Bose Algebras

Lecture Notes in Mathematics Editors: A. Dold, Heidelberg B. Eckmann, Zfirich F. Takens, Groningen

1472

Torben T. Nielsen

Bose Algebras: The Complex and Real Wave Representations

Springer-Verlag Berlin Heidelberg NewYork London Paris Tokyo Hong Kong Barcelona Budapest

Author Torben T. Nielsen Mathematical Institute, ,~rhus University and DIAX Telecommunications A/S F~elledvej 17, 7600 Struer, Denmark

Mathematics Subject Classification (1980): 81 C99, 81D05, 47B47

ISBN 3-540-54041-5 Springer-Verlag Berlin Heidelberg New York ISBN 0-387-54041-4 Springer-Verlag New York Berlin Heidelberg This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in other ways, and storage in data banks. Duplication of this publication or parts thereof is only permitted under the provisions of the German Copyright Law of September 9, 1965, in its current version, and a copyright fee must always be paid. Violations fall under the prosecution act of the German Copyright Law. © Springer-Verlag Berlin Heidelberg 1991 Printed in Germany Printing and binding: Druckhaus Beltz, Hemsbach/Bergstr. 2146/3140-543210 - Printed on acid-free paper

Contents

I

0.

Introduction

1.

The Bose

2.

Lifting

3.

The c o h e r e n t

4.

The W i c k

5.

Some

6.

The

7.

The real w a v e

representation

72

8.

Bose

algebras

of o p e r a t o r s

79

9.

Wave

representations

algebra operators

special complex

to

vectors

ordering

4

F0}(,

23

F]{ in

33

F}[

and the W e y l

relations

53

operators wave

representation

10. A p p e n d i x

I: Halmos'

11. A p p e n d i x

2: G a u s s i a n

of

45

F(][+][*)

len~a measures

66

89 94 96

12.

References

130

13.

Subject

132

index

Introduction The

aim

consequences Though

having

theoretical further.

communication

[15,16,17]

well

Hilbert

by

the

a

eleme n t s with

that

algebra

an

main

is

that

theory,

has

is

a

process

Bose--Fock by

much

filtering

the

spaces

Irving

and

have been an

fixed

E.

of were

Segal

accompanying

combined

is

algebra.

to form

algebra.

Given

algebra

multiplicative

H

extended

operators

scalar

the

commutative

from

the

in the

(in

of

The

product

algebra

literature

treating

the

unit

over

F0H

known the

a

F0H

F0K

multiplication

is

physical

subject.

fermions

their

Weyl

It

having

e, in by

provided

as

the

base

as

important become

which facts

easier

complex

to

are

Bose

space

There

are

inner

product

extra

structure known

in

those

be

directly

relations

the

is

the

closely

to

expose

understand

also

an a

and

can be b e t t e r purely

whereas

in

work

free

without

mathematical

the

with.

product,

Bose

algebra

knowledge

an

algebra

for

Weyl

the

fermion

relations

and

to

areas

Moreover,

a

number

traditional particular

of of

manner the

of for

belong

over

which

reasons

associative

and

introducing objects

the prior

in

physics,

now

role

of

understood.

mathematical

manner

In

this

quantum

no b a c k g r o u n d

to

the

algebra, related.

free commutative, inner

no

compared

Clifford

so

point,

with

in

of

in q u a n t u m

anti-commutator

operators,

not

Introducing

space with

starting

spaces study

with

situation

for

difficult

conjugation

algebras.

to the its

within

the

to m a t h e m a t i c i a n s

as

also

its

by

clearer

creation

counterpart

mathematics

relations

Bose--Fock

originated

In c o n t r a s t

can

and

of

available

intuitively

annihilation

well

of

namely

free

and

to

mathematics,

physics.

the

formalism

linear

K

the

advantage the

becomes

mathematical

the

extends

[26],

in

"the one-boson--space"). The

which

formalism

product

adjoints

H,

spaces

space

object, the

space

derivations

base

[22].

are

space.

application

processing

Bose--Fock

scalar

of

of

description

which

Bose--Fock

in the sixties.

the

operator

extension

with

manner field

the

area

Bose-Fock

society

consider

The

results, of

mathematicians

mathematical

linear

in H are

such

called

the

of

signal

of

both

some

concept

principal

human

(cf.[3])

~,,

the

formalized

the

known

vacuum.

way

and

attention

space

generated

such

new

paper

present

of

digital

and a n n i h i l a t i o n

single

called

a

within

this

to

usefulness

in

and others

In creation

the

in

the

is

origin

appear

and

to

paper

algebraization

physics,

[8]

brought

this

an

their

They

theory

a

of

of

at the

the new are

same

obtained relations. actually

studying

Bose

multiplication

for

on a

time

extending

algebra, There Bose

the

provide are

an

several

algebras

in a

more

or

less

scalar

polynomials 3)

one

natural

also

attach

[6]

which

the is

the

taking

n-fold

very

direct

in

original

to

find

polynomials

generated

a

is

by

rather

natural

with

Hermite obscure),

manner

to

also

admits

LZ--Fock--space

a way

ours,

exponential

based

Hilbert

n-particle

of

products of

the

constructed

correspondence

on

the a

is

in

one

the

Bose--Fock

of

symmetric

of

has,

space

the

the

algebra

for

each

obtained

base

by

space

and

from each other

product.

n-particle

Bose

the

concept

Here

can be o b t a i n e d

these

the

the

elements

elements

to

introducing

which

which

from

of

spaces.

space,

products

order

is

complex

multiplication

algebra

the

will

tensor

the

space

of

of

algebra

structure.

tensor

those

permuting

Hilbert

4)

reader

or

the

Bose

[24],

so-called

identifying

a

algebra

2)

operation

similar

spaces

] ) the

[1 ],

the

Bose a l g e b r a

Hilbert n6~,

in

calculus

In space

way:

as

(here

can

stochastic

by

obvious

product

The

symmetric

spaces,

generated

and

by

is

the

in

base

space. One of the best k n o w n Segal--Bargmann the

elements

construction of

complex

are

elements

the

of

the

conjugate--entire. e lem e n t s

of

vectors,

which

[5]. has

The

view;

here we real

a

wave

representation,

the

exponentials

as

for

laser

is

the

from

into the

one.

the

which

to

the

representation

complete point

In this

following Hilbert

transforms

the

case

optics

L2-space

abstract

(complex)

Contrary

the real w a v e

in

paper

space,

the

constitutes

of

way:

complex

of

the

coherent

appropriate

an

become

quantum

representation.

constructed

constructed

in

paper

of

so-called

and m a t h e m a t i c a l l y

representation

it the real wave

functions

beams

within

contained

a self

the

in w h i c h

exponential

that

known

is the

this

well

gave

real

taking

In

mentioning

surjective.

is

[16],

plane.

the

representation

the

and

space

thus

is i n t r o d u c e d

into

[I]

involves

of not being

call

functor

representation

cf.

complex

[5,20],

states

representation

conjugation

this

are

Schr~dinger

shall

wave

the

cf.

worth

the

wave

Segal

the

space

is

on

functions

space

provide

[15]

of

complex from

It base

complex

In

the

the

base

of the Bose--Fock

representation,

functions

complex

the d i s a d v a n t a g e

account

wave

entire

of

realizations

a

and wave

complex

a unitary

map. The the real

functer

one

that

by a so--called s q u e e z e d a

subject

experiments aspects

of

special with

attract It

transforms

is an o p e r a t o r

turns

out

adjoint

state with

interest

squeezed growing

the c o m p l e x

whose

attention that

infinite

in q u a n t u m

light

the

are

wave

representation

corresponds

q ui t e

energy.

optics. recent

Squeezed The and

into

to m u l t i p l i c a t i o n

first the

states

are

successful theoretical

[13]. so-called

normal-product-algebra

of

creation which adjoint using

and annihilation

we

analyse

f unc t i o n

[21].

of o p e r a t o r s It

The

[11],

important measures principal

to

since

the

be

the

a short

excellent

I Graversen for m a k i n g

would and

Bodil

numerous

the

consisting

of

determined

one

that

real

who

Hilbert

may

the

conjugation

to

the

elements by the

the

the

method

and This

Wigner of

the

conjugation

book we

of

Louisell

apply

without

fits

making

not

Steengaard suggestions

which

are p r o v i d e d

space.

These

algebras.

already

be

spaces,

measures

are With

acquainted can

find

in the a v a i l a b l e

in

express

representations

of Bose

linear

on g a u s s i a n

to

reads

wave

interpretations,

expositions

like

taking

representation.

analogous

the m a t h e m a t i c s

information

appendix

elementary

complex

wave

of operators, of

the

it too

physicist. and

dimensional

necessary

the

real

if

clear

for the t h e o r y

reader,

operation

operators

that

dimensional

importance

in i n f i n i t e

seek

added

complex

infinite

as

a

space

rigorizing

probabilistic on

of

a Bose a l g e b r a The

above.

immediately

the

9.

conjugation

mentioning

for a t h e o r e t i c a l Both

theory

kernels

Bose--Fock

of a p p l i c a t i o n ,

yields

and

construct

a complex

worth

it will

obscure

and

has

described

is

we

produces

representation

8

can now be taken

conjugation

representation

operators

chapter

of an o p e r a t o r

this

spirit

in

have

by g a u s s i a n therefore

of

this

in mind

with

measure

it v e r y t i r e s o m e

literature,

in H i l b e r t

very

spaces,

we

have

b a s e d on

[]8].

my

thanks

for r e a d i n g

to

parts

and corrections.

David of

Adams,

Krista

the m a n u s c r i p t

and

Chapter

A:

]:

The Bose alqebra

The free c o m m u t a t i v e Let

linear

H,

in

the

commutative

alqebra

F0~

be a separable second

algebra

vacuum)

and the Hilbert

space).

We

denote

by

Hilbert

variable.

generated

Let

space with

then

X

the

(called

set

of

the inner product

F0H

by a m u l t i p l i c a t i v e

space ~

FOH,

denote

the

free

o

(called

the

unit

the base or the one-particle

positive

integers.

For

n6~

we

fulfilling

the

define

{

n H0 = span where

ala2...a n

additional

ala2...a n

denotes

linearity

the

}

al,a2,...,aneH

free commutative

,

product,

relation

(t-a + b)a2a3...a n : t.a.a2a3...a n + b.a2a3...a n with

a,a2,a3,...,an,b6~ We

linear

consequently

and

commutative

and

tEC

identify

. elements

operations,

can

which,

be

by

reduced

repeating

to

the

same

where almost all

fn

0

these form.

M o r e o v e r we set 0 = span ~0

{e} = C.o

and co co

F0 H = n=0 @ }{0 n = {

~ fn

fn 6 ~0n '

n=O FOK an a l g e b r a by d e f i n i n g

We make

the addition

:

~ fn + n=0

the m u l t i p l i c a t i o n

:

~ fn n:0

for every

f n , g n 6 Hn0

~ gn = n=0

with

~ gn = n=0

nE~ 0 = ~ U {0}

,

~ (fn + gn ) n=0 ~

~

n=0

f j'gk

j+k=n

and defining

e-f=f-e=f for

f6FoH

.

It is an easy exercise are

associative

and

to show that addition

co~utative,

thus

making

and m u l t i p l i c a t i o n

F0H

a

commutative

algebra

with We

multiplicative

shall

use

the

unit

following

r = i.e.

rk6~ 0 Irl

for

.

notation: n e ~0

(r I , r 2 , . . . , r n )

k=1 , 2 , . . . , n

= rI + r2 +

e

...

,

'

we define

+ rn

= r] !-r2!- . . .-r n.t

r!

r rl r2 rn Irl e-- = e I -e 2 - . . . . e n e ~0 0 e = e , where

{e I , e 2 , . . . , e n }

Proposition an

orthonormal

is a n o r t h o n o r m a l

I .IA:

system

a I ,a2,...,an6}{

every

r 6 ~k

to

It

is

sufficient

.

