This content was uploaded by our users and we assume good faith they have the permission to share this book. If you own the copyright to this book and it is wrongfully on our website, we offer a simple DMCA procedure to remove your content from our site. Start by pressing the button below!
( ! ) = p1/J - a pure state P1/J, where 1/J E L2 (1Rn ) , 111/JII 1 . The projection P1/J is an integral operator with the kernel 'lj; ( q )'l/J( q' ) , and we get from (3. 1 1 ) , =
) r f(p 9.±!t ' 2 )e*p(q-q' dnp 1/J(q)'l/J(q' ) (2-rr n ) n }JRn Introducing q+ = � ( q + q ' ) , q _ = � ( q - q' ) , we obtain 1
=
j ( 21_ , v ) or
=
·
nn rJRn 1/J (q_ + q+ ) 'l/J ( q+ - q _ )eivq+ dnq+ , }
r
j ( u, v ) = lin }JRn 1/J ( q + � nu ) 'lj; (q - � nu )ei vqdnq . Assuming that 1/J does not depend on n, we get in accordance with Corollary 3.3 , 1 r l'l/J (q) l2eivq� v . (u, v ) = p( u , v ) = lim ( 2 j n-+0 (2 7r ) n )JRn 7r n
\)
2.
13 4
Basic Principles of Quantum Mechanics
Thus in the classical limit n --+ 0 the pure state P1/J in quantum mechanics becomes a mixed state in classical mechanics, given by a probability measure dj.L p (p, q ) dnp dn q on JR 2n with the density =
p (p , q )
=
8 (p ) l � (q) l 2 · =
It describes a classical particle at rest ( p 0) with the distribution of coor dinates given by the probability measure l � ( q) j 2 dn q on IRn . When � ( q) = e *PrW r.p ( q) , where r.p( q) does not depend on n, the corresponding density is p(p , q)
=
8 ( p - Po ) l r.p (q) l 2 -
Remark. The Weyl quantization can be considered as a way of defining a function f(P, Q) of non-commuting operators P = ( P1 , . . . , Pn ) and Q = (Q1 , . . . , Q n ) by setting
J (P, Q) (!) . g (p ) + h( q) , then =
In particular, if f (p , q)
=
f (P, Q) = g ( P) + h(Q) . =
pq = p 1 q 1 + · · · + Pn qn
we get , using (3. 1 1 ) , PQ QP f (P, Q) = _ This shows that the Weyl quantization symmetrizes products of the non commuting factors P and Q. In general, let f be a polynomial function, For f ( p, q )
(3 . 13)
f (p, q) =
;
f3 Caf3 pa q ,
L
J a J , I f3 1 :S:N
where for the multi-indices a = ( a 1 , . . . , an ) and (3 = (/31 , . . . , f3n ) , l /3 1 = /3 1 +
and l a l = a1 + · · · + a n , following formula
(3. 1 4 )
if> (!) =
L
J a J J f3 1 :S: N ,
·
·
· + f3n ·
Using (3 . 1 1 ) , we get the
Caf3 Sym (P 01 Q
f3 ) .
Here Sym(P 01 Q f3 ) is a symmetric product, defined by
(3. 15)
(uP + v Q ) k
=
L
JaJ+ If3J = k
�
k' u vf3 Sym(P01 Q f3 ) , a ! ! 01
3. Weyl relations
135
Remark. In addition to the Weyl quantization , consider also the map pings 1 Y'(ll�2n ) -> .2(£) and 2 Y' ( JR 2n ) -> £"(£) , defined by :
:
and
1
=
2 ( ! ) = --( 27r ) n
1
i ii
�
JR2n
j(u , v )e- 2 uvS(u , v) cf"u dn v ,
where j g--1 (!) is the inverse Fourier transform. Though 1 and 2 no longer map Ao into the real vector space do of bounded quantum observ ables, they satisfy all the properties in Proposition 3.2. It follows from (3.5) that for f E Y' ( JR2 n ) the operators 1 (J) and 2 (J) are integral operators with the integral kernels ' j(p, q') e K p( q q ) dnp K1 (q, q') = 2 /i ( )n and 1 (p, q) e K p( q - q' ) cf"p, K2 (q, q' ) = ( 7r Ji) n J 2
:
ln ln
-
respectively. As in the case of the Weyl quantization, these formulas extend the mappings f 1 (J) and f 2 (J) of the space Y'(JR2n )' of tempered distributions on �2n . In particular, if f (p, q) is a polynomial function (3. 13), then l ( f ) = 2: Ca f3 pa Q /3 (3. 1 6) �---+
�---+
lai ,I,BI�N
and (3. 17)
2 (J)
=
2:
lai ,I,BI�N
Ca,B Q 13 p er .
Therefore the mapping f 1 ( f ) is called the pq-quantization, and the mapping f 2 ( f ) - the qp-quantization. Corresponding inversion for mulas are �---+
�---+
(3. 18)
and (3. 19 )
The distribution f(p, q) defined by (3. 18) is called the pq-symbol of an oper ator with the Schwartz kernel K1 (q, q') , and the distribution f (p, q) defined by (3. 19) - the qp-symbol of an operator with the Schwartz kernel K2 (q, q') . The qp-symbols are commonly used in the theory of pseudodifferential op erators. It follows from (3. 18) and (3.19) that if f(p, q) is a pq-symbol of the operator 1 ( !), then f (p q) is a qp-symbol of the adjoint operator 1 ( !) * . ,
136
2. Basic Principles of Quantum Mechanics
Problem 3.6. Prove formula (3. 14) . Problem 3.7. Prove formulas (3.16)-(3. 1 7) . Problem 3.8. Prove that Weyl, pq, and qp-quantizations are equivalent. (Hint: Find the relations between Weyl, pq, and qp-symbols of a given operator. )
3.4. The *-product . The Weyl quantization
: .9' ( JR 2n )
�
ied in the previous section, defines a new bilinear operation * n : .9' ( 1R2 n ) X .9' ( JR 2 n ) .9' ( 1R2 n ) on .9' ( JR 2n ) by the formula h *n h = - 1 ( ( h ) ( h ) ) . This operation is called the *-product 26 . According to ( 3.10 ) ,
2'(£) , stud
�
r r
1 ( h *n h ) (p , q) = 27r 2 n } 2n } !1 ( UI , VI)j2 ( U 2 , V2 ) · ( ) JR JR 2n . e � ( Ul V2 -U2Vl) -i ( ul +u2) p-i ( v1 +v2)q dn dnu dn v1dn V . U1 2 2 The *-product on .9' ( JR 2 n ) has the following properties.
( 3.20 )
1 . Associativity:
h *n ( h * n h) = ( h *n h ) *n f3 . 2 . Semi-classical limit: "1i ( h * n h ) (p , q ) = ( h h ) (p, q ) - � {h, h} (p , q) + O ( n2 ) as n � 0. 3 . Property of the unit: f * n 1 = 1 *n j , where 1 is a function which identically equals 1 on JR 2 n . 4. The cyclic trace property: T ( h *n h ) = T ( h *n h ) , where the ((>linear map T .9' ( JR 2 n ) C is defined by :
� l2n
�
j (p, q ) d p q. T( j ) = ( 2 7r )n Property 1 follows from the corresponding property for the product * n ( see Section 3 . 1 ) , property 2 follows from Proposition 3. 2 , and properties 3 and 4 directly follow from the definition ( 3 . 2 0) . The complex vector space .9' ( JR2n ) Efl C 1 with the bilinear operation *n is an associative algebra over C with unit 1 and the cyclic trace T , satisfying the correspondence principle, lim * ( n->O h * n h - h *n h ) = {h , h}. 26 Also
called
Moyal product in physics.
n dn
3.
137
Weyl relations
Consider the tensor product of Hilbert spaces L 2 (JR.2 n ) 0 L 2 ( JR.2 n ) ::::: L 2 ( JR.2 n X JR.2 n ) , and define the unitary operator ul on L2 ( JR.2n ) 0 L 2 (JR.2n ) by a 181 a U1 = e- 2 ( 8P &ci ) , where n o f) f) f) 0 0 = apk aqk · ap aq It follows from the theory of Fourier transform that for JI , h iii
f;
{ {
E
Y' (JR.2 n) ,
( U1 ( fi 0 f2 ) ) (p1 , Q1 , P2 , Q2 ) = 1 2 (u v u v (2 7r ) n }JR2n }JR2n J1 ( 1 , 1 ) j2 2 , 2 ) · . e T u l v2 - t U1Pl -tU2 P2 - t v l ql -tv2 q2 0 such that
Succinctly,
( 3.23 ) iii e 2 ( u 1 v2 - u2v1 ) into the power series and repeating the proof of Proposition 3.2 gives the result. 0
Proof. Expanding the exponential function
Finally, we get another integral representation for the *-product. Apply ing the Fourier inversion formula to the integral over dn u 1 dn v1 in (3.20 ) , we get
1
{ { fi (p - � v 2 , q + � u 2 ) f2 (p2 , Q2 ) · = ( 2 7r ) 2n J� 2 n }�zn -i u2 p- i vz q+ iuzP2 +i v2 qz � p � q2 dn u dn v 2 , 2 2
.e
and changing variables P l
=
p - � v2 , Ql = q + � u2 ,
�
we obtain
( P 1 , q l ) f2 (p2 , Q2 ) · { *n h ) (p q ) = ( 7r ) 2n { z JR }� n 2 n h . e T2 i (Pl Q - pql +Q1P2 -Q2Pl +pqz - P2 q) � 1 � 1 dn � Q
(h
,
P
Q
2·
p2
Let 6 be a Euclidean triangle ( a 2-simplex) in the phase space IR.2 n with the vertices (p , q) , (P I , Q I ) , and (p2 , Q2 ) · It is easy to see that Pl Q - PQ I + Q1 P2 - Q2Pl + PQ2 - P2 Q
=2
i
w,
3. Weyl relations
1 39
which is twice the symplectic area of /::::,. the sum of oriented areas of the projections of /::::,. onto two-dimensional planes (p 1 , q 1 ) , . . . , (Pn , q n ) . Thus we have the final formula -
� { {
fr (P I , q l ) h (P2 , q2 ) 2 ( 7T ) n }'R2n }'R2n . e � J6 w dnp l dn q l dnp dn q , 2 2
(fr *n h ) (p , q) =
(3 . 24)
·
which is a composition formula for the Weyl symbols.
Remark. It is instructive to compare formulas ( 3 . 23 ) and ( 3 .24) . The latter formula represents the *-product on Y' (IR2n ) as an absolutely convergent integral, and is equivalent to the Weyl quantization. The former formula is an asymptotic expansion of the *-product as n 0, and does not capture all properties of the Weyl quantization. In general, the power series in (3. 23) diverges; for polynomial functions this series becomes a finite sum and gives a formula for the *-product of polynomials. Problem 3.9 (Composition formula for pq symbols ) . Let ft (p, q) and h (p, q ) ---+
be ,
respectively,
pq-symbol of the operator f (p , q)
-
the pq-symbols o f t h e
=
1 (21r n ) n
operators � 1 (ft ) � 1 ( ft ) � 1 (h ) is given by
L2n
and � 1 ( h ) . Show that the
n ft (p, q t ) h (P1 , q ) e f (p -pl ) (q-ql ) d p1 dn q 1 .
(m Uf) ( it Q9 h ) . ) Problem 3 . 1 0 (Composition formula for qp symbols ) (Hint:
Use the formula for
be, respectively,
(Hint:
Use the
. Let ft (p, q ) and h (p , q ) operators � 2 (ft ) and � 2 ( h ) . Show that the � 2 ( ft ) � 2 ( h ) is given by
the qp-symbols o f the
qp-symbol of the operator J (p , q)
o
=
\ L2n
( 2n ) n
formula
for
Problem 3 . 1 1 . Using
dn p 1 dn q 1 . JI (p 1 , q) h (p, ql ) e - f (p -pl ) (q - q l )
(m U2- 2 ) (it Q9 h ) . ) o
( 3 . 24)
prove
that
the
*-product
is associative.
Problem 3 . 1 2 . For classical observable f (p, q) define the *-exponential
analog of the evolution operator) by
exp* f = Compute e xp * (
-itHc ) ,
harmonic oscillator .
3.5.
where
oo
L
n=O
n,-n
1 n.
f *n f *n
Hc (P, q) is
the
n
(the
· · · *n f .
Hamiltonian function ( 2 . 27)
of
the
Deformation quantization. Here we consider the quantization pro
cedure from a formal algebraic point of view as the deformation theory of associative algebras. Let A be a C-algebra (or an associative algebra with unit over a field k of characteristic zero) with a bilinear multiplication map
140
2. Basic Principles of Quantum Mechanics
: A ®c A ----t A, which we will abbreviate as a · b = mo (a, b) . Denote by C [[t]] the ring of formal power series in t with coefficients in linear mapping from Home ( A, A) to Home ( A® 2 , A® 2 ) , which we continue to denote by � ' and � (
..
.. 'P I I ::; a i i AR>.. 'P I I + b i i R >.. 'P I I ::; ( a + m ) ll
0 is arbitrary. Choosing r such that a = r - 1 1 2 (2 7r ) - 3 1 2 C t 1 Vd < 1 completes the proof. 0 =
1 . 3 . The Schrodinger operator of a complex atom, considered in Example 2 .2 in Section 2 .4 of Chapter 2, is essentially self-adjoint on COO (JR3 ( N + l ) ) .
Corollary
Proof. We consider only the special case of the Hamiltonian of the hydrogen
atom,
H=
-
�
-
e2
-, r
r = tx t .
= V1 + V2 , where X 1 is the characteristic Writing = X l V + (1 function of the unit ball B1 = {x E IR3 : t x t -::; 1 } , we have V1 (x) E L 2 (IR3 ) and V2 (x) E L00 (IR3) . 0
V
- x l) V
Another useful criterion applies to real-valued potentials V (x) E LFoc (IRn ) , the space of locally bounded a.e. functions on IR.n . In this case the opera tor H, defined by the formal differential expression ( 1 . 1 ) , is symmetric on C0 (1R.n ) and we have the following result. Theorem 1 . 4. If V ( x) E L �c (IR.n ) is bounded from below, V ( x ) � C a. e. on IRn , then the Schrodinger operator H = Ho + is essentially self-adjoint on
V
Ca (IR.n ) . In
fact, a much more general statement holds.
Theorem 1 . 5 (Sears) . Suppose that the potential x E IR.n satisfies the condition V (x ) � - Q ( t x l ) ,
V (x )
E
LFoc ( IRn ) for all
where Q(r) is an increasing continuous positive function on [0, oo ) such that
roo
l o
dr
jQ(r)
=
oo .
3. Schrodinger Equation
152
Then the Schrodinger operator H = Ho + V is essentially self-adjoint on COO (JRn ) . Problem 1 . 1 . Prove the general case of Corollary 1 .3 ( Hint : Derive the estimate ( 1 . 3) for each term in the corresponding potential energy operator. )
Problem 1 . 2 (Kato's inequality) . Let 7/J E Lfoc (JR. n ) be such that 6. 7/J , defined in the distributional sense, is represented by a function in Lfoc CIRn ). Prove that in the distributional sense, 6. l u l :2': Re 0. (The distribution T if u(x) non-negative tp E Y (JR.n ) . ) =
E
(� 6.u) , where it is assumed that ��:�
Y(JR.n )' is non-negative if
T(�.p)
=
0
:2': 0 for all
1 . 3 . Prove Theorem 1 .4 using Kato's inequality. ( Hint : Show that if L 2 (1Rn) in the distributional sense satisfies ( - 6. + V (x) + C + 1 ) 7/J 0, then 0. )
Problem 7/J 7/J
E
=
=
below potentials V (x) = x and V(x)
Problem 1 . 4 . Prove that one-dimensional Schrodinger operators with unbounded =
-x 2 are essentially self-adjoint on C0 (JR) .
ical problem in quantum mechanics is to describe the spectral properties of the Schrodinger operator H . Here we present some general results charac terizing the spectrum of H . The first basic result is the following. Theorem 1 . 6 . Suppose that V ( x ) E Lk;'c (!R n ) satisfies lim V(x) oo. 1 . 2 . C haracterization of the spectrum. The second major mathemat
=
1:�:1 --->oo
Then the operator H has a pure point spectrum: there exists an orthonormal basis { 7jin } n EN for £ consisting of eigenfunctions of H with eigenvalues A l � A2 � · · · � An � · · · of finite multiplicity, H 7jln = An 7jln , and lim An = oo . n --><Xl Recall that the essential spectrum O'ess ( A ) of a self-adjoint operator A consists of all non-isolated points of 17(A) and of eigenvalues of infinite mul tiplicity. The following result gives a sufficient condition for the essential spectrum of the Schrodinger operator to fill [0, oo ) . Theorem 1 . 7. Suppose that V = V1 + V2 , where V1 (x) E L q (!Rn ) , 2 q � n for n > 4 and q � 2 for n � 4 1 , and V2 (x) E L 00 (1Rn ) satisfies lim V2 (x) = 0 .
1:�:1 --->(X)
Then O'ess ( H ) [O, oo ) , so that 17 ( H) values of H of finite multiplicity. =
1 In the special case n
=
4
one
has q
>
2.
n
(-oo, O) consists of isolated eigen
153
1 . General properties
The next result gives a sufficient condition for the Schrodinger operator with decaying potential to have only negative eigenvalues. Theorem
1.8
( Kato ) . Suppose that V (x ) la:l-> oo
E
L00 (lRn ) and
lim J x ! V (x) = 0.
Then the Schrodinger operator H = Ho + V has no positive eigenvalues.
Recall that the absolutely continuous spectrum and the singular spec trum of a self-adjoint operator A on Y'f' are defined, respectively, by O'ac (A) = cr ( A J£...J and CTsc (A) = cr ( A J�J , where £ac and £sc are closed subspaces for A defined as follows. Let PA be the projection-valued measure for the self-adjoint operator A and let v'l/1 = ( PA 'ljJ, 'ljJ ) be the finite Borel measure on JR. corresponding to 'ljJ E £, 'ljJ =/= 0 . Then £ac consists of 0 and all 'ljJ E Y'f' such that the measure v'l/1 is absolutely continuous with respect to the Lebesgue measure on JR., and £sc consists of 0 and all 'ljJ E Y'f' such that the measure v'l/1 is continuous singular with respect to the Lebesgue measure on R Theorem 1 . 9 . Suppose that the potential V(x) E L00 (lRn) for some
satisfies
V ( x)
= O ( J x J - l -c ) as J x J
----+
c
>0
oo .
Then the Schrodinger operator H Ho + V has no singular spectrum and [0, oo ) . Moreover, cr(H) n ( - oo , 0) consists of eigenvalues of H of finite multiplicity with the only possible accumulation point at 0 . =
CTac (H) =
=
Finally, for the physically important case n 3 there is a useful estimate for the number of eigenvalues of the Schrodinger operator. Theorem
1 . 10
( Birman-Schwinger ) . Suppose that V(x) E L00 (JR3 ) and
H = Ho + V, counted with multiplicities, we have
Then for the total number N of eigenvalues of the Schrodinger operator
1 . 5 . Prove all results stated in this section. (Hint : See the list of refer ences to this chapter. )
Problem
3.
154
Schrodinger Equation
1 .3 . The virial theorem. Let H = Ho + V be the Schrodinger operator whose potential is a homogeneous function on IRn of degree p, i.e., V(ax) = aPV ( x) . The vi rial theorem in quantum mechanics is the relation between the expectation values of the kinetic and potential energy operators Ho and V in the stationary state. Theorem 1 . 1 1 (The virial theorem) . Let '1/J E .Yt', 1 1 '1/J I I = 1, be the eigen function of the Schrodinger operator with the homogeneous potential of de gree p, and let To = (Ho'l/J, '1/J) and Vo = (V'l/J, '1/J) be the corresponding expec tation values of kinetic and potential energy operators. Then
2To = pVo . Proof. I t follows from the Schrodinger equation
-/::}. '1/J + V(x)'l/J = >.'1/J
that
To + Vo = - ( f:). 'ljJ , '1/J) + (V'l/J, '1/J) = >.. For every a > 0 the function '1/Ja (x) = '1/J(ax) satisfies - /::}. '1/Ja + aP+ 2 V (x )'1/Ja = a 2 A'l/Ja ,
so that by differentiating this equation with respect to a at a = 1 and using Euler's homogenous function theorem, we get (1 . 4)
Since �
-/::}. � + V(x)� = >.� + (2>. - (p + 2) V(x))'l/J. E
D ( H) and H is self-adjoint ,
( (H - >.!)� , '1/J) = ( �, (H - >.I)'l/J) = 0 ,
and we obtain from ( 1 .4) that 2>. = (p + 2)Vo ,
which completes the proof.
0
Remark. Since energy levels of quantum systems are related to the closed
orbits of corresponding classical systems, Theorem 1 . 1 1 is the quantum ana log of the classical virial theorem (see Example 1 .2 in Section 1 . 3 of Chapter 1).
Problem 1 . 6 (The Raleigh-Ritz principle) . Prove that ).. is an eigenvalue of a self-adjoint operator H with the eigenfunction 1/J if and only if 1/J is a critical point of the functional F(cp) = ( ( H - >.I)cp, cp) on D (H) . Problem 1 . 7. Derive the virial theorem from the Raleigh-Ritz principle.
2.
One-dimensional Schrodinger equation
2 . O ne- dimensional
Schrodinger
155
equation
The Schrodin-gerhasoperator ger operator the formon the real line - the one-dimensional Schrodin H=
d2 V ( x) ' dx 2 + V (x) E L [0J lR) , -
where the real v al u ed potenti a l the space of l o cal l y inte grable functions on JR. The corresponding eigenvalue problem reduces to the second order ordinary differential equation + where i t is conveni e nt to set In thi s secti o n, we wi l study in detai l the one-dimensional Schrodinger equation case when H 'lj; = >.'lj;
-y11
(2 . 1 )
V (x)y
A
I:
(2 . 2)
Condition
=
=
- oo
0, and are continuous functions on Im k 2: 0, uniformly in x on compact subsets of R ( iv ) The conjugation property:
= !I (x , -k) , h (x, k )
h (x , -k) , Im k 2: 0. (v ) Estimates in part ( i ) hold for Im k 2: 0 . For k =f 0, . e - lm kx O' (x)e ]k[£T ( x) , x k etk (x, ::; j ) f J l jk j elm kx . J h (x, k) - e-tkx j � -J kj-Q- ( x )e ]k[£T(x ) . ( vi ) Asymptotics as j k j oo , Im k 2: 0: e - ikx fl (x, k) 1 + O ( J kj -1 ) , ei kx h (x , k) 1 + O(j k j -1) . !1 (x , k )
=
1
1
--+
=
-
=
Proof. It follows from the method of variation of parameters that differen tial equation (2. 1 ) with the boundary condition limx-.oo e-ikx fl (x, k) = 1 is equivalent to the integral equation (2.3)
JI (x, k) = e ikx
- 100 sin k� - t) V ( t)fi ( t , k) dt.
2.
157
One-dimensional Schrodinger equation
Setting 0 function ka(k) satisfies limk--+ 0 ka(k) = W ( h (x , 0) , h (x, 0 ) ) =/= 0. Therefore a(k) =/= 0 in some neighborhood of 0, and a(k) has only finit ely many zeros in Im k > 0.
2.
One-dimensional Schrodinger equation
161
The case when h ( x , 0) = ch ( x , 0 ) , c f:- 0 , is more subtle. Suppose that there is a converging to 0 subsequence k n = ixn of zeros of a ( k) in Im k > 0 . It follows from part ( i ) o f Theorem 2. 1 that there exists A > 0 such that for all x 2: 0 we have fi (x, ix) > � e-xx for x 2: A and h (x , ix) > �exx for x ::; A, so that -A 1 h (x , ixn ) h (x , O)dx 2: - e -xA . fi (x , ixn ) !I (x , O ) dx, A 4x oo Using integration by parts, we get , as before, x;
loo I:
ft (x , ixn ) !I (x, O)dx
= I:
1 = I:
(fr (x, ixn ) - V ( x ) fi (x , ixn ) ) fi (x, O ) dx
!I ( x , ixn ) (fr (x , O) - V (x ) fi (x, O ) ) dx = 0.
O n other hand , we have
(2. 16)
A oo fr ( x , ixn ) fi ( x , O) dx 0 = { fr (x, ixn )fr (x , O) dx + }A -A +
c · Cn
I:
j
h (x , ixn ) h (x, O)dx.
Since fr (x , k) is continuous on Im k :2: 0, uniformly in x on the compact subsets of JR., we have
jA
lim n -+oo - A fi (x , ixn ) fr (x , O)dx l. 1m
=
j-AA fi (x, 0 ) 2dx 2: 0 .
1. fi (x , ixn ) fi (x, O) 2 C C = C 1m =C = n C n -+oo h (x , ixn ) n -+oo h ( x, 0 ) we obtain from (2. 16) that for large enough n ,
Using
·
0
> 0,
> _1_ e - xn A .
4 xn This is clearly a contradiction, so a ( k) has only finitely many zeros. The proof of local boundness of 1 / a ( k ) in this case is left to the reader. Next , consider the differential equation (2. 1 ) together with the equation obtained from it by differentiating with respect to k:
(2.17) (2. 18)
- y" + V (x)y = k 2 y ,
-i/' + V (x)y = k 2 y + 2 k y .
