence

cj, j = 1,2

to

nO

y E nf + P0"

are defined by

to ]Rx I +]Rx 2

so that

Now

(2.13)

g = 0

nf A P0 = 0

element

w E P0

# 0

w E nf.

Hence (2.13) is established.

Case 2. which implies so that

nf

this implies

Assume

[Wl,W2] = 0.

P0 ~ (n0)f 0"

h0

2 in

Case 3.

space. and of

at

may be chosen in

where

u = w I + /~

since they are necessarily But then we may choose

xI

f

ad w.

P0 ~ no

Since

no

(nf)~

Now since

irreducibly, End P0 so that

w2

n~

and

2 in

and hence

n

h = h0

F.

operates

trivially on

P0

as an abelian group on the 2-dimensional

~u, Cu ~ (p0)~

u = w I - - ~ - w 2.

independent we may choose x2

(n0)f0 = nf ~ P0

is therefore isomorphic

P0

and

and

(f,z>

P0 ~ center n o

has codimension

F'

(2.9).

y - clw I

But since

and hence

which is stable under

[Wl,W2] # 0.

operates,

The commuting ring in

w2 F

Assume F

(n0)f0.

is "half-way" between

defines a positive polarization

it follows that

z EIm

Thus by (2.12) and (2.13) one has

has codimension that

Then

by the relations which implies

since by the relation

(2.9) any non-zero this implies

is such that

cj = (y • f,xj>

so that

to

Furthermore, Wl, w 2

]Rw I e ~ w 2 @ n o = n

and hence

w1

it is clear that

so that

x I = Wl, x 2 = -w 2

becomes

~

are stable under the action

[Wl,W2] = z.

and hence

(2.10)

252

But then one has that n~

P0 N no = 0

so that, since

nf = (no)f0 = nf + P0

~f = (n0)f0.

But then since

~0

~0

by (2.12) and (2.13)

has codimension 2 in

n,

it follows

fails by one dimension of being a maximum isotropic subspace (m.i.s.) of

relative to

Bf.

Now put h = h 0 + ~u Since

and

~0 ~ (n0)E

is a m.i.s, of since

n~

nf ~ h

u E (p0)~

but

h

it follows that

it follows that

~

is stable under

+ h0 + ~u + ~u = (h0 + h0 ) + (p0)~. is a polarization at that

~ + ~

Thus

h

f0"

But

But if

However,

is a subalgebra since

e = (h + ~) N n

f.

so that not only

But now

[(p0)~,(p0)~] = ~z

one has

F.

h + ~ = h0

[p0,~0] = 0

and

h

Also

is a subalgebra since

and since

h0

it follows

z E nf = (n0)f0 ~ h.

we have only to show that

P0 N no = 0 and

Ad Nf.

h0 + t 0

h0 + ~0 ~ (n0)E

is a polarization at But now since

[u,h0] = 0

is a subalgebra stable under the action of

~

is positive.

d = h A n = h 0 N n = h 0 N no = d o .

e0 = (h0 + t 0) A n = (h0 + ~0 ) N N0

then one has

e/d = eo/d0 • (do • po) /do But this is an orthogonal direct sum relative to both clear since Bf.

e0

But also

and

do

are orthogonal relative to

[P0,e0] = 0.

Furthermore

(p0)~. = (p0)~ N h @ (p0)~. N ~ = ~u @ ~u.

eo/dO.

definite on

{[Wl],[w2]} = 0

and

one has

and

Bf0

(do + po)/do

Sf

zation

h2

at

Gg.

Gf. e

{ [w2] ,[w2] } = i.

Now let

J[w2] = -[Wl].

Hence

Sf

gf

functional on

e = glgf-

Thus,

However, if

gf/a ~f/a and

and

(Gf)0

nf = gf N n

is indeed nilpotent. since h2

we put

h = hI + h2

at

But then we may form

hI

at

QED f

which

We assert there exists a positive polari-

(for the identity component

follows from Lemma 3 that

g.

since if

is positive definite.

of

Gf) which is stable

To see this one cannot directly apply Lemma 3 since

necessarily nilpotent. ideal in

since

is positive

(do + po)/do

Lemma 3 shows that there exists a positive polarization

under

j

{[Wl],[Wl]} = (J[Wl],[Wl]) = ([w2],[Wl]) = (f,[wl,w2]>

Similarly

is stable under

Indeed this is

and hence relative to

But by assumption

J[Wl] = [w2] and

Sf.

is stable under

However, it is positive definite on

[wi] = w i + d0,i = 1,2

= = i.

Bf

and

gf

a = Ker f lnf

Furthermore

e

is not then

a

is

induces a linear

G' ~ Nf, the subgroup corresponding to nf, it g exists by passing to the quotient gf/a. But now if

then it follows easily that indG(ng,h)

h

is an admissible polarization

giving the most general irreducible

unitary representation of a simply connected solvable Lie group of type

I.

253 REFERENCES

[i]

Auslander, L. and Kostant, B., "Quantization and Representations of Solvable Lie Groups", to appear (see announcement in Bull. Amer. Math. Soc., 73, 692-695 (1967).

[2]

Bargmann, V., "On A Hilbert Space of Analytic Functions and An Associated Integral Transform", Co,~n. Pure Appl. Math., 14, 187-214 (1961).

[3]

Kirillov, A. A., "Unitary Representations of Nilpotent Lie Groups", Uspehi.

Mat. Nauk., 17, 57-110 (1962). [4]

Pukansky, L., "On The Theory of Exponential Groups", Trans. Amer. Math. Soc., 126, 487-507 (1967).

[5]

Well, A., Yarietes K~hle'riennes, Hermann, Paris (1958).

~TION AND SOLUTION OF AN INFINITE-COMPONENT WAVE EQUATION FOR THE RELATIVISTIC COULOMB PROBLEM by I. T. Todorov

SUMMARY

he aim of these notes is to give a self-contained exposition of the dersolution of an infinite-component wave equation.

They cover some of

s of recent work by C. Itzykson, V. Kadyshevsky, and the author [1,2,3]. First we sketch the derivation of a three-dimensional quasi-potential in momentum space involving integration over the mass-shell hyperboloid We show that for the relativistic Coulomb potential

V(p,q) (p _ q)2

~ation can be written in an equivalent algebraic form in terms of rational ns of the generators of a degenerate ("metaplectic") representation of ~.

The solution of the bound-state eigenvalue problem is carried out by re-

, the representation of of its subgroup

SO(4,2)

S0(3) @ SO(2,1)

with respect to the irreducible representaand by an extensive use of the Bargmann

zation of the discrete series of unitary representations of

S0(2,1).

Institute for Advanced Study, Princeton, New Jersey. On leave from Joint Institute for Nuclear Research, Dubna, USSR and from Physical Institute of the Bulgarian Academy of Sciences, Sofia, Bulgaria.

255

TABLE OF CONTENTS

SUMMARY

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

INTRODUCTION i.

. . . . . . . . . . . . . . . . . . . . . . . . . . . .

256

Quasi-potential Equation for the Relativistic Two-body Problem . . . . . . . . . . . . . . . . . . . . . . . . . . .

256

i.I.

1.2.

1.3. 2.

3.

Old-fashioned Perturbation Theory and Feynman-Dyson Rules . . . . . . . . . . . . . . . . . . . . . . . . .

256

Off-mass-shell'Bethe-Salpeter Equation and Off-energyshell Quasi-potential Equation for the Scattering Amplitude . . . . . . . . . . . . . . . . . . . . . . .