We

define

dim

an

orthonormal

K < n

Then

basis

it

}{

with

find

and

that

=

dimensional }

k6~

a

= n }

f

{e] , e 2 , . . . , e k } to

correspond

,r,

{ a l , a 2 ..... a n

is p o s s i b l e

){ .

such

consider

a finite

in

there

in

{ er

~ = span Choose

fe){~

{e] , e 2 , . . . , e k }

f 6 span

Proof:

To

system

a la 2. ..a n

space

K

,

where

,

•

in t h e

complex

space

K

numbers

with

k =

{t]}i, j

such

that k =

ai

tl..e

.

3

]

for

i=],2,...,n

,

j=1 and

we

get k

al-a2.....a

k

k 1

~

n =

J] =]

j2=I s r •e ~

for

some

s

r

In t h e choose

the

6 C

,

which

case

space

2

n e

• ''

of ~

f

t]]'t]2

"'tin

. . . .

e31

e

]2

3n

Jn=1 with

_r £ ~

evidently being

sufficiently

and

Irl _

= n

is a s u m

of

the

a sum

of

large.

generators

desired for

type.

n }[0 '

we

just

8

The a b o v e

argument

also

Proposition {e],e2,...,ek}

verifies

1.2A:

Let

the

X

an o r t h o n o r m a l

denote

basis

{e~l spans

the w h o l e

B: T h e B o s e

use

alqebra

e+(x)

We

shall

demanding + (x) , i.e.

for to

fulfilling

the

with

H

inner

the

operator

defined

on

the

product

operators

the

e(x)

whole

lemma

for

to

the

Then

the

in p h y s i c s

,

by

x6H

the

whole

dual

and

~+(x)

for e v e r y

determined

set

to

of be

V0H

the a

we w i l l

by

operator

derivation,

f,geFoK

.

by defining

: ]

and e(x)

shall

be c a l l e d

the

creation

and

respectively.

Leibniz

rule,

we o b t a i n

the

recursive

formula

> = <x2x3..Xm,~(x])(ylY2..Yn)> I + y]~(Xl)(Y2...yn

)>

,

relation

1.1B b e l o w

we o b t a i n

= <xl,Yk>e

for

n~m

<x]x2...Xm,y]y2...yn and

H

,

F0K

+ f-e(x)g

~(Xl)(Yk) from

employed

in

= <x2x3..Xm,Y2..Yn~(X])y the

.

and

0 }

product

uniquely

operators

<XlX2..Xm,y]y2..Yn

using

K

space

rule

= g-~(x)f

inner

the a n n i h i l a t i o n

and

space

of m u l t i p l i c a t i o n

the

the Leibniz

Applying

dimensional

k

the n o t a t i o n

The

in the

~c~

the o p e r a t o r

extend

e(x)(f-g) We m a k e

a finite

FOK,

for

x 6 be

proposition.

FOX

To be in a g r e e m e n t often

following

> = 0

n=m n

<XlX2..Xn,ylY2...yn

> : ~ <xl,Yk>'<x2x3"'Xn,YlY2"'Yk-lYk+1"'Yn k:] =

where

~

runs

~ <xl'Y ~ 1 >'<x2,Y~2>'...'<x

through

the

set

of

all

n'

>

Y~n >

permutations

of

the

numbers

{I,2,...,n}

Lemma

I.|B:

For

1)

~(x)(~)

=

2)

e(x)(y)

= <x,y>o

Further,

every

we

have

o

for

n,m6~

we

have

3)

~ ( x ) ( y n)

= n - < x , y > - y n-I

4)

~(x)m

=

yn

x,yEK

n!m ) ! (n-

"<x'y>m'yn-m

for

m1

and

=

c I ,c2, . . . , C n e H

< ~ ( x ) y , c l c 2. • .c n > = < y , x - c l c 2. . . C n >

Thus

,

¢~

on both

,~(x)~

For

=

the

0

.

we get =

= + + 0 = 0 n a(x)y is o r t h o g o n a l to K0 for every

element

n6~

)> )>

,

and we

get ~(x)y As

t = Moreover

= we

have

by

e ( x ) y n = e ( x ) ( y . y n-1 ) = = <x,y>yn-1+ The

last

We

= t-~

for

= <x,y>

easily

often

identify

we

t6G have

. proved

identity

2

induction

( ~ ( x ) y ) y n-I

y. ( n - 1 ) < x , y > y n - 2

identity

,

some

follows

an

by

+ y- ( ~ ( x ) y n-1 ) = n - < x , y > y n-1 induction.

element

and

the

operator

consisting

of

multiplication

b y the

the

for

symbol

x

Given write

x

linear

for

the

the

operator.

be

theory,

To a

the

a

presentation,

element

itself,

operator

e+(x)

operator

x

operator

to

mathematician

,

to

the is

intuitions

will

x6H

in m a t h e m a t i c s

operator

use

for

we

shall

one

will

often

use

.

adjoint

annihilation able

i.e.

gladly

the

and

use

operator adjoint

x to

techniques

this

as

parallel

a rule In

this

the

creation

from

operator

notation

when

w

computing. operator

Hence e(x)

we

,

shall

alternatively

write

x

for the

annihilation

i.e. +

for e v e r y

fEVoH

(x)f

= x-f

~(x)f

= x f

.

Proposition

I.2B:

For

arbitrary

al , a 2 , . . . , a m , b 6 H

we

have

the

identity

Proof:

The

= ~ 0 [ n'.- < a 2 , b > . . . < a n , b >

proposition

follows

by

induction

for for

n~m n=m

and

the

following

calculation.

= =

will

= n..

We

are

now

able

turn

out

to be v e r y

Proposition

to

Hilbert

space

Assuming H

,

and

prove

a result,

which

later

on

useful.

] .3B: To e v e r y ~

Proof:

formulate

= span

that

we w i l l

~

n£~ { an

we have a6H

is a f i n i t e

prove

that

}

dimensional

subspace

of the

K~ = span { an Notice

K n0

that Take

prove

f6K~

that

n

and

dimensional

assume

that

subspace

f 6 { an

as well. a6K

we

}I

have

to

f = 0 .

Choose with

is a finite

J a 6K }

an

k6~

orthonormal

Since

we e x p a n d

the

set

basis {e ~

in a finite

J ~6~

with

a6K

we then

0 =

= ~ ~r.<e[,an>

is a basis

I~I = n }

in

.

rI

= ~ ~r-n!-<e],a>

>.<e2,a>r2>...<ek,a

>rk>

r rl

= ~ ~r'n''al r

r2 "a 2

where

a i = <ei,a>

whole

K

,

space

have

r ,

the

sum f = ~ tr'e[ r

For e v e r y

in

{el,e2,...,ek}

the

rk ...a k

for

> , i=1,2,...,k

variables

As

a

al,a2,...,a k

is r u n n i n g

range

the

through

the

C

and

whole

,

consequently t

= 0

r

for e v e r y

r .

i

As e v e r y

element

n }{0

in

is

finitely

generated,

the p r o p o s i t i o n

holds.

The

above

result

polarization

should

not

be

surprising,

since

the

general

identity n

n! • XlX 2 . . .x n =

~ (-I) n-k k=1

for c o m m u t i n g

variables

Theorem

1.4B:

as usual

[A,B]

operator.

For e v e r y

(The

denote

~

(xil + x i 2 + . . + X i k )n il

= <el,el>

we get by i n d u c t i o n r. r 2 r = <el]e2...enn,e[>

rl.<e(~-11 ),e(~-11)>

=

If

,

instead

we

fulfilling

have

[ # s

rk~s k

,

Without

:

= <e([-l]),e~e[>

rl !.r2!....

then

there

loss

of

*

rk<s k

Because

indices,

we get

the o p e r a t o r s

ek

.

r n.t

=

exists

_r !

a positive

integer

k

g e n e r a l i t y we w i l l a s s u m e that + and (el) c o m m u t e for d i f f e r e n t

rk (r-r k ) s k (~-s k ) < e Z , e ~> = <e k -e ,e k -e > ,e

for

the

:

Hilbert

0

space

which

is

the

completion

of

FOK, The norm

following

proposition

by multiplication,

which

gives

is not

an

estimate

a bounded

of

the

operator.

We

growth shall

in use

the n o t a t i o n Ifl =

Pr__o p o s i t i o n

1.8B:

with

being

the

feFK

.

m f6K and g6K 0 ! n+m 2 ~ ( n ) " If!" Igl '

To e v e r y

If'gl n (m)

for

binomial

coefficient

we h a v e

n! ml(n-m)!

for

positive

13

integers

n,m6~

Before notation.

.

stating

For

the

i,k 6 ~

with

p6~

we

shall

need

a

lemma

and

a

concise

we define

i < k -

proof

if

i.< k.

--

3-

for

j=1,2 .... ,p

]

.

Lemma

inequality

Consider

1.9B:

m ! n .

positive

To every

k6~

m,n6~

integers with

length

fulfilling

I~l = n

the

we have

k! iek k=] and

{ ap

a6H

and

p£N 0 }

spans

-e r }

the whole

r6I

F0H

the p r o p o s i t i o n

,

holds.

Since the set of indices

I

is countable,

the Hilbert

space

FH

is separable.

Definition of the space

|.13B:

The spaces

Lemma of

F~

onto

We define

Kn

1.14B: Kn

and

Denote

Hm

feFK

,

~n

are orthogonal

by

P

n ,

as the closure

subspaces

for

the orthogonal

ner o ,

n~m

Pn(f)

=

we have

f =

~ fln

FK

in

.

n=O

Proof:

To a r b i t r a r y

6>0 If

As

geFoH

,

it is possible

choose -

gl


N

n

f-

,

~ gn n=0

=

~(x)

the L e i b n i z

fer0H =

n=0

domain and

ex p y .

Proof:

By c o r o l l a r y x

by which

*n

~(exp

operators

3.7 we get

(exp y) = <x,y>nexp y ,

x)

are

x)

closable

are for

39

exp x*(exp y ) =

~ x*n(exp y)/n!

=

n=0

y/nZ

~ <x,y>nexp n=0

= e<X,Y>exp y •

Proposition operator

3.1]:

pair

x,yeH

we have

on

F0K

Proof:

~+(exp y) = e<X'Y>e+(exp

First we will show that

It is sufficient

to consider

y) exp x

exp x

elements

is defined

on the whole

of the form

a m. exp z with

a, z6~{ and

mew

By the Leibniz

rule and induction we get n x*n(am-exp z) = ~ (~)-x*k(am)-x*(n-k)(exp

z)

k=0 For

n>m

this reduces

to

x *n (am -exp z) =

m ~ (k)-x*k(am)-exp n k=0

z-<x, z>n-k

,

by which we get the estimate Ix*n(am-exp

m z)l o ) n / n !

, et.cP.exp

c6H

y

N

n=0 for some

x

and some

p6~ 0

i

It is e a s y to p r o v e N

N

~

cP"

tk Yn k--~"n !

n=0 k=0 by w h i c h

it is s u f f i c i e n t N

N

c p- ~

~

n=0k=0 We then have

t k yn

~-T.n!

, cP.et.exp

y

N

to get N

_

c p- ~ (y + < x , y > ~ ) n / n !

•

to e s t i m a t e

n=0

, 0 N

,

it

is

41 N

N

tk.y

n

cP" ~ ~ ~. n! n=0 k=0

N

_

N

cP" ~ (Y + <x'Y>~)n/n' n=0 N

N

t k yn

=cP

=

n

c p- ~

~ (~) tk.yn-k I

n:

n=0 k=0 n=0 k=0 N N t k" yn N n tk yn-k 1 = cP" ~ ~ ~.' n' - cp" ~ ~ k-~'(n-k): n=0 k=0 n=0 k=0 By a standard argument letting the indices run through the diagonal and introducing more terms into the sum, we get the estimate N

N

t k yn

N

cP ~ ~ k--['n!- cP ~ n=0 k=0 n=0 By using proposition 2-N n ~ c p . ~k. y n-k < ! (n-k) !




= ~ n=O co

= ~

n=O 00

= exp

using

the

identity

)go-eXp

b>

exp(x*+a

w

b>

) = exp

x

~%

exp

a

w

on

FOR,.

43

<e+(exp

x)f,g>

Notice

e x p x * (go). e<X 'b>exp b 1 > .

x

(g0-exp

the

exp(x was

exp x

*

.

+a

b)>

b)>

.

identity ) = exp

Now

we

x

know

exp that

a it

is

valid

on

the

whole

as well.