Set y = fi (x , k ) in equation ( 2 . 17 ) , y = h ( x , k ) in equation (2. 18) , and multiply (2. 17) by j2 (x , k) and ( 2 . 18 ) by fi (x , k) . Subtracting the resulting equations, we obtain W ( fr (x , k) , j2 (x , k ) ) '
=
2 kfi (x , k) h (x , k) .
3.
162 Similarly, setting y = h (x, k) in (2 . 1 7) , plying and subtracting, we obtain - W (j1 (x, k ) , h (x, k) )' Therefore
y
Schrodinger Equation
= JI (x, k ) in (2. 18) , cross multi
= 2 kfi (x, k ) h (x, k) .
[A = 2 k I: fi (x, k) h (x , k) dx , A 2 k lx{ A JI (x, k ) h (x, k )dx. - w ( jl (x , k) , h (x, k ) ) l x
(2 . 1 9)
W (fi (x, k) , j2 (x, k) )
=
( 2 . 20)
Now suppose that a (ko ) = 0. We have
W ( fi (x, k ) , h ( x, k) )
=
2i ka(k) ,
so that differentiating with respect to k and setting k = ko gives W (j1 (x, ko ) , h (x, ko ) ) + W ( f1 (x, ko ) , j2 (x, ko ) )
=
2ikoa (ko ) .
Since JI (x, ko) = coh (x, ko) , it follows from Theorem 2 . 1 and Corollary 2.2 that boundary terms in ( 2 . 1 9)-(2 . 20) vanish as A ----+ oo , and we obtain, using that f (x, ko ) is real-valued:
a(ko)
= -i
I:
o
JI (x, ko ) h (x, ko ) dx = - i co l i h ( . , ko) ll 2 .
The eigenvalues of the Schrodinger operator H are simple. Indeed, the function JI (x, ix) is a solution of the differential equation (2. 1 ) for A = -x2 < 0, which decays exponentially as x ----+ oo . Since the Wronskian of any two solutions of (2. 1 ) is constant, the second solution of (2. 1 ) , linearly independent with fi (x, i x) , is exponentially increasing as x ----+ oo. Thus every exponentially decaying as x ----+ oo solution of (2. 1 ) for A = - x2 is a constant multiple of fi (x, ix) . In particular, this proves that the point spectrum of H is simple. This result also follows from the simplicity of zeros of the function a ( k ) and the eigenfunction expansion theorem, proved in the next section.
< An x; < H. Prove that corresponding eigenfunctions fi (x, ix1) have exactly l - l simple zeros, l = 1 , . , n.
Problem 2 . 1 (The oscillation theorem) . Let A 1 = - xi < 0 be the eigenvalues of the one-dimensional Schrodinger operator ·
Problem
2.2.
representation
where
· ·
=
.
.
Prove that for Im k � 0 the Jost solution fi (x, k) admits the
2. One-dimensional Schrodinger equation
Moreover,
t h e kernel
K 1 ( x , t) is
163
differentiable and
l � (x , t ) + i V ( � ) l :S � 0'1 (x) O' ( xtt ) e "'dx) ,
for 4ft (x, t ) . Correspondingly, for t h e integr al representation
with the similar inequality
solution h (x, k)
admits
h (x , k)
where t h e
alent
=
e-ik x +
kernel K2 (x, t ) sat is fies similar to the int egral equ at io n K1 (x, t) = �
1oo
�
with the condition K1 ( x , t) sive approximations. )
Problem
2 . 3 . Show that
tations
1
!00
0 fo r t
Jost
is equiv
t-(s-x)
the method of succes
transition coefficients a(k) and b(k) 1
the
100 V (s) i t+ ( s - x ) K1 (s, u)duds
V(s)ds + � =
Im k :::; 0
0
the
b(k)
transition
log ( l
+ lb(p) 1 2 ) dp p-k
=
have
� /_: B (t) e-iktdt,
2k
coefficient
} rrn l=l
the represen
a (k)
s at is fi es
the
k - i xt . k + i xt
2 . 2 . Eigenfunction expansion. Here we explicitly construct the resol vent kernel for the one-dimensional Schrodinger operator H , and show that it is self-adjoint with the domain consisting of functions '1/J E £ 2 (�) , which are twice differentiable on �, and such that -'1/J" + V (x)'I/J E £ 2 (�) . Using the method of complex integration we derive the eigenfunction expansion theorem for the operator H, which generalizes the corresponding result for the operator Ho of a free quantum particle, consi ere in Section 2 . 3 of Ch apter 2 . For A E C \ [0, oo ) let
(2.21)
R>, (x, y) =
d d
{
_
JI (x, k ) h (y, k) 2ika (k) JI (y, k) h (x , k ) 2ika (k)
if x
2:
y,
i f X :::; y,
where the branch of the function k J:\ on C \ [0, oo ) is defined by the condition that Im k > 0. For fixed x and y the function R>.. (x, y) is mero morphic on C \ [0, oo ) with simple poles at Az = - xf , l = 1 , . . . , n. For fixed =
3.
1 64
Schrodinger Equation
A =/= Az the function R;.. (x, y ) = R;.. (y, x) is continuous in x and y , and it follows from part ( v ) of Theorem 2 . 1 and (2. 15) that e - I m kl x-yi (2. 22) , C > 0. I R;.. (x , y ) l ::::; C
lkl
The kernel R;.. (x , y ) for A =I= Az defines a bounded integral operator R;.. in L 2 (JR) by the formula ( R;.. 'I/J ) ( x)
=
j_:
R;.. ( x , y )'I/J (y )dy .
Indeed, it follows from (2. 22) that for 'ljJ E L 2 ( JR ) ,
l k i 2 II R;.. 'I/J I I 2 ::::; C 2
=
C2
i: i:
j_: (j_: i:
e - Im k(IYl l + ly2 1)
e - I m k l x-yi i 'I/J (y) l dy
)
2
dx
1 '1/J ( x + Y I )'I/J ( x + Y2 ) l dx dy1 dY
2
::::; 4C2 (Im k)- 2 I I 'I/J I I 2 .
In particular,
2C I I R;.. II ::::; l k Im k l Lemma 2 . 1 . The operator H is self-adjoint and R;.. = (H - AI) - 1 for A E C \ { [0, oo ) U {,\1 , . . . , An } } , where I is the identity operator in L 2 ( JR ) . Proof. Let g E L 2 ( JR) . As in the variation of parameters method, it follows from ( 2 . 2 1 ) that for A E C \ { [0, oo ) U {..\ 1 , . . . , An } } the function y = R;.. g E L 2 ( JR ) is twice differentiable a.e. on lR and satisfies the differential equation Thus (H - AI) R;.. ���. Let
=
-y
"
+ V (x) y = AY + g (x) .
I and, in particular, Im(H ± ii)
=
L 2( JR ) , so that H is
no be the Hilbert space of C 2 -valued functions
. ) ( '1/J, cp) /_:'1/J (x )cp (x )dx -I; /_: cp (x ) (fooo (u 1 (x , k) /_: '1/J( y ) u 1 ( y , k )dy + u ,(x, k ) 1: ,P( y ) u 2( y , k)dy )dk )dx + t � ' (,P, ,p, ) (,p, ,
. E C \ [0, oo ) set g ( x, >.) = ( R;.. 'I/J ) (x ) . The function g(x, >. ) for fixed x is meromorphic in >. E C \ [0, oo ) with simple poles at >. = >.1 , l = 1, . . . , n . It follows from ( 2 . 2 1 ) and Proposition 2 . 1 that
Res ;.. = ;.. 1 g (x , >. ) = Since so that
( = - lf���� 'I/J ( x ). /_: (::z , ixt)'I/J(y)d h(x, ixz) h( y y z a )
HR;.. = R;.. H I + >.R;.. , we have -g" ( x, >.) + V ( x ) g ( x, >. ) = =
'1/J (x) + >.g(x , >.) ,
g ( x, >.) - ).1 '1/J(x) ).1 ( R;.. cp ) ( x) , where cp = -'1/J" + V (x )'I/J E Co (lR) . From (2.22) we get
+
=
I ( R;.. cp ) (x) l ::; c I Y>:I
so that ( 2 . 27)
as
g ( x, >. ) = - � '1/J (x) + 0 ( 1 >. 1 - 3 / 2 )
1 >. 1 -
It follows from Proposition 2 . 1 and ( 2 . 2 1 ) that g(x, >.) = 0 ( 1 >. 1 - 1 / 2 ) as ( 2 . 28)
as
>.
oo ,
1 >. 1 __,
0.
oo .
For 0 < c < 1 < N let C Ce, N be the contour consisting of the following parts: (i) the arc Ce of a circle >. = c:eie , c ::; I arg 0 I ::; 1r, oriented clockwise; (ii) the arc CN of a circle >. = Ne i9 , eN ::; I arg tl l ::; 1r , oriented anti-clockwise, where N sin c N = c sin c:; (iii) the segments I± of the straight =
2.
One-dimensional Schrodinger equation
167
lines Im ,\ = ±E sin E which connect the boundaries of the arcs. Choose E and N such that all poles A t of g (x , .-\ ) are inside C , and consider the integral 1 I _ g (x, .-\ ) d.-\ . 21r2 lc On one hand, by the Cauchy residue theorem .
=
I=
{
n n z l: Res A=A 1 g (x , .-\ ) = L ('I/; , '1/J 2) '1/Jz (x) . I I 'I/J l=l
z ll
l=l
On the other hand , it follows from (2 . 27) and (2.28) that
N-+limoo � JcNr 21r2
g (x, .-\ ) d,\
=
- '1/; ( x)
and
Thus we obtain '1/J (x) ( 2 29 )
t I I 'I/J1 I I l=l
.
l
=
where
2
('1/J, '1/Jz )'I/Jz (x)
1 = lim lim -. c-+0 N -too 2 1r2
lim
r g (x, ,\ ) d.-\ = 0. � 21r2 Jc£
(}If7.
g (x , .A)d.A - f g(x, .A)d.A
c-+0
+
)
}I7.
-
2�i fooo (i: ( RA+io (x, y) - RA -io (x , y) ) 'l/; ( y) dy) d.-\ , RA±io (x, y ) = lim RA±ic (x , y) . c-+0
To compute the difference RA + io (x , y ) - RA -io (x , y) , observe that on the cut ,\ 2: 0 we have y',\ + = k 2: 0 and y',\ = - k � 0. It follows from (2.21) that for x 2: y 1 !I (x , k) h (y, k ) + JI ( x , - k ) h ( y , - k ) RA + t. O ( x, y ) RA-t. O ( x , y ) ' i 2 k a ( - k) a (k) and using the equations 1 b ( - k) fi (x , k) = - h (x, - k) !I (x, -k) , a ( - k) a( k) 1 b ( -k ) h ( y, k) , h ( y, - k ) = a ( fi (y , k) + a (k) k) which follow from (2. 1 1 ) and (2. 13) , we obtain 2 RA +io (x, y) - RA-io (x , y) = 2 (JI (x , k) JI (y , k) + h (x, k) h ( y, k ) ) 2 k i a (k ) l
iO
_
_
_ _
=
(
iO
)
i (u 1 (x, k )ui (Y , k) + u2 (x , k )u2 ( y, k) ) ,
2k 2 where ,\ k . Substituting this into (2.29) and using the symmetry R A (x, y) RA ( y , x) of the resolvent kernel, we get the eigenfunction expansion (2.25) . Let %'ac be the restriction of the operator %' to the subspace £ac = (I - P)£. It follows from the completeness relation that the operator %'ac =
=
168
3. Schrodinger Equation
is an isometry, so that Im %' = Im %'ac is a closed subspace of Sj o . Thus to verify the orthogonality relation %' %' * = Io , it is sufficient to show that Im %' = Sjo . Using integration by parts we easily get that a self-adjoint operator %'ac H %'a�1 is a multiplication by A operator in Im %' with the domain %' D (H ) . Moreover, %' Rll- = R�0 ) %' , f.L E tC \ [0, oo ) , o
where R� ) is the resolvent of the multiplication by A operator in Sj 0 , so that o Im %' is an invariant subspace for R� ) . Now it follows from Lemma 2 . 2 in Section 2 . 2 of Chapter 2 that there are Borel subsets E1 , E2 � [0 , oo ) such that Im %' , �= E Sjo . 4? =
(��)
= { (�!:��)
}
If, say, Ei = [0, oo ) \ E1 has positive Lebesgue measure, then for A = k 2 we have u 1 (x , k ) 'l/J (x ) dx 0 for all '1/J E C6 (IR) , so that u (x , k ) Sj o .
I: =
E
Ei
=
0 for all x
E
lR
- a contradiction. Therefore, Im %'
0
Let o//o £ Sjo be the corresponding unitary operator for the Schrodinger operator Ho of a free particle, constructed in Section 2 . 3 of Chapter 2 ( with fi 1 and m ! ) , and set U = %' * %'o : .Yt' -t �c · :
-t
=
=
Corollary 2 . 4 . The restriction of H to the absolutely continuous subspace �c is unitarily equivalent to Ho , HI = UHoU-1 . 'JU) v.'lj;.
E £ we have ( W± cp, '1/J)
=
tEToo ei>..t ( e -itHocp, '1/J) = 0
2. One-dimensional Schrodinger equation
171
by the Riemann-Lebesgue lemma, as we have seen in Section of Chapter thatAssuming that wave operators exist and satisfy it is easy to prove for anyitmeasurabl e functi on fthios property Indeed,foraccordi ng toonstheJ spectral thefor orem, i s suffi c i e nt to prove the functi >. all In this case, immediately follows from the identity by passing to the limits Using and we conclude sounithat the wave operators tary restri equivcaltieoncens ofbetween and to the subspace .ffac establish the The physi c al meani n g of the wave operators i s the fol l o wi n g. The one-parameter group particle of uniin tary operators describesfitheeld.evoFor lluargetion of thethe quantum parti c l e wi t h posi t i v e energy moves far away from the cen ter and as whichitcorresponds s evolution tois descri bedmotiby othen. Mathemati one-parameter groups is the free c al l y thi expressed there exists a vector such that by the fact thatlim for every 0, and such a vector issatigisvfienes by Similarly, for every the vector lim 0. Asis expla physi calbyintheterpretati on, fororthogonal ity ofbound Im statesto theare subspace £pp a i n ed fact that al l ti m es l o cal i z ed near the potential center, whereas the free quantum particle goes to infinity as 2.3
2.
(2.33) ,
j ( H ) W±
(2 .34)
T E
n R
R
=
W± J (Ho)
( )
(2.34) t
--t
±oo.
(2.3 4 )
=
e ir>.
(2.32)
W± HW± = Ho ,
H I J""" A, 1/J 1 ( x , t) = 1/Jl (x, t)
=
fooo fooo 'Pl (k) (eikx-ik2 t + S21 (k) e -ikx-ik2t ) dk,
when
x
< -A.
Using the method of stationary phase ( see Section 2 . 3 in Chapter 2 ) we get as t --+ -oo, when x > A, when
x
< -A,
when
x
> A,
and as t --+ oo,
when x < -A.
1 74
3. Schrodinger Equation
Assuming that the neighborhood Uo is "sufficiently small" , we see as in Section 2 . 3 of Chapter 2 that as t ---+ - oo the solution 7/J 1 ( x , t) represents a plane wave with amplitude I 'P 1 (ko ) l moving from x = - oo to the right, to ward the potential center, with velocity v 2ko . When t ---+ oo , the solution 7/J1 ( x, t) is a superposition of two plane waves: the transmitted wave with amplitude 1 81 1 (ko ) l times the original amplitude, located to the right of the potential center and moving toward oo with velocity v , and the reflected wave with amplitude factor I 821 ( ko) I , located to the left of the potential cen ter and moving toward -oo with velocity - v . Corresponding transmission and reflection coefficients are T = 1 8u (koW and R l 821 ( ko ) l 2 . Analogously, we have as t - oo , =
,
7/J2 ( x t) 7/J2 (x, t)
=
=
�l t l 'P2 (-�) 2t
v
=
---+
e
1 O(ltl- ),
i:t2 -� +
O(ltl -1 ) ,
when x > A, when x < -A,
and as t ---+ oo , 7/J2 (x, t )
=
7/J2 (x , t )
=
---+
� (; ) (; ) i:t2 + � (-; ) ( ; ) e -� + o (t - 1 ) , 'P 2
812
t
'P 2
t
822
t
-
e
t
- !f i:t2
O (c 1 ) ,
when x > A, when x < -A.
As t - oo , the solution 7/J2 (x, t) is a plane wave with amplitude I 'P2 (ko) l moving from x = oo to the left, toward the potential center, with velocity - v . When t ---+ oo , the solution 7/J2 (x, t) is a superposition of two plane waves: the transmitted wave with the same amplitude factor l 822 ( ko ) l , located to the left of the potential center and moving toward -oo with velocity - v , and the reflected wave with amplitude factor l 81 2 ( ko ) l , located to the right of the potential center and moving toward oo with velocity v . Formulas ( 2.40) show that transmission and reflection coefficients for the solution 7/J2 ( x, t) are the same as for the solution 7/J 1 ( x, t ) , and are independent of the direc tion of travel. This is the so-called reciprocity property of the transmission coefficient.
The solutions u 1 (x, k) and u2 (x, k) of the stationary Schrodinger equa tion have the property that corresponding solutions 7/J1 (x, t) and 7/J2 (x, t) of the time-dependent Schrodinger equation become the plane waves as t ---+ - oo . In accordance with this interpretation, we denote them, respec tively, by u i-) ( x, k) and u �-) (x, k) . Solutions of the stationary Schrodinger equation, u i+) ( x, k ) = u2 (x , k) u2 ( x , - k ) and u�+ ) ( x , k) = u1 ( x , k) u1 (x, - k) , admit a similar interpretation: they correspond to the solutions of the time dependent Schrodinger equation which become the plane waves as t ---+ oo . =
=
2. One-dimensional Schrodinger equation
Introducing
U+ (x ' k)
=
(
ui+) (x, k) u 2( + ) ( x, k)
)
1 75
(
(x, k) and U ( x, k) - ui-) -) ( u (x, k) _
_
2
)
we get by a straightforward computation using (2. 1 1 ) and (2. 13) that (2 .4 1 )
The following result establishes equivalence between stationary and non stationary approaches to scattering theory.
Theorem 2 . 5 . For the one-dimensional Schrodinger operator H = Ho + V with potential V(x ) satisfying condition (2.2) the wave operators exist and
are given explicitly by
W± o/t'± OU'o . Here o/t'± £ --> SJo are the integral operators given by =
:
o//± ( 7/l ) :
=
-->
1
rn= Y
100 7/J(x) U± (x, ..f>.) dx,
2 7r - oo
and OU'o £ SJo establishes the unitary equivalence between the operator Ho in £ and the multiplication by A operator in SJo, OU'o ( 7/l ) =
00 ( 1 J2-rr 7/J (x ) 1
- oo
e - iv'>:x e iv'>:x
)
dx .
The scattering operator S in £ and the multiplication by the scattering matrix S ( ../).) operator in SJo are unitarily equivalent: Proof. The formula for the scattering operator immediately follows from the definition of the operators o//± , formula (2.41) , and orthogonality relation o/t'_ o/t'� Io ( see Theorem 2.3) . To prove that W± = OU'± OU'o , it is sufficient
(��)
=
to show that for all
0, we have X + 2 t - X + 2a t x rJAoo I loroo ry(k ) e -i(kx+k2t) dk l 2dx :::; c2 }Aroo (x +d2at) 2 A + 2 a( :::;
+
:::;
+
2.
177
One-dimensional Schrodinger equation
Choosing t > �
2C2 , we obtain J�+) ( t) a€
< E.
Other integrals are treated simi0
Corollary 2 . 6 . The wave operators satisfy orthogonality and completeness relations W± W± = I and W± W± = I - P.
Proof. The result of Theorem 2.3, proved for the operator %'
=
%'_ , ob viously holds for the operator %'+ . Thus relations (2.32) and (2.33) follow from the corresponding orthogonality and completeness relations for the op erators %'± . 0 Remark. The orthogonality relation (2.32) is trivial once the existence of
the wave operators is established. Thus Theorem 2.5 gives another proof of the orthogonality relation in Theorem 2.3. Problem 2. 7 ( The Cook's criterion ) . In abstract scattering theory, prove that the wave operators W± exist if for all
cp E
.Jit',
loo I I Ve- itHo 'P i i dt
2a, and V(x) V0 > 0 for 0 � x � 2a. Show that when E varies from 0 to Vo , T increases from 0 to (1 + V0a 2 ) - 1 - the penetration of a potential barrier by a quantum particle. =
2.4. Other boundary conditions. Here we consider two examples of the
Schrodinger operator
H = d 2 + V (x) x with the potential V(x) having different asymptotics as x -
d2
Example 2 . 1 . Suppose that the potential
j_� (1 + ixi) j V (x) - c2 i dx
0. The Schrodinger operator H has negative discrete spectrum, consisting of finitely many simple eigenvalues ..\1 < · · · < An < 0, and ab solutely continuous spectrum [0, oo ) , which is simple for 0 < ). < c2 and is of multiplicity two for ). > c2. This qualitative structure of the spectrum is determined by the structure of the level sets Hc (P , x) ). of the classi cal Hamiltonian function H (P x ) = p2 + V (x ) : the eigenvalues could only appear for compact level sets, when the classical motion is periodic, while the values of ). with non-compact level sets, where the classical motion is c
,
=
3.
1 78
Schrodinger Equation
infinite, belong to the absolutely continuous spectrum. The absolutely con tinuous spectrum has multiplicity one or two depending on whether the corresponding classical motion is unbounded in one or in both directions, i.e . , when 0 < A < c2 or A > c2 . These results can be proved using methods in Section 2 . 1 . Namely, set A = k2 and define the function k1 = v'k 2 c2 by the condition that Im k1 � 0 for Im k � 0. In particular, sgn k1 = sgn k for k real, l k l > c. The differential equation -
- y" + V (x) y = k 2y
(2.4 3 )
for real k has two linear independent solutions ft (x , k) , ft (x, - k) = ft (x, k) , where ft (x, k) has the asymptotics ft (x, k) = eikx + o ( 1 )
x ---> oo .
as
For real k , l k l > c, equation (2.4 3 ) also has two linear independent solutions h (x, k ) , h (x, -k) = h (x, - k ) , where h (x, k) has the asymptotics - oo . h (x , k) = e - i k1 x + o ( 1 ) as x ( Actually, these solutions satisfy estimates similar to that in Theorem 2 . 1 . ) A s i n Section 2 . 1 , for real k , l k l > c , we have ___,
h (x, k) = a (k) ft (x , - k) + b (k)ft (x, k) ,
where (2 .44) a ( k ) =
1
2i k
W ( ft (x , k ) , h (x , k) ) , b ( k ) =
1
2i k
W ( h (x , k ) , ft (x , - k) ) ,
and a ( -k) = a ( k ) , b( - k ) = b (k) . However, W ( h (x, k) , h (x , - k ) ) = -2ikl , so that for real k , l k l > c, we have h ( x , k) =
k k a ( k) h (x, - k) - b ( - k) h (x , k ) . kl kl
From here we obtain the normalization condition i a ( k W - l b (k) l 2 =
�1 ,
l k l > c.
For fixed x solutions ft (x, k) and h ( x , k) can be analytically continued to the upper half-plane Im k > 0. For -c < k < c solution h (x, k) is real valued and satisfies h (x , k) = a (k) ft (x, - k) + a( k ) ft (x , - k) ,
where a ( k) is still given by the same formula (2 .44) . The function a (k) does not vanish for real k and admits meromorphic continuation to the upper half-plane Im k > 0 where it has finitely many pure imaginary simple zeros i;.q , . . , i xn which correspond to the eigenvalues A1 = - xi , , An = - x� . .
. . .
2.
1 79
One-dimensional Schrodinger equation
The absolutely continuous spectrum of the Schrodinger operator H fills For 0 < A = k 2 < c2 the spectrum is simple and
[0 , oo )
.
1
u(x, A) =
a(k)
h (x, k) ,
0 < k < c,
are the corresponding normalized eigenfunctions of the continuous spectrum. For A = k2 > c2 the spectrum has multiplicity two and u1 (x, A) =
1
a (k)
k > c,
h (x , k ) ,
are the corresponding normalized eigenfunctions of the continuous spectrum. Denoting by 'l/Jz ( x) the normalized eigenfunctions of H corresponding to the eigenvalues Az , we get the eigenfunction expansion theorem: for 'lj; E L2 (IR) , 'lf; (x) =
+ where Cz
C(k) = Example
Cz'l/Jz (x) +
100 (C1 (k)u1 (x, k)
( 'lj;, 7/Jz ) , l = 1 ,
=
� .
.
.
, n,
1: 'lb (x)u(x, k)dx, 2.2.
decays as x
+ C2 (k)u 2 (x, k) ) dk,
and
Cj (k) =
1: 'lj; (x) uj (x, k)dx,
j = 1, 2.
Suppose that the potential V (x) grows as x �
� oo :
100
and there exists x o problem -y"
lac C(k)u(x , k) dk
(1
E
+
- oo
and
lx i ) I V ( x) l dx < oo for all a ,
IR such that the spectrum of the Sturm-Liouville
+ V(x)y = Ay,
- oo
<x
:S
xo ,
and y' (xo)
=
0
is bounded below and discrete. The latter condition is a quantitative formu lation of the property limx-+ - oo V(x) = oo . Then for every real k there exists a solution f ( x, k) of the differential equation (2.43) with the asymptotics f (x, k) = ei kx + o ( 1 ) as x
�
oo ,
and there is a function s ( k) such that u(x, k) = f (x, - k) + s (k) f (x, k)
180
3. Schrodinger Equation
is square integrable over ( - oo , a ) for any real a. The Schrodinger operator H has simple absolutely continuous spectrum [0, oo ) and discrete spectrum con sisting of finitely many negative eigenvalues. Functions u ( x , k) are normal ized eigenfunctions of the continuous spectrum and the corresponding eigen function expansion theorem has the following form: for every 'ljJ E L 2 (JR) , ,P (x ) =
t,
C1,P1 ( x)
where Cz = ( 'l/J , 'l/Jz ) , l = 1 , . . . , n , and C (k) =
/_:
+
J.oo
C ( k) u ( x, k ) dk ,
'lj; ( x ) u ( x , k )dx .