259

A Simple Model:

263

Algebraization

The Scalar Coulomb Problem . . . . . .

of the Relativistic Coulomb Problem

.....

264

. . . . . . . . . . . . . . . . .

264

2.1.

Introductory Remarks

2.2.

A Remarkable Representation

of the Conformal Group.

2.3.

Algebraic Form of Equation

(2.2) . . . . . . . . . . . .

Solution of the Coulomb Eigenvalue P r o b l e m

•

265 268

.........

269

3.1.

Group Theoretical Treatment of the Algebraic Equation

3.2.

Calculation of the Energy Eigenvalues

APPENDIX:

A.

254

.

.........

270

DIFFERENT REALIZATIONS AND PROPERTIES OF THE EXCEPTIONAL REPRESENTATION R 0 OF S00(4,2) . . . . . . . . . . . . .

The Set of Conformal Transformations in Space-time as a Global Realization of R 0 . . . . . . . . . . . . . . . . . . as one of the Metaplectic Representations

of

SU(2,2)

273

273

B.

R0

C.

Quadratic Identities in the Enveloping Algebra of the Metaplectic Representations . . . . . . . . . . . . . . . . . . .

REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

269

• •

274

275 277

256 INTRODUCTION

This paper consists of three parts.

First, I will try to persuade

you that the equation we are going to solve has something to do with physics.

We

will consider a class of relativistic quasi-potential equations for the two-body problem and will single out a simple equation of this class corresponding to the scalar Coulomb interaction.

Second, we shall show that our simple equation is

equivalent to an infinite-component wave equation written in terms of the generators of a unitary representation of the conformal group

S0(4,2).

Finally, we

shall solve the arising eigenvalue problem by applying some known tools of the theory of representations of the pseudo-unitary group. In Section i we will have to use, without much explanation, physicists'

some of the

jargon (which is introduced in the first few chapters of any textbook

on quantum field theory).

The rest of my talk (Sections 2,3) is practically self-

contained and does not require any special knowledge of physics.

i.

i.i

QUASI-POTENTIAL EQUATION FOR THE RELATIVISTIC TWO-BODY PROBLEM [i~2~3]

01d-fashioned Perturbation Theory and Feynman-Dyson Rules

We will be concerned in what follows with the scattering and bound-states problems of two relativistic particles. Let us have two equal-mass particles of initial (4)-momenta final momenta

pl,p 2.

ql,q 2 and

Taking into account the energy-momentum conservation

(Pl + P2 = ql + q2 )' we can express

Pi

and

qi

in terms of three 4-vectors:

the

center-of-mass momentum P = Pl + P2 = ql + q2

'

(i.i)

and the relative momenta i P = ~(Pl - P2 )' On the mass-shell,

i.e., for

i q = 2(ql - q2 ) "

Pl2 = P22 = ql2 = q22 = m 2

pP = qP = 0,

(1.2)

we have the identities

1 p2 p2 1 p2 q2 m 2. 7 + = 7 + =

(We use the system of units for which

c = ~ = i

throughout these notes.)

In the

framework of quantum field theory, to each particle one usually makes correspond a local field operator. fields

~l(x)

and

So, we associate with particles i and 2 the complex scalar

~2(x), of mass

m

and assume that their interaction is given

by the local Hamiltonian density ~x)

= -g(:~(x)~1(x):+:~(x)~2(x):)~(x)

,

(1.3)

257

where

(I0>

: :

is the sign for the Wick "normal" product

:~*(x)~(x): = lim [~*(x + y)~(x - y) - ] , y÷0 is the "free vacuum") and ~(x) is a hermitian field of mass ~. Then, the

scattering amplitude can be written as a (formal) power series in the coupling constant

g.

There have been two different presentations of this formal expansion:

the old-fashioned

(non-covariant)

covariant technique.

perturbation theory and the modern Feynman-Dyson

The second one is much more familiar nowadays.

Each term of

the series is represented in this approach as a sum of multiple integrals corresponding to the so-called Feynman diagrams

(see Figure i).

P/

i 1 Pl t

£"

ql

Pl ~"

la


01

0

for

k0 < 0

(1.4)

(represented by solid lines) is not conserved,

the conservation law in each vertex taking into account the energies of the dotted lines.

For instance, to vertex 1 of the diagram in Figure 2a corresponds a factor -

where

n

$

6 ( q I + k - Pl +

(El - [U(A)F](u) = F(A-Iu)

.

(2.13)

That is the Majorana representation of the complementary series of unitary representations of

SO0(4,1), i.e., the only representation of the complementary series

which can be extended to a representation of mark that the representation

S00(4,2).

(2.13) in the space

~-2

To see this we first rewith scalar product (2.12)

is equivalent to the representation given by the same formula (2.13) in the space ~-I

of homogeneous functions of degree of homogeneity

-i, equipped with scalar

product (F,G) I = -i ~ F(u) i G(v) 6(u 0 - l)~(v 0 - l)d(u2)~(v2)d5udSv 27 4 (uv) 2

.

(2.14)

We mention that the integral in (2.14) is in general divergent because of the singularity for to

N

u = v.

It has to be defined by analytic continuation with respect

of the hermitian form

replaced by

(F,G) N

2N+!F(rN) (uv) -3-N) 3 ~7/2F(- N - ~)

(in which

_ i (uv)-2 2~ 4

(cf. [14]).

in the integrand is

The scalar product defined

through this analytic continuation is positive-definite if and only if NN (uN0 ,u0)

The normalization is chosen in such a way that S0(4)

invariant vector in

which maps

$~-i onto

~

(up to a factor)).

= i

(T-IF)(v) =

The intertwining operator

T

in the space

(2.15)

i ~ F(u) 8(u0 _ i)6(u2 ) d5u 2~ 2 uv

The action of the five additional generators S00(4,2)

is the only

~C_2 and its inverse are given by

(TF)(u) = -i ~ F(v)6(v0 _ i)6(v2 ) d5____~v, 2z 2 (uv) 2

of

N (F = u O

N(N + 3) < 0.

~-2

is defined by

Fa

(a = 0,1,2,3,5) of the Lie algebra

268

v (FaF)(u) = [T(UaF)](u ) = -i ~ a F(v)~(v 0 - l)6(v2)dSv 2~ 2 (uv) 2

.

(2.16)

It can be verified by a straightforward

computation that these operators satisfy

(together with the generators

S00(4,1))

Fab

of

the commutation relations

(2.4).

In particular, i-"1(Ua -~- i[Fa,F b ]

Ub ~Ua

(-iu 0 - ~ It is easily seen also that the operators scalar product (2.12). in the Appendix.

for

a,b = 1,2,3,5

for

a = 0, b = 1,2,3,5 .

(2.17)

Fab

(2.16) are hermitian with respect to the

Some further property of the representation

(In particular, we show that

tion of the Lie algebra of

S00(4,2)

of the group; the global form

T0

~0

are given

defined so far as a representa-

can be in fact integrated to a representation

of the representation

coincides with the familiar

realization of the conformal group in space-time which leaves invariant the 22 D'Alembert equation Of(x) = ( ~2)f(x) = 0.)

2.3.

In the space

Algebraic Form of Equation (2.2)

3{ of functions

~(p)

the operators

Fa

(2.16) assume the

form (r ~)(p) =

2 ~ q~ - 72 [(p _ q)212 $(q) s(q0)6(q0 - l)d4q

(rS~)(p) = _ 2__ ~ i ~(q)c(q0)6(q0 ~2 [(p _ q)212 Comparing

(2.18)

- l)d4q .