Example coherent yields

3.14:

vectors a

momentum

(cf.

yield

physical

[11])

states,

one

As

mentioned

which

particle

minimize

state,

i.e.

the

the

the

normalized

uncertainty.

{x 1 =

I

,

we

If

xeH

have

the

operator p = 2-½.i.~-(e+(x)

and

earlier

- a(x))

position I

q = 2--2"(e+(x) with

~

denoting We must

Planck~s

constant

+ a(X))

divided

by

, 2~.

calculate Ap-Aq

on

the

the

coherent

square

of

states

the

in

FH

standard

a coherent

state.

=



=

It is n o t

hard

to s h o w

=

that

the

(Ap) 2 =

More

e-½lzl2-exp



specific,

_

2

denotes

let

z

T h e n we h a v e

,p2 (~z)>



= - ( x + y *)n-1

identities

are

easy

to v e r i f y

by using

the operator

46

identity [An,B]

that holds

A n-k. [A,B]'A k-1

k=1 operators

for a r b i t r a r y

Proposition

n ~

=

4.4:

For e v e r y

A

and

x,y£H

B

and

we h a v e

n6~

[in] *n

1)

(x+y)

= n!-

~

(i) k *)n-2k) : k!- (n-2k) l : (x+y

k=0 [in]

2)

:(x+y*)n:

k

= n!.

(-½) (x+y)n-2k k ! - ( n - 2 k ) .t k=O

Proof: follow

We will

at once,

prove

where

I)

by induction.

n>1

n=0

and

n=1

From

the

by d e f i n i t i o n

A 0 = I = the i d e n t i t y Take

The cases

operator.

T h e n we h a v e * n

(x+y) Assuming induction

n

hypothesis

= (x+y*) to

be

n-I

odd,

and lemma

+

(x) + (x+y*)

we

get

[

n-1

y

*

n-I 2

]

4.3 we get

n-1 2

(x+y

*)n-la+

(x)

(~) k

= ( n - 1 ) !-

k!- (n-l-2k)

!

: (x+y*)n-I -2k) :~+(x )

k=O n-1 2 (i) k =

(n-1)

1-

+

k!- (n-1-2k)'

* n-I -2k (x)

: x+y

}

:

k=0 n-3 2 (~) k kl-(n-1-2k)!

+ (n-l)!-

(n-1-2k).-

:(x+y

* n-2-2k )

k=0 n-1 2

(i) k kZ-(n-1-2k)!

(n-1)!-

a+

n-1-2k (x)

: x+y*)

k=0 n-1 2

+ (n-1)! i=l

)i_ 1 (i(-~),(n+1-2i

!(n+]-2i):(x+y

* n-2i )

47 n-1 2 = n!- ~ n~2k k!.(n-2k)! (½) k

~+(x)

:(x+y*)n-l-2k:

k=O n-1 2 + n! ~

i (½) i-I *)n-2i i!. (n-2i) ! : (x+y

i=I n-1 2 = n!- ~ n-2k (½) k + * n-1-2k n k!- (n-2k) ! (x) : ( x + y ) k=O n-1 2 + n! ~ 2-i (½) i * n i!. (n-2i) I : (x+y n-2i: i=O

We return to the first identity (x+y*) n = ( x + y * ) n - ~ + ( x )

+ (x+y*)n-ly *

n-| 2 = n'. ~ n~2k (~) k n-l-2k • k!-(n-2k)! ~+(x) :(x+y*) : k=O n-1 2 + n' ~ 2-i (½) i * n-2i )n-ly* • n i!.(n-2i)! :(x+y : + (x+y* i=0 and by using the induction h ~ o t h e s i s on the last t e ~ , we get n-1 : n!-

2 n~2k (1)k l ~ * n-l-2k . ~+,x, :(x+y ) k!-(n-2k)! L k=O n-1 2 2-i (½) i :(x+y*) n-2i + n!n i!. (n-2i)! i=O n-1 2 (½) k : (x+y*)n-1-2k) : + (n-1)'- ~ kl. (n-1-2k) I k=O

48 n-1 2 = n !- ~

n - 2 k (½)k ki.in_2k)! n

~+(x)

:(x+y*)n-1-2k:

k=0 n-1 2 2-i (½) l n i!-(n-2i)!

+ n!- ~ i=0 n-1 2

:(x+y*)n-2i

I< )k * n-1-2k * n - 2 k (z y , x > :(x+y ) : y n k:. (n-2k) !

+ n!k:0 Summing

the f i r s t

and

last t e r m s g i v e s

n-1 2

= n!- ~

n - 2 k (I~ k . [ (n+m-2k)! !]½ (n-k)! [(n-k)!-(m-k) -(nV--(-(-(-(-(-(-(~-k)!-V(m-k)!k=O Ixln-k. lal m-k m k .V(n+m_2k),.ixln-k. lalm-k (n-k)!

n=m k=0 m =

~ [ (~) j)n nl = ~ Ep "{~e+(exp

x) exp y*

= e-½.a+(exp

-

).We+f

f).exp-e*.exp-f*

exp-(e+f) ,

= e -½(<e ' f> - ).e-½1e+fl 2 -~+(exp(e+f)) (<e,f>

f) exp(-f*)

3.~2 we get

= e -½ (lel2+IfI2+2<e'f>)-a+(exp(e+f))

-~

we get

)

= e-½(lel2+Ifl2).e+(exp

= e

formula

exp-(e+f) *

= e-i'Im(<e'f>).We+f

52

Lemlaa

4.10:

For

every

e6K

we

have

on

FIH

, W

= W

e

W e W_e

Proof:

For

<Wef,g> The

other

is

unitary we

get

is

now

transformation

of

clear FIH

Weyl

Theorem

e)

of

the

F]H

FH

exp(-e*)f,g> of

that

onto

I

have

a consequence

transformation the

we

= < e - ~Il e l 2 - e + ( e x p

relation

It

f,g6FiH

=

-e

.

onto

lemma

=



4.9.

operator We

extend

FK

.

We We

As

a

is by

a

unitary

continuity

corollary

to

to

lemma

a

4.9

relations.

4.11 :

For

every

e,f6H

we

have

on

FH

the

Weyl

relations We-W f =

Example element

holds

e6H

on

the

4.12: .

Then

whole

e-i'Im(<e,f>).We+f

Consider it

FH

is

.

a

unitary

easy

to

show

FU-W e

"FU* =

operator that

W(Ue)

the

U identity

of

and

an

Chapter

5:

In the

Some

this

chapter

operator

6_

we

special

are

This

operators

primarily

operator

representation

introduced

in c h a p t e r s

A:

in Hilbert

spaces

Conjuqation

Definition

5.1A:

conjugate--linear

A

concerned

links

the

6 and

7.

conjugation

in

with

complex

a Hilbert

investigating and

real

space

wave

~

is

a

mapping : K

,H

fulfilling I)

is a n i n v o l u t i o n

2)

<x,y>

=

Just

after

for a l l

proving

procedure

can

be

we

the

conjugation

extend

abandon

in t h i s

extension holds

is

them

applied

case

.

proposition to

the

x,yeH

conjugate to

standard

multiplicative

on

2.1

we

mentioned

linear

isometries

a conjugation

on

notation

as t o o

the

F-

subalgebras

the

F0H

that

the

as w e l l . whole

same Hence

FH

.

We

complicated.

The

and

,

and

in

the

FIK

invariant.

An element

feFK

will

be c a l l e d

real

if

f = f

Theorem Hilbert

5.2A:

space

H

Consider Then

the

{ exp x is t o t a l

in

FH

Proof: space

K

,

,

i.e.

Choose

and

the

the

r>0

and

a

conjugation

set

I x6H

a real

consider

fixed

span

is r e a l of t h e

orthonormal set

and set

Ixl "..." ( < X , e n > ®

+ en)

r

rl = ~ ar. r

. -...

>r <x,e n

r1 n.~>

= ~ ar.<X,el > r

rn -....<X,en>

Letting x = s]e I + s2e 2 +...+ where

n6~

and

Sl,S2,...,Sn6~

are

2

sI + s we

get

by

setting

Sk=

0

for

such

+...+s~

Sne n that

< r

2

,

k>n r]

0 =

,

ar-S ]

rn "...'s n

r Since

(Sl,S2,...,Sn)

run

over

the

ball

a

= 0

for

all

with

radius

r

we

conclude

that

by which

B:

The

f = 0

functors

h*

and

,

6*

denote

a

Hilbert

--

conjugation

r

.

(H,)

Let

r

space 00

and

an

orthonormal

basis

{en}n= I

with

a

fixed

chosen

55 Definition

5.1B: We define the m a p p i n g h_ : FI~

,

V~

by the series co *

h* = ~ a(en)e(en)

=

n=l

The

lower

index

--w

~ en en n=l

-

points

out

that

the

definition

depends

on

the choice of the conjugation. We shall need the following properties

Theorem depend

5.2B:

The

operator

on the choice of the basis

I)

F0H

is an invariant

2)

h~

(a

3)

h

(amexp

TM)

well-defined

a m-2

is well H , h_

for

a,z6H

and

does

not

depend

h_

and

does

not

and ,

i.e.

aeg

and

h_(FOH ) c F0K

.

men 0 + a

]exp

z

m6~ 0 .

I) is evident and it follows and

defined

= [ m ( m - 1 ) < a , a > a m - 2 + 2 m < a , z > a m-1

with

Proof:

in

space under

= m(m-1) z)

h

of the o p e r a t o r

on

the

from 2) that the operaZor chosen

orthonormal

basis.

prove 2) notice that ~n*(a

TM) =

ma m-1

which implies W

--W

en en

(am )

= m ( m - 1 ) < e n , a > < e n , a > a m-2

and h~(a m) = (~ < ~ , e n > < e n , a > ) m ( m - 1 ) a m - 2

= m ( m _ 1 ) < ~ , a > a m-2

n=l

3)

is similar,

Theorem

5.3B:

thought a little more technical.

For all

x6H

[(~-h~)m,~+(x)]

and

m6~

the c o m m u t a t i o n

.lh*.m-1 = m(~ _)

~*

relation

is To

56 holds

on the w h o l e

Proof: f=ak.exp

z

Take

with

*-* enen(xakexp

FIN

.

m=1

a,z6K

is

It

and

k6~ 0 .

-

sufficient

consider

elements

We compute,

-

z) = < X , e n > < e n , a > . k - a k lexp z + < X , e n > < e n , Z > - a + <en,a><en,x>-k.ak-]exp -

k

exp

z

z *-*

+ <e n , z > < e n , x > - a k e x p which

to

k

z + e+(X)enen(a

exp z)

implies

*-* + enen~ ( x ) [ a k e x p

+ *-* k z] - ~ (X)enen[a exp z] =

= kak-lexp

z. [<X,en><en, a> + < a , e n > < e n , X > ]

+ akexp

z-[<X,en><en,Z>

+ <en,X>]

+

Since

a (x)

½(h:e+(x) which

is a c l o s e d

- e+(x)h:)-akexp

operator

z = k<x,a>ak-lexp

over

z + ak<x,z>exp

n6~

z ,

gives [½h:,a+(x)](akexp

It

we get by s u m m a t i o n

is

easy

to

check

that

for

z) = x * ( a k e x p

arbitrary

z)

operators

A,B

and

operators

x*

m6~

we

have m = ~ A m-k

jAm,B]

[A,B]

A k-1

k=1 Using commute

this

identity

we get on

and

the

fact

that

the

and

h*

define

the

FIN k

[(~lh*_).k ,a+(x)]

Definition

=

5.4B:

~ (½h:) k-j x* j=l

(cf.

[20]

_J' J - ] (2i h *

paragraph

transformation w

5_ : FOH-----o

FOH

by the s e r i e s

5_ = (-~h_)n/n! n=O

=

k(~lh*_) k - 1 ~ *

2A)

We

57 The

operator

6_

can

be g e n e r a l i z e d

[20]) by taking in place of a conjugation L : K which

is

conjugate

real-self--adjoint

linear.

has a dual o p e r a t o r

a+(6L ) ,

-

a continuous

6L

(cf.

mapping

; H

Then

Hilbert--Schmidt

to an o p e r a t o r

in

the

strict

case

when

contraction,

L

is

the operator

which is d e n s e l y defined.

a 8L

In physics

the

elements 6 L = a+(6L).~ when n o r m a l i z e d

provide

called u l t r a c o h e r e n t

Proposition and

m6~

the so-called

squeezed

states;

in [20] they are

vectors.