Problem 2 . 1 0 . Give an explicit form of the eigenfunction expansion theorem for the potential V ( x)
=
ex .
0.
Problem 2 . 1 1 . Find energy levels for the Morse potential V(x) a >
=
e-2x - 2e-x ,
2 . 1 2 . Give an explicit form of the eigenfunction expansion theorem for the potential V (x) Fx, describing the motion of a quantum particle in a homo geneous field; according to Problem 1 .4, the corresponding Schrodinger operator is self-adjoint. (Hint: Solve the Schrodinger equation explicitly in the momentum representation, and express normalized eigenfunctions of the continuous spectrum in the coordinate representation in terms of Airy-Fock functions, defined in Section 6.2 . )
Problem
=
Problem 2 . 1 3 . Give an explicit form of the eigenfunction expansion theorem for the potential V ( x) = - � kx 2 , where k > 0 ( according to Problem 1 .4, the corresponding Schrodinger operator is self-adjoint ) .
3 . Angular momentum and 80 (3) 3 . 1 . Angular momentum operators. In Chapter 1 ( see Example 1 . 10 in Section 1 .4) for a classical particle in JR 3 we introduced the angular mo mentum vector Me x x p with components =
Mel = X2P3 - X3P2 , Me2 = X3P l - XlP3 , Me3 = X1P2 - X2Pl · According to Example 2 . 1 in Section 2.6 of Chapter 1 , they have the fol lowing Poisson brackets with respect to the canonical Poisson structure on T * JR3 :
(3.1) {Mel , Me2 } = - Me3 , {Me2 , Mea } = - Mel , {Me3 , Mc l } = -Me2 · The square of the angular momentum M; M'A + M-;2 + M;3 satisfies =
{M; , Mc1 } = { M; , Me2 } = {M; , Mea } = 0 .
3. Angular momentum and S 0 ( 3 )
If the Hamiltonian function
Hc (p , x )
181
= 2pm2 + V (x )
is invariant under rotations, V ( x ) = V ( I x I ) , then components of the angular momentum are the integrals of motion {He , Mel } = {Hc , Mc2 } = {Hc, Mc3 } = 0 and {Hc , M; } = 0. This can also be verified directly using Poisson brackets ( 3 . 2 ) {Mcj , pk } = -Ejk lPl and {Mcj , X k } = -Ejkl X l , i, j, k
=
1 , 2, 3,
=
where Ejk l is a totally anti-symmetric tensor, E 1 23 1 . Correspondingly, i n quantum mechanics the components o f the angular momentum operator M = Q x P are defined by
M1 = Q 2 P3 - Q 3 P2 , M2 = Q3P1 - Q1P3 , M3 = Q 2 P3 - Q3P2 , where Q = ( Q1 , Q 2 , Q3) and P = ( P1 , P2 , P3) are, respectively, coordi nate and momentum operators. Since operators Qi and Pk commute for i i=- k, there is no ordering problem when defining quantum angular momen tum operators. It ollows from Heisenberg commutation relations that their quantum brackets are the same as the corresponding Poisson brackets ( 3 . 1 ) :
f
( 3. 3)
{M1 , M2 }n = -M3 , Equivalently,
{M2 , M3 }n = -M1 ,
( 3.4)
{M3 , MI }n = -M2 .
[M1 , M2] = iliM3 , [M2 , M3] ihM1 , [M3 , M1 ] = ihM2 . The operator of the square of the total angular momentum M 2 M'f Mi + M§ satisfies =
=
+
Correspondingly, for the Hamiltonian operator p2 H V (Q)
= 2m +
with spherically symmetric potential V(x) V(lx l ) operators M1 , M2 , M3 and, therefore, M 2 , are quantum integrals of motion: =
This can be verified directly by using quantum brackets
{Mj , Pk}n = - Ejkt Pz , { Mj , Q k } n = -Ejkl Qz , j, k , l = 1 , 3, which are the same as Poisson brackets (3.2), and follow from Heisenberg commutation relations. (3 . 5)
2,
3.
182
Schrodinger Equation
2
In the coordinate representation Yf' L (JR3 , d3x) the operators of an gular momentum are given by the following first order self-adjoint differential operators: =
(3.6) (3 . 7) (3.8)
They have the property that M1 'ljJ = M2 '1/J = M3 '1j; 0 for any spherically symmetric smooth function '1/J (x) = '1/J ( I x l ) . In other words, angular momentum operators act only on the angle coordinates in JR3 . Namely, let x 1 = r sin '!9 cos 0. Then for every ( 4 . 6)
H = -� + V(x) V (x)
k E JR3 the Schrodinger equation
u(±)(x, k) satisfying the following asymptotics as l x l --+ oo: e±ik-r + o ( 1 ) , u(±) (x, k) = eik:c + f(±) ( k ,w, n )(4. 7) -;. r where k = k w, x = rn. Asymptotics (4.7) are called Sommerfeld's radia tion conditions. To prove the existence of solutions u (±)(x, k), one should consider the following integral equations: u(±)(x, k) = eik:c + }[{3{ Q(±)(x - y, k ) V ( y)u(± ) ( y , k) d3y , (4 .8) has two solutions
4.
187
Two- body problem
where c( ± ) (x, k) =
1
e ±ikr
41!'
r
- - -- .
Integral equations ( 4.8) are equivalent to the Schrodinger equation ( 4.6) with Sommerfeld's radiation conditions ( 4. 7) , and are called Lippman-Schwinger equations. Their analysis uses Fredholm theory and Kato's Theorem 1 .8. Solutions uC±l (x, k) are called stationary scattering waves. They are analogous to the solutions uj± l (x, k) for the one-dimensional case, where the vector w E 8 2 replaces the index j 1 , 2. The absolutely continuous spec trum of H fills [0, oo) and has a uniform infinite multiplicity, parametrized by the two-dimensional sphere 82 . Solutions uC ± l (x, k) are normalized eigen functions of the continuous spectrum. In general, the operator H has finitely many negative eigenvalues At < 0 of finite multiplicities m1 , l 1 , . , n . In terms of the operators fl'± £ £ given by =
=
:
fl'± ('l/J) (k)
=
--->
. .
r 'l/J (x)u(±) (x, k)d3 x,
(211') - �
JJR3 completeness and orthogonality relations take the form
0. Show also that in this case the Schrod inger operator
m this that
Problem 4.2 (Born's approximation) . Show that -t oo ,
and deduce fro
f(k, n , w) � JJR3r uniformly on n, w S 2 . +
E
471'
u( + l ( x ,
eik(n-w)x v ( x ) d3 x = o( 1 )
as
k) = eikx k --+
+
o(1)
as
oo ,
4 . 3 . Prove unitarity of the operator S using Schrodinger equation ( 4.6) , radiation conditions (4. 7) , and the Green's formula.
Problem
Problem 4.4 ( The optical theorem ) . Show that
fs2 1 f (k , n, w Wdn � Jmf(k,w,w). =
( Here the left-hand side is the total cross-section in direction E =
k2. )
w
at the energy
Problem 4 . 5 . Let Az = -x[ < 0 be the eigenvalues of H with multiplicities m1 , l 1 , . . . , n . Prove that for fixed x and solutions u(±l (x, k) admit a meromorphic continuation to the upper half-plane > 0 w i th poles of orders m1 at ix1 , l 1 , . . . , n.
wImk
=
=
Problem 4 . 6 . Prove that the wave operators W± exist using the non-stationary approach. ( Hint : Show that when V (x) satisfies (4.5) , Cook's criterion, formulated
in Problem 2 . 7, is applicable. )
4 . 3 . Particle i n a central potential.
Schrodinger operator H
The eigenvalue problem for the
n2
V (x) 2p, simplifies, when H commutes with the 80 ( 3 ) -action in Yt' L 2 (JR3 , d3 x) : [H, T(g)] = 0 for all g E 80 (3) . This reduces to the conditions (4 . 9 ) [ H, Mi ] 0, i = 1 , 2, 3, and is equivalent to the property that the potential V is spherically sym metric, V (x) = V(r) , r = l x l . In particular [H, M3 ] = [H, M 2 ] 0, where M2 = M'f + Mi + Mj , and operators M3 and M 2 are commuting quantum integrals of motion for the Hamiltonian H . As follows from results in Section 3.2, one can look for the solutions of the eigenvalue problem H'lj; = E'l/J =
--� +
=
=
=
4.
189
Two-body problem
satisfying
. . . , l.
M2'1j; = n2 l ( l + 1) '1j;, m = -l, Using (3. 9 ), n,2 o ( o ) M H = - - - r 2 - + -2 + V (r), 2p,r2 or or p,r2 so that in accordance with the decompositions (3 1 4) and (3.15), we look for the solutions in the form rn, '!f;( x ) = Rt(r) Yim ( n ) , where Yi m are normalized spherical functions. Separating the variables, one gets the following ordinary differential equation for the function Rz ( r): � :!__ ( dRz ) n2 l ( l + 1) r Rl + V ( r)Rl = ERl · 2 p,r dr 2 dr + p,r 2 Introducing ft(r) = rRt(r), we obtain the so-called radial Schrodinger equation _ 2np,2 dd2r�� + n,2ip,rl � 1) ft V(r)fz = Eft. (4. 10) Since for the continuous potential V ( x ) the solution 'lj; ( x) is also continuous, equation (4.10) should be supplemented by the boundary condition ft ( O ) M3 'lj; = m'lj; and we get
.
X =
2
+
=
0.
The radial Schri:idinger equation looks similar to the Schri:idinger equa tion for a one-dimensional particle, if one introduces the so-called effective potential - v( + n l l + Veff r 2 where the second term is called the centrifugal energy. However, since is defined only for r > 0, equation is equivalent to equation with the potential satisfying = oo for < 0, which describes the infinite potential barrier at = Since the radial Schri:idinger operators ..:!!__ n2 l l + 1 ) H 2 are obtained from the three-dimensional Schri:idinger operator
r) 2 (p,r 2 1) ' ft (4.10) (2.1) (x) x V x 0. z - - 2n,p,2 dr2 + V (r ) + (p,r 2 n,2 + V (r) H = --� 2p, by separation of variables, the operators Hz with boundary condition fz (O) = 0 are self-adjoint in L2 ( 0, ) whenever H is a self-adjoint operator in L 2 (IR3 ) . TT
oo
()
1 90
3. Schrodinger Equation
In particular, it follows from Theorem 1 . 9 in Section 1 . 2 that when the bounded potential V satisfies V (r) = O(r-l-c) as r - oo for some c > 0, then Hz are self-adjoint operators with simple absolutely continuous spectrum filling [0, oo ) , and negative eigenvalues with possible accumulation point at 0. If V(r) = O(r-2-c) as r - oo, then the operators Hz have only finitely many negative eigenvalues, and for l large enough have no eigenvalues at all. The same conclusion holds if
100 r j V(r) i dr
( 4. 1 1 )
0 for E > 0, and where > 0 for E < 0. When r 0, the most singular term in the radial Schri::i dinger equation is given by the centrifugal energy. Since the elementary differential equation !" l ( l + 1) f r2 has two linearly independent solutions r-Z and rZ+l , the solution fz (r) sat isfying the boundary condition fz (O) = 0 has the asymptotics (4. 1 3 ) fz (r) = Cr Z +l + o ( 1 ) as r - 0, and is uniquely determined ( up to a constant ) . Since fz (r) = Cd1+ (r) + C2 f1- (r) , for some constants C1 and C2 depending on E, the differential equation (4. 10) has no square-integrable solutions for E > 0. The corresponding so lution fz (r) is bounded on [0 , oo ) , and is an eigenfunction of the continuous spectrum. This agrees with the description in Section 1 . 6 of Chapter 1 , since for E > 0 the classical particle in the central potential Veff ( r) goes to in finity with finite velocity. For E < 0 the equation C1 (E) = 0 determines the eigenvalues of Hz , which are simple. This also agrees with the classi cal picture, since classical motion is finite for E < 0. For a short range potential the equation C1 ( E ) = 0 has only finitely many solutions. When -
=
4.
191
Two-body problem
Veff(r) > 0 for r > 0 - the case of repulsive potential - the operator Ht has no eigenvalues. Remark. When V (r) = 0, the radial Schrodinger operator Ht has only simple absolutely continuous spectrum filling [0, oo ) . The substitution
2�-tE > o , Jf, J (O , � = n and k = lxl = v� reduces differential equation ( 4 1 0 ) to the Bessel equation
ft (r)
kr
.
=
2 2 � 2 ddeJ + � dJ d� + (�
1/
2) J = 0
of the half-integer order v = l + ! · The corresponding solution regular at 0 - the Bessel function of the first kind Jl + ! (� ) - is given explicitly by
Jl + ! ( � ) 2
and
-
2
=
( � d� ) l sin� �
( - 1 ) l v{2 �l + l � �
;(,
(4.14)
It can be shown that
satisfy the normalization condition (2. 14) in Section 2.2 of Chapter 2:
(
)2
1 r>O k+b. li�0 � Jo Jkr !Et (r)da(E) dr = 1 , where k = .j2jiE and da(E) dE dk. The corresponding eigen function expansion theorem is the special case of the classical Fourier-Bessel transform for integer l, which generalizes the Fourier sine transform: for every f E £ 2 (0, oo ) , (4 . 15)
=
�
=
f(r) = fooo ct ( E) fEt (r)da(E) , Ct (E) =
.
100 f(r ) fEt (r) dr.
In general, for every l = 0, 1 , . . , denote by !Et ( r ) the solution of the radial Schrodinger equation ( 4 . 1 0 ) with the following asymptotics: {2 kr l1r - 2 + 8t + o(1) as r -t 00 . (4. 16) fEt ( r) = v � sin fi:
(
)
The function 8t (k) , k = .j2j)E, is called the phase shift. It follows from (4.14) that for the free case 8t ( k) = 0. The eigenfunctions of the continu ous spectrum !Et (r) satisfy the same normalization condition (4. 1 5) as the
192
3. Schrodinger Equation
eigenfunctions for the free case. The function Sz (k ) = e2iod k ) plays the role of the scattering matrix for the radial Schrodinger operator Hz . It admits a meromorphic continuation to the upper half-plane Im k > 0 with simple poles k = ixkz = i J-2J.1Bkz , where Ekz are the eigenvalues of Hz . Let Eoz < E2 z < < EN1 - 1z < 0 be the eigenvalues of Hz , and let fj z ( r ) , j = 0, . . . , Nz - 1 , be the corresponding normalized eigenfunctions. By the oscillation theorem, the eigenfunctions fjz ( r ) have j simple zeros in (0, oo ) . The functions ·
·
·
fjz r ) (n) , 1/Jj z m (x) = -r Yzm (
m =
-1, . . . , l ,
are normalized eigenfunctions of the Schrodinger operator H with eigenval ues Ejz , and the functions fEz ( r ) Yzm n ) , m = - 1 , . . . , 1 , 1/JEZm ( X ) = -( r are normalized eigenfunctions of the continuous spectrum. The correspond ing eigenfunction expansion theorem for the Schrodinger operator H with spherically symmetric potential states that for every 1/J E L 2 (R3, d3 x) , oo
Z
1
00
oo
Z N1- l
1/J(x) = L L Czm (E )'I/J Ez m (x)da(E) + L L L Cjz m'I/Jjz m(x), Z=O m= - Z j =O l=O m=-Z O where
function 1/Jjz m(x) the radial quantum number and denote it by nr . The pa rameter l is called the azimuthal quantum number, and the parameter m - the magnetic quantum number. This terminology originated from the old quantum theory, where to each value Ekz there corresponds a classical orbit. The parameter n = nr + l + 1 is called the principal quantum number, so that nr = n - 1 - 1 is always the number of zeros of the corresponding radial eigenfunction fjz ( r ) . Remark. In physics, it is traditional to call the parameter j in the eigen
Remark. In general, the eigenvalues
Ejz of a Schrodinger operator H with
spherically symmetric potential have multiplicity 21 + 1. For special poten tials, due to the extra symmetry of a problem, there may be "accidental degeneracy" with respect to the azimuthal quantum number l. This is the case for the Schrodinger operator of the hydrogen atom, considered in the next section. Problem 4. 7. Prove all results stated in this section. (Hint: See the list of refer ences to this chapter. )
5.
193
Hydrogen atom and S O ( 4)
Problem 4 . 8 . Find the energy levels of a particle with angular momentum l = 0 in the centrally symmetric potential well V ( r) = V0 < 0 when 0 < r < a and V (r) = O when r > a.
-
4.9.
Problem V (r)
=
ar- 2
+
Find the spectrum of the Schrodinger operator with the potential br 2 , a, b > 0.
Prove that the scattering wave u(x, k) = u
reduces (5.9) to
( 2 (l : 1) - 2i k) F{ + ( � - 2ik(�+ 1) ) Fl 0 . The solution of equation (5.10) satisfying Fz(O) = 1 can be explicitly written as Fz( r ) = F ( l 1 i>-., 2l + 2 , 2ikr) , where F ( a , ')' , z) is a confluent hypergeometric function, defined by the ab solutely convergent series (5 . 10)
F{' +
=
+
+
F(a, ')', z) =
� f (a + n)f(')') zn f='o f (a) f ( 'Y + n ) n!
for all a and 'Y i= . . . The confluent hypergeometric function is an entire function of the variable z and satisfies the differential equation
0, -1, - 2 , . zF"
+
8 This integral was evaluated
by
(!' - z ) F' - aF = 0 .
Sommerfeld in 1916.
3. Schrodinger Equation
198
For o: = - k and 'Y = p + 1 , where k, p = 0, 1 , 2, . . . , the confluent hypergeometric function reduces to associated Laguerre polynomials Qpk ( x ) = (k +p .1 p) ! F ( - k , p + 1 , x ) , considered in the previous section. For Re 'Y > Re o: > 0 the function F(o:, "f, z) admits the integral representation (5. 1 1 )
f('Y) { 1 ta- 1 ( 1 - tp-a- 1 e z t dt F(o: ' 'Y ' z) = r (o:)r('Y - o:)
lo
'
obtained by the Laplace method. The confluent hypergeometric function satisfies the functional equation (5. 12) F(o:, "f , z) = e z F( 'Y - o:, "f, -z) and has the following asymptotics as z ---> oo : (5. 13)
�=
( ) 1 ( - z ) -a ( 1 + O ( z - ) ) F(o:, "f , z ) = r o:) ( r( ) + 'Y ez za-�' ( 1 + O(z- 1 ) ) . r (o:)
It follows from (5. 12) that fz (r) r 1 +1e - i k r F (l + 1 + i>., 2l + 2 , 2ikr) is real-valued and it follows from (5. 13) that the function =
e 2 k lf(l + 1 - i>.) i fz ( r) !Ez ( r) = v'21f 2?T(2l + 1 ) ! has the following asymptotics as r ---> oo : l?T ( 5 . 15) fEz ( r) = y ; sm k r + >. 1og 2 k r - 2 + 6z , (2k)l+1
(5. 14)
where (5 . 1 6 )
[2 .
61 (k)
=
"
)
(
arg f(l + 1 - i>.) ,
>. =
� k
=
1 __
V2E '
is the phase shift. The partial S-matrix for the Coulomb problem is r(z + 1 - i>.) . s1 (k) = e 2i.)
It admits meromorphic continuation to the upper half-plane Im k > 0 and has simple poles at k = ixjz , which correspond to the eigenvalues Ekz of the radial Schrodinger operator H1 • The radial eigenfunctions of the continuous spectrum (5. 14) satisfy normalization condition (4. 1 5 ) . Remark. It is instructive to compare asymptotics (5. 15 ) for the Coulomb potential, which is long-range, with the corresponding formula ( 4. 16) for
the general short-range potential. The long range nature of the Coulomb interaction manifests itself by the extra logarithmic term ).. log 2kr in (5. 15 ).
5. Hydrogen atom and S0 ( 4 )
199
The eigenfunction expansion theorem for the Schrodinger operator H of the hydrogen atom has the same form as in Section 4.3: the eigenfunctions oo, and the eigenfunc 'lj;kzm (x) are given by ( 5 . 4 ) where k 0, 1 , . . . , Nz tions of the continuous spectrum are given by !Ez ( r ) ( 'lj;Elm ( x ) = -- Yz m n ) , l = 0 , 1 , . . . ' m -l, . . . l =
=
=
r
,
.
5 . 3 . Hidden SO ( 4) symmetry. As we have seen in Section 1.6 of Chapter
1 , classical system with the Hamiltonian function He (P, x)
=
p2
2m
a
- -:;: '
in addition to the angular momentum Me, has three extra integrals of mo tion given by the Laplace-Runge-Lenz vector
ax · p Me - We = x r m
According to Example 2 . 2 in Section 2 . 6 of Chapter 1 , the integrals Me and We for the Kepler problem have the Poisson brackets {Wej , Mek } = -Ejkl Wcl , { Wej , Wck } = 2HeEjkt Mcl ,
{Mcj , Mek } = -Ejkl Mcl ,
where j, k, l = 1 , 2 , 3 and c1 2 3 1 . The quantum Kepler problem i s the Coulomb problem. In the coordinate representation .Ye = L 2 (IR3 , d3x) , its Hamiltonian is =
H=
p2
-
2
-
a
-
r
'
where r = l x l and we put m = 1. The quantum Laplace-Runge-Lenz vector - the Laplace-Runge-Lenz operator W - is defined by aQ ' W = 21 ( P X M - M x P) - -:;:or in components Wj
aQ · = "21 Ejkt ( PkMz + Mz Pk ) - ----;:-
,
j = 1 , 2, 3.
Here Qi are multiplication by X i operators, and it is always understood that there is a summation over repeated indices 1 , 2 , 3. Using commutation relations (3.5) , we also have (5. 17)
. . = MXP-a Q + znP, a Q - znP W=PX M-r r
where the term i nP plays the role of a "quantum correction" . The following result reveals the hidden symmetry of the Coulomb problem.
200
3. Schrodinger Equation
Proposition 5 . 1 . The Schrodinger operator H of the hydrogen atom has
quantum integrals of motion M and W,
[H, Mi ] satisfying
W·M=M W ·
= =
[H, Wi ]
=
i
0,
=
1 , 2 3, ,
0 and
(5. 18) Moreover, self-adjoint operators M and W have the following commutation relations: We
=
know that [H, Mj] 0. To prove that Wj are quantum inte grals of motion, we first compute P 2 , . It follows from Heisenberg ' s commutation relations that Proof.
[ �j]
(5. 19)
2
so that using Leibniz rule and relations r = x i + x � + x � , [Mz , r ] Qj Pk - QkPj = Ejkt Mz , we obtain
[p 2 r ] Qj
, -
=
Q 2 Zn3 ·� j
r
+
=
0, and
Qk 2 ZIUo · � �jkl --:f Ml ·
r
Now using the first equation in (5.17) and (5. 19) , we get
2
It is easy to prove the relation W · M W1 M1 + W2 M + W3 M3 Indeed, it follows from the definition of M and commutativity of Pk and Qz for k f. l that =
M P ·
=
P·M=M
·
=
0.
Q Q M = 0, =
·
and using the first equation in (5. 1 7) and commutation relations for the components of the angular momentum, we immediately get W M 0. To prove that M · W one should use the second equation in (5.17) . =
0,
·
=
5.
201
Hydrogen atom and 80(4)
Verification of (5. 18) is more involved. We have
W 2 = Wj Wj
=
(
Ejkl Mt Pk -
� + )(
Ci j
ifiPj
= Ejkl ( EjmnMt Pk PmMn - a Mz Pk Qjr + Ci 2 =
Qj Qj Qj + . 't; - pj 2r
'lCin
r
+
· 't; _
- i/iMzPkPj ) - Ci€jm n
[ �J. j
Since operators Mj and Pj commute, the identity is still applicable and we get EjklEjmn Mz PkPmMn
=
- inPj
)
Qj PmMn q
Qj 't; 2 . 't; pj D + n pj pj 'llU;.jmn pj £m Mn - 'lCin
a 2 I + n2 P2 + EjktEjmn Mt PkPmMn - CiEjkl
- ian Pj ,
�
Ci j
EjmnPmMn -
r
( � +� MtPk
j
j
Pk Mt
)
( a X b) 2 = a2 b2 - ( a · b?
2 - (M x P) · (P x M) = M P2 - ( M · P ) 2 = M2P2.
Using (5. 19) we easily obtain (summation over repeated indices)
Putting everything together, we get (5.18) . It is also not difficult to establish commutation relations between Mj and Wk . Using (3.5) and properties of Ejkl , we get [Mj , Wk ]
=
[
Mj , Ekmn PmMn -
�
a k
- i liPk
]
= i li(EjmpEkmn PpMn + €jnp€kmnPmMp) - i/icjkl = ili(Pj Mk - MkPj ) - i licjkt
( Ci�t + )
( a;z + inPz ) = i ficjkl Wz .
i/iPz
Finally, to establish commutation relations between components of W, we use the representations (5.20)
aQ . a Q = P2 Q - (P · Q)P - W = Q P2 - P ( Q · P) - r r
The first formula in (5.20) follows from the first formula in (5.17) by using Cjkt PkMt = €jkt€lmn PkQmPz =
=
Pk ( PkQj - Pj Qk )
Ejkt ( Etjk PkQj Pk + Clkj PkQkPj ) =
Qj P 2 - Pj ( P
·
Q) - 2i liPj ,
3.