(2.19)

(2.18) with (2.19) we see that (p~)(p)

= (i__ F ~)(p) . F5

Taking into account that for any analytic function F(Fs)(F ~ -+ r~5) = ( r and using Equations

F

(2.20) of

F5

we have

-+ F~s)F(F s _+ i)

(2.21)

(C.9), (C.10) (see Appendix C) we can verify that for

% = 0

the operators i P~ = ~ 5 F satisfy the identities

(2.22)

[P~'Pv! = 0, p~p~ = i.

On the other hand, one can check directly (or by using (2.15)) that (!_ F5 ~)(p) Inserting

I 1 ~ ~(q) s(q0)6(q2 - l)d4q • 2~2 (p _ q) 2

(2.22) and (2.23) in the quasi-potential

Equation

(2.23)

(2.2) we find the fol-

lowing algebraic equation for the relativistic Coulomb problem

269

F~l [F0(E - -~51 F0 ) + ~]#E(p ) = 0 .

(2.24)

Before going to the solution of Equation (2.24) we will make the following general comments. (i)

The prescription (2.22) for the algebraization of the (free) 4-mo-

mentum does not depend on the interaction under consideration. (2)

The simple algebraization of the potential based on Equation (2.23)

is peculiar to the case of zero mass exchange.

The relativistic Yukawa potential

$2 V(p,q) =

(2.25) (p _ q)2 _ ~2

leads already to considerable complications (see Section 111.2 of Reference [2]). The reason is that the kernel in the scalar product (2.8) in tO the relativistic Coulomb potential.

3£ is closely related

If on the other hand we adapt the scalar

product in our representation space to the potential (2.25) for

~ > 0, the sim-

plicity of the free Hamiltonian will be lost. (3) lem: 3£ of

We can use Equations (2.18-20) and (2.23) to solve the inverse prob-

given a d h o c ~0

an infinite-component wave equation in the representation space

(see References [13,15,16]) to reconstruct an equivalent integral equa-

tion in momentum space.

3.

3.1.

SOLUTION OF THE COULOMB EIGENVALUE PROBLEM

Group Theoretical Treatment of the Algebraic Equation

In order to get rid of the inverse powers of multiply it from the left by

rsr~ir s

F5

in Equation (2.24) we

and put

~E "= F0fE " This leads to the following equation for

(3.1)

fE:

[(r 0 - ErS)r 0 - ~ r5]f E = 0 . First of all we observe that the operators the Lie algebra of

(3.2)

F0, F 5

and

[r0,r05] = ir 5, [rs,r 0] = ir05, [r05,r s]

momentum

generate

= -iF 0 .

Equation (C.12) of Appendix C shows that for the representation operator of

F05

S0(2,1):

S0(2,1)

is equal to the Casimir of

R0

(3.3) the Casimir

S__0_0(3). Hence, for fixed angular

£ r~ - F~ - F~5 = L2 = £(~ + l) .

Since Equation (3.2) is obviously an eigenvector of

L2, say

fE£"

S0(3)

invariant, we will require that

(3.4) fE

is

270

Equation (3.4) and the positivity of

F0

imply that we have to deal with

one of the discrete series of unitary representations of Bargmann[17]

(see also [14] Chapter 7).

SO(2,1)

described by

Each irreducible representation

this series can be realized as a group of coordinate transformations able multiplier)

in the space

~l DI

:

of

R~ £)

(with a suit-

of analytic functions on the unit disk {z 6 ~ ,

Izl < 1 }

(3.5)

.

is considered as a Hilbert space with scalar product 21 + 1 ~ ~DI(I - zz--)2£g--~f(z)d2z

(g,f)£

The generators of the representation with respect to

RI1) are

.

(3.6)

first order differential operators

z: d

r 0 : z7~+

l+

i, r s : ( l +

l)z + ~

1

(z 2 + l)

d

7~z (3.7)

1 d r05 : i[(/ + 1)z + ~ (z 2 - i) ~ z ] . It is easily seen that the operators

(3.7) satisfy the c o ~ u t a t i o n

relations

(3.3)

and the identity (3.4). Inserting (3.7) in (3.2) we get the following second order (linear) differential equation for

fEl(Z):

{zQ d2

~ + [(/ + 2 + ~ ) Q

+ (l + l)Q'z + ~ L m z]

d

dz 2

(3.8)

+ (l+

i)[(£ + I)Q' + 7

z]}f : 0

where E

Q =-~ (z 2 + i) - z, Q' : Ez - i

3.2.

Calculation of the Enersy Eisenvalues

The eigenvalues of fE£

E

have to be determined from the condition that

be regular in the unit disk.

(3.8) are

z = 0, z = ~

J

The possible singular points of any solution of

and + !E A z = z± = !E -

Among these four points only two

z = 0

and

- E2

z = z_

.

(3.9)

belong to

D I.

They are both

"weak singularities" of the differential Equation (3.8) and there are regular solutions

f0

and

f

in the neighborhood of any of them.

In order to ensure that

these two solutions are analytic continuation of one another, it is necessary to assume that the branch points at

z = z+

and

z = z

are of the same type (so

that one could consider a single-valued solution of (3.8) regular in the cut z-plane with a cut between

z+

and

~

which does not cross the unit disk).

27q

For z ÷ z+

the a s y m p t o t i c

form of (3.8)

d2 d + B ~ [A(z - z+) dz 2

is

C]f+ : 0

+

(3.1o)

a

A = /i - E 2 z+, B = z+[/l

wi th

- E 2 (£ + i) + ~ ] .

For

the singular

z ÷ z+

solu-

~+ f+

tion

of

(3.10)

behaves

like

%

:

(z - z+)

l

B

X

-

:

-I

where (3.11)

-

- E "2

2E~I

For

z ÷ ~

Equation

(3.8)

d2

is equivalent

to

d (3£ + 4 +-i-~)z ~ z + 2(£ + i)(£ + 1 + T ~ ) ] f a

+

[Z 2

(3.12)

= 0 .

dz 2 The relevant

solution

of

(3.12)

is ,J

The branch ~

- v+

points

at

z = z+

is an integer.

f

oo

and

= z

=

-£

z

=

a

the eigenvalues

1

-

a 2E

(3.13)

"

are of the same

oo

type if and only

if

So, w e put i

- %=~f Thus,

-

with

E

of

n

-

i)

-

i

:

n

-

l

.

(3.14)

(/I_E 2 E

are determined =

2n

~ ~ (En + ~ n )

from the equation - E2 n

(3.15)

or 9

E 3 +--~ E 2 - (I - -~-~-)En - -~ : 0 n n n 4n2 n Only one of the three real roots an expansion

in

=

__

of

(3.16)

satisfies

In order

a

_ ~2 + n

n

to find

the range*

3 a3 _ 3an4 + ... 2 n

of the solution

of the quantum

of Equation f(z)

In view

of (3.8)

the coefficients

f

=

.

number

of the coefficients

*

f

of the power for large

This p r o b l e m was not touched

:~.

n

we look at the power

(3.18)

~ f z~ ~=0 satisfy

the following +

series

(3.18)

Dividing

in R e f e r e n c e

[2].

recurrence

[(~ - i)(~

+ 2(£ + i)(£ + 1 + B)]f _ 1 = 0, ~ = 0, i, 2 . . . . . of c o n v e r g e n c e

(3.17)

""

(3.8):

(v + i)(~ + £ + 2 + 6)f + 1 - 2(~ + £ + l ) 2 c h l f

The radius

as

an - 2n "

i a2 + a3 17 ~4 + En = 1 - ~ n n ---8 n

expansion

It can be w r i t t e n

(3.15).