5.5B: The o p e r a t o r

is well d e f i n e d and for

6

a6K

we have [m/2]

I

8_(a m) = .

(_½)n

m! (m-2n)!

<ara> n m-2n n! a

n=0 and e s p e c i a l l y

~(~) 6_(x)

Proof:

=

= x

for all

xeK

.

It is enough to prove that [m/2] 6~(a m) =

~ n=0

(_½)n

We start by proving that for all m! (m-2n)!

(h~) n a m = {

m! (m-2n)!

<ara> n am-2n n!

m,n6~ n am-2n

for m>2n for m1

and assume

that

the result

We have

(hl) n a m = hi[

(h~) n-1

= hi[ {

am

]

m! (m-2n+2)! o

n-1

a m-2n+2

for m>2n+2 for mexp(x

we s h a l l c a l c u l a t e h~exp

z = exp

+ y)

6_(exp

z)

Since

z

we get (-½h~)nexp

z = (-½)nexp

z ,

thus g i v i n g

-½ 6*(exp

z)=

~ n=0

Hence we get

FIK

f,geFiK

fog :

Remark

the

(-½)nexp

z/n!

= e

exp z .

a n d the

64

6 ~ ( e x p x ~ e x p y) = (6[exp x ) ( 6 1 e x p

y)

-½ -½ = e

e

exp

(x+y)

-½( + ) = e

exp(x+y) -I < x , x > + < y , y > )

+½<x+y,x+y>

= e which

, 6_(exp(x+y))

.e

implies

+½( (exp x)~ e x p y)

+ )

= e

exp(x+y)

= e<X,Y>exp(x+y)

.

Slnce e<X'Y>exp(x+y)

= (exp x) e < X ' Y > (

e x p y)

= (exp x) e x p x * ( e x p y) =

the o p e r a t o r

:exp(x+x

): e x p y

,

identity (exp x)~ = : e x p ( x + x

holds

on the c o h e r e n t This

identity

Let us d e f i n e

vectors

in

FH

f,geFoK

a new inner product

with

a~(x) e(x)

respect

with .

o for all

):

we

denote

+ a o ( X ) = x~

by

to the B - m u l t i p l i c a t i o n .

respect

to

=

nk

•

....

,

and

•

e~[z]

e~[z]

exp(-Izl2)dz

=

~k nI =



nk ..

m -

mk - e x p ( - I z l 2 )dz

I ..

k k ~ = ]~-

i=l Without

_Zn i "Z m i - e x p ( - { z i { 2 ) d x i d Y i

C

loss

;

of g e n e r a l i t y

we

assume

~n-zm'exp(-Izl2)-dxdy

that

= 2~.



= = z> = ~ ar" <e--,exp r

r rI =

~r.<el,z >

r

r2 .<e2,z >

r "''''<en'Z>

n

r rl =

ar'Zl

r2

rn

"z2

" " "'Zn

'

r where

z i = <ei,z>

L2--space function

L2(C) on

,

the

the

on

Notice

i6~

function

elements

representation

functions

for

Identifying f[- ]

can be

K

as

considered

the

weiqhted

as an e n t i r e

~ .

Therefore wave

6C

are

of

the

often

image

of

FH

under

the

conjugate

continuous

functions

called

the

complex

holomorphic

H that

we

extend

some

on

H

to

71

continuous

functions

on

the measure-theoretical on a measure

zero

Example pointed

out

traditional Considering Fourier

set

6.6:

enlargements point

of v i e w

This

is

a

Fourier

transformation

transform

of

f

f

6

in

that

amounts space

to Cn

in the

the

complex

FC n z6C n

composition

wave of

set

L 2 ( R n)

= f[i~]

awkward

functions

find

space then

defined

2.11.

coincides the

from

2).

example

and

It

was

with

the

L2(R n )

evaluating

the

that

,

representation with

is

appendix

of

in

we easily

f[- ]

It

F(--i) ~

=

•

(cf.

operator

(~f)[~] i.e.

K

continuation

that

function

the

of

since we extend

to a f u l l m e a s u r e

earlier

a

~

Fourier

a 90 d e g r e e s

transformations rotation

of

the

Chapter

7:

The real wave representation

This by S e g a l

chapter

f

the

representation

[15]. We a p p l y the t e c h n i q u e

Definition of

concerns

in

xeK

7.1:

For

every

developed

feFiK

we

7.2:

the

V-value

f_(x)

For

f,g6FiH

of lemma

= <exp x,6_f>

we h a v e t h a t

= f_(x)-g_(x)

From chapter

hence by virtue

*

= (6_f)[x]

(f~g)_(x)

f*g_(x)

[20].

define

*

Proof:

in

originally

setting f_(x)

Lemma

introduced

for all

xeH

.

5 we h a v e t h a t

6.2 w e get

= (61(f,g))[x ] =((61f)(61g))[x ] = (61f)[x](61g)[x ] : f_(x)-g_(x)

For f_(')

arbitrary

in d i r e c t i o n

aeH a

and

7.3:

= ~-~ d f_(t-a

For f i x e d

feFIH

d d-~ f _ ( t - a

Proof: sufficient

Since

to p r o v e

the

define

the

derivative

of

and

+ - )it= 0

a,b6K

*

+ b)

(a f ) _ ( t . a

operators

a

we h a v e + b)

and

6

commute,

that

d <exp(t.a dt This

we

setting

Oaf_(')

Lemma

f6FiH

can be r e d u c e d exp(h-a)-~ h

+ b),f>

= <exp(t-a

to p r o v i n g a

+ b),a

f>

that in

FH

for

h--~0

it

is

73 since the operator

is closed.

a+(exp x)

We compute co

exp(h-a)-e h

--

a

=

exp(h-a)

h- -~ a- h

= hl [ 2 (h'a)nn! - ~ - h-a ] n=0

co

_ I 2

co

(h'a)k+2 (k+2) .'

h-a) n 1 ~ ( n' -h

k=0 n=2 a k+2 h k = ~ (k+2) !.h . k=O We have for Ihl -< 1 co

co

I

~ I

exp(h-a)-e h

ak+2 hk (k+2)! h I

I

- a

k=O co


f_(x) - (b f)_(x) Since b= Re(b) + i-Im(b) , where

Re(b) - b +2 ~

we can assume that

and b6K

Im(b) = b2.i-~ ' is real and that

without loss of generality llbll=1

We have {bf,g) = (2~)-½k~ (bf)_(x).g_(x).exp(_~Ixl2).dx ~k = (2~)-½k[ [<x,b>f_(x) - (b*f)_(x)]g_(x)-exp(-½1xI2)-dx = (2~)-½k[ f_(x)g_(x).exp(-½1xl2).dx

&k

-- (2~)-½k!k(b*f)_(x)g_(x).exp(-½1xl2).dx = (2~)-~k~ f_(x)g_(x)-exp(-½1xl2)-dx ~k Choosing the first basis vector

eI = b

- (b*f,g)

.

we get

(f,b*g) = (2~)-½kF f_(x)(b*g)_(x).exp(-½1xl2).dx = (2~)-½k!kf_(x). [d~ g_(t'b + x)lt=o-eXp(-½ Ix,2)'dx = (2~)-½k~ ~k

f---~-~'O~ g--(x)'exp(-½1xl2)dx1"''dXk I

Since f_(x-----7.~ig_(x).exp(_½,x,2)dXl

=.

~ ~a-Xl 0 [f~-(x)"exp(-~ Ix I2 )]g-(x)dXl

and and -

a ~-~ig_(x)'exp(

=_

-½

Ixl2)dXl

=

~ ~-~1[f_(x)'exp( a _½ Ixl 2)]g_(x)dx I

76

--T

[a

f_(x).exp(_½1xi2),

xlf_(x).exp(_½1xl2)]g_(x)dxl

= ~ xlf-(x)'g-(x)'exp(-½1xl2)dXl

returning (f,b , g) =

T ~--~1f_(x), g_(x) •exp(-½ Ix 12)dx]

to the integration

I ~ 2~) -~k

over

~k

,

we get

a _½ ix I2)d x f_(x).~-~ig_(x).exp( ~k

=

2v)-½k~

xlf_(x) • g_(x) • exp(-½ Ix 12)dx ~k

_ (2v)-½k~ =

a f_(x), g_(x), exp(_½ 1xl 2)dx ~k ~--~I

2v)-½k[~k <e I 'x>f-(x) "g-(x) "exp(-½ Ix 12)dx

- (2v)-½k!k(eTf)_(x).g_(x)-exp(-½1xl2)dx =

2=)-½k[

f_(x) • g_(x) • exp(-½ 1xl 2)dx

-- (2~)-½k[ =

(b*f)_(x)-g_(x).exp(-½1xl2)dx

2v)-½k[ f_(x)-g_(x).exp(-½1xl2)dx

- (b*f,g)

= (bf,g)

We

introduce

Hilbert-Schmidt

the

enlargements

is done in appendix

functionals

continuous If

~H

=

~H

of the real Hilbert

defined on

as

in chapter

on finite

'

sitting

space

H ,

on

as it

,

functionals

6 we

dimensional

extend

continuous

subspaces

of

linear H

,

to

H .

is a finite dimensional

the conjugation to extend

way

functionals K

H

measure

2.

In a similar real

gaussian

and if

K

of the form

subspace

denotes

of

H ,

its real part

invariant ,

under

then we have

77

-~(dz)

.

K

The theorem

subscript

N

in

~'}N

refers

to the

normal

ordering

(see

8.8). Take

P = af*

and

Q = bg*

Since

f exp

z = f[z]exp

z ,

we

get <exp(-z).P.exp(z),exp(-z).Q-exp(z)>

Lemma

8.2: F o r

a,b,f,g6FoK (af

Proof:

It

representation

is

,bg }N = < a , b > < f , g >

an

easy

consequence

of

the

complex

wave

we o b s e r v e

that

~'}N

and

~,~

are e q u a l

on the

P e

e(FoX)

H+K

Lemma within

that

a n d the above.

In p a r t i c u l a r , subspace

we h a v e

= f[z]g[z]

8.3:

FOK,

Let

P

denote

the

conjugation

in

e

(S

Proof:

of

) P

6 O(FOH )

O(FoH),(,} N ,

i.e.

*

,T~ N = ~T

A complex

conjugate--linear

operator

The operation O(F0~ ) 9 P

is a c o m p l e x

dual

for all

conjugation

involution <x,y>

This we check

,S} N

S,T 6 O ( F o H )

,

as d e f i n e d

in c h a p t e r

fulfilling =

for all

on the g e n e r a t o r s

x,y

af*,bg*

. 60(FoH

)

5, is a

81

/(af

) , b g )N = ~ifa , b g ~'N = < f , b > < a , g > =

Definition the

creation

8.4:

For

= (af

,gb ~ N = ( ( b g

every

pair

x,y6K

) ,af ~ N

"

we define

operator +

*

a (x+y)

: O(FoK )

' O(rOK)

by e+(x+y*)P and

the

annihilation

=

:(x+y*)P:

for P e O ( £ 0 K )

operator

~(x+y ) : O(FoK ) as the dual of

Notice (x+y) e+(x+y*) c(x)

is

and

,

and

+

a

e(x+y*)

8.5:

(x+y)

the

case

generators

af

y

=

0

multiplication ,

the

operators

by

operators ~+(x)

and

an e x t e n s i o n .

£ O(FoH )

we h a v e

that

(x-a)f

elements

af

60(FoH

*

+ a(y-f)

I = 0 (x * a)f * + a ( y * f) *

We o n l y lemma

verify

2).