2 02
Schrodinger Equation
(5 .20) P Q Q PEj kt[Wk , Wz] = 2EjmnWmWn aQn aQm 2 = 2 Ejmn ( P 2 Qm - ( P · Q ) Pm - - ) ( Qn P - Pn ( Q · P ) - - ) 2M1 ( -P 2 ( Q P ) + ( P Q ) P2 + � ( Q P) - ( P Q ) �) = -2inM1 ( P 2 - �a ) = - 4 i nM1 H , i where we have used [M, P · Q ] 0 and [p Q , �] = : . This proves that [Wk , Wz ] = -2ihcjkl Mj H. Let £a = P H ( 0)£, where PH is the projection-valued measure for
and relation · 3 i fil . The second formula in follows from · the first by using Heisenberg's commutation relations. Now we have =
r-
=
·
·
·
=
r-
·
0
·
- oo ,
the Schrodinger operator H. Since self-adjoint operators M and W com mute with H, the subspace £a is an invariant subspace for these operators. The Schrodinger operator H is non-negative on £a , so that on this subspace the operator is well defined. Now on £a we set
( - 2 H) - 112
J(±) = !(M ± ( - 2H ) - 1 1 2 W ) . 2
5.1 that self-adjoint operators Jj±) on £a satisfy [Jj±) , Jk± ) ] = i!icj kl Jl( ±), [Jj+ ) , Jk- ) ] = 0
It follows from Proposition the commutation relations
(5.21)
and
(5 . 22)
(5. 2 1)
Equations are commutation relations for the generators of the Lie al gebra .so (4 ) , which correspond to the Lie isomorphism .so (4) .so (3) EB .so (3) , and exhibit the hidden SO ( 4) symmetry of the Coulomb problem! Together with ( 5 . 22 ) , they allow us to find the energy levels pure algebraically. Namely, the eigenvalues of the operators (J(+) ) 2 and are, respec tively, and that = b = and it follows from so that the corresponding eigenvalue of H is �
h2 h (h + 1) h2 l2 (l2 + 1), En = - 2 fia22n2 '
Assuming that
(5. 2 3)
£a �
EB
l E � Z?:o
(J
with the initial condition
'lj;n (q , t ) l t = O = cp ( q ) e K s (q ) .
It is assumed that the real-valued functions cp(q) and s (q) are smooth, s (q) , cp (q) E C00 (lR) , and that the "amplitude" cp(q) has compact support. The substitution ( 6 .2 ) is 'lj;n,(q, t ) = e K S (q ,t,n) , and differential equation ( 6.3) takes the form
(6 . 6 )
as + _...!_ (as) 2 at
2m
aq
+
V (q) =
.!!!__
a2s .
2m aq2
To determine the asymptotic behavior of S (q, t, n) as n ---> 0, we assume that as n ---> 0 S (q , t , n )
00
= 2 ) -intSn (q , t) , n=O
6. Semi-classical asymptotics
-
I
207 (6. 6 ).
and substitute this expansion into Comparing terms with the same powers of n we obtain that t ) satisfies the initial value problem __.!_ V( ) t and t ) l t=O whereas t) satisfies the differential equation
So (q, 8So + ( 880 ) 2 q = 0 (6. 7) 8 2m 8q + ' (6.8) So ( q , s (q) , S1 ( q, 8S1 + 1 8So 8S1 = - 1 82So (6·9) 8t m oq 8q 2m 8q2 ' the so-called transport equation , and (6. 10) SI (q , t) lt=O = cp ( q ) . 1 satisfy non-homogeneous differential equa The functions Sn ( q , t ) for =
(6.9).
n
>
tions similar to is the Cauchy problem for the The initial value problem Hamilton-Jacobi equation with the Hamiltonian function
(6. 7)-(6.8)
Hc (P, q)
considered in Section Section of Chapter of characteristics,
2. 3
=
p2 + V( q ) , 2m
2.1,3theof solution Chapter 1. According to Proposition 2. 1 in of (6. 7 )-(6. 8 ) is given by the method
So ( q , t) = s ( qo ) + lot L (T'(T))dT. Here L ( q , q ) ! mq2 - V ( q) is the Lagrangian function, and I ( T ) is the characteristic - the classical trajectory which starts at qo at time T = 0 with the momentum Po = �: ( qo ) , and ends at q at time t, where qo is uniquely determined by q. ( We are assuming that the Hamiltonian phase flow satisfies the assumptions made in Section 2. 3 of Chapter 1. ) It follows from Theorem 2 . 7 in Section 2. 3 of Chapter 1 that along the characteristic, 8So (q , t) = m dr (t) , oq dt
(6. 1 1)
T =
9t
so that
(6. 12) (6. 9)-(6.IR,10)defined for the transport equa in Section 2. 3 of
Now we can solve the Cauchy problem tion explicitly. Consider the flow 7rt : lR
�
3.
208
Schrodinger Equation
'Y(Q, q T Q = 1rt ( q ) T = t ; ) q Q 8So oQ (Q , t) = m O"'fat (Q , q ; t ) with respect to q we obtain o2So (Q t) aQ = m o2'Y (Q q t) = m d ( aQ ) , 8Q2 , oq oqot , ; dt oq so that (6.9) can be rewritten as d 1d aQ dt 81 (Q , t) = - 2 dt log oq ' and using (6. 10) we obtain l2 s1(Q, t) = cp( q) l aQoq (q) � -
q
Chapter 1 , and denote 11 by the characteristic connecting points at 0 and at ( under our assumptions the flow 7rt is a diffeomorphism and the mapping is one to one ) . Differentiating the equation
T=
t-t
Therefore,
1
'1/Jn (Q , t ) cp (q ) I�� (q) � -2 e * (S(Q,q;t)+s (q)) ( l O ( n) ) , where S( Q, q; t ) is the classical action along the characteristic that starts at q at time T = 0 and ends at at time = t. =
(6. 13)
Q
+
T
The rigorous proof that (6. 13) is an asymptotic expansion as n ----> 0 uses the assumptions made in Section 2.3 of Chapter 1 , and is left to the interested reader. Here we just mention that asymptotics (6.13) is consistent with the conservation of probability: for any Borel subset E C IR,
l cp (q) l 2 dq
1 '1/Jn (Q , tW dQ
r + O ( n) = r jE jEt as n ----> 0, where Et 7rt (E) . Remark. When assumptions in Section 2.3 of Chapter 1 are not satisfied, the situation becomes more complicated. Namely, in this case there may be several characteristics 'Yj which end at Q at having as their corresponding initial points. In this case,
=
T=t
(T)
'1/Jn( Q ,
qj
1
t) = L cp( qj ) I�� (qj) � -2 e K (S(Q,qi;t)+s (qj )) - '¥tti (1 + O ( li) ) , J
where J.Lj E Z is the Morse index of the characteristic the number of focal points of the phase curve
'YJ .
It is defined as
( q (T) , p(T)) with initial data
1 1 There should not be any confusion with the quantum coordinate operator
Q.
6. Semi-classical asymptotics
-I
209
�:
= ( qj ) with respect to the configuration space R It is a special case of a more general Maslov index. qj and Pj
The case of n degrees of freedom is considered similarly. Under the as sumptions in Section 2.3 of Chapter 1 , the solution '1/J n ( q, t) of the Schrodinger equation (6. 1) with the initial condition '1/Jn ( q , t ) lt= O = . of SymN , constructed as follows. Let Y>. be the Young diagram associated with A E Par(N) � a collection of N boxes arranged into n left-justified rows, with the first row containing AI boxes, the second row containing A 2 boxes, etc. Thus the diagram
corresponds to the partition 10 = 5 + 3 + 2. A Young tableau A associated with the Young diagram Y>. is an assignment of N integers 1, 2, . , N to the N boxes such that each box gets a different integer; denote by 'I>. the set of all Young tableaux associated with the partition A E Par(N) . A canonical Young tableau A>. E 'I>. is obtained by numbering the boxes consecutively along the rows from left to right. For a Young tableau A E 'I>, , say the canonical one, define two subgroups of SymN : . .
7l'
Row(A) = {7r E Sym N : preserves the rows of A} , Col(A) {7r E SymN : 7l' preserves the columns of A} . =
4.
230
Spin and Identical Particles
Now let Qt. = .. = Qt. · ITA , and the representation T>.. : SymN ---> End G>., , where Sym N acts by left multiplication. The following result is fundamental. Theorem 3 . 1 . Each T>.. is an irreducible representation of SymN , and rep resentations T>.. and Tf.l- are not isomorphic if A =1- f.l · The dimension of the
representation T>.. is given by the Frobenius formula N! d>., = ! . . . n ! h l
n ui - lj ) ,
ti = Ai + n - i , i = l , . . . , n,
t<J
and IT� = d>.,ITA . Every irreducible representation of S ym N is isomorphic to the representation T>.. for some partition A E Par ( N) . Remark. It follows from the representation theory of finite groups that
R=
so that
>..
E9
r
E Pa ( N )
d>., G>..
(Qt. =
N! =
E9
r
>.. EP a ( N )
L
>.. E Par( N )
End G>.. as algebras) ,
d� .
There is another expression for dimensions d>.. as a product over the boxes of the Young diagram Y>.. , given by the so-called hook-length formula. 3 Another choice of a Young tableau
A E 'I>. gives a SymN-module equivalent to G.>. .
3.
231
System of identical particles
Remark. One can explicitly construct the bases in the subspaces G>. by the
following procedure. Let A be the standard Young tableau for a partition A E Par(N) : a Young tableau with the condition that in each row of Y>. , the numbers in the boxes are strictly increasing from left to right, and in each column - strictly increasing from top to bottom. Let 7f E Sym N be the unique permutation such that A 1T'(A>. ) and put eA = IIAe11' . Then the vectors e A, where A runs through all standard Young tableaux for the partition A, form a basis of G>. . In other words, the basis elements are ob tained by successive symmetrization on the rows of Y>. with a subsequent anti-symmetrization on the columns. =
=
The partition .A (N) corresponds to the Young diagram with only one row and produces the trivial representation of SymN . The partition .A ( 1 , . . . , 1) corresponds to the transposed Young diagram - a dia gram with only one column - and produces another one-dimensional rep resentation, the alternating representation 7f f--f ( - 1 )c-( 71') . In general, let X = (.A� , . . . , .A�) be the partition conjugated to the partition .A, which is defined by transposing the Young diagram Y>. by interchanging the rows and columns (.A� is the number of terms in the partition .A which are greater than or equal to i ) . Then (3.6) Other partitions correspond to the symmetries of mixed type. The next result will play a crucial role in determining the symmetry properties of coordinate wave functions. =
3 . 2 . The tensor product T>. 0 TJlo contains the trivial represen tation of Sym N if and only if 11 = .A, in which case it has multiplicity one. In terms of a basis { e i } f� 1 in G>. such that the representation T>. is given by orthogonal d>. x d>. matrices t>. (7r)ij ,
Proposition
T>. ( n' ) e i =
d>,
L t>. (7r)jiej ,
j= l
i=
1 , . . . , d>. ,
the one-dimensional subspace of the trivial representation is spanned by the vector I: f�1 e i 0 ei E G >. 0 G >. . Correspondingly, the tensor product T>. 0 TJlo contains the alternating representation of SymN if and only if 11 A', in which case it has multiplicity one. In terms of a basis { ea f�1 in G N such that the representation TN is given by the matrices tN (7r )ij = ( - 1)e(1T')t>. (7r)ij , the one-dimensional subspace of the alternating representation is spanned by the vector I:f� 1 e i 0 e � E G>. 0 GN . =
Problem 3 . 1 . Prove all the statements in this section. (Hint : See the list of references to this chapter) .
4.
232
Spin and Identical Particles
In addition to the diagonal action of the Lie group SU ( 2 ) in the tensor product (C2 ) 0 N , discussed in Section 3. 1 , there is also a left action of a symmetric group SymN by permuting factors4 : T( 11' ) ( v1 0 · · · 0 VN ) v11'- l (I) 0 · · · 0 v11'-l ( N ) • 11' E SymN. 3.3. Schur-Weyl duality and symmetry of the wave functions.
=
For every partition A E Par ( N ) denote by II>. the Young symmetrizer corresponding to the canonical Young tableau A>. , and by S>.C2 the image of the operator T(II>. ) in (C2 ) ® N . The subspace S>.C2 is zero-dimensional when the Young diagram Y>. has more than two rows, and is an irreducible U ( 2 ) -module, called the Weyl module , when the Young diagram Y>. has one or two rows. For the partition A = ( N ) the Weyl module S>.C2 Sym N C2 and is isomorphic to the highest weight module V>., N with the dominant weight ( N, 0) , defined in Section 1 .2. For the partition A = (AI , A 2 ) the Weyl module S>.C2 is isomorphic to the highest weight module V>., N with the dominant weight A. Remark. This construction is a special case of the general construction given by H. Weyl, which applies to the symmetric group action in the ten sor product V® N , where V is a finite-dimensional vector space. The Weyl modules S>. V associated with the Young diagram Y>. are zero-dimensional when the number of rows is greater than n = dim V , and are isomorphic to the highest weight modules of GL(V) with the dominant weight A when the number of rows is less than or equal to n. The association V t--t S>. V is called the Schur functor. =
The following result, the Schur- Weyl duality , establishes explicit pairing between irreducible representations of U ( 2 ) and SymN that appear in the decomposition of the representation of SU ( 2 ) x SymN in (C 2 )0 N into the irreducible components. Theorem 3 . 2 ( Schur-Weyl duality ) . There is a direct sum decomposition N into irreducible components of the representation R� x T : lf"2 ) ® N ( \L.
--
ffi \I7
>.EPar ( N, 2 )
s>. \L. !f"2
10. '. '
where Par ( N, 2 ) is the set of partitions of N with number of terms less than or equal to 2 .
Setting >. 1 = If + s , A2 = � - s ( cf. Section 1 . 2) , from the Frobenius formula we obtain N N (3. 7) . - N d>. = dim G >. N =
(
2-s
) (
2-s-1
)
4It moves the vector in the i-th place of the tensor product to the 7r( i )-th place.
3. System of identical particles
233
Now return to the main problem of finding total wave functions satisfying ( 3.5 ) . It follows from the Schur-Weyl duality that there are ( 2 s + 1 ) d>. linear independent spin wave functions xmj ( a1 , . . . , aN) of spin s satisfying S2Xmj = s s + 1)xmj and Sa Xmj = mxmj , ( 3.8 ) where m = - s , - s + 1, . , s - 1 , s and j = 1 , . . . , d>. . Here d>. corresponds to the partition .A = s, If - s ) or .A = N ) for s � ) and is given by ( 3.7 ) . The spin s representation Ps of SU 2 ) acts on the index m of the spin wave function Xmj (al , . . . , aN), and the representation T>. of Sym N acts on the index j . The total wave function has the form
( (If + .
(3. 9)
.
( ( (
=
d>..
q,m (e l , . . . , eN) = I: wj (x b · · · , xN)Xmj (a b · · · , aN) , j=l
where m = -s, . . . , s. It follows from Proposition 3 . 2 that the total wave function q,m (el , . . . , eN) is totally anti-symmetric if and only if the coordinate wave functions Wj (Xl , . . , xN) transform according to the conjugated representation TN of Sym N , .
d>..
(3. 10 ) (P11" W i)( x � , . . . , XN) = 2:::> >- {rr)jiWj (Xl , · · · , x N), j=l
i=
1, . , d). . .
.
We summarize these results as follows.
Wj (xl, . . . , XN) associated with the total wave function of spin s of a system of N identical particles of spin � are those solutions of the N -particle Schrodinger equation (3.4) that transform according to the representation TN of the symmetric group Sym N , where .A' is a conjugated partition to .A ( If + s, If - s) . The spin wave functions Xmj (a l , . . . , aN) satisfy ( 3.8 ) , and corresponding total wave func tions q,m (eb . . . , eN) are given by (3. 9) . In general, for a given energy level E there are 2s + 1 linearly independent total wave functions of spin s, 0 :::; s :::; If , and s is an integer. Theorem 3.3. Coordinate wave functions
=
If -
Remark. For the highest value of total spin s = If the coordinate wave function w (xl , . . . ' XN) is totally anti-symmetric. Since (N)' = ( 1 , . ' 1) tJ. Par ( N, 2) when N > 2, coordinate wave function for the case N > 2 is never totally symmetric. .
.
Using explicit description of the Young symmetrizer in the previous sec tion, we obtain the following result. Corollary 3.4 (V .A. Fock ) . The coordinate wave function
associated with the total wave function of spin following symmetry properties.
s
W ( x1 , . . . , x N)
is characterized by the
4.
234
Spin and Identical Particles
(i) w ( x 1 , . . . , X N ) is anti-symmetric with respect to the group of argu ments x1 , . , X k , k If + s . ( ii) 'lt ( x1 , . . . , X N) is anti-symmetric with respect t o the group of argu ments x k + 1 , . . . , x N . (iii) 'lt ( xl , · · · , xN) satisfies k w ( x � , . . . X N ) = L w (x1 , . . . ' Xi- 1 , Xk+b Xi+ I , . ' Xk , Xi , Xk+2 > . . . ' XN ) . .
'
.
=
.
.
i= 1
The case of N identical particles of arbitrary spin s is treated similarly. The symmetry properties of coordinate wave functions can be obtained by using Schur-Weyl duality for the action of U(l) X SymN in C1 , where l = 2s + 1 . It has the form ( c1y)9 N = EB s>- c 1 0 c>. , >.EPar(N,l) where Par(N, l) is the set of partitions of N with number of terms less than or equal to l , and S>.
helium atom ( N
=
2;
the case of the
illustrates the difficulties of the problem . For this purpose var
3)
ious approximate methods like the one-electron approximation, the Hartree-Fock method and its modifications, have been developed (see details and
[FY80]
for a clear introduction) .
A
[LL58, Foc78]
for the
correct description of the proper
ties of atoms and an explanation of D . l . Mendeleev 's periodic table of elements, based on these methods , has been a maj or triumph of quantum mechanics. For a detailed account we refer the interested reader to
[LL58, Foc78]
and
[FY80] .
The N-particle Schrodinger equation, in principle, also describes5 all properties of molecules studied in chemistry. However , this "reductionism" does not work due to the complexity of the quantum N-body problem, and approximate methods like Hartree-Fock approximation play a central role in quantum chemistry. The vector space
IC2
0
IC2
describes t he spin degrees of freedom of two elec
trons , and the spin states which are not represented as a tensor product are called
entangled.
u
0
v
Entangled states are used to describe the Einstein-Po dolski
Rozen paradox, a quantum experiment , which apparently violates the principle of locality (see its description in
[Bel87, Sak94] ) .
The fact that quantum mechan
ics is correct and allows for an accurate description of the microworld is beyond any doubt , and is confirmed by numerous physical experiments. The Einstein Podolski-Rosen paradox is only a "paradox" because classical intuition does not always correspond to the physical reality. Moreover, it follows from the
equalities [Bel87]
that any attempt to turn quantum mechanics into
a
Bell in
determin
the web site http : I I en . wikipedia . orglwiki iEPR_paradox for further references.
istic theory by introducing the so-called The vector space
IC2
0
·
·
·
0
IC2 ,
hidden variables
is fallacious ; we refer to
describing the spin degrees of freedom of several
5In this sense, quantum mechanics is a "theory of everything" for chemistry.
236
4.
Spin and Identical Particles
identical particles of spin � , also plays a fundamental role in quantum computa tion theory. This is a rapidly developing field, and we refer the interested reader to the monograph [KSV02] for the mathematical introduction, and to the web site http : I I en . wikiped i a . orglwikiiQuantum_c omputer for updates on the special ized literature. Another rapidly developing and exciting new application of quantum mechanics is quantum cryptography, reviewed at http : I I en . w ikipedi a . orglwiki I Quantum_crypt ography.
Section 3.2 is a crash course on the representation theory of the symmetric group SymN - a beautiful piece of classical mathematics which has many inter esting applications in modern combinatorics and representation theory. Our goal here was to give a concise and clear presentation of the basic necessary facts, and we refer to the classic text [Wey50] and to [FH91] for detailed accounts of rep resentation theory, and to [ G W9 8] for the Schur-Weyl duality. In Section 3.3 we emphasize the role of Schur-Weyl duality in studying the symmetry properties of coordinate wave functions. Physics textbooks like [LL58] , [Dav76] and [Sak94] carefully treat the basic examples, but are somewhat vague in describing the form of the total wave function of N particles. The monograph [ Fo c 78 ] is very precise in this regard, but in his presentation V.A. Fock is trying to reduce the usage of group theory to a minimum. Our goal in Section 3.3 was to fill this gap and to give a clear description of the symmetry properties of coordinate and spin parts of the total wave function by using the language of representation theory. For the case of N particles of spin � this problem was solved by V.A. Fock in 1940; see [Foc78] , as well as in yet another classic [Wig59] . For more applications of group theory to quantum mechanics we refer to the latter text, and to [Wey50] .
Part
2
Funct ional Met ho ds
and S upersymmet ry
Chapter
5
Pat h Integral Formulation of
Q uantum Mechanics
1.
Feynman path integral
Here we present Feynman's path integral approach to quantum mechanics, which expresses the propagator of a quantum system with the Hamiltonian H - the kernel of the evolution operator U(t) = e - k tH - as a "sum over traj ectories" of the corresponding classical system. 1 . 1 . The fundamental solution of the Schrodinger equation. Recall
( see Section 1.3 of Chapter 2) that in terms of the evolution operator U(t) = e - k t H, the solution 'lj;(t) of the initial value problem for the time-dependent
Schrodinger equation
. d'lj; zndt ( t) = H 'lj; (t), 'lj; (t) l t= O = 'lj;
( 1 .1) (1 .2) is given by 'lj;(t)
=
U(t)'lj;. For the Hamiltonian operator H
=
p2
2m
+ V (Q)
of a quantum particle in .!Rn moving in the potential field V ( q ) , the initial value problem (1. 1 ) - ( 1.2) in coordinate representation becomes the following 239
24 0
5.
Path Integral Formulation of Quantum Mechanics
Cauchy problem: 81/J
n2
Ll 'l/J + V (q) 'l/J , = at - 2m 1/J(q, t) i t =O = 1/J (q) , ( 1 .4) where Ll is the Laplace operator on IRn . Under rather general conditions on the potential V(q) ( i.e., when V E L�c (IRn) is bounded from below ) the Cauchy problem ( 1 .3) - ( 1.4) has a fundamental solution: a function K(q, q', t) which satisfies partial differential equation ( 1 .3) with respect to the variable q in the distributional sense ( i.e. , K(q, q' , t) is a weak solution of the Schrodinger equation ) , and the initial condition ( 1 .5) K(q, q1, t) l t = O = 8(q - q ' ) . The solution of the Cauchy problem ( 1 .3 ) - ( 1 .4 ) can be formally written as
( 1 .3)
in
1/J(q' , t) = { K(q ' , q, t) 'lj; (q)dn q, J.JF.n where the integral is understood in the distributional sense. For 1/J E L 2 (1Rn ) formula ( 1 .6) should be understood as 1/J(q', t) = l.i.m. { (q ' , q, t) 'lj; (q)dnq = l.i.m. { K(q' , q, t) 'lj; (q)dn q, }JF.n R-+oo }lqi5, R K where l.i.m. stands for the limit in the £ 2 -norm. In general, 1/J(q, t) is only a weak solution of the Schrodinger equation (1 .3) , and it is a regular solution when 1/J belongs to the Carding domain Do ( see Problem 1 . 7 in Section 1.3 of Chapter 2) . The fundamental solution K(q, q1, t) is the distributional kernel ( in the sense of the Schwartz kernel theorem ) of the evolution operator U(t). Using the group property U(t + t') = U(t) U(t') , we can rewrite ( 1 . 6 ) in the form ( 1 .6)
r K(q' , t'; q, t)'lj;(q, t)�q, }JRn where K(q', t ' ; q, t) = K(q', q, t' - t) . The function IK(q', t'; q, t ) J 2 has a physical meaning of conditional probability distribution of finding a quan tum particle at a point q' E IRn at time t' provided that it was at the point q E IRn at time t. Remark. In physics terminology, the distributional kernel K(q' , t' ; q , t) of the evolution operator U ( t' - t) is called the complex probability amplitude, or simply the amplitude or propagator. In Dirac's notation, K (q', t'; q, t) = (q', t' l q, t) .
( 1 . 7)
1/J(q ' , t')
=
Finding the propagator of a given quantum system is a fundamen tal problem of quantum mechanics. When the spectral decomposition of the Hamiltonian operator H is known, the propagator can be obtained in
2 41
1. Feynman path integral
a closed form. Namely, suppose for simplicity that H has a pure point spectrum, i.e. , there is an orthonormal basis of the Hilbert space £ L2 (lRn dn q) consisting of the eigenfunctions { �n ( q ) };;:o= O of H with the eigen values En . We have =
,
00
� ( q ) = L Cn �n ( q) , n =O
so that ( U ( t ) � ) ( q)
=
00
L e- * EntCn�n (q)
n =O
and
where the series and integrals converge in the £ 2 -sense. If the change of orders of summation and integration was justified, we could write ( 1 .8)
K ( q' , t; q, t)
=
00
L e- * EnT �n (q') �n (q) , n=O
T
=
t' - t .