.

/i - E 2 = n

series

(3.16)

o

a B = ~

+

2 +

6)

(3.19)

•

is determined

the left-hand

3£ +

relation

by the behavior

side of (3.19)

by

2?2

+ i

1 -- we obtain the following asymptotic

and neglecting the terms of order

form for the recurrence relation (~ + £ + 2 + ~)f~+l - E2 (~ + 21 + l ) f

+ (~ + 31 + B)f~ 1 = 0.

(3.20)

It corresponds to a first order differential equation which can be obtained by multiplying by

z~

and summing over

v.

The result is

E E zQf' + {(2£ + I)Q + ~ [£(z 2 - i) + B(z 2 + l)]}f = ~ (£ + 1 + ~) . (We have used the initial conditions (3.21) regular

f-i = 0, f0 = f(0) = i.)

(and normalized to i) for

£ + i + ~ Cz _ z .

z = 0

(3.21)

The solution of

is

/I-E

f(z) = zl + i + B

[(z - z+)(z - z_)]

0

k~ - z_/ (3.22)

[(~ - z+)(~ - z_)]/-Id~ We c a n d e f i n e

f(z)

cut along the real

as analytic semi axis

.

single

z ~ z+

valued provided

B + n, /I

in accordance with (3.15) only if

n ~ ~.

fied that for

For

1 = 0

n = i,

2,

model

equation.) 1 ~ i.

for

lution of Equation (3.8) for the l

s

z = z

n m i; it is easily veriz = 0.

(This

[6] there is no limit of "maximal

The present argument cannot exclude

We observe that (3.22) gives the exact so-

waves

the correct behavior

the exact range of the quantum number S0(4)

a

(3.23)

It is regular for

we actually have to require

however the values

n = 1

with

...

E = 0, Equation (3.8) has no solution regular for

binding" in our quasi-potential

the familiar

cut z-plane

that

is defined in (3.19)).

shows that contrary to the W i c k - C u t k o s k y

it has for all

in the

E2

-

(B

function

(1 = O)

(z - z+) ~+ n

is always

but not for as

z ÷ z+).

1 ~ i

(however,

We expect that

n ~ I + i, which would give

degeneracy of the energy levels of the non-relativistic

hydrogen atom (as well as of the Wick-Cutkosky model).

We mention that the second

order term in Equation (3.17) reproduces precisely the Balmer formula for the nonrelativistic Coulomb energy levels as it should be in any consistent relativistic generalization of the Coulomb problem.

273

APPENDIX A

DIFFERENT REALIZATIONS AND PROPERTIES OF THE EXCEPTIONAL REPRESENTATION R 0 OF S00(4,2)

A.

The Set of Conformal Transformations in Space-time as a Global Realization of R 0

Consider the space f (x)

X

of negative frequency solutions

i (2~)3/2 'ff ~(~) e-iX$6+(~) d4 , ~0+(~) = 0(~0)~(~2 )

(A.1)

of the D'Alembert equation of(x) -

-

f(x) = 0

~$(x) ~x 0

~(x) ~x 0

(A. 2)

with scalar product (f,g) = i

~(~(x)

) g(x) d3x

x0=t

(A. 3)

= ~ 7(Og(O~+COd4~

.

The representation of the conformal group acting in

X

which leaves Equation (A.2)

and the scalar product (A. 3) invariant is generated by the following transformations: •

(i)

/

Polncare transformations [U(a,A)f](x) = f(A-l(x - a))

(A. 4)

Dilations

(ii)

(iii)

(U(%)f)(x) = %-if(%-ix)

(A. 5)

[U(R)f](x) = i___ f(-x) . x2 x2

(A.6)

Inversion

The inversion

(Rx)p

=

-

x -~P x 2 does not actually belong to the connected component

of the identity of the conformal group, but the set of non-linear transformations x [R{b,l}Rx]~ = belongs to

S00(4,2)

- x2b P

d(b,x)

~

'

d(b,x) = 1 - 2bx + b2x 2

(A.7)

and generates the so-called special conformal transformations 1 I x - x2bp~ [U(R{-b,Z}R)f] (x) = d(--ffT~,x ) f~- d T b ~ 7 "] "

(A. 8)

The (hermitian) infinitesimal operators of the subgroups (A.4), (A.5), and (A. 8) are given by

27#

P

= i~ , M

= i(x 3v - x ~ ), ( ~

D = -i(l + xp~ ~) , K

~ F,

P ~ F

'

(A.9)

= i ( 2 x + 2xpx 3v - x2~ ) .

These operators are related to the generators M

z ~-) ~x ~

+ F

Fab

and

Fa

D< =~ FS, K ~ F

used in Section 2.2 by

- Fp5

(A.10)

This well-known representation of the conformal group (related to the O-spin O-mass particles) is equivalent to the representation intertwining operator

V

which maps

R0

$C onto

X

defined in Section 2.2.

The

can be written down explicitly:

+(p) ~ f(x) = ~ f D ~-)(p + x) e(p0)6(p2 - l)¢(~)d4p

~

(A. II)

where D~-)(x) =

i f e-iX~6~(g)d4~ (2~)3

-i i (2~)2 (x 0 _ i0)2 _ ~2

(A.12)

is the Lorentz invariant negative frequency solution of Equation (A. 2). tribution

D~-)(x)

(The dis-

appears in quantum field theory as the two-point function of a

zero mass field.)

The realization of the representation

R0

in

X

displays its

irreducibility with respect to the Poincar~ subgroup of the conformal group.

B.

R0

As One of the Metaplectic Representations of

The metaplectic series of unitary representations of

SU(2,2)

SU(2,2) can be con-

structed in infinitesimal form starting with the 4-dimensional representation (2.5) of the Lie algebra. spinor

To do this, we introduce the 4-component operator valued

~0 satisfying the canonical commutation relations N

here

6

ments

~

~,B = 1,2,3,4, ~ = ~p*6 ;

(B.I)

is the hermitian matrix satisfying (2.7) and normalized by the require-

det 6 = i, By0

is positive definite.

It is easy to verify that the set of

operators

PAB = ~ A B ~

(B. 2)

obeys the commutation relations (2.4) since (B.I) implies that

[FA~'FcD] = ~[~'A~'~'CD ]~ "

(B. 3)

The metaplectie series of the so-called ladder representations of

SU(2,2)

corre-

sponds to the (star) representation of the canonical commutation relations (B.I) in the Fock space in

F

F

defined in the following way.

There exists a unit vector

I0)

(defined up to a phase factor) for which (70 + 1)qo[0) = ~(Y0 - 1)

The vector

[O)

so defined is

10) : O, (F0

SU(2) x SU(2)

[0) :

[0))

.

invariant.

In order to label the irreducible representations of the metaplectic series, it is convenient to extend the representation defined by (B.2) to a

(B.4)

275

representation of

U(2,2)

by introducing a 16th generator, i,~

c = ~ ~. C

(B.5)

belongs to the center of the enveloping algebra of the Lie algebra

17(2,2) and

hence, should be a multiple of the identity in each irreducible subspace of is easy to verify that the spectrum of C

=

I

-

i,

It can be proved that for fixed

C

% =

C

in

(B.6)

%) the ladder representation

in the corresponding invariant subspace

F%

It

is given by

_+ i, ±i, ...

0,

(or

F

F.

of

F

Rk

is already irreducible.

acting All

elements of the center of the enveloping algebra of the metaplectic series are functions of

%.