For

the

= ~(xa)f*

+ a(yf)

)

we

+

, b g )N =

it

of W i c k

known

is m e r e l y

*

=

(a+(x+y*)af*,bg*~N

Then

of

to the w e l l

the notation

*

(x+y)af

by using

in

reduce

On the

~ ( x + y * )af * =

get

that

operator

I = x+y

*

Proof:

is the

*

~(x+y*)

2)

obvious

*

+

e

c+(x+y*)

therefore

Lemma

I)

a+(x+y *)

that

It

~ 0(FoH)

that

~(x+y

)bg

b>

+

= ~af * ,(x * b ) g * + b ( y * g) * ) N

= (x b ) g

+ b ( y g)

"

82

Care expression the

should for

be

taken

a(x+y

same as

f*a+(y)



=

W

= af

+ af

It is o b v i o u s

operator

in

the

I .

*

= (x+y

*

that every operator

operators

*

,u+v )N-af

~+(x+y*)

f r o m the a l g e b r a ,

H e n c e we h a v e the f o l l o w i n g

Theorem inner product

- (vy f)

+ a[(y*v)f]*

*

polynomial

(y*(vf))

8.8:

O(FOH )

equipped

~')N

is a B o s e a l g e b r a

x,y6K

the

annihilation

We

creation

operators

denote

O(FoK)'~'~N

operators ~(x+y

by

the

identity

with

the W i c k m u l t i p l i c a t i o n

with

the v a c u u m +

;

and

are

is a

theorem.

*

~+H

,

D(VOH )

~

I

and

a n d the b a s e

*

(x+y)

with

the

dual

)

O ( F o K ) '{'}N

the

completion

of

the

space

-

Consider

for

x+y

6 H+H

the c o h e r e n t

:exp(x+y in the H i l b e r t

space

By v i r t u e

):

~(FOK )

of p r o p o s i t i o n

:exp(x+y

vectors

*

~

): =

4.7 the i d e n t i t y

:(x+y

* n * ) :/n! = e + ( e x p x ) e x p y

n=0 holds

on e v e r y e l e m e n t :exp(x+y We

i.e.

the

denote

by

from

FIK

,

and h e n c e we get

): = e + ( e x p x ) e x p y O(FIH )

the

extended

s p a n of e l e m e n t s

:P-exp(x+y ) :

in

O(FOH )

Bose

algebra

of

O(ro~)

,

84 for

P 6 0(FoH ) , x+y

6 H+K

Then we have

O(FI~ ) = span } ~+(a-exp x)(b-exp y)* = span

Our goal called

a+(f)g *

f,g6FiH } .

is to construct

a Bose algebra

and an inner product

§ ,

new algebra is the algebra Let

{an}n6 ~

a,b6FoH , x,y6~ }

~,>

denote an orthonormal

basis in

Let

the

O(FIH )

which

are

--

symbol

§

is an orthonormal real

with

denote

the

respect

to

the

in

/5_

inverse

of

conjugation

relative

to

the

algebra

By using the real basis written above

* ~( En)2 + ~(Fn)2 = 2.~(en)~(en)

we get

co

§ = exp(-½ ~ [~(En)2+~(Fn)2]) n=l the

consisting

complex

00

and

H+H*

W

an)

basis

operator

and the conjugation

and the fact that

We define

2 -~- (e n + en) .

elements

H

*

F n = 2 -~ (-i)" (e n I nEN}

of this

O(EOH),:-:,{,) N

En

{En,F n

a multiplication,

such that the ,-picture

=

Here

with

operator

= exp(- ~ ~(en)~(en) ) , n=1

§-1

§-i = exp(+ ~ ~(en)e(en) ) n=l

Then we define the new multiplication p§Q = §[:(§-Ip)(§-~Q):] according

to theorem

inner product

Since product H+H

(,}

(,}

§ for

5.7C and definition of

P,Q e O(F]K)

in

O(FI}{ )

setting

p,Q e O(FI~ ) 5.8C.

Moreover

we define

the

setting

(p,Q} = ( §-Ip,§-IQ }N " §-i is the identity operator on K+K* , , restricted to H+H and the original inner

the

inner

product

on

are identical. For

product

(,)

the algebra

O(FOH )

as an algebra

we use the notation O(FIK),§,~ ,}

with

multiplication

V0(H+~* )

and write

§

and

FI (~+H*)

inner for

85

We F0(~+H

shall

),(,)

write

, the

)

for

the

base

8.9: . H+H

F0(H+H ,

the

),§,(,}

creation

is a Bose

their

dual

annihilation

the

space

Proof:

We m u s t

algebra

with

the v a c u u m

- ~(y+x*)

operators

M(x+y

4(x+y

of

operators

M + ( x + y *) = e+(x+y*) and

completion

.

Theorem I

F(H+H

) = a(x+y

)

s h o w the following:

) I = 0 *

[4(x+y

+

*

),4

4 + ( x + y *)

*

(u+v)]

= (x+y

generates

*

,u+v ) I

from the v a c u u m

I

the w h o l e

rO(~{+){ )

O(~o~{ )

=

and f i n a l l y S+(x+y*)

and

From

4(x+y

lemma

)

are dual

8.5 we have M(x+y

and

from

the

*

[4(x+y

conlmutation

+

operators

on

F0(~{+K

) I = ~(x+y

relation

in

) I = 0 ,

theorem

8.7

we h a v e

.

(U+V)]

= [ ~ ( x + y * ) , ~ + ( x + y *) - ~ ( y + x = [~(x+y*),a+(x+y*)] = (x+y

Since

•

that

*

),4

),(,)

4+(x)a

,u+v

= xa

)]

= (x+y~,u+V*)N.I

)I

and

N+(y

)f

=

(yf)

for all

x,y6H +

a,feFoH

,

it

the i d e n t i t y

is

easily

seen

operator

I

that

the

generate

creation

the w h o l e +

It operators

remains on

to

F0(K+K

w

prove ),(,~

6*e+(X) which

in

the

algebra

that

F1 ( H + H )

algebra

the

fact

that

the

= O(FIK )

and

and

4(x+y

commutation

- e(x))6

§

F0(H+K

for corresponds

4(x+y

and

)

*

§ M + ( x + y *) = (4+(x+y *) - 4 ( y + x * ) ) § By u s i n g

N (x+y)

*

~ (x+y)

We n e e d

= (~+(X)

operators

and

*

)

from

x6H

)

are

dual

theorem

5.6B

, to

. commute

and

applying

86

the o p e r a t o r

a+(x+y *

we get + . (x+y)§

= §~+(x+y*)

,

and h e n c e § - I M + ( x + y * ) = ~ + ( x + y * ) § -I For

P,Q 6 F0(H+H

)

and

x+y

£ H+K

-

(M+(x+Y*)P,Q)

= (§

I M + ( x + y * ) P , § - X Q ) N = (a

= ~§ =

Definition of a f i n i t e

-1

P,e(x+y

*

)

5-1

Q)N

~

~§

*

(x+y)§-IP,§-~Q) -

8.]0:

We

denote

~(H+H

by

N

IP'M(x+Y*)§-~Q)N

{§-~P,§-~M(x+y*)Q) N = (P,M(x+y*)Q)

number

)

.

the

set

of

polynomials

of v a r i a b l e s

T h e n we d e f i n e

we h a v e

+

and

where



a,b6H

.

the t r a n s f o r m a t i o n : F0(K+K

, 9(K+K

) = 0(FOK )

)

by (~P)[z] where

z,(exp-z)P(exp

for all

z)>

z6H

,

p 6 F0(H+H* )

It t u r n s function,

cf. Using

out

If H

extensions

~P[. J

the g e n e r a t o r s

we to

that

correspond

to t h e W i g n e r

distribution

[21,25].

(~P)[z]

from

= <exp

= <exp

extend

the

P = af

z,(exp-z)af functions

Hilbert-Schmidt

by

we h a v e t h a t

~(H+H*) ,

(exp z)> = a [ z ] - f [ z ]

~P[-]

6 ~(K+~*)

enlargements

we g e t

H

and

in denote

the the

usual space

that 1

! where

~

is the g a u s s i a n

Theorem

8.11:

measure,

F o r all

defined

on

P,Q 6 F0(H+H* )

l i p , Q~ =

7rP[ z ] .TrQ[ z ] . ~ ( d z ) K

~

.

we h a v e t h e i d e n t i t y

way of

87

where then

for

simplicity

of

automatically

understood

to a H i l b e r t - S c h m i d t

Proof:

notation

we

have

that

enlargement

the

H

omitted

the

functions

usual

are

It is

extended

from

H

.

We define =

(P,Q)'

~P[z].~Q[zl-~(dz) K

Thus

for

af

,bg

e F0(K+H

v(af

)[z]-v(bg*)[z]

)

w e get

that

= a[z]f[z]-b[z]g[z] = a[z]g[z]-b[z]f[z]

and

then

by v i r t u e

of

the

complex

(af Using ~x+y

the

elements

* , U + V *~'

=

from

+ <xv,®>

<X,U>

= <x,u>

It operators

remains with

generators

af

*

*

(M+(x+y*)af

+

to

,bg

*

+

,bg )'

= (e

H+H

that

6 F0(H+K

*

)

= <xag,bf>

= (af

,u+v }

M+(x+y*)

and

We

prove

*

For

x+y 6K+H *

- ~(y+x

+

b)f>

,(x b)g

get

+

*

(x+y)af

we

,

we get

(,}' * *

=

to

(ag[z]).(bf)[z]

=

= (x+y

show

respect

wave

,bg }'

the b a s e

=

+

+ (af

,b(y g)

= (af * , M ( x + y * )bg*}

While

proving

Corollary

theorem

8.12:

On

8.11

the

the

following

generators

af

that (af

,bg } = < a g , b f >

corollary

,bg

was

e F0(H+~

shown:

)

we

have

88 Lemma

8.13: The operation F0(H+H

is a complex

conjugation (S

,T>

) 9 P in

,P

£ F0(H+H

(F0(H+K),(,))

= (T*,S)

for all

,

)

i.e.

S,T 6 F0(~+H

)

Proof: As usual we check the identity on the generators af

,bg

6 F0(H+H

)

We have ((af

)

,bg

)

= l[fa

,bg

= ~af

,(gb

Notice

> = ) )

=

= ((bg)

that the same result

,af

>

concerning

the algebra

0(FOK),(,) N

has been proved in lemma 8.3. w

From the d e f i n i t i o n

Theorem complex

same,

8.14:

conjugation

In

of

F0(H+H

the

Bose

the

)

we get the following

algebra

~--product

and

F0(H+H the

),§,(,)

Wick

product

results.

with

the

are

the

thus P~Q = :PQ:

Theorem

for all

8.15: The Bose algebra

the Bose algebra

F0(H+H*),(,)

p,Q 6 F0(H+~

8(FoH),(,> N

)

is the ~--picture of

Chapter

9:

In

Wave

this

F(H+K

)

,

lemma

8.] 3.

representations

chapter

equipped

In c h a p t e r

we

with

shall the

:exp(x+y for the c o h e r e n t

vectors

Let us d e n o t e

in

*

study

complex

8 we d e r i v e d

of

F (H+K)

some

wave

representations

conjugation

,

discussed

of in

the e x p r e s s i o n

+

): =

(exp x ) e x p

y*

[(FOH )

by co

exp§(x+y*)

=

~ (x+y*)§n/n! n=0

the c o h e r e n t

vectors

of

(x+y*)§n

F(~+H*)

For

P 6 F0(~+~* ) = O(F0~ )

= §(:(§-~(x+y*))n:)

we have

= §(:(x+y*)n:)

and thus c0

(exp§(x+y),p)

=

~ ((x+y*)§n/n!,p~ n=O

=

~ ~§ ( : ( x + y * ) n : / n ! )

, p~

n=O =

~{: (x+y

*)n:

/n!,§- ~P}N

n=0 = {:exp(x+y

):'§-~P)N

= (§:exp(x+y):,P}

Hence exp§(x+y In c h a p t e r

) = §:exp(x+y

8 we c o m p u t e d

the r e d u c e d

§ = exp(-

): expression

for the o p e r a t o r

e(en)e(en) ) ,

n=l where

{en}ne ~

is an o r t h o n o r m a l

basis

in

K .