The series (1 .8) converges in the distributional sense, and gives a represen tation of the propagator in terms of the spectral decomposition of H . A similar representation of the propagator exists when the Hamiltonian H has absolutely continuous spectrum. Consider the simplest case of a free quantum particle with the Hamiltonian operator p2 Ho = - . 2m The corresponding Schrodinger equation can be solved by the Fourier method (see Section 2.3 in Chapter 2 ) , and we obtain
� (q1, t1) = l.i.m. ( 2 tr n) � { -
}!Je,n
e * (q'p- i:,; T ) ,j; (p , t ) dnp
= l.i.m. }{!Je. n K( q ' , t' ; q, t ) � ( q, t ) dn q , where ( 1 .9 )
1
K( q' ' t'·' q ' t ) = ( 2trn) n }{!Je.n e k (p(q' - q) - i:,; T) �p '
100
Using the classical Fresnel integral formula etax dx e sgn(a) ( 1 . 10) - oo
.
2
=
1ri
4
�
-,
lal
T
=
t' - t .
242
5. Path Integral Formulation of Quantum Mechanics
and completing the square in ( 1 .9) , we obtain the following expression for the propagator of a free quantum particle: m 2 im ( ' ) 2 K(q 1 t · q t ) -- e 2nr q-q ' (1.11) ' ' ' 2n'ifiT where i 2 e- 4""" and T > 0. n
Remark.
equation
1
1rin
=
=
(
)
n
Formally replacing the actual "physical" time t in the Schrodinger n,2
. a'l/J zn8t = - 2m for a free quantum particle of mass m by the Euclidean time ( or pure imag inary time ) -it, we obtain the heat ( diffusion ) equation
_!_ au
D at
=
�'lj;
�u
on �n with the diffusion coefficient D = _!!.__ . It is quite remarkable that 2m in agreement with this formal procedure the propagator ( 1 . 1 1 ) for a free quantum particle is obtained from the heat kernel on �n by the analytic continuation T - i T . 1--t
Problem 1 . 1 .
Show that lim
R -> oo
{R sin x2dx
Jo
by integrating the function
Problem 1 . 2 . 'l/J (q' , t)
=
Problem 1 . 3 .
e-z
2
=
lim
over an appropriate "pizza-sliced" contour.
Show directly that
(e- -ktHo'l/J) (q')
{R cos x 2 dx = �2 {if V2
R->oo Jo
=
l.i.m.
� lrrtn{ e �'{, (q-q' ) 2 'l/J (q) dn q . (�) 2mnt
Give an expression for the propagator when
H
operator with rapidly decreasing potential, considered in Section
is a Schrodinger
2 in
Chapter
3.
1 . 2 . Feynman path integral in the phase space. For a general Hamil tonian operator H = Ho + V, where V = V(Q) , there is no simple formula for the propagator K (q1 , t 1 ; q, t) like ( 1 . 1 1 ) . This is because operators Ho and V do not commute, so that e - i t H -1- e - k t Ho e- itV . It was Feynman's fundamental discovery that there i s another represen tation of the propagator for a quantum system in terms of the corresponding classical system. We start by describing Feynman's approach for the case of a quantum particle with one degree of freedom. It is based on the so-called Lie-Kato-Trotter product formula, which allows us to express the exponen tial ei(A+ B ) of two non-commuting self-adjoint operators in terms of the individual exponentials eiA and e i B .
1 . Feynman path integral
243
(Lie-Kato-Trotter product formula) . Let A and B be self adjoint operators on £ such that A + B is essentially self-adjoint on D (A) n D(B) . Then for 1/J E £,
Theorem 1 . 1
ei ( A + B ) ?/J = n---+ limoo (e * A e * 3 t 1/J.
Proof. We consider only the special case when A and B are bounded op erators, which was already proved by Sophus Lie. Set en = ei ( A+B ) / n and Dn = eiA/ n eiB/n. We have the telescopic sum enn Dnn = enn enn - 1 Dn + enn-1 Dn enn - 2 Dn2 + . . . + Cn Dnn 1 Dnn _
_
= L_,; """"'
n-1
k =O
_
enn - k -1 ( en
-
_
k Dn ) Dn'
and since by the Taylor formula l i en - Dnll :::; c2 for some constant c n we obtain l i e� - D� l l :::; � n This proves the result with convergence in the uniform topology.
>
0, 0
We will always assume that the Hamiltonian H Ho + V is essen tially self-adjoint on D (Ho) n D (V) , so that the Lie-Kato-Trotter formula is applicable. (According to Theorem 1 . 2 in Section 1 . 1 of Chapter 3 , for the case n 3 we can assume that V V1 + V2 , where V1 E L 2 (ffi.3) and V2 E L00 (ffi.3 ) . ) Applying the Lie-Kato-Trotter formula to A = � Ho and B - � V , we obtain that in the strong operator topology ; i�t i� T e - li T H n---+ limoo ( e - T H0 e - Tt v t , b.. t (1 . 1 2 ) =
=
=
-
=
n
= -,
=
i�
t is where T = t' - t. In the coordinate representation the operator e - TV K a multiplication by e - V ( q ) f:l. t operator, and the distributional kernel of the i�t operator e - T Ho is given by ( 1 . 9 ) , where T is replaced by b.. t . Thus the distributional kernel K ( q1 , q; b.. t ) of the operator e - i�t Ho e - i�t v is given by
(1.13)
K (q' , q; b.. t )
=
1
oo * ( ' ( e p ( q - q) - � +V ( q )) f:l.t ) dp. j t. _ 00
27fn
Since the kernel of a product of two operators is a composition of corre sponding individual kernels, for the distributional kernel Kn ( q ' , t'; q, t) of i�t i�t the operator (e - T H0 e-TV) n we obtain the following integral representation:
( 1 . 14 )
Kn ( q1 , t' ; q , t ) =
I I kIT� n- 1
·
·
·
�-1
K ( qk+l , qk ; b.. t )
n-1 IT dqk , �1
244
5. Path Integral Formulation of Quantum Mechanics
where qo = q , qn = q ' . Now replace each factor K ( qk + l , qk ; Llt ) in ( 1 . 14) by its integral representation ( 1 . 13) , where the corresponding variable of inte gration is denoted by Pk , k = 0, . . . , n - 1 . Changing the order of integrations in the resulting (2n - 1 )-fold integral and using ( 1 . 12), we obtain the follow ing remarkable representation of the propagator of a quantum particle as a limit of multiple integrals when the number of integrations goes to infinity: ( 1 . 15) im Kn ( q1 , t' ; q , t ) K(q ' , t' ; q , t ) = nl-+oo n- l . n -l exp L (Pk ( qk+l - qk ) - Hc (PkJ qk ) b.t) = nl�� k =l k=O Ja>.2n - l 2 Here Hc (P, q ) = ; + V ( q ) is the classical Hamiltonian function, and qo = m q , qn = q'. Heuristically, formula ( 1 . 15) admits the following interpretation. To ev. t ( o . ' ery pom P , PI , . . . , Pn -l , q l , . . . , qn -1 ) E T!l) 2n -1 assign a piece-wise lm2 ear path a in the extended phase space JR x IR of the classical particle, defined by the following time slicing procedure . Let t k = t + k b. t and a ( T ) = (p( T ) , q ( T ) , T ) , where - k p( T) = Pk, q (T) = qk + ( T - tk ) k + l k+1 - k ' and T E [t k , tk +l ] , k = 0, . . . , n - 1 . Then for the Riemann integrable potentials V ( q ) we have n -1 ( 1 . 16) L (Pk ( qk + l - qk ) - Hc (Pk , qk ) b.t ) = S (a) + o ( 1 ) , as n __, oo , k=O where
I · · · I {*
} �:� IT d�:d:k .
ll'\,.
.
�
S(a)
=
1
(pdq - HcdT )
=
•
�
1 (p(T)q(T) - Hc (P(T) , q (T)) )dT t'
is the action functional of a classical system with the Hamiltonian function Hc (P , q ) (see Section 2.2 in Chapter- 1 ) . This suggests to interpret ( 1 . 15) as a kind of "integral" over the space P ( IR2 ) �'t':t of all paths a ( T ) = (p( T ) , q (T) , T ) in the extended phase space JR2 x IR such that 1 q ( t ) = q and q( t' ) = q', which was used in the formulation of the principle of the least action in the phase space (see Section 2.2 in Chapter 1 ) . Thus we put
( 1 . 1 7)
K(q', t'; q, t )
=
J
P(Ja>. 2 )q:e q ,t
e*S(a ) �p�q,
1 Note that there is no condition on the values of p ( r ) at the endpoints.
1 . Feynman path integral
245
-
't '
where the "measure" �p�q on P(IR2 ) �;t is given by
o n- 1 q �p � q = nr dp IT dpkd k . .:..� 21r'!i k= l 21r'li
( 1 . 18)
Representati on propagator ofisa thequantum famousparticle. It expresses the prop for the agator K( q' , t ' ; t) as the "weighted sum" over all possible "histories" of - 2 ) ; , where each path has a comtheplexclweiassigchtalexppartiHcS(u)}. le, the paths u P(IR qt On one hand, thi s representati onquantum clearly shows the fundamental di ff erence between cl a ssi c al mechani c s and mechan ices.s,Thus clathessicalcrimechani particles)moves alactiongonclfuncti assicalonaltrajecto riwhereas whicinhinarequantum tmechani ical poicncs,tss theal(extremal of the ( u), ble paths contribute to the com plex probabilicltyearlamply poiitudents K(at theq', t'rel;l q,possi t) . On the other hand, representati o n a ti o n between quantum mechani c s and cltheassiweicalghtsmechani cs in the semi-classical limit 0. Namely, as 0 exp {-k S(u)} become rapidly oscil ating and their contri b uti o ns tolatter happens wil mutual l y cancel, unl e ss the acti o n i s al m ost constant. The nts, whieschofgiavepartithecmajor contrifrombutiiotns toquantum descri This pnear itis ohown,thewhiclcriachssitiwiccalall poibetrajectori l e emerge discussed in detail in Section The Feynman path i n tegral i n the phase space i n not an inte gral in the sense of abstract integration theory, since the formal expression �p �q does not define a measure2 on the path space P( IR2)�; t . Besi des, the frespect unctionalto eanyp{Kmeasure} hasf..t absol u te val u e and cannot be i n tegrabl e wi t h 2 The rig d with the property J.L( P ( JR )� ;r ) orous mathemati c al meani n g of i s the ori g i n al f o rmula expresses e integrals"paradox" as the number ofwhiquantum inchtegrati omechani ns goesthecspropagator tois defiinfininedty.entiasThiarelsliymexpliint ofaclinamulssis thectaliplapparent that terms by It i s not no "intrinsic" definwhiitiocnh ofalso requiinresclathessicspeci al terms, besicedesof thetheso, siapproxi tinmcee there slimcinatigisoprocedure a l choi n of by the Riemann sums. Using the Fresnel i n tegral ( 1 . 10) for the integrati o n over p i n ( 1 . 13), we obtain the following formula for the distributional kernel of the operator Feynman path integral in the
( 1 . 1 7)-( 1 . 18)
q,
phase space
E
q't'
u
S
'li
( 1 . 1 7)-( 1 . 1 8)
'li
-t
-t
( 1 . 17)
( 1 . 17) .
6.1.
Remark.
-
x
S (u)
't '
1
= oo.
( 1 . 1 7)-( 1 . 18)
( 1 . 1 5) ,
( 1 . 1 7) .
( 1 . 1 7)
( 1 . 1 6)
1 .3.
( 1 . 15) , S( u)
Feynman path integral in the configuration space.
� eH� t
-V( q )�t}
.
2 In abstract measure theory the measure is non-negative and countably additive.
il>t
5.
246
Path Integral Formulation of Quantum Mechanics
Repeating the time slicing procedure from the previous section, instead of ( 1 . 15) we now get
K(q1 , t ' ; q, t ) = nl�� Cn��t ) ; ( q'+�� qk ) V(q, ) �� n
( 1 . 19) x
t/ { �� ( exp
Assuming that in IR, where qo
where
S('Y)
=
1
t'
2
' -
) }IT dq,.
Qk q(tk ) , k 0 , . for some smooth path "f (T) = q(T ) q and Qn = q1, we obtain as =
=
. . , n,
n
=
----t
oo ,
t'
L ( 'Y' ( T))dT = 1 L (q( T) , q(T) ) dT, L ( q , q)
=
� mq2 - V ( q ) ,
is the action functional of a classical particle of mass m moving in the potential field V( q ) ( see Section 1 . 3 in Chapter 1 ) . This suggests to interpret the limit of multiple integrals ( 1 . 1 9 ) as the Feynman path integral in the configuration space ( 1 . 20)
K ( q' , t'; q, t) =
1
e i 8b) � q .
P(IR )q�t' q,t
Here P(IR)�:f is the space of smooth parametrized paths 'Y in the configura tion space JR. connecting points q and q1, and the "measure" � q on P(IR)�:f is given by (1.21)
� q = n--+oo lim (
n-1
:. ) IJ dqk· 27r � t k =l n
2
2
As the Feynman path integral in the phase space, the Feynman path in tegral in the configuration space is not actually an integral in the sense of integration theory, and the correct mathematical meaning of ( 1 . 20)- ( 1 . 2 1 ) is given by ( 1 . 19) . Formula ( 1 .20) expresses the propagator K(q', t'; q, t) as the sum over all histories in the configuration space of classical particle. - paths � ')' E P(IR)�;t , by assigning to each path ')' a complex weight exp U S ('Y ) } .
247
1. Feynman path integral
Convergence in ( 1 . 19) , as well as in ( 1 . 15), is understood in the distributional sense. In particular, for every 'ljJ E L 2 (IR) ,
Remark.
x
( e - i TH 'ljJ) ( q) = nlim ---> oo
��: j exp { � � ( ; ( Qk+�� Qk )
where all integrals are understood as f�oo - in the £ 2 -sense.
( 2 7r:.�t ) �
) } tJ
Z
'
-
=
V(qk ) �� �(qn)
dqk ,
limR---. oo � i � R ' and all limits q
It is said in physics textbooks that the Feynman path integral in the configuration space is obtained from the Feynman path integral in the phase space by evaluating the Fresnel integral over �p,
Remark.
. t'
e K ft
(1. 22 )
(p¢.- Hc (p,q) )dr�p�q.
Note that the symbol � q has two different meanings: in the left-hand side of (1 .22 ) it is defined by (1. 2 1), whereas in the right-hand side it is defined as a part of ( 1 . 18) . p2
1.4. Several degrees of freedom. H
Let
= 2m + V ( Q)
be the Hamiltonian operator of a quantum particle in ffi.n moving in the potential field V ( q) . As in the case of one degree of freedom, by using the Lie-Kato-Trotter product formula, the propagator K ( q , t'; q, t ) is expressed as a limit of multiple integrals when the number of integrations goes to infinity, '
(1. 23)
2 Here He (p , q) = !!__ + q ) is the classical Hamiltonian function, and Qo = 2m q, Q N = q1 • This representation is symbolized by the Feynman path integral in the phase space .41 = T*ffi.n , e K fo- (vii - Hc (P,q) )dr �p � q (1 .24 ) K ( q', t' ; q , t )
V(
=
f
P ( J� ) qq :t' ,t
,
248
5. Path Integral Formulation of Quantum Mechanics
where �P � q
=
r N �oo
N- 1
IT
dn Po
dnPk dn qk
( 2 7rn)n k=1 ( 27rn)n
and P(.4')�:f is the space of all admissible paths CJ in the extended phase space .4' x � connecting points (q, t) and (q' , t') (see Section 2.2 in Chapter 1). Equivalently, the propagator K ( q' , t'; q, t) can be written as ( 1 .25) X
K(q' , t' ; q, t) =
lim
J { � � ( ; ( k+�� L
N--+oo
q
exp
�
(2 : t) 7!' z
Qk
)
' -
nN 2
V (q• )
) } IT At
� q• .
This formula is symbolized by the Feynman path integral in the configura tion space, ( 1 .26)
K(q ' , t ' ; q , t ) =
J
.
t'
e k ft
L(q,q)d-r
�q,
where L(q , q) = !mq 2 - V (q) is the corresponding Lagrangian, �q
= J�oo (
nN
N- 1
IT dn qk , 27l'�� t ) k=1 2
and P(M )�:r is the space of smooth parametrized paths in the configuration space M connecting points q and q' . The precise mathematical meaning of formula ( 1 .26) is the same as of ( 1 .20) . Remark. Formally, definition ( 1 . 2 4 ) of the Feynman path integral in the phase space can be extended to the case (.4', w , He ) , where .4' = T* M and w = dO, the canonical Liouville 1-form on .4' . Namely, one can obtain a distributional kernel K (q' , t'; q , t) by using the same time slicing as in (1.24) and replacing the 1-form pdq by 8, and the volume form dnpk dnqk by an appropriate multiple of the Liouville volume form on .4'. Similarly, repre sentation (1.26) can be formally extended to the case of general Lagrangian system (M, L ) , where the configuration space M is a Riemannian manifold, by using the same time slicing and replacing dnqk by the Riemannian volume form on M. However, in general it is not clear for which unitary operators these Feynman integrals are actual distributional kernels. Moreover, even in the case M = �n with the standard Euclidean metric, representations (1.26) and ( 1 .24) , where He is a Legendre transform of L, do not necessarily agree if L is not of the form "kinetic energy minus potential energy" . However,
2.
Symbols of the evolution operator and path integrals
249
when L = !9J.L v (x ) i;J.Li; v - V ( x ) (cf. Example 1 . 7 in Section 1 .3 of Chap ter 1 ) , then with appropriately defined time slicing, Feynman path integrals (1.26) and ( 1 . 24) agree, and are equal to the propagator for the Hamiltonian operator 2 H fi L\ + 2 g V. Here L\ 9 is the Laplace operator of the Riemannian metric on M, introduced in Example 2.4 in Section 2 . 4 of Chapter 2.
=
Formula ( 1 . 24) could serve as a heuristic tool which enables us, in some cases, to quantize the classical Hamiltonian system ( .4, w, He ) · In general this is a rather non-trivial problem, especially when the phase space .4 is a compact manifold. Remark.
2.
Symbols of the evolution operator and path integrals
In Sections 2 . 7 and 3 . 3 of Chapter 2 we introduced Wick, pq- , qp- , and Weyl symbols of operators. For the evolution operator U(T) = e-KTH these symbols can also be represented by Feynman path integrals. Here we consider only the case of one degree of freedom, since generalization to several degrees of freedom is straightforward. 2 . 1 . The pq-symbol. Let F1 (p, q, T) be the pq-symbol of the evolution operator U(T) . Using ( 1 . 15) and formula (3. 18) in Section 3.3 of Chapter 2 - the relation between the pq-symbol of an operator and its distributional kernel - we get
H (p, q , T) = I: K(q - v, T; q, O) eKPvdv
( 2. 1 )
= J��
J J { � k=O L (Pk (qk+l - qk) - Hc (Pk , qk ) L\t ) } IT dp��1:qk , k=1 exp
···
. n- 1
n- 1
JR2n - 2
where Pn - 1 = p and qo = qn = q. Formula (2. 1 ) can also be obtained from the composition formula for pq-symbols ( see Problem 3.9 in Section 3.4 of Chapter 2) . Indeed, it is sufficient to observe that the pq-symbol of the operator e - --,;:- Ho e- --,;:- V is f (p, q ) = e- --,;:- Hc (p, q) , and to represent the pq-symbol of the operator ( e - --,;:-Ho e - --,;:- V ) n as the multiple integral iLl.t
iLl. t
i Ll. t
i Ll. t
JJR n-2J f(p , qn-I ) f (Pn-2 , qn2
···
i Ll. t
2) · · ·
f (p2 , q2 ) f (po
,
q)
5. Path Integral Formulation of Quantum Mechanics
250
where qo = q. In the same spirit as ( 1 . 15 ) , formula (2. 1 ) can be written the following Feynman path integral for the pq-symbol: (2 . 2)
F1 (p , q , T )
= J
o(p,q) ( JR2 )
e k S ( a ) �p� q ,
S (a) =
as
1 pdq - Hc dt , )
(
(p, q)
where n (p ,q ) (JR2 ) = {a : [O, T] -+ JR2 : a (O) = a(T) = E JR2 } is the space of parametrized loops in the phase space JR2 which start and end at the point (p , q ) . The mathematical meaning of (2.2) is the original formula ( 2 . 1 ) with a special choice of the approximation of S( a ) by Riemann sums. Remark.
2 . 2 . The qp-symbol. Let F (p , q, T) be the qp-symbol of the evolution 2 operator U(T) . In this case instead of formula ( 1 . 12) , convenient for pq symbols, one should consider the equivalent representation (2.3)
The distributional kernel k ( q', q; b. t ) of the operator e - i�t v e by
K ( q', q; b. t) = 2�n
(2.4)
and as in Section 1 . 2 we obtain (2 . 5)
=
J!_.�
K ( q' , t' ; q, t ) = nl� . n- 1
i
t
� Ho is given
i: e * (p(q'-q)-( � +V(q') )�t)dp,
j J k�II K (qk+ , Qk i b.t) �1II dqk n- 1
···
n- 1
1
p-1
n- 1
J Jexp { � kL=O (Pk (Qk+ 1 - Qk) - Hc (Pk . Qk+ I ) b.t)} �:� II d�:d:k . ···
k=1
]R2n - l
Using formula (3.19) in Section 3.3 of Chapter 2 , we get from (2.5) , (2.6) = nl!_.�
J J ]R 2 n-2 · · ·
i:
F2 (p , q, T) = K ( q, T; q + v, O)e*pvdv n- 1 . n- 1 d exp L (Pk (Qk + l - Qk ) - Hc (Pk , Qk +I ) b. t) II k=O k=1
{�
·
p�-:ndQk ,
}
where Po = p and qo = Qn = q. Formula (2.6) can also be obtained from the composition formula for qp-symbols ( see Problem 3 . 1 0 in Section 3 .4 of Chapter 2) by observing that the qp-symbol of the operator e - T V e - T Ho is again e - THc (p,q) . In the same spirit as ( 1 . 15) , formula (2.6) can be written iLl.t
i Ll. t
i Ll. t
2.
Symbols of the evolution operator and path integrals
251
as the following Feynman path integral for the qp-symbol: =
F2 (p, q, T)
( 2.7 )
J
n (p,qJ (JR2 )
e x S(u) ?Jp?Jq.
This expression looks exactly like ( 2.2 ) , so that the formal path integral rep resentation does not distinguish between pq- and qp-symbols! Of course, the mathematical meaning of ( 2.7 ) is the original formula ( 2 .6 ) , which requires a special choice of the approximation of S(CJ) by Riemann sums, which is different from the one used for the pq-symbol. Problem 2 . 1 . Deduce formula (2.6) from the composition formula for qp-symbols.
Let F (p, q; T) be the Weyl symbol of the evolu tion operator U(T) . It follows from the Weyl inversion formula - formula ( 3.12 ) in Section 3.3 of Chapter 2 - that 2.3. The Weyl symbol.
F (p, q , T)
=
i: K (q - !v, T; q + !v, O) e xPvdv.
However, for general potentials V ( q ) one cannot perform integration over v by using either ( 1 . 15 ) or ( 2 .5 ) , and we choose another approach. Namely, de note by U ( tl.t ) an operator3 with the Weyl symbol e - K Hc (p , q) l:l.t, and suppose that lim U ( t:J.tt. (2.8) U(T) n->oo =
Using the composition formula for the Weyl symbols, formula ( 3.24 ) in Sec tion 3.3 of Chapter 2, we can express the Weyl symbol as . n- 2 (2 .9) ( 2 L ( (Pk - �k ) (TJk+l - TJk) exp F p , q , T) = nl!_,� k=O
JJR4(n-1)J { * ( ·
·
·
n-1
n-1 ""' } ) dpkdqkd�k -1d1Jk- 1 , - ( qk - TJk) (�k+l - �k) ) Hc (Pk , qk)tl. t II L..J
1rn k=O k=1 where �o = p, 1Jo = q and �n- 1 Pn - 1 , TJn- 1 qn- 1 · As for formula ( 1 . 1 5 ) , formula ( 2. 9 ) can be written as the following Feynman path integral for the Weyl symbol: ( 2. 10 ) F (p, q, T) =
=
J eX
3 There is
=
f0T {2 ( p(t ) - W )) i]( t ) -2 (q(t ) - ry ( t ))�( t )-Hc (p ( t ) ,q( t ))} dt
no easy way to
express it in terms
?Jp ?Jq ?J� ?J1] ,
of e- � Ho and e- � V .
252
5. Path Integral Formulation of Quantum Mechanics
where 1M
1M
::u p ::u q
IM(: ::U 1M ::u.., TJ
-
n- 1 'llm IT dpk dqk d�k- 1 drJk - 1 n --+oo k =1
to
1rn
'
and "integration" goes over the space of real-valued functions p(t) , q(t) , � (t) , and ry( t) on [ 0 , T] satisfying � ( 0 ) = p, ry( O ) = q and � (T) = p ( T ) , ry (T)
q (T) .