In particular,

the second order Casimir operator

C2

of

SU(2,2)

is given by 1 c 2 = ~ tAB tAB

=

3(% 2

-

i)

.

(B.7)

It has been shown explicitly in Reference [18] that the metaplectic representations

RX

so defined are equivalent to the representation of the conformal group

in space-time, for

corresponding to zero-mass particles of helicity

% = 0, we recover the representation

R0

%.

In particular,

described in Section 2.2 and Appen-

dix A. The ladder representations representations References

RX

are closely related to the two metaplectic

of the real symplectic group

[19,20].

Namely, if

valued representation of

R (0)

Sp(4,R)

Sp(4,R)

in 8-dimension described in

is the single-valued and

R (I)

acting in the same Fock space

%=0,±i,±2 .... R (I) =

the double-

F, then

(B.8)

~ • R% +i 3 %= _~,_+~ ....

More about the different realizations of the ladder representations alence is said in Appendix to Reference [2].

and their equiv-

The term metaplectic and the first

mathematical description of the metaplectic representations

of

Sp(n,R)

is due to

Weil [21]. (See also Mackey [22].) The description of the metaplectic representations of

U(2,2)

in terms of creation and annihilation operators was first given

by Kurs,unoglu [23].

C. Quadratic Identities in the Enveloping Algebra of the Metaplectic Representa~ons We shall collect in this section a set of quadratic identities which hold in the enveloping algebra of the metaplectic representation of

U(2,2).

They can

be derived by using (B.I), (B.2), and the identity ~ ~ + 2~°TB (Ya)B~ ( Ya) 6B = 6Bd6 a=0,1 2,3,5 where

e

B oB T~ '

is the completely ant±symmetric unit tensor in 4-dimension

(C.l)

276

(s 1234 = i)

and

BYabB - I (the

=

superscript

B

is defined

- t Yab' t

(up to an irrelevant

(a,b = 0,1,2,3,5)

tB

= -B,

(B-b

= ~i

B~

~

e~TB

(C.2)

to the left of a matrix stands for transposition).

Each of the metaplectic stricted

,

sign) by

representations

to any of the 5-dimensional

their second order Casimir operators

RX

remains irreducible when re-

rotation subalgebras

of

are functions of

only.

X

S0(4,2).

Hence,

A direct

calcula-

tion gives i F

ab

F ab = 2(% 2 - i)

1 = ~ F

F p~ + F F ~ P

(C.3)

(repeated upper and lower indices have to be summed over the range p,v = 0,1,2,3).

Comparing

a,b = 0,1,2,3,5;

(C.3) with (B.7), we find

r r ~ = rsrS~

2 _ i .

= x2 +

(C.4)

F5

We also have i F (with

Fij = aijkLk,

Fg v = _L2 _ _N2 = %2 _ i - D 2, __LN = -%F 5

Foj = Nj, i,j,k = 1,2,3).

More generally,

(C.5) the following ten-

sor identities hold: {r 5,Y v} - {Y ,F 5} = 2 F s F {FCA,FCB} = (X 2 - I ) ~ ( A , B As mentioned before, ducible when restricted K~

and

M

-

Xs

(C. 6)

TF a T

= 0,1,2,3,5,6)

each of the representations

. RX

to the Poincar~ subgroup generated by

; see (A. 10)).,

The scalar product of

also irreM p~

(or

(C. 8)

P

and

K

=

_

2(F 5

•

1)Fpv - XSp~OT

is a function of

X

F~T

and

.

The Casimir operators X:

of the

SO(4)

(C.10)

subalgebra are expressed in terms of

3 j=l ~ (L23 + r~ 5) = r2° * X2 - I, L rsll = Xr 0 .

From (C.4) and (C.II), it follows

(C.9)

?5:

KP = (PK)* = 2[X 2 + (F 5 + i) 21 .

and

and

(C. 6) implies P~K~ - P Kp

F0

remains Pp

This gives P P~ = K K p = 0, P L = PO X . P

Equation

(C.7)

(C.II)

that =

-

(C.12)

277

REFERENCES

[11

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[21

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[3]

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[4]

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[7]

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[8]

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[9]

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Bargmann, V., "Irreducible Unitary Representations of the Lorentz Group",

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Mack, G., and Todorov, I. T., "Irreducibility of the Ladder Representations of U(2,2) When Restricted to Its Poincar& Subgroup", J. Math. Phys., i0, 2078 (1969).

[19]

Itzykson, C., "Remarks on Boson Commutation Rules", Commun. Math. Phys., 4, 92 (1967).

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Weil, A., "Sur Certains Groupes d'Operateurs Unitaires", Acta Math., iii, 143 (1964).

[22]

Mackey, G., "Some Remarks on Symplectic Automorphisms", Proceedings Amer. Math. Soc., 16, 393 (1965).

[23]

Kur§unoglu, B., Modern Quantum Theory, W. H. Freeman and Co., San Francisco (1962), p. 257.

TENSOR OPERATORS FOR THE GROUP

SL(2,C)

by W. RHhI*

INTRODUCTION

This talk consists of three parts:

some selected topics of a purely mathe-

matical theory of irreducible tensor operators,

the adaptation of this theory to the

decomposition of the current density operators of elementary particle physics restricted to single-particle spaces, and an application of this formalism to a phenomenological analysis of certain scattering experiments.

i.

i.i.

IRREDUCIBLE TENSOR OPERATORS

Notations and Some Known Facts About the Representations of

SL(2,C)

We shall mainly adhere to the notations of Gel'fand and Naimark.

[i]

In

particular we make explicit use of matrices like the following ones a =

all al

E SL(2,C); u =

\a21 a22! where

~ SU(2) ~ K, k =

~ =

\u21 u2e!

X, ~, z~ are complex numbers and

SL(2,C). SL(2,C)

Ull u12

K

~

,

stands for '~aximal compact subgroup" of

One possibility to realize the principal series of representations of is on a space of measurable functions

f(z)

which have finite norm with

respect to the scalar product P

(fl,f2) = j fl(z)f2(z)dz We denote this space

L2(Z).

.

(i.i)

The group operations are introduced by T~f(z) = ~X(z,a)f(Za)

(1.2)

with

~a = k~a; ~a =

*

za

, k =

X(z,a)

"

European Organization for Nuclear Research, Geneva, Switzerland.

(1.3)

280

where

p

is real and

m

is an integer.

We characterize the representation

X

by

the pairs of numbers -m i +~+~p

X = (m,p) = {nl,n2} ; nl, 2 =

(1.4)

and use -X = (-m,-p)

if

X = (m,p) •

We call this realization of the principal series the "noncompact picture". Another realization of the principal series is obtained in a space of measurable functions

~(u)

on

K

L~(K)

satisfying the constraint

~(u(~)u) = elmer(u)

(1.5)

which have finite n o ~

with respect to the scalar product (~I,~2) = f @l(u)~e (u)dp(u)

We introduce the operators

TX a

"compact picture".

X

and

(1.6)

by the definition

(Z

Ta~(u)X = a X ( u , a ) ~ ( U a ) ; ua = kUa, k = and a relation between

"

ax

as in (1.3).

(1.7)

This realization is denoted the

We arrive at the principal series in this compact picture if we

use the technique of induced representations and induce from one-dimensional unitary representations

~(k)

of the subgroup of triangular matrices

k

$(k) = I x l i P ( . ) -m and identify the cosets of this subgroup in group

U(1)

in

SU(2)

SL(2,C)

with the cosets of the sub-

by a=~

.