T h e n we get

90 co

co

e(en)e(en):exp(x+y

): =

n=l

e(en)~{en,X+y

)N:exp(x+y

):

n=l co

=

~ (en, x+Y*)N" (en, x + Y * ) N :exp (x+Y*) : n=]

=

~ <en, x><en,Y> :exp (x+Y*) : n=l

=

~ <en,X>:exp(x+Y

*) :

n=l

= :exp(x+y

):

and hence *

exp§(x+y P 6 F(H+H

For

)

) = e - < y , x > . .exp(x+y

the value

P[x+y and

hence

the

complex

of

P

in

representation

):

x+y*

] = (expg(x+y),P~

wave

*

6 H+K

is

, with

p,Q

6

F(H+H

gives

)

the i d e n t i t y (P,Q} where

~

examine of

transformation

~

.

1

of

)

the H+H*

the

P 6

omit

the

FO(•+?{

functions

equipped

as the d o m a i n

with

,

are

correspondence

extended

of integration.

between

by

real ,

wave

and the

the

real

part

of

with

H+H

respect

to

the

, i.e.

)

notation

) = §P[z+z

the

the c o n j u g a t i o n

we

compute

P.(z+z

)

I zeK }

the from

qJ-value chapter

of 7 and

P

z+z

in

use

P(z+z

We have P(z+z

to

.

~ = { z+z* For

1

] - Q [ x + y * ] ' ~ ( d x ) ' ~ K ( d~y )

that

closer

F(H+H

denote

conjugation

P[x+y

enlargement

representation

We

I

indicates

Hilbert--Schmidt

We will

=

] = ~.expg(z+z

= (:exp(z+z

):'P)N

),§P)

= ~P[z]

= < e x p

= ~(af

) ,

z,f>

)[z]

representation

of

F(H+H

)

yields

for

P,QeF(K+K*) (P,Q) where

the

functions

=

P(z+z

are e x t e n d e d

)-Q(z+z

)-~(z+z

) ,

to a H i l b e r t - S c h m i d t

enlargement

of

over which we integrate. Let with

H~

denote

the s p a c e

H

considered

as a real H i l b e r t

space

inner product ~

T h e n the u n i t a r y

=

Re()

=

½(

+

)

mapping ]

*

HA 9 z transforms

the

measure

Hilbert--Schmidt (P,Q)

~

enlargements

=

P(z+z

of

F(~+H

,

)

9.1:

We denote

The Weyl

the

H ,

).~N(z+z

fundamental

measure

i.e.

for

) =

get

for

the

operator

we h a v e

~P[z].~Q[z].~(dz)

~

extends

to a u n i t a r y

mapping

b y the s a m e symbol.

We ,

e6K

,

on

FIH

is of the f o r m ,

(exp e) e x p ( - e

Campbell-Baker-Hausdorff )

)

on

(~)

the e x t e n s i o n

p 6 FI(K+H

sitting

I

W e = e-½1]e][ by

,

P,Q 6 F 0 ( H + H

2. + Thus

~

theorem.

The t r a n s f o r m a t i o n L2

onto

of

).Q(z+z

and we get the f o l l o w i n g

Theorem

into

and

z6H

formula

and

) some

calculations

we

92 ~P[z]

= <exp

z,(exp-z)P(exp

= = < e , e x p

z ((exp-z)P(exp

P exp

z>

vP[z]

= < e , W _ z P W zO>

z))>

= Thus

by which

~P

operator

W

in

--Z

On t h e *

since

is the s o - c a l l e d

vacuum

expectation

v a l u e of the

Z

generators

af

*

~(af and

P W

zeH

,

6 FO(?~+H

)

we have

*

"k

) [z] = ~ ( f a the

)[z]

elements

af

~P For

P,Q

7rP[z]-TrQ[z]

= f[z]-a[z] are

= ~P

6 I" 1 ( K + K )

= P(z+z

and

)-Q(z+z

:PQ:(z+z

in

for all z6H

F(?{+H

)

P 6 F(K+K

)

l-§Q[z+z

] = §(:PQ:)[z+z

) = ~:PQ:[z]

= ~(af ,

)[z]

,

we get

we have

) = §p[z+z

= (§P)§(§Q)[z+z =

total

= a[z]-f[z]

]

]

,

and h e n c e ~(:PQ:) We

finish

introduced

the

x+y

Using

chapter

in c h a p t e r exp§(x+y

for

= (~P)'(~Q)

for all

returning

8. C o n s i d e r

*

P,Q e F I ( K + K to

the

operators

§

and

§-~

the i d e n t i t y

) = §:exp(x+y

*

): = e

-

:exp(x+y

*

):

6 K+K

the

fact

that

:exp(x+y*):

Campbell-Baker--Hausdorff

=

a+(exp

§-le+(exp

x) e x p y

Approximating

f,g£FiK

anti-normal

to

the

operator, §-1(fg*)

exp

*

= e 0 n

llEnflll = I ( E n f ) ( ~ ) ' ~ n ( d ~ ) n An a r b i t r a r y

f

can be w r i t t e n

f+

and

f-

= J f(s)-~(ds) ~

uniquely

are n o n - n e g a t i v e

assume

= llfllI

as ,

integrable

f+-f-

i.e.

and

f = f+ _ fwhere

function,

functions

such

that

= 0

Ifl = f+ + f-

.

Then we have llEnfUl = IIEn(f + - f-)lll ~ llEn(f+)lll + llEn(f-)lll = llf+lll + llf-llI

= IIf÷÷ f-lll = llifllll = ilfltl

Definition there

exists

an

4A: A n6~

function

f

defined

such that f = fOPn

,

where

~n Pn

: ~

on

~

is c a l l e d

tame

if

101 denotes

the

natural

Notice the

first

projection.

that

n

if

f6L1 (~)

variables,

is

a tame

function,

depending

only

on

sets

in

then E f = f n

Lemma

5A:

Proof: ,

Since

every

of

of

tame

the

Borel

combinations functions

The

cylinder

function

6A:

sets

are d e n s e

sets can

indicator

cylinder

Theorem

functions

form

be

(Jessen)

L I (~)

a base

for

the

approximated

functions are

in

of

by

cylinder

Borel finite

sets.

linear

Indicator

tame.

For

f6L I (~)

Enf

~ f

we have

in

that

L 1 (~)

n~

Proof:

Consider

6>0

Find

a tame

I] f - g l l l As

g

first

is tame, N

there

variables.

exists For

an

n_>N

function

g6L1(7)

such

that

< 6/2 N6~

such

we h a v e

that

g

depends

only

on the

that

Eng = g

and HEnf

- fl] l
S2m + 2Sm(Sn-Sm) we can continue n _> ~ ~ [s2 + 2Sm(Sn-Sm)]~(dx) m= I Qm as the inverse images by the We construct the sets Qm ' m6~ I

function

104

f(x) = the smallest

k6~

= inf { k6~

such that

ISk(X)[

]Sk(X) l > £

> 6 }

Hence Qm : f-] (m Notice coordinates

that

the

= { x6~

function

I f(x) = m } .

IQm

only

function

only

on

the

Since the functions

sm

first

m

Xl,X2,..,x m .

We return to the integration. depend

depends

on

the

(Sn-Sm)

first depends

m

variables

Xl,X2,...,x m

only on the variables

and

,

and

IQ~ the

Xm+1,Xm+2,...,x n

,

we get by the Fubini theorem

f

Sm(Sn-Sm)~(dx)

= ~ (IQm'Sm)(Sn-Sm)~(d~) N~

Qm

= ~ IQm'Sm~(d~) Sn-S m

being linear in the variables We estimate

then

f(x) = m ,

= 0 ,

Xm+],Xm+2,..,x n

the second expression using the fact that if and hence s~(dx)

n

~ (Sn-Sm)7(d~)

Sm(X) 2 ~

> 6 . 62~(dx)

xeQ m ,

Then we get = 62~(Q m)

~Qm Qm we return to the first estimate n n n

~ a~ ! ~ k=1 m=1

s~(dx)

!

m

~ 62~(Qm)= m=1

62~(m~i

Qm)

= 62~( ~ x_6~~

there exist

m = x n <X

on

and

is t r u e

en = where

n=!

Let

..

us d e f i n e

,0,I,0,

the

n-th

for

-x =

a

linear

..

) ,

position

,

and

( x 1 ' x 2 ' ..,x n '" .)6~

measurable

functional

k

defined

have

I k ( e n ) I 2 < co

n=1

i 2)

k ( x_ )

co k ( e n ) < X_, e n >

=

a.e.

in

N

n=1 Moreover,

the

Before the

representation

proving

translation

the

is u n i q u e .

2

theorem

we

need

some

preparations.

We

define

operator T

Z

for y6R ~

by setting (T f ) ( ~ ) where

f

denotes

a

function

defined

: f(~-Z) on

N

, Moreover,

we

introduce

a

107

subspace ~0

of

=

~

,

x6~

there

Theorem following

5B:

exists

an

Consider

a

statements

are

~(X

Proof: where

~

is

It

is

the

such

measurable

that

xn

set

B

=

0

for

C

}

n>N

~

Then

the

equivalent,

(I) (2)

N6N

~(B)

> 0

.

+ B)

> 0

for

all

to

see

that

easy

gaussian

measure

Z£~ 0

the

~n

theorem

in

the

holds

finite

in

the

case

dimensional

space

~n Let are

us

cylinder

first sets.

prove

the

Consider

restricted

the

form

of

theorem

5B,

where

B

set co-- n

C = B ~ ~ for

an

ne~

arbitrary consider m result

and y6~ 0

cases.

< n

:

> n

Borel

there

two

Then

follows m

a

by :

the

We

B •

an

regard

y

finite

~m-n

is

prove

the

=

a

B C

exists

rewrite C

where

we

set

B

Assume

m6~

as

such

being

dimensional the

set

~-n

~

~n

Borel

C

in

that

in

~(C)

>

y6~ m

the

space

~n

0

.

We

have

,

and

For to

the

case. as

= B ~ ~m-n

set

that

8) E ~ - m

Rm

Then

we

, refer

to

the

case

m

~n To integrable an

m6~

function such

that

full f

version

: ~

y6~ m

~

.

If

of

the

theorem,

Consider n > m

we

have

TMEn(f ) = EnTz(f ) , where

(Enf)(zl,z

2 .... z n)

= ~

f(~+~)~-n ~-n

and

z = (zl,z2,..,Zn) By

the

Fubini

theorem

the

function

(d~)

[6~ 0

take ,

a

then

non-negative there

exists

108 E

is i n t e g r a b l e

over

~n

with

n

f

: ~n

respect

(Enf)(Z)~n(dZ)

, C

to = ~

~n Notice

that

~n

and

f(x)~(dx) ~

f 2 0

implies

that

Enf 2 0

Assume that

f(£)~(dx)

> 0

R

Hence (Enf)(z)~(dz)

> 0

for all

n6~

.

~n U s i n g the t h e o r e m

for the

finite dimensional

case and the r e l a t i o n

TzEn(f ) = EnTz(f ) , we get t h a t

~mn

(EnTxf)(z)~(dz)

for all

> 0

n6~

.

Hence 0 < ~n(EnTxf)(£)~(d~) Setting

f = IB ,

Corollary

the t h e o r e m

6B:

= ~mTZf(~)~(d~)

follows.

For a r b i t r a r y

linear

measurable

X

we

measure.

We

functional

have that

~0 c ~(x) where

~(k)

denotes

Proof: will prove

~(k)

the d o m a i n

of

contains

linear

a

,

k

subset

E

of

full

that ~0 c E c ~(k)

Assume define

that

there

exists

an

~0 £ ~0

\ E .

the sets E t = tx0 + E = {tx0+ x

] ~6E}

For positive

t

we

109 Since

E

is a linear

different

indices.

set,

the

sets

Et

measures.

is

a

family

of

This contradicts

Proposition N

. T h e n we h a v e

are

pairwise

disjoint

for

Using theorem 5B, we get ~(Et)

Hence

Et

> 0 .

pairwise

disjoint

the fact that

7B: Denote by

h

~(~)

sets

with

positive

=

a linear measurable

functional

on

that

tk(en)l 2


0

we get n

exp(i-u-h(~))~(d~)

= ~ exp[i.u

R

k=l

~

= k~=nl ~ e x p [ i . u . h ( e k ) t -

n

it2 ]

dt.