Remark. Expression (2. 10) looks very different from (2.2) and (2.7) . How ever, the variables �(t) and ry (t) appear only linearly in the exponential in (2. 10) , and the integration over them can be performed explicitly. Indeed, using integration by parts ( and ignoring subtleties with the boundary terms ) we can transform the integral over �� to
j
e ¥- J[ �(t) (q(t) -2iJ(t)) dt �� '
which equals the product of delta-functions J(27](t) - q(t) ) over [0, T] and allows us to replace 27](t) by q(t) in (2. 1 0) . Thus we obtain the formula F (p, q , T) =
J
e k foT (Pti-Hc (P ,Q )) dt �p � q ,
which looks exactly like (2.2) and (2.7) ! This heuristic argument shows that the Feynman path integral, understood naively, "erases" the difference be tween various symbols of the evolution operator. Of course, one should al ways specify particular finite-dimensional approximation of S ( a ) in order to get correct results. Problem
2.2.
Find the integral kernel of the operator U(flt) .
Problem
2.3.
Derive formula (2.9) .
2.4. The Wick symbol.
Instead of the variables z and z, used in Section vlfi z and a = vlfi z, n which introduce explicit -dependence in the calculus of Wick symbols. Let H ( a, a ) be the Wick symbol of a Hamiltonian operator H. Here we derive a formula for the Wick symbol U ( a, a; T) of the evolution operator U(T) e- t T H. Denote by U(b.. t ) an operator with the Wick symbol e - k H ( a, a) l:>. t , and as in the previous section, suppose that 2 . 7 of Chapter 2, it is convenient to use variables a
=
=
(2. 1 1)
U (T)
=
U(b.tt . nlim --+oo
Using the formula for the composition of Wick symbols ( see Theorem 2.2 and Problem 2.18 in Section 2.7 of Chapter 2 ) , we get the following expression
2.
253
Symbols of the evolution operator and path integrals
for the Wick symbol Un (a, a; T) of the operator U( b.. t ) n : ( 2. 12 )
Un (a , a; T)
=
·· ·
j j exp { � (a(an -l - a) - iH (a , an-l)b..t cn - 1
n-l d2 k n-l + I )ak (ak-1 - ak) - iH (a k , a k_1 )b.. t ) ) } II 1r� , k=l k=l where a o = a . Setting an = a, we can rewrite the sum in ( 2 . 1 2 ) as n L ) a k (a k-l - ak) + a( an - a) - iH (a k. ak-I)b.. t ) , k =l and obtain (2 . 1 3 )
U(a, a ; T) = n�oo lim Un (a, a ; T) n
exp
=
lim j n--+oo n .
.
.
c -1
j
n-1 d2 a k 1rn . k=1
{ !i1 ( L)ak (ak-1 - ak) + a( an - a) - iH (ak , ak_I ) b..t ) ) } II k=1
As in the previous examples, we can represent formula ( 2 . 13 ) by the following
Feynman path integral for the Wick symbol: (2 . 1 4)
where
U(a, a; T)
=
J
e * f0T ( iaa- H (a, a ))dt + i a( a (T ) - a ) ::ga::ga
{���?:!}
,
n- 1 d2 ::ga::ga = n->oo lim II �k . k=1 1rn
Here integration goes over all complex-valued functions a(t) and a(t) sat isfying boundary conditions a(O) a and a(T) a, and such that a(t) is complex-conjugated to a(t) for 0 < t < T. =
=
Remark. It should be emphasized that in ( 2 . 14 ) the values a(O) and a ( T ) are not complex-conjugated to the fixed boundary values a (O) = a and a(T) a, but rather are "variables of integration" . This should be compared with formula ( 1.17 ) , where the boundary values for the function q ( r ) are fixed, and the boundary values for the function p ( T ) are free. The difference is that at t = 0 we fix the boundary value of the function a(t) , while at t T we fix the boundary value of another function a(t) . =
=
5. Path Integral Formulation of Quantum Mechanics
254
The Wick symbol U ( a, a ; -i T) of the operator e- � TH can also be represented by
Remark.
U ( a, a ; - i T)
=
lim U a, ; i n� oo n ( a - T )
=
lim
n� oo
n
exp
j j ·
·
=
U ( -i T)
·
cn - 1
n -1 2
d ak { n1 ( I )ak (ak- 1 - ak) - H (ak , ak_ I ) �t) + a ( an - a) ) } II 7rn k =1
(2 . 1 5)
I
{!��?::}
e - � for ( aa +H ( a,a ) ) dt+�a(a ( T ) - a ) �a �a ,
k=1
which is the so-called Euclidean path integral - the Feynman path integral with respect to the Euclidean time, which will be discussed in Section 2.3 of Chapter 6. In particular, if the Hamiltonian H has a pure point spectrum and e - � TH is of trace class, then using the relation between the trace and Wick symbols ( see Problem 2.17 in Section 2.7 of Chapter 2 ) , we obtain from ( 2. 15) Tr
( 2.16 )
e - l TH = 1i
{ aa (OO )I==aa ( T)T) } () (
where now integration goes over all complex-conjugate functions a( t) and a ( t) satisfying periodic boundary conditions a(O) a(T) and a ( O) a (T) , and =
=
Indeed, it follows from the identity n- 1
( 2 . 1 7) =
L {ak (ak-1 - ak ) + (ak+l - ak)ak } + 2 {a(an -1 - a) + (a 1 - a)a}
1n 1
2
L iik (ak- 1 - ak) + a(an - 1 - a)
k =1
1
k =1
that when a and a are variables of integration , one can put a n a and ao = a, which together with an a and ao a imply periodic boundary conditions. Thus n-1 n 2:: ak (ak-1 - ak ) + a(an -1 - a ) = 21 2:: { a k (ak -1 - ak ) + ( ak+1 - ak ) ak } , =
=
=
k =1
k =1
3.
255
Feynman path integral for the harmonic oscillator
which in the limit n -t oo becomes
{T
( - aa + aa ) dt lo due to the periodic boundary conditions. � 2
= -
{T
lo
aadt
In the holomorphic representation, the matrix symbol K ( a , a; T) of the evolution operator plays the role of a propagator K ( q', T; q , 0) in the coor dinate representation. Using the relation between matrix and Wick symbols ( see Lemma 2.4 in Section 2 . 7 of Chapter 2 ) , we obtain K (a, a; T)
( 2. 18 )
=
J
e k f[ (iaa-H(a, a))dt + ka a ( T) pa,Pa .
{�l�?::}
Problem 2.4. Find the relation between the Wick symbol of the evolution oper ator and the propagator. (Hint: Use the analog of formula (2.45) in Section 2 . 7 of Chapter 2 . ) 3.
Feynman path integral for the harmonic oscillator
As a first impression, one may think that Feynman path integrals are not very practical. Indeed, they are defined as limits of multiple integrals when the number of integrations goes to infinity, and it seems difficult to evaluate them. In fact, this is not so: Feynman path integrals turned out to be very useful for various computations, and for several important cases they can be evaluated exactly. Here we consider the basic example4 , the Feynman path integral for the harmonic oscillator. 3 . 1 . Gaussian integration. Evaluation of Feynman path integrals sim plifies when corresponding finite-dimensional integrals can be computed ex actly for every n ( or for large enough n ) . This is the case for the important class of Gaussian integrals.
metric n
x n
Lemma 3 . 1
( Gaussian integration ) . Let A be a positive-definite, real sym
matrix. We have
{ e- � ( A q,q)+ (p,q) dn q � e � ( A - lp,p) ' Jdet A Ja n where ( , ) stands for the standard Euclidian inner product in JRn . =
( 3.1)
eiA (e*A)n, and formula ( 1 . 14) gives the same answer
4 The simplest case of a free particle provides a trivial example, since the Lie-Kato-Trotter product formula reduces to every
n.
=
as
(1.11) for
5. Path Integral Formulation of Quantum Mechanics
256
Proof.
Completing the square we obtain
so by the change of variables q x + A - 1p the integral reduces to the standard Gaussian integral fJRn e - 4 (Ax,x)dn x, which is evaluated by diago nalizing the matrix A and using the one-dimensional Gaussian integral =
loo
-oo
e
_ !2 ax 2 dx =
ff1r
-,
a
a>
0
0.
Formula (3. 1 ) is one of the fundamental mathematical facts which is used in many disciplines, from probability theory to number theory. For applications to quantum mechanics, one needs a version of Lemma 3 . 1 when the decaying exponential factor is replaced by the oscillating exponential factor, as in ( 1 . 10) . Corollary
3.1.
Then
Let A be a real, non-degenerate symmetric
n
x
n
matrix.
(3.2)
where the integral is understood in the distributional sense as lim R_, oo � q i.:;; R and v is the number of negative eigenvalues of A .
Formula (3.2) follows from (3. 1 ) by analytic continuation. It also can be proved directly by completing the square, diagonalizing the matrix 0 A, and using the Fresnel integral formula ( 1 . 10) .
Proof.
There is also a complex version of Gaussian integration, given by the following analog of Lemma 3 . 1 .
Lemma 3 . 2 ( Gaussian integration i n complex domain ) . Let C be a complex matrix such that its Hermitian part ! ( C + C*) is positive-definite. We
n x n
have (3.3)
where ( , ) stands for the standard Hermitian inner product in en , 2 - d 2 z 1 . . . d Zn ·
!T"n n ll.... , an d d 2 z Problem
_
3.1.
Prove Lemma 3.2.
a,
b
E
3.
Feynman path integral for the harmonic oscillator
257
3.2. Propagator of the harmonic oscillator. The classical harmonic oscillator with one degree of freedom is described by the Lagrangian function L ( q, rj) = ! m (rj 2 - w 2 q 2 ) . The corresponding Hamiltonian operator for a quantum harmonic oscillator is given by
The following result is an exact computation of the propagator K( q', t'; q, t) for the harmonic oscillator, using the Feynman path integral in configuration space. Proposition 3 . 1 . The Feynman path integral for the harmonic oscillator
is explicitly evaluated as follows: im
J
rt'
e 2ii" Jt
· 2 w 2 q2 ) dT !»q =
(q
where for - = Tv < T < Tv + l =
2 2 m.,... w_-= � e 2 1i s m w T { (q +q' ) cos wT-2qq'} ' : ...,....., 21rifi sin wT
1rv
7r(v + 1)
w
w
mw --- = _ I!i4 _ rriv -2 e 27ri n sin wT
When T 'TriLl
-->
, v
E N, we have
mw . n 21r l sin wT I
Tv, the right-hand side converges in the distributional sense to 1f i V
e - 2 8( q - q') for even v, and to e - 2 8(q + q1) for odd v .
In this case the ( n - 1 ) -fold integral i n ( 1 . 19) is Gaussian and can be computed exactly using (3.2) . Namely, we have Proof.
n- 1
L) ( qk+l - qk ) 2 - c 2 q� ) = ( An - l q , q ) - 2 (p , q ) + l + q' 2 , k=O where c = w!:l. t, q = (q1 , . . . , qn - d , p = (q, 0, . . . , 0, q') are vectors in lRn - 1 , and An - 1 is the following three-diagonal ( n - 1 ) x ( n - 1) matrix: 0 0 0 2 - c2 - 1 c 2 -1 2 -1 0 0 - 1 2 - c2 0 0 0 An- 1 =
0 0
0 0
0 0
2 - c2 -1
-1 2 - c2
5. Path Integral Formulation of Quantum Mechanics
2 58
It follows from ( 3.2 ) that
m
(3.4)
e 2�';;;t { q2 + q' 2 - ( A;;-_: lp,p) }
'
27riiL�t l det A n - 1 1 where v lln -1 is the number of negative eigenvalues of the matrix An -1 · It is easy to find det An -1 and ( An - I P , p ) . Namely, p ut an det An . Expand ing det An with respect to the last row, we obtain the three-term recurrence =
=
relation ( 3.5 )
1.
with the initial conditions a- 1 = 0 and ao = Recurrence relation ( 3.5 ) has two linearly independent solutions z n and z- n , where 2 - c:2 = z + z - I , and a solution an satisfying these initial conditions is given by Un =
z n+ 1 z - n -1 z - z- 1 _
From here it is easy to obtain that the eigenvalues of the matrix An are
given by
Ak = z + z- 1 - 2 cos n
Since c: = w:f' we have z = ei0 where
.0 sm
7rk
--
0
+1
=
1
c:
,
+
k = 1,
. .
.
, n.
O (n- 2 ) and
sin wT ( + 0 ( n _1 )) aS n 001 wut and for n large enough the matrix An - 1 has exactly v negative eigenvalues whenever Tv < T < Tv+1 · In order to evaluate the inner product (An-1P, P) we only nee d to know the corner elements of the inverse matrix B A;;-� 1 , which are given by _ Un 2 - sin ( .n - 1 ) 0 ' B1 n - 1 = Bn -11 = 1 = sin-nO B 1 1 Bn -1 n - 1 -. SID 0 sm n 0 Un Un - 1 Thus we get 2 q 2 + q' 2 - (A;;-� 1 p, p) = �O 2 sin � cos ( n (q2 + q' 2 ) - 2 sin 0 qq' . 2 2 sm n Using these formulas and passing to the limit n -+ oo in (3.4) , we obtain the expression for the propagator for the case T i- Tv . The l i mit T Tv is evaluated by using the following standard formula in the theory of sin nO det An - 1 =
=
A
-t
=
-
_
- .
_
(
1)0
)
-+
distributions:
i(x-y)2 1ri 1 lim -- e -2t - = e 4 o(x - y ) .
t-+0 v'2irt
0
3.
Feynman path integral for the harmonic oscillator
259
Remark. It follows from Proposition 3 . 1 that in the limit w ---+ 0 the prop agator of the harmonic oscillator turns into the propagator ( 1 . 1 1 ) of a free particle.
For even v, special values T are integer multiples of the period 2 7!" w of the harmonic oscillator ( see Section 1 .5 in Chapter 1 ) , so when t' - t = Tv , the extremal connecting q at time t and at time t ' exists i f and only if q' = q. Correspondingly, when t ' - t = Tv for odd v, the extremal connecting q and exists if and only if = - q . For the general case t' - t I- Tv , the extremal connecting q at time t and q' at time t ' exists for all q and q' . The integer v is the Morse index of the trajectory T) - the number of negative eigenvalues of the corresponding Jacobi operator .7 , v
Remark.
q'
q'
q'
.7
-m
=
d2 2 dT
q(
-
mw 2 ,
t
:S
T :S
t' ,
with Dirichlet boundary conditions ( see Problem 1.7 in Section 1.3 in Chap ter 1) . Problem 3.2. Compute Weyl, pq, and qp-symbols of the evolution operator for the harmonic oscillator by using: ( a) formula ( 3 . 2 ) and p ath integral representations from Section 2; (b) formula (3.6) and formulas (3.12) , (3.18) and (3. 19) in Section 3.3 of Chapter 2. Problem 3.3. Show that the matrix symbol of the evolution oper ator for the harmonic osci ll at or is K(a, a ; T) exp{aae- i w T - �wT} by using: ( a) series ( 1 .8) in holomor p hic representation; ( b ) Lemma 3 . 2 and path integral representation from Section 2; ( c ) formula (3.6) and result of Problem 2.4. =
3.3. Mehler identity.
( 3.6 )
It is instructive to compare the closed expression
K(q', t ' ; q, t) =
mw -_ ___ :-
2/i sm wT { (q2 +q' 2 ) cos wT- 2 q q ' } e�
2ni!t sin wT for the propagator of the harmonic oscillator with the series ( 1 .8) . Putting x= y = q' and using the explicit formula for the normalized eigenfunctions
ffq , ff
1/Jn (q)
=
- � q2 1 e Hn (x) ;n,;;-:r n y 2 n! ,
corresponding to the eigenvalues En = !tw(n + � ) where Hn (q) are classical Hermite-Tchebyscheff polynomials ( see Section 2.6 in Chapter 2) , we obtain the series
260
5. Path Integral Formulation of Quantum Mechanics =
which converges in the distributional sense. Setting z e- iwT and compar ing with (3.6) , we get the formula 2 x y z - ( x2 + y2 ) z 2 1 2 exp ( 3.7) � � H ( ) H ( ) X Y n LJ 2 n n I n 1 z2 ' Y� .L - z n=O . where z i= ± 1 , and the square root in the right-hand side is understood as in Proposition 3. 1 . When l z l < 1 , formula (3.7) is the classical Mehler identity from the theory of Hermite-Tchebyscheff polynomials. Thus we obtained a distributional form of the Mehler identity for l z l = 1 by computing the propagator of the harmonic oscillator in two different ways. Formula ( 3. 6) shows that the propagator K ( q' , t' ; q, t) is a smooth func tion of q, q' and T t' - t whenever T i= Tv , and it is singular at T Tv. Corresponding eigenvalues of the evolution operator U ( Tv ) are e - 1T"i v ( n+ � ) , so that when v is even we have U ( Tv ) = e- "�.., I, and therefore
{
=
-
=
}
=
i IS ( q - q' ) , + Tv ; q, t) = e T in perfect agreement with Proposition 3. 1 . For odd v using the series ( 1.8 ) we obtain v 00 * HT .., 1T" e e � L ( - 1 ) n Pn , K (q' , t
-
.
=
-
7r v
.
n =O
where Pn are projection operators on the eigenspaces C'l/Jn of H . Since Hn ( - q) = ( - I ) n Hn ( q ) , in this case we have
i IS (q + q' ) , + Tv ; q, t) = e- T which again agrees with Proposition 3 . 1 . 7T" l/
K (q' , t
4.
Gaussian path integrals --->
was already mentioned in Section 1 that in the semi-classical limit n 0 the leading contribution to the propagator K ( q , t' ; q, t) is given by the classical trajectory qct (T) . This suggests to represent the paths 1 = q (T) E 't' P (JR )�: t as q ( T ) qc1 ( T) + y ( T ) , where y ( T) - the quantum fluctuation part - satisfies Dirichlet boundary conditions y ( t) y (t' ) 0. It follows from the principle of the least action ( see Section 1 . 2 in Chapter 1) that It
=
(4. 1)
S ( qcl
+ y)
= Set + �
=
1 (mii t'
V" (qc i (T) ) y2 ) dT
=
+ higher order terms in y ,
where Sc1 = S (qc� ) is the classical action. Similarly, for the case of several degrees of freedom, (4.2)
S ( qcl
+ y)
= sci
+
�
1t'
.J (y ) y dT
+ higher order terms in y ,
4.
26 1
Gaussian path integrals
where .:J is the corresponding Jacobi operator ( see Problem 1 . 7 in Section 1 . 3 of Chapter 1) . It is remarkable that the Gaussian path integral
J {y(t')=O} y(t)=O
(4.3)
over the fluctuating part can be evaluated explicitly in terms of the reg ularized determinant of the second order differential operator .:J. Here we do this calculation for the case of the free particle and harmonic oscillator, and give another interpretation of formulas ( 1 . 1 1 ) and (3.6) for the prop agators. The Gaussian path integral ( 4.3) also plays a fundamental role in the semi-classical asymptotics which we discuss in Section 6. 1 . In general, formula (4.2 ) , with higher order terms in y , and Gaussian integration over �y, constitute a basis for the perturbative expansion of the propagator. 4. 1.
Gaussian path integral for a free particle.
free quantum particle is (4 .4)
;-rn:-
K(q', t' ; q, t ) = y �
e
im(q-q1) 2
The propagator of a
2n T
and the corresponding classical trajectory is
q' - q qci(T) = q + ( T - t) ---r , T t' - t. Using the decomposition q(T) = qc� (T) + y(T) , where y( t ) = y( t' ) = 0, we =
obtain where
S(q) = �
1t' mildT
t' · r q 2l d Sci = 2 jt m c 1
T
=
Sc1 + S(y) ,
(q - q')2 =m 2T
Assuming that �q = � y under the "change of variable" q rewrite the Feynman path integral for a free particle as
=
qcl + y, we can
K(q', t ' ; q, t ) = e K 8cl
{y(t') y(t)=O=O} i
im(q-q' )2
Remarkably, the classical contribution e K 8c i = e 2n T exactly reproduces the exponential factor in the propagator for a free particle. The integral over the fluctuating part - the Gaussian path integral for a free particle - does not depend on q and q1 and, as we know, coincides with the prefactor in (4.4) .
5. Path Integral Formulation of Quantum Mechanics
262 A
more conceptual way to interpret this result is the following. Let
_!!_ dT ' be the second order differential operator on the interval [t, t'] with Dirich let boundary conditions y(t) = y (t') = 0. The operator A is self-adjoint on L 2 (t, t') . For any real-valued, absolutely continuous function y(T) satis fying Dirichlet boundary conditions and y, iJ E L 2 (t, t') , we have by using integration by parts D
( Ay , y) = -
=
' ijy dT t' y2 dT. t 1 =1
The "integrand" in the fluctuation factor
e �� Jt iidr �y J {y(t')=O}
y(t)=O
is the exponent of the quadratic form of the operator A, and in accordance with the finite-dimensional formula (3. 1) it is natural to expect that this Gaussian path integral is proportional to ( det A ) - 2 . Of course, the problem here is to understand what we mean by a determinant of a differential op erator. Clearly, it should be defined by some regularization of the divergent infinite product A n , where A n are non-zero eigenvalues of A. The most natural and useful regularization is given by the so-called operator zeta-function. Namely, let A be a non-negative self-adjoint operator in the Hilbert space £ with pure point spectrum 0 :S A 1 :S A 2 :S . . . , such that for some a > 0 the operator ( A + I) -a is of trace class. Then the zeta function (A ( s) of the operator A is defined for Re s > a by the following absolutely convergent series: 1
fl�= l
(A ( s ) = L A s . A >O n 1
n
If (A ( s ) admits a meromorphic continuation to a larger domain contain ing the point s 0 and is regular at s = 0, then we define a regularized determinant of A by ( 4.5 ) (0) det ' A = exp =
{ - dJ: }·
Here the prime on the symbol det indicates that zero eigenvalues are ex cluded from the definition of an operator zeta-function. In the special case when 0 is not the eigenvalue of A, it is customary to denote the regularized
4.
Gaussian path integrals
263
determinant of A by det A . We will also write det A = II' A n , I
..\n> O
where the prime indicates that the infinite product is regularized by the operator zeta-function. We have (cA( s ) = C 8 (A( s ) for c > 0, so that det ' cA = c...r - 2A , 2 v" A
;:
c sin det ( A - AI ) =
T
,
=
and comparison between formula det(A + w2 I) det Aiw and Lemma gives c = 2 . Thus we have proved the following result.
4. 1
5. Path Integral Formulation of Quantum Mechanics
268
Lemma 4 . 2 . The characteristic determinant det ( A - ).. ! ) of the operator A = - D 2 on the interval [t , t'] with Dirichlet boundary conditions is well defined, and is an entire function of >.. . It is given explicitly by
det ( A - ).. J ) = det A
IT ( 1 - �) n= l An
=
2 sin vfA T . vfA
Problem 4 . 2 . Prove the Jacobi inversion formula. (Hint: Use the Poisson sum00
mation formula
where
f
E
Y (IR) and
00
L f (n ) = � L ] ( 21rn ) ,
n= - oo
n = - oo
j is the Fourier transform of f.)
Problem 4.3. Prove formula (4.9) . Problem 4.4. Let L
=
m ( ±2
2
+
]/
+
i2 )
+
eB
y- x 2c (x y )
be the Lagr angian of a classical particle moving in the constant uniform magnetic field B (0, 0, B ) . Show that the propagator of the corresponding quantum particle in Section 2.4 of Chapter 2) is given by (see Example =
2 .3 ( m ) � wT exp im2 { (z - z1 ) 2 w cot wT [( x - x1) 2 K ( 1 , t1 t ) fi T = 21rinT sin wT +(y - y1)2] 2w(xy1 - yx1 ) } · r
;
r,
+
+
(Hint: Use Corollary 4. 1 . ) 5.
Regularized determinants o f differential operators
Here we study the characteristic determinant det ( A Liouville operator A = -D 2 + u(x) ,
D=
>.. I )
of the Sturm
d dx '
on the interval [0, T] with Dirichlet or periodic boundary conditions, and its generalizations to the matrix case. 5 . 1 . Dirichlet boundary conditions. Suppose that u(x) E C1 ( [0, T] , JR) . The operator A is self-adjoint on L2 (0, T) with domain D ( A ) = {y (x) E W 2 ' 2 (0 , T ) : y (O)
=
y (T)
=
0},
where W 2, 2 (0, T) is the Sobolev space. It has a pure point spectrum with simple eigenvalues ).. 1 < ).. 2 < · · · < < · · · , accumulating to oo. Moreover , as n -> oo , 1 {T (5. 1) u(x)dx . where c
An
=
T
Jo
5.
269
Regularized determinants of differential operators
Consider first the case A > 0. Putting
't? A ( t )
=
we have for Re > � '
00
L e - An t ,
=
n I
s 1 roo d (A ( s ) = f ( s ) lo 1JA(t ) t 8 tt
(5.2)
=
( 5.3 )
dt . s dt + 1 r I 't? A ( t ) t s t ) A (t t 1J t f ( s ) lo f(s ) }I roo
1
Since 't? A ( t ) O(e- .X. 1 t ) as t oo , the first integral in (5.3) converges absolutely for all E C and represents an entire function. Using asymptotics ( 5.1 ) and the Jacobi inversion formula we get as t ---> 0, =
--->
s
't? A (t ) = � e-ct where
( ( ;; ) - 1 ) '!?