The equivalence of the compact and the noncompact picture is easily established. Following Gel'fand spaces

DX

[i] we consider a set of closed topological vector

which are dense in the Hilbert spaces

itely differentiable functions cal for a space of type

K

the corresponding spaces

~(u)

~(K).

satisfying (1.5) and possess a topology typi-

in Gel'fand's notation [2]. DX

They consist of infin-

In the noncompact picture

consist of infinitely differentiable functions

(considered as functions of two real variables) which possess an as~ptotic

f(z) expan-

sion

f(z) around

z = ~.

spaces

DX

DX

ip-2/ z V m

~ C k,i=o

z-k'fz -&

(1.8)

The topology is simply carried over from the compact picture.

are invariant subspaces under operation of

are continuous. spaces

Izl

The

T~, and the operators

T Xa

We emphasize that in the compact picture the definition of the

is independent of the parameter

p.

One space

used simultaneously for the definition (1.7) of operators

DX T~

may therefore be with fixed

m

but

281

variable

complex

P.

Completing

the spaces

DX

for fixed

m

with respect to the

scalar product norm (1.6), we obtain the original Hilbert spaces fashion we can make

L~(K)

which are nonunitary

if

to carry representations

Imp # 0.

Translating

picture we obtain representations

L2(K). In this m for arbitrary complex p,

X

this construction

in Hilbert spaces

L 2 (Z) P]

into the noncompact

with the scalar product

defined by (fl'f2) = ~ fl(z)f2(z)(l For any fixed L 2 (Z) pI

onto

X

with arbitrary complex

L~(K)

which intertwines

The spaces n2

DX

(1.4) are positive

spanned by polynomials

+ Izl2)Pldz' p

Pl = Imp .

there is an isometric mapping from

the bounded operators

possess invariant closed subspaces integers. in

z

In the noncompact

and

~

T X. a E X if both

picture the spaces

of maximal order

nI EX

and are

n I - i, respectively

n 2 - i.

Therefore dim E X = nln 2 , In addition the space negative integers. DX

DX

possesses an invariant

This subspace is denoted

FX

(1.9) subspace if both

nl, n 2

are

and consists of all functions of

whose momenta

( i , 10)

/ zk~f(z)dz vanish for all orders 0 0

integral

(1.30)

We want to investigate the condition under which the integral in the invariant subspace

EX3 •

(1.28) lies entirely

With the notation I A i = ~ (° i + ~i ) i B i = ~ (o i - pi )

(1.31)

a necessary condition for this to happen is A 3

:

-v

-

1

(1.32)

B 3 = -p - i I v,p = 0, i, 2, "'" . If in (1.29)

f3(z3)

lies not entirely in

F

-X 3 the trilinear invariant functional has a pole in Res B(f3,f2,fl)

~.Ip!

(replace o3

X3

by

in (1.29)),

-X3

with the residue

I dz3f3(z3) lJ 2

z3z

~z v ~ x fl(z)d z .

(1.33)

286

In order that the inner integral polynomial

in

in (1.33) defines an element of

z, z, we must implement

EX3 , i.e., a

(1.32) by the requirement & n3 - I (1.34) ~ n~ - i

which can be obtained by inspection

from

o I + o 2 = io 3 - 2 = n 3 + n~ - 2 =

~i + ~2

As an example we consider

I

-m3 = n3 - n3 " the four-vector

~3 in which case integral

v, ~

with

= n~ = 2

range over the values

0

and

i

only.

We write the inner

in (1.33) as f f2(z)A(v,~Iz,z3)fl(z)dz

and get the following

operators

A(v,~Iz,z3)

(1.35)

(in a new normalization)

A(0,01z,z3)

= (z - z3)(E c ~3)

A(I,OIz,z3)

= (z - z3)(E - E3) ~ z - (nl - I)(E - E3)

= (z - z3)(E - E3) ~ - (n{ - l)(z - z3) 22 A(I,IIz,z 3) = (z - z3)(~ - ~3 ) ~z~E (n I - I)(E - ~3) A(O,llz,z3)

-

The representations

XI

and

(n{

-

X2

l)(z

-

z3) ~ - ~ +

are restricted

(n I

-

l)(n{

-

(1.36)

i)

.

by (1.32) to

m I = -m 2 - 2(v - ~) (1.37) 01 which relation we abbreviate Finally we consider that

DX 3

possesses

=

as

-02

-

2i(v

+

~

i)

XI = (-X2)v ~.

the case that both

an invariant

-

subspace

FX 3

n3, n~ .

are negative

integers,

so

In this case the integrals

£k f dz3z3z 3 / N ( z 3 , z 2 , z l l - X 3 , X 2 , X l ) f 2 ( z 2 ) f l ( Z l ) d Z 2 d z can easily be evaluated by elementary methods of functions

fl,2

belonging

I

(1.38)

and shown to be zero for any pair

to regular representations

Xi,2

and for all

0 ~ £ ~ -n 3 - i O < =k < - , =- i n 3 provided

A3

and

B3

(1.31) are not simultaneously

integers

on the half axis

A 3 ~ -n 3 - 1 (1.39) B 3 < =This means

n 3' - i

that with the sole exception of the cases

fines an element of

FX3.

Only in the exceptional

(1.39) the integral

cases

(1.39)

(1.28) de-

the components

287

in the finite dimensional space

D×3/Fx3

are nonzero.

If we implement (1.39) by

the further condition A 3 ~ 0, B 3 ~ 0 we obtain finite irreducible tensor operators.

(1.40) In fact, it is possible in this case

to decompose Naimark's kernel into two intertwining operators that map onto

each

D_XI,2, the finite irreducible tensor operator obtained earlier which maps the

tensor product of E_X 3

DXI,2

into

D_X 1

and

D_X 2

into

E_X3, and the intertwining operator from

2.

CURRENT DENSITY OPERATORS

Dx3/Fx3.

2.1.

Vertex Functions

First we introduce a special realization of a unitary irreducible representation of the group tive energy. SL(2,C)

SL(2,C) x T 4

for a particle of mass

M

and spin

S

and posi-

We define a Hilbert space of measurable, vector valued functions on

Cq(a), -S =< q =< S, S - q

integral, which have finite norm with respect to

the scalar product (~I,¢2) = f dp(a) ~ ¢l(a)¢2(a) q q q dp(a)

is the Haar measure on

SL(2,C)

normalized

(2.1)

(in the notations of Section i.i)

as

a = ~k, dp(a) = (2~)-4dzdkd~ In addition, we require that the functions

~q(a)

•

be covariant on right cosets of

SU(2), i.e., ~q(Ua) = ~ID~qr(U)~qr(a) The matrix

DS

describes a unitary irreducible representation of

We call this Hilbert space {M,S}

.

L2(M,S).

(2.2) SU(2)

of spin

S.

In this space we define the representation

by Ua~q(a I) = ~q(ala) Ux~q(a) = exp{½ iMTr(~a+a) }~q(a)

where

x

is a two-by-two matrix constructed from the translation four-vector X

=

~

(2.3) x

as

XO

P=0,I,2,3 p p Here

o°

is the unit matrix and

Ok, k = i, 2, 3, are the familiar Pauli matrices.

We mention that the four-momentum vector used in physics is related with the argument

a 6SL(2,C)

by

1 Po = ~ MTr(a+a) (2.4) Pk = -

½ MTr(aka+a) "

288

The unitary irreducible representation but reducible on the subgroup

SL(2,C).