~

exp(i.U.kn(X,)v(dx )

;

= -~ exp(-~u2[X(ek)l k=] Elementary

~ h(ek)xk]exp(i-U-kn(~))~(d~)

2)

computation

exp(i-U'Xn(~))~(d~) ascertains

that

110

# exp[i.u.h(ek)t

exp(-½u21A(ek)l 2

- ½t 2] dt

2~T Using

this we get that n

0

By the d o m i n a t e d

lim ~ exp(i.lk(x))~(dx) iR~ n--~

= ~

convergence

lim iit ~

= ~

theorem

we get

exp(i.~A(x))~(dx)

n------~

1.~(dx)

= 1 ,

0o

IR

which

is a c o n t r a d i c t i o n .

Proposition

8B: C o n s i d e r k(en)

a linear m e a s u r a b l e

= 0

for all

n6~

functional

h

•

If

,

then h = 0

Proof:

k(en)

= 0

for all

a.e.

n6~

implies

that

u

Let

Since

E

denote

X

the d o m a i n

is linear,

is s y m m e t r i c ,

of d e f i n i t i o n

and d e f i n e

I

h(x)

~ 0 }

E- = { x6E

1

h(x)

~ 0 } .

we have

that

E + = -E-

, and

since

the m e a s u r e

we get

it is o b v i o u s

=

x

w(E-)

.

that

~(E +) + ~(E-) Since

k ,

E + = { x6E

w(E +)

Then

for

+ E+ = E+

for

> ~(E)

every

= I

x6~ 0

and

likewise

for

E-

,

111

both

I

and

I

E+ of

variables.

either

0

are

constant

From

or

the

1 ,

Kolmogorov

and

likewise ~ ( E +)

and

with

respect

to

every

finite

number

E-

zero-one with

law

E-

= ~(E-)

we

get

that

w(E + )

is

Then

= I ,

hence ~({ x e E

We

are

Proof remains

to

now

k(x)

ready

(Theorem

prove

2),

= 0 })=

to p r o v e

4B):

~(E+

theorem

Since

I)

D E-)

= I

4B.

amounts

to

proposition

7B,

it

i.e. o~

k(x)

=

~ h(en)<X,en >

a.e.

n=l

Since

by p r o p o s i t i o n

7B

IX(en)I

2

< ~

,

n=]

theorem

3B a s c e r t a i n s

that

k(en)<X,en > n=l

converges We m u s t

a.e.

prove

in

N~

that

and

A =

h

A(ek)

defines o

=

For

a linear keel

we

measurable

functional

A

.

have

~ A ( e n ) < e k , e n > = A(ek) n=l

By p r o p o s i t i o n

8B we

conclude 00

h(x)

= A(x)

h ( e n ) < X_, e n >

=

a.e.

n=l

For measure

of

arbitrary sets

of

functionals

the

form

kl,..,k n

,

we wish

to

calculate

the

112

{ x6~ ~ where

ak,b k

I ak< k k ( X ) < -

are r e a l n u m b e r s . [A>c]

Lemma

in

~

Consider

9B:

with

values

to a f u n c t i o n

in

bk

= { x6~ ~

If

} ,

often use the shorter

I ~(x)>c

a sequence •

, k=1,.-,n

We shall

}

{f n}n= I

{fn}n~ I

notation

of m e a s u r a b l e

converges

almost

functions everywhere

f , i.e. fn

t h e n to e v e r y

n

c6~

~ f

there

pointwise exists

for a.e.

a sequence

ck

_x6~ ~

,

{Ck}k6 ~

fulfilling

~ c k

and ~[f>c]

Proof : zero--measure

Assume

= lim lim ~ [ f n > C k ] k n

that

[ f=c ] co

set on w h i c h

{fn}n= I

=

0

does

and not

denote

converge

by

to

f

M .

the T h e n we

get [fn>C] pointwise

on

the

By u s i n g

set

~

\

(M U

the d o m i n a t e d

n

[f>c]

[f=c])

,

convergence

~[fn>C]

i.e.

almost

everywhere

on

t h e o r e m w e get

~ ~[f>c] n

The

case when

W e find a s e q u e n c e

the

set

{Ck}ke ~

ck

~ c

and

[f=c]

has p o s i t i v e

of real n u m b e r s ~[f=ck]

= 0

measure

now

follows.

fulfilling

for e v e r y

ke~

.

k This

is

possible,

uncountable

family

contradicting

the

of

since

there

disjoint

fact t h a t the m e a s u r e

would

sets ~

with

otherwise positive

is finite.

exist

an

measure,

113

By sets

applying

[f>ck]

the

above

and using ~[f>c]

The measure

established

the dominated

= lim v[f>ck] k--~

to

convergence

the

zero--measure

theorem,

we get,

= lim lim ~[fn_>Ck] k--~ n--~

of the sets ~[f1>cl , f2>c2

where

result

fl ' f2''''fm

are

,.., fm>Cm ] ,

measurable

functions~

can

be

calculated

in a

similar way. We shall ~

, denoted

apply

by

the

k .

lemma with

a linear

It has been proved

earlier

cO

X(_x)

by

Then by lemma

AN

the

=

anX n

with

in

that

an

n=1

sum of the terms

with

indices

from

I

to

N

.

9B we get v[k>c]

where

functional

cO

n=1 We denote

measurable

the sequence

{Ck}k6 ~

We now calculate n ~[kn->Cm ] = ~{ x6~m

1

= lim lim V[hN>_Ck] k---~ N---~ converges

to

c6~

,

.

~[hN_>Ck]

~ akx k -> c m } k=1

= (2~) -½n ~

lM'exp(-½(x2+'''+x2))dx1"'dXn

'

~n where n M = { xE~n By

choosing

spanned

an

orthogonal

[

~ akXk _> Cm } . k=l transformation in

sending

the

line

by ( al,a 2 .... a n )/lla_nll2 , n into

IIaoII2 = vector

k=1 in ~n

,

the

line

spanned

by

the

we get by using the transformation

first

natural

theorem

basis

114 co

=

exp(-½t2)dt

= (2~) -~

exp(-½t2)dt

.

Cm/I] -an II 2

{ t_>Cm/I[--an ]I2 } 0o

We

choose

k

with

Ilxl12

2

=

1

,

.

i.e

~

a n2

1

=

By

first

letting

n=l

n ,

and

m

afterwards

,

go to

infinity

the

above

expression

reduces

to ~[k>c]

= (2=) -~ ~ exp( -½t 2 )dt

,

C

with

k

being

a

linear

measurable

functional

in

Nm

with

m

IX(en)l 2 =

1

n=]

The e x p r e s s i o n

can e a s i l y

~[kl>Cl .... km>Cm]

be e x t e n d e d

= (2~)-½m i

... i e x p ( - ½ ( x ~ + . . + x ~ ) ) d x l

cI where

kl,..,k m

denote

linear

to -.dx m ,

cm

measurable

functionals

in

Nm

all

,

with 0o

Iki(en) I2 =] and

the

i

One

.

vectors hereby

A property

simple

n=l i

{a n obtains

set

for

= Xi(en)}n6~ an

of the m e a s u r e

orthogonal

orthogonal

theoretical give

argument

kl,..,h m

= (2~) -~m ~

are m e a s u r a b l e Iki(en)

different in

together

tRN

with

with

the

i {a n = k i ( e n ) } n £ ~

N_~m .

additive

bm . .. ~ e x p ( - ½ ( x 1 2+ . . + x ~ ) ) d X l . . d X m am

functionals

12 =1

for

in

~

i=1,..,m

with ,

n=1 and

indices

us the e x p r e s s i o n

aI where

for

transformation

bI ~[a]12

n=1

the extension

We have

measurable

os

amn2 =

the

a weak

as

by mn

the

is

taken

~ 62

domain

a Since

A

are

transformation

: 62

uniquely

extending

given

then

I

L2

of

~

A Then

n

if e l e m e n t s

{amn}m,n6~

transformation

A

number

,

k=1 , . . , n

.

k=1 .... n }

.

sets

of e q u a l

We define

measure.

117

bI bn ~(B) = (2~)-½n ~ .. ~ exp(-~(x1+..+Xn))dx I 2 2 I ..dx n aI

an

and that U(B) = (t ~ 6 ~ where

ak
= <Jbn,X> = hn

for

Let us define co

t2 = { X 6 C ~

~ ,Xn,2 < ~ } n=l

~2 = { X 6 ~ ~

~ kn,Xnl2 < m } n=l

x6K

(cf.

[23])

and the positive

121 By i d e n t i f y i n g ~

t 2

~A~2 via

the

orthonormal

measure

~

on the

,

a

basis =

½,1

corresponding

{bn}n6 ~

in

on the

Borel

,

Borel

sets

in

~(~) a

Since

unitary

weakly

measurable

is i n v a r i a n t under

the

spaces. the

under

the

showing

we h a v e basis to

show

selected

Hilbert--Schmidt

that

HI

enlargements

then

the

get

62

and

the

unitary

between

does

measure

not

two

~

can

different

Hilbert

depend with

does

This

~

is i n v a r i a n t

for i d e n t i f y i n g

gaussian

orthogonal measure

7~

maps ~

to

the

the m e a s u r e

denote a

extend

~2

enlargement.

exist

gaussian u the m e a s u r e

as

then

in

used the

H2

there

and

~

the m e a s u r e

K

that

and

of

that

in

can d e f i n e

of

onto

transformations,

like

if

62

extensions

we

2) = I

these

orthonormal would

of

,

transformations

In p a r t i c u l a r ,

chosen

= ~(Z

, sets

linear

corresponding

We on

transformations

~2

H

not be

on

62 depend

done

by

Hilbert-Schmidt

Hilbert--Schmidt

enlargement

fulfilling c H.

i=I,2

.

1

If in H2

C

,

f

denotes

and

fl

and

respectively,

almost

then

Hilbert-Schmidt is f i n e r

extends

f2

are fl

function

continuous

and

f2

are

defined

on

H

with

extensions

of

f

to

equal

on

~

,

values KI

hence

and equal

everywhere.

Definition

KI

a continuous

|F:

Let

Hi, I

enlargements

than

K2

to a c o n t i n u o u s

of

if the

and

a Hilbert

and space

identity I : H -

, H

one-to-one

map

I : H], I

~ H2, 2 .

H2, 2 H,

We

denote say

that

122

It is o b v i o u s a constant

C > 0

that

such

if

HI

is f i n e r

K2 ,

than

then

there

exist

that

JIxJl2 ~ c-JJxlJl for

all

norm

of

xeH I

.

The

in

smallest

C

fulfilling

the

above

is

the

operator

~(Kl,K2)

Lelmaa 2F:

Consider

seminorms

{]]-][n}n6 ~

and define

ll'li*

in

well

known

setting co

n=] If a s e q u e n c e

fulfills

{Xp}pe ~

IIXp - XqII.

, 0 p,q

and

for every

nE~

JlXpitn

. 0 , P

then

llXpl].

, 0 P

Proof: method

of

The

proof

verification

amounts

that

the

to

an

adjustment

countable

direct

sum

of of

the

Hilbert

spaces

is c o m p l e t e .

Lemma K,

.

3F:

Let

HI, I

denote

is a c o n t i n u o u s enlargement H2

2)

k

Hilbert--Schmidt

enlargement

of

If k =

])

a

linear

H2, 2 is

finer

functional, of

than

is c o n t i n u o u s

H,

: H

, C

then

there

such that

HI in t h e m e t r i c

of

l

<en,-> I = Expanding

aeK

we

[lenll~-<en,->

•

get

a =

~ an-e n

with

=

an

<en,a>

and

lan 12




= n=]

numbers

Choose

a

with

m]

strictly

increasing

= I

that

such

n=1 For

x,yeH

we

then

<xcy>.

sequence

{mn}n6 ~

of

k=m~

define

m~+~-1

m0+~-1

k=mn

k=m~

= n=1

and

mn+~-] n=1 Since

by the

Cauchy--Schwarz

k=m0

inequality

÷

II x II .2

=

i 2n. mi n=l



k=mn

[ak 12 < °°

natural

124

2

in

for

<x,y> 2 = <x,y>. + <x,y>] and the corresponding

For

norm,

H

setting x,y6H

I["I12 '

2 IIx II~ = IIx I1.2 ÷ fixrlI consider the expression

xeH

o0

k(X) =

~ an.<en,X> n=1 mn+~-I

co

ak'<ek'x> = n=l k=mo

mn+

~ (v~)-n" (v~)n ~ ak'<ek 'x> n=1 k=m~

Using the Cauchy--Schwarz

inequality, we get mN+1-1 h(x)l -< ~ (v~)-n" (v~) n ~ ak'<ek ,x> n=1 k=m~ -< [ ~ 2-n]~" n=]

= .e.

k

llxJl.