( 5.4 )
a
( 1 + O (t) )
T
1 = --
2 fo '
- "2
a!t
=
+ ao
+ JA (t ) ,
1
ao = - 2
and JA ( t ) = 0( vt) . Thus for the second integral in ( 5.3 ) we have a 1 dt 1 rI 1 rI ao d 1JA ( t W tt = Jo 1JA (t ) t s t '
( s - !)r( s ) + sf ( s ) + f ( s )
f ( s ) Jo
-
s
so that it admits a meromorphic continuation to the half-plane Re > and is regular at = 0. Therefore we can define
s
I
det A =
OO IT
=
I
An
n I
=
{
d exp - (A
ds ( 0 )
-
�
}·
Now repeating verbatim the arguments in the proof of Lemma 4.2, we see that for Re ( A. N - A. ) > the truncated zeta-function (;t-.!i) (s) admits a meromorphic continuation to Re s > - � , and is regular at = 0. Defining the regularized product Jl�= N +1 ( A n - A. ) by the same formula 4. 1 1 ) , we see that
0
det (A - A. I )
s
oo
N IT (A. k - A. ) ITI
(A.n - A.)
(
n= N +I k= I is an entire function of A. with simple zeros at A n· To remove the assumption A > 0, replace A by A = A + ( a - A. I )J > 0, where a > 0. Then6 det ( A - A.I ) det ( A - ( A. + AI - a ) I ) , =
=
6 The definition of det'
A
does not depend on the choice of a > 0 .
5. Path Integral Formulation of Quantum Mechanics
2 70 and d et ' A
=
{
det A
if 0 is not an eigenvalue of A, 1 I lim.>.- o _A- det ( A + .A ) if 0 is an eigenvalue of A.
Let Y l (x, .A ) be the solution of the second order differential equation ( 5.5 ) -y " + u (x) y = .Ay on [0, T] , satisfying initial conditions ( 5.6 ) Y l ( 0, .A ) = 0, y � ( 0, .A ) 1 . It is known that for every 0 ::S x ::S T, the solution Y l ( x , .A ) is an entire function of of order ! and as oo , =
A
( 5. 7 )
Y l ( x , .A )
=
A
----+
:;
sin
x
+ 0 (I.AI-le/ Re -/.\/ x )
.
The entire function d ( .A ) Yl (T, .A ) has simple zeros at the eigenvalues A n of the operator A, and has the following Hadamard product representation: .A ( 5.8 ) d ( .A ) = c x' II .A n A #O =
n
(1 - ) .
1 if 0 is an eigenvalue of A, and 8 = 0 otherwise. Theorem 5 . 1 . The characteristic determinant det ( A - .AI ) is given by the
Here c is a constant,
8
=
simple formula
det ( A - .A I )
=
2d ( .A ) .
Moreover,
( �) An '
det ( A - .AI ) ( -.A ) 6 II 1 det ' A = .An# O
_
which fixes the constant in ( 5.8 ) as c = ! ( - 1 l det ' A. Proof.
Since both functions are entire, it is sufficient to prove the equality ( det A - .A I ) = 2 d ( .A ) for Re ( .A1 - .A ) > 0. In this case, using dt 1 �' ':.A .>.I ( s ) __ r>e' Tr e - (A - .>.I) tt s
for Re s
>
,
-
=
f ( s ) Jo
t
! and differentiating under the integral sign we get 1_ r= � A- ) - (A-.>.I)t t s dt, S a.A ( .>.J ( - f ( s ) lo Tr e _
_
5. Regularized determinants of differential operators which is now absolutely convergent for Re s respect to s at s = 0, we obtain
>
271
- ! . Differentiating with
�i ( 0) = r)Q Tr e- (A ->..I )t dt = Tr ( A .X I) - 1 8s8.X '> A - >..I It follows from (5. 1 ) that the operator R>.. (A - .XI) - 1 - the resolvent of A - is of trace class. Thus all our manipulations are justified and we arrive at the following very useful formula:
}0
-
·
=
d log det(A - .XI) d.X
= - Tr R>.. , which generalizes the familiar property of finite-dimensional determinants. To compute the trace in (5.9) , we use the representation of R>.. for .X of. An as an integral operator with the continuous kernel R>.. (x, �) (cf. formula (2.2 1 ) i n Section 2 . 2 o f Chapter 3) . Namely, let y2 ( x, .X ) b e another solution o f (5.5) with boundary conditions Y2 ( T, .X) 0 and y� ( T, .X) = 1 , so that W ( y1 , Y2 ) (.X) = Y� (x, .X) y2 (x , .X) - Y1 (x , .X ) y� (x, .X) = - d ( .X ) . Using the method of variation of parameters, for the solution of the inho mogeneous equation -y" + u (x) y = .Xy + f (x) , A of. A n , satisfying Dirichlet boundary conditions we get = T R>.. , �)f 0 d� (5.9)
=
y (x)
where
lo
(x
(
,
if X � � '
(5 . 1 0)
if X � � -
Since R>.. is a trace class operator on £ 2 ( 0, T) with the integral kernel R>.. ( x, �) , which is a continuous function on [0, T] x [0 , T] , its operator trace equals the "matrix trace" , Tr R>.. = T R>.. (x, x)dx = Y (x .X) y2 (x, .X) dx . d .X) T 1 ,
lo
t lo
We evaluate the last integral by the same computation used in the proof of Proposition 2 . 1 in Section 2 . 1 of Chapter 3 . Namely, put iJ( x , .X) = (x, .X) , and consider the following pair of equations: -iJ� + u (x) y 1 = .X i;1 + Y1 , -y� + u(x) y2 = AY2 ·
��
5.
272
Path Integral Formulation of Quantum Mechanics
Multiplying the first equation by y2 (x , .X ) , the second equation by iJ1 (x, .X) and subtracting, we obtain (" ' Y1Y2 = Y1 Y2 - Y1 Y2 = W Y1 , Y2 ) , •
II
-
· II
so that
(5. 1 1) This formula is valid for any two solutions of the differential equation (5.5). Using boundary conditions for the solutions Y 1 and y2 , we finally get
loT Y1 (x, .X)y2 (x, .X)dx
(5. 12)
= iJ 1 (T, .X ) .
Thus we have proved that for Re ( .X 1 - .X ) > 0 , d (5. 13) Tr R>. = - .X log d(.X) , d which implies that
(5. 1 4 )
det(A - .X I) = C d( .X )
(5. 15)
as
for all A E C and some constant C. It follows from ( 5. 7) that
J-L
___.
+ oo.
Thus in order to determine the constant C in (5. 1 4) , it is sufficient to com pute the asymptotics of det(A + J.Ll) as J-L +oo. We have ___.
- 1 roo UA ( t ) e -j.Lt t s dt + 1 r 1 UA ( t ) e -J.Lt t s dt · '> A +J.Ll ( s ) T r(s) }1 t r(s) lo The first integral is an entire function of s whose derivative at s = 0 expo nentially decays as J-L + oo . For the second integral we have 1 1 1 1 dt dt rJA ( t) e - J.Lt t s rJ A ( t)e-J.Lt t s = r ( s) 0 t r(s) 0 t 1 dt 1 { e -J.Lt t s · + + T r(s) }0 vt .a
r
-- 1
___.
.a
-- 1 ( a-� ao )
Since JA ( t) = 0( vt) as t 0, the first integral is absolutely convergent for Re s > - � and its derivative at s = 0 is O (J-L- � ) as J-L +oo. For the ___.
___.
5. Regularized determinants of differential operators
273
remaining integral we have
r�s) l (":Ji ao) e -"' t' �t = "-;(.� -' (r(s - � ) - f e -'t•- l �t ) t + a�(s�s ( r ( s ) - 100 e-t t s � ) . +
It is elementary to show that the s-derivative of this integral at s = 0 has asymptotics -2y'1ra_ ! fo - ao log J-L + 0(e - �LI 2 ) as J-l -t + oo . Using ( 5.4 ) we finally obtain 2
as J-l
det(A + J-LI) = -t
efoT ( fo
1
1 + O (J-L- 2 )
+oo, and comparison with ( 5.15 ) gives C = 2 .
)
D
Remark. When zero is not an eigenvalue of A, its inverse A - 1 is a trace
class operator and
det(A - >.. I ) = detp ( I _ >..A _ 1 ) det A where detp is the Fredholm determinant.
'
Remark. In Section 6.1 we will use Theorem 5.1 for evaluating the fluc tuating factor in the semi-classical asymptotics of the propagator, and in Section 3. 1 of Chapter 6 - for evaluating Gaussian Wiener integrals.
A similar result holds for the matrix-valued Sturm-Liouville operator with Dirichlet boundary conditions. Namely, let U(x ) = { uij (x) }i,j = 1 be a C 1 -function on [0, T] which takes values in real, symmetric n x n matrices, and consider A = - D 2 In + U(x) , where In is the n x n identity matrix. The differential operator A with Dirich let boundary conditions is self-adjoint on the Hilbert space L 2 ( [0, T] , en ) of e n -valued functions, and has a pure point spectrum accumulating to oo. Its regularized determinant det' A and characteristic determinant det (A - >.. I ) , where I is the identity operator in L 2 ( [ 0 , T] , en ) , are defined as in the n = 1 case. Let Y(x, >.. ) be the solution of the differential equation -Y" + U (x)Y = >.. Y satisfying initial conditions Y(O, >.. ) = 0, Y ' (O, >.. ) = In , and put D(>.. ) = det Y (T, >.. ) . The entire function D(>.. ) has properties sim ilar to that of d(>.. ) , and the following analog of Theorem 5.1 holds.
274
5. Path Integral Formulation of Quantum Mechanics
Proposition 5 . 1 . The characteristic determinant det(A the formula
>..I ) is given by
det(A - >.. I ) = 2 n D(>.. ) . Moreover,
( �)
det(A - >.. I ) = ( - >.. ) 8 IT 1 det' A >-n #O
_
An
'
o is the multiplicity of the eigenvalue ).. = 0. Problem 5 . 1 (Gelfand-Levitan trace identity) . Prove that � ( \ \ ( o ) c) )d u ( O) + u(T) ( where
L....,
where
A�O)
n= l
=
_
"n
_
"n
=
_2_ 1T 2T
(":; ) 2 and C = � faT u ( x )dx .
0
u X
X
_
4
'
5 . 2 . Periodic boundary conditions. As in the previous section, we as sume that u ( x ) E C 1 ( [0, T] , �) . The Sturm-Liouville operator A -D 2 + u ( x ) with periodic boundary conditions is self-adjoint on £2 ( 0, T) with the domain D ( A ) { y ( x ) E W 2 • 2 (0, T ) : y (O ) = y (T) and y' (O ) y' (T ) } . It has a pure point spectrum with the eigenvalues < A 2 n - l :S A 2 n < · · · >.. o < >.. 1 :S >.. 2 < accumulating to oo . Moreover, as n ---+ oo , 4 7r2n2 (5. 16) A 2 n-l = --y;2 + c + O ( n -2 ) , =
=
=
·
·
·
where c is the same as i n (5. 1 ) . Replacing, i f necessary, A by A - (>.. o + a )I with a > 0, we can always assume that >.. o > 0, and define 00
n =O Using asymptotics (5. 16) , we get that as t ---+ 0, t ( 1 + O(t) ) = '19A (t) = e- ct79
�
(� )
a1 + O (Ji) ,
where a _ � = as in (5 . 4) , but ao 0. This allows us to define the 2 regularized determinant det' A and the characteristic determinant det(A >.. ! ) exactly as in the previous section. Since ao = 0, we now get as J.L ---+ + oo det ( A + J.LI ) = e foT 1 + O (J.L - ! ) (5 . 1 7) =
(
)
5. Regularized determinants of differential operators
275
( see the end of the proof of Theorem 5 . 1 ) . Here we denote 7 by Yl (x, .\) and y2 (x, .\) solutions of the Sturm-Liouville equation ( 5.5 ) satisfying initial conditions Yl (O , .\) = 1 , Yi (O, .\) = 0 and Y2 (0, .\) = 0, y� (O , .\) = 1 . Solutions Y l and Y2 are linearly independent for all A and the matrix Y ( X , .\ )
-( _
Y l (x , .\) Y2 (x , .\) Yi ( X , A ) y� ( X , A )
)
satisfies the initial condition Y ( O , .\) = !2 , where h is the 2 x 2 identity matrix, and has the property det Y(x, .\) = 1 . For fixed x the matrix Y (x, .\) is an entire matrix-valued function of ,\ having the following asymptotics as .\ --+
oo :
( 5.18 )
Y (x, .\)
=
(
� sin
cos vf\x
vf\ sin �x
�x
cos �x
)(
12
+ O ( l .\ 1 - l e i Re y';\lx ) )
.
By definition, the monodromy matrix of the periodic Sturm-Liouville problem is the matrix T (.\) = Y(T, .\) . The monodromy matrix satisfies det T (.\) = 1 and is an entire matrix-valued function. The following result is the analog of Theorem 5.1 for the periodic boundary conditions . Theorem 5 . 2 . One has det (A - .\I) = -det 2 (T( .\ ) - !2 ) = Y I (T, .\) + y� (T, .\) - 2 , where det 2 is the determinant of a 2
(
x
2 matrix. Moreover,
)
- ,\ 6 det ( A -; .\I) = ( ) IT 1 � , An det A An.,.-0 where { An }�= O are the eigenvalues of A, and 0 the eigenvalue A = 0 . _J_
_
�
o
�
2 is a multiplicity of
Proof. The proof follows closely the proof of Theorem 5.1, and we will assume that A > 0. First, in exact analogy with (5.9) we obtain that for .\ i= A n , d d.\ log det ( A - .\J) = - Tr R.>- ,
where R.>- = ( A - .\I) - l . To get a closed expression for the integral kernel R(x, �) of the operator R.>- , we use the same variation of parameters method as in the previous section, but now for periodic boundary conditions. As a 7 There should
be
no confusion with the notation in the previous section.
276
5. Path Integral Formulation of Quantum Mechanics
result, we obtain that the symmetric, continuous kernel R>. ( x , e) is given for X :S e by the formula R>. ( x, e)
=
-
(Yl (x , -\ ) , y2 (x , -\ ) ) ( T( -\ ) - I2 ) -1 T( -\ )
(-���t�))
= -Tr 2 { (T (-\ ) - I2 ) - 1 T (-\) Z (x, e; -\ ) } , where Tr 2 in the last formula is the matrix trace and z ( x , e ·' -\ ) = Y l (x , -\ )y2 (e , -\ ) Y2 (x , -\)y2 (e, -y1 (x , -\ ) yl (e , -\ ) -y2 (x , -\ ) y1 (e, -\ ) ·
,\))
(
As in the proof of Theorem 5 . 1 , we need to compute J0T Z(x , x ; -\ ) dx . It readily follows from formula ( 5 . 1 1 ) and the definition of the monodromy matrix that Therefore
d d� log det(A - -\I ) = Tr 2 (T( -\ ) - I2 ) - 1 d T( -\ ) = d� log det 2 ( T (-\) h ) -\ and det ( A - ,\I ) C det ( T( -\ ) - I2 ) . To determine the constant C we set A = -p, ---> +oo and compare asymptotics ( 5. 17) with the asymptotics
(
=
- ,
)
=
1
2 - Tr T(- p, ) = 2 - 2 cosh foT ( 1 + O(p,- 2 ) ) = - e foT ( 1 + O(p,- � ) ) , 0 which follows from (5 . 1 8) . Thus C = - 1 . Remark. In Section 3.2 of Chapter 6 we will use Theorem 5.2 for calculating Gaussian Wiener integrals over the loop spaces. det2 (T(-p,) - I2)
In the special case u( x )
=
0, we have
Yl (x, -\ ) = cos �x and Y2 (x , -\ ) =
so that
:;x ,
sin
�T ·
-
det (- D 2 ,\I ) = 2 ( cos �T - 1 ) = -4 sin2 Setting A = -w 2 < 0, for the operator A iw = -D 2 + w 2 we get T . h2 w 2' det A iw = 4 sm ( 5 . 20) ( 5 . 1 9)
and also
(5 . 2 1 )
det' Ao =
det ( A - I ) -\ - >.->Olim A
=
lim w-->0
det' Aiw w2
=
T2 .
5.
277
Regularized determinants of differential operators
Remark. Using that the spectrum of Aiw consists of double eigenvalues >-n (w) =
c;nr, n = 1 , 2 , , and of the simple eigenvalue we can derive formulas (5.20)-(5.2 1 ) directly, as was done in Section 4.2 for Dirichlet boundary conditions. There is also an analog of the heuristic computation in Section 4.2: 2 4 oo oo det Aw ) IJ IJ 2 ( 1 + 4 n = smh -2 . ' (0) = det'A = w An . . .
o
w2 ,
A n (w) 2
2
w2
n= l
.
w 2 T2
-
7r 2
n= l
2
T2
2
wT
similar result holds for the general second order differential operator A = -D 2 + v (x)D + u ( x ) on the interval [0, T] with periodic boundary con ditions. Namely, repeating the proof of Theorem 5.2, we have the following Theorem 5 . 3 . For the differential operator A = -D 2 + v (x)D + u ( x ) one has det(A - >. I) = e � f[ v ( x) dx det2 ( T ( >. ) - h ) A
-
=
-
e � J[ v (x)dx -
(Yl (T, >. ) + y; (T, >.)
_
where solutions Y 1 , 2 ( x , >.) and the monodromy matrix the same formulas as for the case v (x)
=
0.
1
_
e g v (x) dx ) ,
T ( >. )
are defined by
In particular, the following result will be used in Section 2.2 of Chapter
8. Corollary 5.4.
det' ( D 2 + wD) = -
2T .
-
w
wT
smh - . 2
Proof. The proof is an elementary computation, using Theorem 5.3, an explicit form of the solutions Y 1 , 2 ( x , >.) , and the formula 2 1.1m det ( D + w D - >. I ) . O det ( D 2 + w D) , >.-+0 '
-
=
-
-
A
+ w D on [0, T] with periodic bound 2 + iw ary conditions has simple eigenvalues A n ( a ) = = - oo , , oo , we can also repeat the heuristic computation in Section w 2 T2 det' ( D2 + w D ) _!._ sinh wT . = + = 47r2n2 wT det ' ( D 2 ) n= l Remark. S i nce the operator . . .
-
-
- D2
IJoo (1
)
( 2;n )
2
( 2;n ) , n 4.2:
A similar result holds for the matrix-valued Sturm-Liouville operator with periodic boundary conditions. Namely, let U (x) {uij (x)}f,j = l be a C 1 -function on [0 , T] which takes values in real , symmetric x matrices, and consider A = D 2 In + U(x) . =
-
n n
5. Path Integral Formulation of Quantum Mechanics
278
The differential operator A with periodic boundary conditions is self-adjoint on the Hilbert space of en -valued functions, and has a pure point spectrum accumulating to oo. Its regularized determinant det' A and characteristic determinant det ( A - ).. I ) , where I is the identity operator in are defined as in the n = 1 case. Let ).. and Y2 (x , ).. be the solutions of the differential equation = + satisfying, respectively, the initial conditions and ).. ) = The monodromy matrix is defined as the following 2n x 2n block matrix: ).. ) Y2 (T, ).. T ( ).. ) ).. ) Y� (T, ).. ) and is a matrix-valued entire function. The analog of Theorem 5 . 2 is the following statement.
£2([0, T], en )
£2 ( [0 , T], en ) ,
Y1 (x, )
)
-Y" U(x)Y )..Y Yi.(O , ).. ) = In , Y{ ( O , ).. ) = 0 ¥2(0, 0 , Y; (o, ).. ) = In . T().. ) )) (YY1 ((TT, - { , _
'
Proposition 5 . 2 . The characteristic determinant is given by the formula
det ( A - ).. I ) =
( -l ) ndet2n (T().. ) - hn ) ,
where det 2n is the determinant of a 2n
x
2n matrix, and
( 1 �) An , where 8 is the multiplicity of the eigenvalue ).. = 0. det ( A ; ).. I ) det A
=
( - ).. ) 0
IT -J.
_
AnrO
Problem 5 . 2 . Prove Theorem 5.3.
Problem 5.3. Derive Corollary 5.4.
Here we continue to assume C1([0 , T], JR) , and consider the first order differential operator A = D + u( x) on the interval [ 0 , T] with periodic boundary conditions y ( O ) = y(T). The 5.3. First order differential operators.
that u (x)
E
equation
y' + u (x) y ).. y has an explicit solution y (x) = e .\x f;' u (r) dr , which is periodic if and only if ).. = where =
An ,
C
-
and uo =
1 { T u ( x ) dx . T Jo
Thus the spectrum of the operator A coincides with the spectrum of the operator Ao = D + uo .
2 79
5. Regularized determinants of differential operators Proposition 5.3. For u0
> 0,
det(D + u(x) ) and det ' D = T for uo Proof.
series
=
1 - e -uoT ,
=
0.
The zeta-function of the operator A with uo > 0 is given by the
00
(A (s ) = L ,\8 , n =- oo n 1
where A;;-8 = e-s log >- n with the principal branch of the logarithm. This series is absolutely convergent for Re s > 1 . Introducing the Hurwitz zeta-function ((s, a) =
00
1
� (n + a)s '
where Re a > 0 and Re s > 1 , we can rewrite (A (s) as
( 211") -s (e-2 ( (s, a ) + e2 ((s, rr i s
rr i s
211"
1 uoT . , a = 1 - -- t. ug It is well known that the Hurwitz zeta-function admits a meromorphic con tinuation to the whole s-plane with single simple pole at s 1 with residue 1, and 1 1 8( ((O, a) = 2 - a, !:l (O, a) = log r (a) - - log 21r.
(A (s )
=
T
a) ) +
=
2
us
Using the classical formula r ( 1 + z ) r ( 1 - z)
we obtain
=
1fZ
. -
Sin 1fZ
,
d(A ( 0) = log lr (a) l 2 - log u0 T + uo T ds uo T � - log( e � 2 -e 2 )+ =
-
2 -2-,
so that det (D + u(x) ) = 1 - e- uo T . Finally, (v (s ) = limu0_. o ((A (s ) and we get det' (D + uo ) . det ' D = lim =T +0
uo-
UQ
- u0 8 ) , 0
Remark. For uo < 0 one should use the branch of the logarithm with the cut along the positive semi-axis, and the above arguments give
det(D + u(x)) = 1 - e uoT.
5.
280
Path Integral Formulation of Quantum Mechanics
Remark. One can also consider the operator A = D + u ( x ) on the inter val [0, T] with anti-periodic boundary conditions y ( O ) = -y (T) . The corre sponding eigenvalues are ,
An
-
_
uo
+
1ri ( 2n + 1 ) T
, n E Z,
and the passage from periodic to anti-periodic boundary conditions amounts to replacing uo by uo + � . It follows from Proposition 5.3 that for uo > 0, det ( D + u ( x ) ) = 1 + e - uoT .
Remark. Proposition 5 . 3 is very useful for calculating Gaussian path in tegrals in the holomorphic representation, discussed in Section 2. 4 . As an example, consider the harmonic oscillator with the Wick symbol H(a, a ) = w(aa + ! n) . Formula (2. 16) expresses the trace Tr e- k TH as a path integral in the holomorphic representation. On the other hand , using the explicit form of the eigenvalues En = w n ( n + ! ) , we immediately get
Comparing with (2. 16) , we obtain that
( 5. 22)
f {ii(O)=ii(T)} a(O)=a(T)
1 det ( D + ) ,
e- k foT (iiit+waa) dt 91a91 a =
w
which should be considered as a special analog of the finite-dimensional Gaussian integration in the complex domain - formula (3.3) . Problem 5.4. Give a direct proof of formula (5.22) . 6.
Semi-classical asymptotics -
II
-
Here we consider the semi classical asymptotics the asymptotics of the propagator8 Kn( q ', t'; q , t) as n 0. We compare the heuristic method, based on the Feynman path integral representation ( 1 .26) , with the rigorous analysis, based on the short-wave asymptotics, derived in Section 6. 1 of Chapter 3. -
8 Here the dependence on the Planck constant
li
is introduced explicitly.
6. Semi-classical asymptotics
-
II
281
6 . 1 . Using the Feynman path integral. Start with the Lagrangian L(q, q) = � mq2 V(q) for a classical particle with one degree of freedom. The propagator Kn(q', t ' ; q, t) is given by the Feynman path integral (1.20) , and we will formally apply the stationary phase method to investigate its behavior as n - 0. As in Section 4, we assume that there is a unique classical trajectory Qc! ( T ) connecting points q and q1 at times t and t' , set q (r) = qc� (r) + y (r) , and consider the expansion (4. 1 ) , i.e. ,
-
where u(r)
=
S(q)
=
Sc1 + � m
� V" ( qc� (r)) and Sc1 =
1t' (i/ - u (r) y2 ) dr
+ O( y 3 ) ,
1t' ( �mq;1 - V(qc� ) ) dr = S (q' , t' ; q , t) .
According to the stationary phase method ( see Section 2.3 in Chapter 2) , the leading contribution to the Feynman integral (1.20) as n - 0 comes from a critical point of the action functional - the classical trajectory Qcl ( T ) . Thus we obtain as n - 0, Kn ( q ', t'; q, t)
(6 . 1 )
�
e Ksci
J
e ��
Jt (iP-u(T)y2)dT �y
{ y(y(t')t) =O=O } = V7ri li :et A exp { * S (q', t' ; q, t) } .