{M,S}

of

SL(2,C) x T 4

is unitary

In order to reduce this representation we

embed the space

L2(M,S)

in the space

L2(SL(2,C))

representation.

A canonical way of doing this is

which carries the right regular

~q(a) = / d~(ulD~,(ul~(u-la) where

~(a), is any element of

SL(2,C)

L2(SL(2,C)).

If we apply the Plancherel theorem of

to the right regular representation, we obtain the direct integral decom-

position L2(M,S) =

where

L2(X)

~ m=-2S

~ dp(m 2 + p2)L2(x)

(2.5)

0

carries the principal series representation

realization we may use for example the space

L~(K)

X

of

SL(2,C).

discussed in Section i.

emphasize that this decomposition is free of degeneracies.

We

If we restrict a current

density operator to single particle spaces, say its domain is in range in

As a

L2(MI,SI)

and its

L2(M2,S2) , it decomposes together with the two spaces, and it is this de-

composition which we are interested in. In order to define suitable vertex functions for a given current density operator

j~(x)

acting between the spaces

L2(MI,2,SI,2) , we consider the matrix

elements (~21j~(011~l > = (~2,j~(01~l) (2.6) = NIN 2 where

~i E L2(Mi,Si).

The

~

f

qlq2 Ni

d~(al)d~(a2)*~2(a2)F>(a2,a 1)q2ql~ql(aI )

are normalization constants.

The normalization

customarily used in physical literature is such that for the matrix element of the ~:e$ectromagnetic current between proton states we have F (e,e)q2q I = ~ ~ B ~o qlq2 (e

is the unit element of

times charge of the proton

SL(2,C)) which is achieved by N = (2S + i)~ 2M

~

8~2M 2 .

\(2~)31

Of course the domain of

jp(0)

is not the whole Hilbert space

eral, but at least it is not smaller than the space ferentiable functions with compact support on (2.2).

Cc(MI,SI)

SL(2,C)

L2(MI,SI)

in gen-

of infinitely dif-

that satisfy the constraint

Under the Fourier decomposition (2.5) this space goes over into a space of

functions (for the realizations

L2(K) these functions can be written m satisfying the constraint (1.5)) which are entire in p.

@(u,m,p)

The definition (2.6) is not yet unique, we complete it by requiring covariance on right cosets of

SU(2)

then a vector valued function.

in (2.12).

The vertex function

Let us define

lal 2

= Tr(a+a)

.

£ (a2,al)q2,q I

is

289

The "four-momentum transfer"

q

q = P2 - PI' Pl,2 = P(al,2) (see (2.4)) lies in the domain 2

ala~ 1 2

2

From field theory we know that below a "threshold mass"

Mth

_~ < q2 < M 2 th may equal two pion masses, for example) the vertex function is analytic as a

(Mth

function of the real variables on

SL(2,C) × SL(2,C).

In the worst case, namely

when 2 < (M I _ M2)2 Mth there is a finite

q2

interval on which we have no analyticity.

But a physicist's

intuition lets us expect that in this interval we have at most a finite number of singular points due to additional thresholds with continuity at these points and continuous differentiability

in between.

The harmonic analysis of the vertex func-

tions is consequently beset with at most a complication due to their behavior if tends to infinity.

q2

We may try to handle this complication by means of a regulariza-

tion procedure. In order to formulate the four-vector covariance of the vertex function and the covariance on the right cosets of basis in

L~(K),

SU(2)

the "canonical basis".

it is advantageous to introduce a

We use the functions

~qJ(u) = (2j + i) ½ D1 ~ mj, q (u)"' -j ~ q < = j ' j = 21 [ml + n, n = 0, i, 2, ... where

DS

is the same unitary matrix as in (2.2).

ness of this basis in

L~(K)

basis lies in the spaces spaces

EX

and

F X.

note its elements by fJ (z) q

DX

(2.7)

The orthonormality and complete-

follows from the theorem of Peter and Weyl.

This

and a subbasis can be used to span the invariant sub-

It can be carried over to the noncompact picture where we def~(z).

If the operator

TXa

in

DX

acts on a basis element

we obtain the "coordinate functions" TxfJ(z) = j,qr ~ D~3'qtJq(a)f~.r~(z) " a q

In particular we have

(2.8)

Jl D~ 6.. D (U) 31qlJ2q2 (u) = 3132 qlq2 (2.9) D~

,

31q132q2 (d) where the matrix

d

= d

d~

.

(~)

qlq2 3132q

is defined by d =

, ~ ~ 0 .

(2. i0)

e-2D/ Finally, we switch from the vector labels with respect to the canonical basis in the space

= 0, i, 2, 3

to components

E X, X = (0,-4i), which carries the

290

vector representation J FQ(a2,al)q2ql: J = Q = 0 and J = i, Q = +i, 0, -i 1 1 1 - F0 ' F0i = _(~)7 F 3, F~I = ±(~) 2(r i ¥ iF 2) • r~ = ~2

(2.11)

Then the covariance properties of the vertex functions are expressed by the formulae rJ(u2ae,ulal)q2q I Q

=

J [ D $2 ~,(u2)D S1,~ (u-i I )FQ(a2,al)q~q$ _r_r ~iHl ql q2 M~H~ L L

(2.12)

J -l -i = [ _(0,-4i)..~J' FQ(a2a ,ala )q2ql j,Q,UjQj,Q, ta)xQ,(a2,al)q2ql

(2.13)

J t~l

2.2.

The Decomposition of a Vertex Function with Covariance in the Principal Series

In (2.12), (2.13) the covariance was formulated in so general terms that we may immediately modify these equations and study vertex functions which transform as a representation X

X

of the principal series.

To avoid confusion we add the label

to the arguments of the vertex functions, the coordinate functions in (2.13)

D (0'-4i)

are replaced by

D X.

The main tool of the Fourier decomposition of the

vertex function obtained in this fashion is Naimark's theorem. tinue in

X For

analytically until we arrive at the point X

At the end we con-

(0,-4i)

in the principal series complex conjugation maps

again. DX

onto

D_X

(independently of the two pictures), in particular f~(z) X ( D_X •

(2.14)

Denoting analytic continuations of the complex conjugate off the principal series by (''')*, we have from (2.14) and (1.2) (f~(z)X) * ( D_X

(rafq(Z)× J X)~ = T]X(f~(z)X)*

(215)

The unitarity of the principal series representations implies ( D .X

. 2q2(a))*

31q13

=

D~ -I) 32q2Jlql (a "

(2.16)

The bilinear invariant functional (1.14), (1.15) enables us to introduce a matrix calculus by

.

.

B((fq2)32*,Afql)31 = (x2;j2q21A]Xl;jlql> provided f For

A = ~

this gives

Jl ( ql

, f32 6

DX1 q2

DX2, ADX1 c DX2

(2.17)

29~1

{x;j2q2111X;jlql> = ~jlJ2~qlq2 Y

whereas

A

=

T~ a

(2.18)

leads us back to the coordinate functions (2.8) (x;j2q21T~IX;jlqI> = D~ . (a) . J2q2]lql

(2.19)

A similar notation can be used for the trilinear invariant functional (1.29) for the representations

X1 , -X2, -X3 B((f~)*, (f32),,fJ i) = (X 2 ;jzq 2 IA~(×3) I×1;JiqI> q2 ql

for any

and

(2.20)

Jl J2 fql E DXI, fq2 ~ DX2' fQJ ~ DX3

X3

in the principal series, say.