[ ~ 2n n:1

~ ak'<ek ,x> k=mn

in the metric of

Since ~0

~0

[Iep[l 2. = p=1

]

_< IIxH2 ,

is continuous

CO

mn+

I --I

2

~ ~ 2 n" ~ ak" <ek, ep > p=] n =1 k=m~ ~ mn.~-] 2

:

p=1 n=1 ~

n=l p=l

I --I

k=m~ mn+~-1 k=m~

n=1 p=1 k=mn ~o mn+1-1 2

125 2

IIep II2 Then of

the H,

completion

H2

is a H i l b e r t - S c h m i d t

enlargement

.

It r e m a i n s that

< ~

p=1 H, 2 of

to p r o v e

that

H2

is finer

than

It is o b v i o u s

HI

the i n c l u s i o n m a p p i n g I : K, 2

~ H, I

is c o n t i n u o u s . To p r o v e is s u f f i c i e n t

that

the

continuous

to s h o w that

for each

[]Xp

1)

-

of

sequence

..$Xp~pe~ C H

xqll2

,

I

is o n e - t o - o n e ,

extension

0

P,g and

2)

IlxpU]

~ o

w

~ o

.

P we

have

11xp II 2 P

From

2) it follows

that

for e v e r y

k6~

<ek,Xp> l

~ 0 P

and in p a r t i c u l a r

that

<ek,Xp> for e v e r y

ke~

= l]ekll?2-<ek,Xp>1

By l i n e a r i t y

we get that

m~+~-I 2n" For

n6~

and

x6~

~ k=m~

we d e f i n e

2

ak" <ek'Xp> [

~ 0 P

the s e m i n o r m s

[Ix I[n2 = 2n"

[I" []n '

mo÷~-1 ~ ak" <ek'x> k=m n

Then by lemma

> 0 P for e v e r y

2F w e get that

ILXpll.

~ 0 P

2

n£~

such

that

it

126

and t h e r e b y

II Xp II 2

0

,

P

Lemma

4F:

Hilbert-Schmidt a

fixed

denotes

{Kn,n}ne~

enlargements

Hilbert-Schmidt

Hilbert--Schmidt n6~

If

of a H i l b e r t

enlargements

enlargement

~,~

finer

sequence

K,

space

K0, 0

a

all

then

,

than

of

finer

there

Kn,n

than

exist for

a

every

.

Proof: the i n c l u s i o n

Take

an

orthonormal

mappings

from

basis

~, II" II to

{en}n6 ~ ]{, If- IIn

in

K,

Since

are H i l b e r t - S c h m i d t ,

we

have 00

Cn : It follows

that

~o

that

llxlln Note

2

Ilepll n )~
with

K,

IIenlI]-<en,->

lemma

K,

x,yeH

in

the

in

inner

a

hence H

.

Hilbert--Schmidt H2, 2

and

enlargement

such

that

the

H3, 3 product

= <x,y> I + <x,y> 3

norm

IJxjj : IJxlJ + IJxjj Then

we d e f i n e

K

to be the

~llenll 2 = n=1 which

makes The

the

inner

inclusion

completion

~

l'en'l~ +

n=] product

well

of

~

H,~

.

llenll~ < "

We h a v e

,

n=] defined.

mappings I : H,



H],]

I : K,



~ K3, 3

and

are

obviously That

continuous. their

extensions

are

one-to-one

follows

from

the

fact

that

129

the

functionals

the and

If- U3 hence

-

norm

dense

When a complex

{hn}ne ~ and

~n

Hilbert

continuous these

gaussian

space 7~

in b o t h

functionals

(Hi,i)'

considering

and

H,

will

then

where

we

a real

z =

denote

• e x p ( - ( I Z l 12 +

(Zl,Z2,...,Zn)

6 Cn

in

let

and

(H,)'

H,

denote

space.

the m e a s u r e

...

+

with

dz = dxldYldX2dY2...dxndY

In a s i m i l a r

n

in

[Znl2))dz

~2 c

given

way

~H

denotes

,

zk = x k + i.y k 6

indicates

the

the

in

Lebesgue

~2n

measure

in

~2

c

given

on

by I (2v)-~n-exp(-½(x~

where and

dense

always

Hilbert

integration

~n

are

If-IfI - n o r m

(H3,3)~

measures

and

the

by -n

and

that

in b o t h

The m e a s u r e on

are

x =

(Xl,X2,...,Xn)

dx = dxldx2..0dx n

+

...

2 + Xn))d ~

,

6 ~n indicates

the L e b e s g u e

integration

in

~n

I

Observe produced

that

the m e a s u r e

by considering ~

}{

~C

as a real

= ½(

on a c o m p l e x Hilbert

+ )

=

Re

Hilbert

space

with



.

space inner

}{

is

product

References

[]]

V. Bargmann, Associated

On a H i l b e r t Space of A n a l y t i c F u n c t i o n s and an

Inteqral T r a n s f o r m Part I,

Pure Appl. Math.

[2]

Sci.

14 (]961), page

O. B r a t t e l i and D. W. Robinson, Statistical Mechanics

[3]

]87--2]4.

O p e r a t o r A l g e b r a s and Q u a n t u m

II, S p r i n g e r Verlag,

(1981).

J. Cook, T h e m a t h e m a t i c s of second q u a n t i s a t i o n , Math.

Soc.

74

Trans. Amer.

(1953), page 222--245.

[4]

W. Feller, A c k n o w l e d q m e n t ,

[5]

R.J.

Proc.

Nat. Acad.

Sci.

USA, Vol 48

(1962) page 2204. Glauber,

Field,

[6]

A l a i n Guichardet, Topics,

[7]

C o h e r e n t and I n c o h e r e n t States of the R a d i a t i o n

Physical R e v i e w

R.L.

13](1963),

page 2766--2788.

Syn~metric H i l b e r t Spaces and R e l a t e d

L e c t u r e Notes in M a t h e m a t i c s

26], S p r i n g e r V e r l a g

Hudson and J.M. Lindsay, A non--commutative m a r t i n q a l e

r e p r e s e n t a t i o n t h e o r e m for non--Fock Q u a n t u m B r o w n i a n Motion, Journal of F u n c t i o n a l A n a l y s i s

[8]

R.L.

Hudson and K.R.

S t o c h a s t i c Evolutions, [9]

page

301-323.

R.L.

Hudson and R.F.

61

(]985), page 202--221.

Parthasarathy, Commun.

Streater,

rule w i t h W i c k orderinq,

Q u a n t u m Ito's F o r m u l a and

Math.

Phys.

93

(1984),

Ito's formula is the chain

Physics Letters,

vol 86A, n u m b e r 5,

page 277-279.

[10]

P. Kristensen,

L. Mejlbo,

E.T. Poulsen,

b u t i o n s in i n f i n i t e l y m a n y d i m e n s i o n s I

[11]

W.H. Wiley

[12]

: Commun.

Math

II

: Math.

III

: Commun. Math.

Louisell,

Scand.

.Phys.

Tempered distri-

I,II,III

I(1965)

]4(]964) Phys.

6(]967)

page

175--214.

page

129--]50.

page

29- 48.

Q u a n t u m S t a t i s t i c a l P r o p e r t i e s of Radiation,

]973.

J.E. Moyal, ~ u a n t u m M e c h a n i c s as a S t a t i s t i c a l Theory, Proceedings

of the C a m b r i d g e P h i l o s o p h i c a l

S o c i e t y 45

(1949)

page 99--124. []3]

Klaus M e l m e r and W o j t e k Slowikowski,

A new H i l b e r t space

a p p r o a c h to the m u l t i m o d e s q u e e z i n q of liqht, J.Phys.

[14]

A:Math.Gen

E. Nelson,

21

(1988) page 2565--2571.

The free M a r k o f f field,

J. Funct. Anal.

]2(1973)

page 211--227.

[]5]

I.E. Segal,

T e n s o r a l q e b r a s over H i l b e r t space I, Trans.

Amer.

Soc.

Math.

81

(]956), page

]06--134.

131

[]6]

I.E. Segal, The complex--wave representation of the free Boson field, Topics in Functional Analysis,

Adv.

in Math.,

Supplementary Studies Vol 3 (]978), page 321-345. [17]

I.E. Segal,

J.C Baez and Zhengfang Zhou,

Introduction to

alqebraic and constructive quantum field theory, Univ.

[]8]

Press,

to appear.

G.E. Silov and Fan Dyk Tin, Nauka, Moscow

(1967),

[19]

B. Simon,

[20]

W. S~owikowski,

The P{#)~ Euclidean

W. Slowikowski, Weyl theory,

Quantization of Questionnaires,

W. S~owikowski,

Concerninq pre-supports of linear probability

Lecture Notes in Mathematics

W. S~owikowski, Analysis,

794, Proceedings

1979, Springer.

Measure Theory Applications to Stochastic

Proceedings,

Oberwolfach,

Germany 1977, Lecture

Notes in Math. vol 695, page 179--191, Springer [25]

A. Unterberger, Notes in Physics

[26]

L.V.

1978.

Les operat6urs metadiffer6tiels,

Lecture

126 (1980), page 205-241.

Zyla and Rui J. P. deFigueiredo,

Nonlinear System

Identification based on a Fock space framework, Control and Optimization, page 931--939.

Math.

Vol 10 No. 9 (1988), page 697--709.

Measure Theory Oberwolfach

[24]

Advances

to appear.

W. Slowikowski,

measures,

in Bose alqebras,

9 (]988) 377-427.

Bose alqebras of operators and the Wiqner-

Comput. Modelling.

[23]

(Quantum) Field Theory,

(1974).

Ultracoherence

in Applied Mathematics

[22]

Inteqral, Measure and Derivative,

(Russian).

Princeton University Press

[21]

Princeton

SIAM J.

Vol 21 N0. 6 (November 1983),

Subject

i n d e x

real wave representation 2,3,53, annihilation operator ],3,6,8,10, 72,77,90,91 21,25,61,64,79,8],83,85,92 anti-commutator I second quantization 30 anti-normal 92 Stone 29,30 base space 1,2,4,]0,23,65,83,85, tame 100,101,102 total 16,53,66,67,92,118,120 87 Bose albebra 1,2,3,4,6,]0,11,20, ultracoherent 57 vacuum 1,4,10,20,23,65,83,85,92 22,23,65,79,83,84,85,88 - extende 36,65,83 value of 66,90 weak measurable linear trans-- Fock space 1,2,3 formation 115,116,121 Campbell--Baker--Hausdorff 50,5], 59,91,92 Weyl 1,45,52,91 coherent 2,33,34,43,44,63,64,66, Wick 22,45,59,79,81,83,88 83,89,92 Wiener 16,20 commutation 9,10,11,12,45,55,59, Wigner 3,86 60,65,82,85,93 complex wave representation 2,53, 66,69,70,71,78,80,87,90 conjugation 1,2,3,53,54,55,57,69, 73,74,76,77,80,84,88,89,90 creation operator 1,3,6,8,10,2], 25,64,79,8],83,85,92 cylinder set 96,97,]01,107 derivation 1,6,10,]],30 Fourier transformation 25,31,32,7] free commutative algebra 1,4 - product 1,4,18 gaussian content 96 measure 3,68,76,78,80,86,96,99, 107,115,116,117,118,120,121,]22 Halmos 27,94 Heisenberg 33,43 Hermite 2,31,32 Hilbert--Schmidt enlargement 68,71,76, 80,86,87,90,91,118,120,121,122, 125,126,127,128 Kolmogorov extension theorem 97 -- inequality 103,105 -- large number theorem 104 - zero one law 102,111 Leibniz rule 6,30,31,36,37,39,61,82 linear measurable functional 102,103, 106,108,109,110,111,113,114,115, I]7,122,123,128,129 Nelson 27 one-parameter group 29 4-- picture 65,84,88 product 59,61,62,63,64,65,78,88 value 72,90 -

---