The linearity and continuity of the func-

tional implies B((fJ) '(mX2 aT .J2,, q2' mXl~Jl, = Z DX1 ,, -1 Q * j {q{j~q~ 31q131ql (a I )Dj2q2j~q~(a2) x

..t r AQ(X3) t r> • {X2,j2q21 J IXl ;jlql

With i2.16), (2.19) and matrix calculus we can continue this equation = (x2;j2q 2

TX2AJ ,

.~X1

a2 QtX3)~a~ 1Xl;jlql>

.

(2.21)

From (2.15), (2.16) and the invariance of the trilinear functional we have X3 . jr {x2;j2q 2 T X2a_IA~'QtX3)TXIlX1;jlqI>a = jSQDjQj,Q,(a)(x2;j2q21A~t(X3)IxI;jlql > .

(2 .22)

Comparing (2.21), (2.22) with (2.12), (2.13) we recognize that the vertex function FQ(a2,al I J X)q2ql

has the same covariance properties as the matrix element X2 J XI (x2;S2q2ITa2AQ(X)T _llXl;Slql > a1

where

X1

and

X2

are arbitrary.

This fact suggests that we decompose the vertex

functions into such matrix elements. In fact we define a Fourier transform by J M(X2,XI;X ) = f dp(a2a7 I) ~ £Q(a2,a11Xlq2ql JQ TX2.j, . X 1 x {x2;S2q21

when all three

XI,2

and

X

(2.23)

a2AQtX)TaTIlx1;Slql>*

are in the principal series.

(2.23) can be verified to be independent of

ql

and

q2"

The left-hand side of The main tool in the in-

version of this Fourier transformation is Naimark's theorem in the form (note that the product

X x X2

is decomposed into

~@ dXl )

292

f dXI<X2;J~q~IA~I(X) IXI;JlqI>*<x2;J2q21A~(x)IxI;Jlql} Jlql

(2.24) J232 q2q2 Jd'~QQ '

and the Plancherel theorem for

SL(2,C)

in a similar matrix version

i / dX Here

6(al,a2)

~ D~IQIJ2Q2(al)(D~IQIJ2Q2(a2))* JIQIJ2Q2

is the delta-function on

Haar measure (see (2.1)).

SL(2,C)

= ~(al,a2)

(2.25)

normalized with respect to the

From (2.12) and (2.131 we have

; dXldX2M(x2,Xl;X)<x2"S2q2Jr~2Ai(x)IXl;Slql} ' a2 q

2

= ~

NxIdx2 jlJ2q{q~ x / d~(a~)j!Qr<X2;j2q~

X 2 jt r * jr r Ta~AQr(X) BxI;jlqI> FQr(a2,elX)qEq 1

where we exploited the covariance to set

a I! = e.

Inserting the two formulae

(2.24), (2.25) into this expression we obtain the desired result

F~(a2,elX)q2ql.

By means of covariance we can extend this result to the general inversion formula J rQ(a2,al]X)q2ql

1 = g ; dxIdX2M(x2,xI;X)

× <x2;S2q 2

TX2 J . XI a2A~(X)TaTIIXI;SIqI>

.

(2.261

It does not make much sense to discuss the convergence of (2.23) and (2.26) using the information on the vertex function supplied by field theory.

We mention

only that a sufficient condition for the proper convergence of both (2.23) and (2.26) is infinite differentiability and rapid decrease (that is, faster decrease than any J power of Hal) of PQ(a,elX)q2ql on SL(2,C). We call such vertex functions "smooth". 2.3.

The Decomposition of the Four-vector vertex Function

For a smooth vertex function we write (2.23), (2.26) in the form J FQ(a2,allX)q2ql

i = ~ ; dXldX2 ; d~(a~a~ i)

jrQr

jr XI x PQ,(a~,a~IX)q2ql<X2;S2q21TX2AJ (x)T IIxI;SIqI> a2 Q a7

(2.27)

a,~ilxi;Siqi>*

X2AJt. ,TXI x <x2;S2q 2 Ta ~ Q,(X) which we want to continue in

X.

Because of the smoothness of the vertex function

we may handle the expression (2.27) rather freely without facing problems of convergence.

When we reach the point X = {n,nr}, n = n t = 2

295 T

we define

F~ = 0

stricted to

J ~ 1

for

J > i.

The first bracket

('..>

in (2.27) with

J

re-

is finite everywhere except that a pole occurs whenever

×i = (x2)~ (see (1.37), remember that ('''>*

X2

has been replaced by

-X2

in (2.20)).

The brackmt

of (2.27) is in turn zero everywhere except at the same position, where it

assumes a finite limit.

Both the residue of the pole and the finite limit are

matrix elements of vector operators (1.36). gration over

XI

and

X2

A careful analysis shows that the inte-

picks up just the residue of the pole, and of the two

integrations only one is left.

One of the two representations

the principal series, for convenience we choose

X1.

function accounts for the nonunitarity (polynomial increase) of i variable X = ~ mp and S = min(Si,S 2) we get J 8i FQ(a2,a i)q2q I = ~--

XI,2

is pushed off

The smoothness of the vertex XI.

With the new

2S +i~ ~ ~ f dX ~,~=0,i m=-2S -i~

(2.28)

x Mg~(m,X)(x;S2q2'T~2A~(n,~)T~'X~z;SIqI> and the Fourier transform M(m,X)

-

J

= ~ I d~(a2all) (-l)J(2J + i) JQ FQ(a2'al)q2ql

(2.29)

•S iql> * x (x;S2q21T~2A~( I - ~,i - ~)TXI-~'I-~IXI-~,I-~ ' I a1 As long as

X

is in the principal series

(.--)*

means the complex conjugate.

Formula (2.29) for the Fourier transform can easily be simplified.

We set

first =

a2 u2d(n2)a a I = Uld(-nl)a (d(nl,Z)

~1,2 ~ 0, n = n I + n 2

as in (2.10)) and can decompose the measure correspondingly du(a2a7 I) = (4~)-IdU(Ul)d~(u2)sh2~d~ .

Inserting this into (2.29) we obtain M ~

(X) = [4z(2S 1 + I)(2S 2 + i)] -I 7 dnsh2n 0 × I (-l)J( 2J + l)F~(d(n2),d(nl)-l)q2ql JQJiJ2qlq2

(2.30)

(x*)l - IX* × ds2J2q2(n2)d JISIq~' (NI)(X*;J2q21A~(I - Z,I - ~)I(X*)I_z,I_~;JIqI> where we used the notation X* = (m,-p) In (2.30) we may set tions drops out. momentum

nI

or

n2

for

X = (m,p) •

equal to zero in which case one of the two d-func-

We recall that by (2.4) the matrix

d(n)

corresponds to the four-

294

p = (Mchn,O,O,-Mshn)

.

The vertex function entering (2.30) has therefore been brought into a "collinear" frame of inertia.

In these frames it is easy to express vertex functions by some

conventional kind of form factors.

The momentum transfer

q2

is

q2 = M~ + M~ - 2MiM2ch ~ . In physical applications the vertex functions have to be regularized to fulfill the assumption of smoothness.

One of the basic premises in standard appli-

cations to physics is that the removal of the regularization can be accounted for by a mere change of the integration contours in (2.28).

Typical statements arrived at

in such applications involve asymptotic expansions of vertex functions.

The deriva-

tion of such asymptotic expansions is always based on the following "Weyl symmetry relations" which reflect the existence of an intertwining operator for two representations

X

and

-X:

J

J

d~ . (~) = B l(-h)B 2(h)dTX. (~) J132 q J132 q