Here A is the corresponding Jacobi operator - a second order differential operator -D2 - u(r) on the interval [t, t'] with Dirichlet boundary condi tions. ( We are assuming that the potential V (q) is sufficiently smooth so that u (r) E C1 ( [t, t'] ) .) Formula ( 6. 1 ) is a remarkably simple expression which shows a deep relation between semi-classical asymptotics of a quantum me chanical propagator and classical motion. Remark. When u ( T ) =
.2_ V" ( Qc! ( T ) ) , the regularized determinant of the m differential operator A = -D2 - u(r) on the interval [t, t'J with Dirichlet boundary conditions can be expressed entirely in terms of the classical tra jectory Qcl ( r ) . Namely, differentiating Newton's equation mij V ' (q) , (6.2) with respect to T, we find that the function T ) = iJ.c1 ( T ) satisfies the differ ential equation Ay = 0. When y (t) = 4ct (t) = 0, the function9 = -
Yl ( T ) = yy((r)t )
9 Here we assume that y(t) "I 0, so that V' (q) "I 0.
y(
5.
282
Path Integral Formulation of Quantum Mechanics
satisfies initial condition (5.6) ( where the interval [0 , T] is replaced by the interval [t, t'] ) . According to Theorem 5 . 1 , we have in this case, (t ) det A = 2 YI ( t') = - 2 m qcl ' . (6 . 3) V' ( q )
In order to find the solution YI ( r ) of the differential equation Ay = 0 for the case y(t) =f. 0, observe that the Wronskian y1y - YYI of its two solutions is constant on [t, t'] . Using (5.6) we get Y I Y - YY l y(t) , and solving this differential equation we obtain =
YI (r) = y(r)y(t) Thus we get the formula
(6.4)
det A = 2y(t) y (t')
d
lr yt(s)
1 t l y �) 2
,
·
y(r) = qc� (r) ,
which expresses the fluctuating factor in the semi-classical asymptotics of the propagator in terms of the classical motion. Similarly, for the case of n degrees of freedom, when L = ! mq 2 - V(q) , and V (q) E C3 (lRn , JR ) , we obtain ffi 1 e ifi S(q ,t ,q, t ) Kfi(q', t'; q, t) � . (6.5) 7r t li v'det A as 1i --t 0, where A = -D2 - U(r) and n 82 V 82 V . U(r) � (qc�(r) ) = � (qc�(r)) q� u qJ uq i,j=l
( )�
I
}
{
=
I.
Let Kfi ( q , q', t) be the fundamental solution of the Schrodinger equation - the solution of the Cauchy problem ( 1 .3) and ( 1.5 ) . Since Kfi(q' , t' ; q, t) = Kn, ( q' , q, T) , where T t' -t, we need to find the asymptotics of a fundamental solution Kfi(q, q' , T) as li --t 0. The solution of this problem can be divided into two parts. 1. Find the short-wave asymptotics - asymptotics as !i --t 0 of the solution '1/Jn,(q, T) of the Cauchy problem for the Schrodinger equa tion 6 . 2 . Rigorous derivation.
=
. 8'1/J t !iat
li2 82'1/J 8q 2
= - 2m
with the initial condition '1/J fi (q, t ) l t =O =
+
V (q )
cp (q ) e * s (q) '
'l/J
6. Semi-classical asymptotics
2.
-
II
283
where s(q) , rp(q) E C00 (1R., JR.) , and the amplitude rp(q) has compact support. Using the representation 5(q - qo ) =
rp(q)
100 e H (q-qo ) d�,
21r n _ 00 where
1(t) - 1(0) establishes the isomorphism between the space of C ( [O, 1r] , JR.) of functions orthogonal to 1 and C( [O, 1r] , JR.; 0) , and use the Fourier sine coefficients for the embedding C ( [O, 1r] , JR.) '----> 1/.)
Problem 2 . 3. Prove that the Wiener measure
=
2.2. Conditional Wiener measure and Feynman-Kac formula. Let fl q , q'
=
{1
E
11
t�;r:,t'
in : 1 (t)
=
Q , / (t ' ) =
q' }
be the space of all parametrized paths in in which start at q E !Rn at time t and end at q ' E !Rn at t' , and let 'ffq , q' be the corresponding subspace of continuous paths. The conditional Wiener measure P, q , q' on fl q, q' is defined
2.
2 97
Wiener measure and Wiener integral
similarly. We replace a positive linear functional l on C ( O) by a positive linear functional lq,q' on C (Oq,q' ) , which for 'P E Cfin (Oq,q' ) is defined by
lq,q' ('P) = { . . . { F (qi . . . . , qm)P(q', qm ; t' - tm) . . . }R,n }P.,n · P(ql , q; t1 - t)cJ:Iq1 . . . dnqm, where t � t1 � · · · � tm � t' and 'P (;) = F( ; (t l ) , . . . , ; (tm)) . Then ·
·
zq , q' ( 'P ) = r 'P dpq,q' . lnq , q'
As in the case of the Wiener measure J.lw , the conditional Wiener measure J.lq,q' is supported on continuous paths and J.lq,q' ('"t'q ,q' )
=
P(q', q; t' - t ) . p2
Let
H = Ho + V = 2 + V ( Q ) m 2 be the Schrodinger operator on L (IR.n, dn q) with continuous, real-valued and bounded below potential V(q). Denote by Ln(q ' , t'; q, t ) , t' > t, the t 1-t heat kernel - the integral kernel of the diffusion operator e - -��- H . Here is the main result of this section. Theorem 2 . 1 (Feynman-Kac formula) .
Ln(q' , t'; q, t) = r�
J'ifq ,q'
1
e - x ft V(-y(T) ) d'T dpq,q' (;) , t'
where /-lq,q' is the conditional Wiener measure. Proof. We have by the Lie-Kato-Trotter product formula e- � TH
=
lim ( e - t;,t Ho e - t;,t v ) N ,
N-+oo
T t = ' b.. N
where T t ' - t . Let L t ) 2 yn 1 -->oo ydet An following n x n matrix: -1 -1 0 0 1 0 0 a1 0 0 - 1 a2
J J ...
JRn
} tJ
dy,
= nlim
Here Yo
=
Yn and An is the ao -1 0
An =
0 -1
0 0
0 0
-1
an -2
-1
an - 1 where a k = 2 + u ( t k ) ( b.. t ) 2 , k 0, 1 , . . . , n - l . We compute det An by the following elegant argument . First, note that the real >. is an eigenvalue of An if only if the difference equation (3 . 1) - (Yk + l + Yk- 1 - 2 yk ) + u ( tk ) (b.. t ) 2 Yk = >.yk , k = 0, . . . , n - 1 , with initial conditions y_ 1 and y0 has a "periodic solution" - a solution { yk }k l satisfying Yn - 1 = Y-1 and Yn = YO · For given >. denote by Vk1 ) (>.) and vk2) (>.) solutions of (3. 1 ) with the corresponding initial conditions v�i (>.) 1 , va1) (>.) = 0 and v�i (>.) = 0, va2) (>.) = 1 , and put =
=
=
T,
(
(>.) v�2� 1 (>.) n (>.) - v�1�1 � l ) v (>.) v�2 ) (>.) _
)
·
It is easy to show that the discrete analog of the Wronskian Vk� 1 ( >.)vk2 ) (>.) 1 Vk ) (>.)vk� 1 (>.) does not depend on k, so that det Tn (>.) 1. Since every solution Yk of the initial value problem for (3. 1 ) is a linear combination of the solutions Vk1) (>.) and vk2) (>.) , we have =
From here we conclude that >. is an eigenvalue of the matrix An if and only if det(Tn (>.) - /2 ) = 0, and the multiplicity of >. is the multiplicity of a root of this algebraic equation. Since v��l (>.) = O(.xn - 1 ) , v�2) (>.) = ( - >.) n + O ( .xn - 1 ) as ). oo , �
304
6. Integration in Functional Spaces
we obtain det(An - >.. In ) = - det (Tn ( >.. ) -
I2 ) = v�� l (>..) + v�2) (>.. ) - 2.1 i\ It remains to compute limn -. oo det(An - >.. In ) · Denote by y >.. ) and of the difference equation) (3.1) with yi2\ >.. ) , correspondingly,�1ftwo solutions ) 1 9f b b the initial conditions y (>.. ) = Y (>.. ) = 1 and y (>.. ) = 0, Y 2 (>.. ) = !:l t . We have Vkl) (>.. ) = Yil ) (>.. ) - �t Yi2) (>.. ) and vi2 ) (>.. ) = �t Yi2) (>.. ) , so that (2) (>.. ) (2) (>.. ) � � Y n �:n-l 2. det(An - >.. In ) = y l (>.. ) + (3.2) Now it follows from the method of finite differences that i1) (>..) = y1 (t, >..) , lim yi2) (>..) = y2 (t , >.. ) lim (3.3) y k,n -.oo k , n-.oo as limk , n-. oo t k = t, where Y 1, 2 (t, >.. ) are two solutions of the differential equa tion -y" + u (t) y = >..y with the initial conditions Yl (0, >.. ) = y� (0, >.. ) = 0 and Y2 (0, >.. ) = 0, (3.2) - (3 . 3) and Theorem 5.2 in Section 5.2 of Chap yter� ( O5,, >..we) finallyUsing obtain lim det ( A n - >.. In ) = Yl( T, >.. ) + y�( T, >.. ) - 2 det(A - >.. I ) . n->oo Example Corollary 2.2 and Theorem 3.2 can be used to compute � Tr e-* TH for the harmonic oscillator H = 2m ( P2 + m2w2Q2 ) . We have -
1,
= 1.
=
0
3.1.
1 Tr e _ln TH - �:::: 7= :;:::
where Aw = -D 2 + (3 . 4)
w2,
- Jdet Aiw ' and by formula (5.20) in Section 5.2 of Chapter 5,
Tr e-* TH =
1
2 sinh wT
.
2
Of course, since the eigenvalues are En = !iw(n + ! ) , we can get the same result by using geometric series
Similar results hold for the general second order differential operator A = - D2 + + on the interval [0, T] with periodic boundary conditions.
v(t) D u(t)
4.
305
Notes and references
Theorem 3.3. Suppose that det A > 0, where A =
Then
r e - ¥k J[ (v(t)y(t)y(t) +u(t)y2 (t)) dt dp}(:,op ( y )
=
- D 2 + v (t)D + u (t) .
1
v'det A . Example 3.2. Let A = -D 2 + wD. It follows from Theorem 3.3 and (2.6) that for E > 0 1 r e- ¥k J[ (wy(t)y(t) +cy 2 (t) ) dtdp,�op ( y ) Jdet ( A + c l ) lc
lc
=
v 27r 100 = �l nT
m
- oo
T
=
e - ".;'[ cx2 dx r e- ¥fi g (wyo (t)yo (t) +cy5 (t)) dt dp,�op ( yo )
}
£0
o
e- � f0T (wilo (t)yo (t) +cy5 (t ) ) dt dp,�op ( yo ) , =
where we have used the decomposition y (t) = x + yo (t) , J{ yo (t)dt 0 . Here £o is the subset of the free loop space £ which consists of loops with zero constant term in the Fourier series expansion. Using Corollary 5.4 in Section 5.2 of Chapter 5, we obtain
rle e - ¥k foT wyo (t)yo (t) dt dp,�op ( Yo ) = lim JdetT( A..fi+ c:I) o E:--+0
(3. 5 )
T
(!r
sin "f We will use this result in Section 2.2 of Chapter 8. �
Problem 3 . 1 . Derive formula (3.4) using (2.5) .
Problem 3 . 2 . Prove Theorem 3.3. 4.
Notes and references
There is a vast literature on Wiener integration theory and Brownian motion, and here we present, in a succinct form, only the very basic facts . Material in Sec tion 1 is standard, and our exposition follows the exercise section in (Rab95] . For necessary facts from probability theory, including Kolmogoroff ' s extension theo rem and an introduction to the stochastic processes, we refer to the classic treatise [Loe77, Loe78] and recent text [Kho07] . In particular, the statement J..L ( Yt') 0 in Proposition 1 . 1 follows from the strong law of large numbers. =
Th elegant construction of the Wiener measure in Section 2 belongs to E. Nel son [Nel64] , and our exposition, including the proof of the Feynman-Kac formula, follows [RS75] . We refer the reader to Ito-McKean's classic monograph [IM74] for the construction of the Wiener measure from the probability theory point of view; along this way Problems 2 . 1 and 2.2 get solutions. The classic book [Kac59]
306
6. Integration in Functional Spaces
and lectures [Kac80] by M . Kac are another excellent source of information on Wiener integration and its applications in different areas of mathematics. The re lation between Wiener and Feynman path integrals in Section 2.3 belongs to E. Nelson [Nel64] . Problem 2.6 is taken from R.H. Cameron's paper [Cam63] ( see also [RS75] ) ; this result shows that there is no complex-valued analog of Wiener measure associated with the complex diffusion coefficient with Re D > 0 ( as op posed to the statement made in [GY56] ) . Theorems proved in Section 3 serve as a rigorous foundation for our discussion of the Gaussian Feynman path integrals in Section 4 of Chapter 5. Our proof of Theorem 3. 1 in Section 3 . 1 , which uses the method of finite differences, follows the outline in [GY56] which attributed it to [Mon52] . The proof of Theorem 3 2 in Section 3.2 seems to be new. .
The Euclidean quantum mechanics, obtained by replacing the physical time t by the Euclidean ( or imaginary) time -it, is ultimately related to the theory of stochastic processes. It can be formulated by a set of axioms of the one-dimensional Euclidean quantum field theory, and we refer the interested reader to [Str05] for the detailed discussion.
Chapter
7
Fermion S ystems
1.
Canonical anticommutation relations
1 . 1 . Motivation. In Sections 2.6 and 2.7 of Chapter 2 we have shown that the Hilbert space £ -:::: L 2 ( JR, dq ) of a one-dimensional quantum particle can be described in terms of the creation and annihilation operators. Namely, the operators 1 1 1 . a* = -- (Q - iP) and a = 1M ( Q + z P) v 2/i J2n satisfy the canonical commutation relation
[a, a*] = I
on W 2 • 2 (JR) n W 2 • 2 (JR) , and the vectors
'lj;k
(a* ) k
=
1
Jkf 7f;o , k = 0, 1 , 2, . . . ,
where 7j;0 (q) = (7rn) - 4 e- 2 h q2 E £ satisfies a 'lj;0 0, form an orthonormal basis for £. The corresponding operator N = a*a is self-adjoint and has an integer spectrum, 1
=
N'lj;k = k 'lj;k , k = 0, 1 , 2 , . . . .
Similarly, for several degrees of freedom £ -:::: L 2 ( :1Rn , dn q) , the creation and annihilation operators are given by 1 and a k = 1M ( Q k + z. Pk) , k = 1 , . . . , n , v 2 /i 1 Here
in comparison with Section
2.6
of Chapter
2 we put
w =
1.
-
307
308
7. Fermion Systems
and satisfy canonical commutation relations ( 1 . 2) [a k , at] = [ a k , a i ] = 0 and [a k , ai] = Okzl , n
1
k, l = 1 , . . . , n .
2
The ground state, the vector 'lj;0 ( q ) = (7rn) - 4 e- 2n q E £, has the property a k'l/Jo = 0, k = 1 , . . . , n , and the vectors ( a i ) k l . . . ( a� ) kn = '1/Jo , k 1 , , kn = 0, 1 , 2 , . . . , 1 1 'l/Jk1 , ... ,kn Jk 1 · · · · kn · form an orthonormal basis for £. The operator N = L�=l aka k is self adjoint and has an integer spectrum: N 'l/Jk 1 , . . . ,kn = ( k1 + · · · + kn ) 'l/Jkl , ... ,kn ' and the Hilbert space .Yt' decomposes into the direct sum of invariant sub spaces ·
( 1 .3)
·
.
k =O
- the eigenspaces for the operator N. However spin operators, introduced in Chapter 4, satisfy algebraic re lations of different type. Namely, consider the operators a± = � S± = � (81 ± iS2 ) , where 81 and 82 are spin operators of a quantum particle of spin ! (see Section 1 . 1 of Chapter 4) . Using the explicit representation of spin operators by Pauli matrices, we get
The operators a± are nilpotent, ai = 0, and satisfy the anticommutation relation (]'+(]'- + 0'- G'+ = h , where h is the identity operator in C 2 . Introducing the notion of an anti commutator of two operators, [A, B] + = AB + BA, we see that the operators a a_ and a * = a+ satisfy canonical anticom mutation relations [a , a] + = [a* , a* ] + = 0 and [a , a*] + = h . The vector eo ( � ) has the property ae o = 0 , and together with the vector a * e o = ( fi ) they form an orthonormal basis of C 2 . The matrix =
=
N = a* a = ! (a3 + h ) =
(� �)
1 . Canonical anticommutation relations
309
has eigenvectors eo and a"eo with eigenvalues 0 and 1 . Thus in complete analogy with the previous discussion, we say that a and a* are fermion creation and annihi lation operators for the case of one degree of freedom. The Hilbert space of a Fermi particle is .Yt' = C 2 , and the vector e0 is the ground state. It is straightforward to generalize this construction to the case of several degrees of freedom. Namely, canonical anticommutation relations have the form ( 1 . 4) [ak , azl+ = [a'k , aiJ+ = 0 and [ a k , alJ + = 8kzl, k, l = 1 , . . . , n , where I is the identity operator, and creation operators aj are adjoint to annihilation operators aj in the fermion Hilbert space .Yt'F . Canoni cal anticommutation relations ( 1 .4 ) can be realized in the Hilbert space .Yt'F = (C 2 ) 18m = C 2 n as follows: (1 .5) a k = "-v-' 0"3 ® · · · ® 0"3 ® a ® h ® · · · ® /2 , k-1 a'k = � ® a* ® /2 ® · · · ® h , ( 1 .6) k-1 k = 1 , . . . , n. The ground state, the vector '1/Jo = eo ® · · · ® eo E .Yt'F , satisfies ak'l/Jo = 0, k = 1 , . . . , n , ( 1 . 7) and the vectors k (1 .8) 'l/Jk1 , ... ,kn = ( a i ) 1 . . . ( a�) kn 'l/Jo , k 1 , . . . , kn = 0, 1 , form an orthonormal basis for .Yt'F . The operator n N = l: a'kak k=1 is self-adjoint and has an integer spectrum: N'lj;k1 , . .. ,kn = ( k1 + · · · + kn ) 'l/Jk 1 , . . . ,kn ' and the Hilbert space .Yt'F decomposes into the direct sum of invariant sub spaces ( 1 . 9)
n
- the eigenspaces of N. Remark. The fermion Hilbert space .Yt'F is isomorphic to the spin part of the Hilbert space of n quantum particles of spin � , discussed in Section 3. 1 of Chapter 4. The corresponding fermion creation and annihilation operators can also be used for describing spin degrees of freedom. The fundamental importance of the fermion and boson Hilbert spaces - .Yt'F = ( C 2 )®n and
310
7. Fermion Systems
L 2 (1Rn , dn q ) - manifests itself in quantum field theory, which for mally corresponds to the case n oo , and describes quantum systems with infinitely many degrees of freedom. Yl'B
=
=
in the fermion Hilbert space Yt'p is irreducible: every operator in Yt'p , which commutes with all creation and annihilation operators a k and a k , is a mul tiple of the identity operator.
Lemma 1 . 1 . Realization ( 1 .5) -( 1 .6) of canonical anticommutation relations
Proof. It is sufficient to show that if {0} -=/= V � Yt'p is an invariant subspace for all operators a k and a k , then V = Yt'p . Indeed, every non-zero 'ljJ E V can be written in the form 'ljJ = L Ckl , ... ,kn 'l/Jkl , . .. , kn ' k1 , ... , kn=O,l and let Ck1 , ... , kn 'l/Jk 1 , . .. , k n be any of its non-zero components with maximal degree k 1 + + kn . Using canonical anticommutation relations and ( 1 .7) , we obtain . J, _j_ a nkn . . . a k1 1 .1,'P - CtpQ , c- C k , ... ,k n r 0 , so that '1/Jo E V. Applying creation operators to '1/Jo , we get V Yt'p. 0 ·
·
·
1
_
=
fermion Hilbert space Yt'p is analogous to the representation by the occupa tion numbers of canonical commutation relations, discussed in Section 2. 7 of Chapter 2 , and is called representation by the occupation numbers for fermions. It should be emphasized that the algebraic structure of the for mer relations allows their realization in a finite-dimensional Hilbert space, while the algebraic structure of the latter relations warrants the infinite dimensional Hilbert space. Remark. The realization of canonical anticommutation relations in the
Analogously to (1 . 1 ) , coordinate and momentum operators for fermions are defined by
f§_ ( ak + ak ) ,
{§.
Pk = -i ( a k - a k ) , k 1 , . . . , n . As follows from ( 1 .4) , they satisfy the following anticommutation relations: ( 1 . 1 1 ) [Qk , Q zl + = [Pk , Pz] + Mkz l and [Pk , Qzl + = 0 , k, l 1 , . . . , n - a fermion analog of Heisenberg commutations relations, introduced in Section 2 . 1 of Chapter 2. The following result is a fermion analog of the Stone-von Neumann theorem from Section 3 . 1 of Chapter 2 . ( 1 . 10 )
Qk
=
=
=
=
Theorem 1 . 1 . Every irreducible finite-dimensional representation of canon
ical anticommutation relations is unitarily equivalent to the representation by occupation numbers in the fermion Hilbert space Yt'p .
1.
Canonical anticommutation relations
311
Proof. Let V be the Hilbert space which realizes the irreducible represen tation of canonical anticommutation relations ( 1 .4) . First of all, there is t.po E V, III.Po ll = 1, such that a 1 t.po = · · · = an t.po = 0 . Indeed, choose any non-zero t.p E V; if a1t.p =I 0, replace it by the vector a 1t.p, which obviously satisfies a1 (a1t.p1 ) = 0. If a2 (a1t.p) =I 0, replace it by a2a1t.p, which is annihilated by a 1 and a2 , etc. In finitely many steps we arrive at a non-zero vector rj; annihilated by the operators a 1 , . . . , an , and <po = r{;/ l l r!; l l · Now consider the subspace Vo of V, spanned by the vectors k (a�) kn t.po , k 1 , . . . , kn = 0, 1. I.Pk 1 , .. . ,kn = ( a i ) 1 It follows from ( 1 .4) that Vo is an invariant subspace for all operators ak and a"k , so that Vo = V. Since operators a"k and ak are adjoint with respect to the inner product in V, it is easy to see, again using canonical anticommuta tion relations ( 1 .4) , that vectors t.p k1 , ,kn form an orthonormal basis for V. The mapping V 3 t.pk 1 , . .. , kn 'l/Jk 1 , ... , kn E £p establishes the Hilbert space D isomorphism V £p . •
�--+
�
•
•
•..
syl
of the k-th particle in the system of n spin � particles (see Section 3 . 1 of Chapter 4) in terms of fermion creation and annihilation operators in £F .
Problem 1 . 1 . Express the spin operators
1 . 2 . Clifford algebras. We have seen in Section 2 . 1 of Chapter 2 that the Heisenberg Lie algebra is the fundamental mathematical structure as sociated with canonical commutation relations. Similarly, the fundamental mathematical structure associated with the canonical anticommutation re lations is Clifford algebra. Let V be a finite-dimensional vector space over the field k of character istic zero, and let Q V k be a symmetric non-degenerate quadratic form on V, i.e. , Q( v) = (v, v), v E V, where V ®k V k is a symmet ric non-degenerate bilinear form. The pair ( V, Q ) is called quadratic vector space. :
-+
:
-+
Definition. A Clifford algebra C(V, Q ) = C(V) associated with a quadratic vector space ( V, Q ) is a k-algebra generated by the vector space V with relations v2 = Q(v) · 1 , v E V.
Equivalently, Clifford algebra is defined as a quotient algebra C(V) = T ( V ) / J, where J is a two-sided ideal in the tensor algebra T(V) of V, generated by the elements u ® v + v ® u - 2 (u, v) 1 for all u, v E V, and 1 is the unit ·
7.
312
Fermion Systems
in T ( V ) . In terms of a basis {ei } f= l of V, the Clifford algebra C (V) is a k-algebra with the generators e 1 , . . . , e n , satisfying the relations [ei , ej ] + = ei ej + ejei = 2 1P (ei , ej ) · 1 , i, j = 1 , , n . When k = C ( or any algebraically closed field of characteristic zero ) , there al ways exists an orthonormal basis for V a basis { ei } Z: 1 such that IP ( ei , e k) = 6ik · In this case for every dimension n there is one ( up to an isomorphism ) Clifford algebra Cn with generators e 1 , . . , e n and relations eiej + ejei = 26ij · 1 , i, k = 1 , . . . , n . Remark. If k = JR, there exist non-negative integers p + q = n and an isomorphism V � JRn such that Q ( x ) = x � + · · · + x; - x;+l - · x�, x E JRn . This classifies Clifford algebras over JR. Definition. A left module S for a Clifford algebra C ( V ) is a finite-dimensional k-vector space S with the linear map p : C ( V ) 0 S S such that p ( a b 0 s) = p ( a 0 p ( b 0 s) ) for all a , b E C (V ) and s E S. . . .
-
.
·
·
-
_,
The fermion Hilbert space £F, introduced in the previous section, is an irreducible C2 n -module. Indeed, it follows from canonical anticommutation relations ( 1.4) that self-adjoint operators (1.12) 'Y2 k-1 ak + ak , (1.13) 'Y2 k = - i ( a k - ak ) , k = 1 , . , n , satisfy the relations 'YJ..L 'Yv + 'Yv 'YJ..L = 26J.Lvi, f..L , 1/ 1, . , 2n, (1.1 4 ) where I is the identity operator in £F . We define the action of the Clifford algebra C2 n on £F by setting p ( 1) = I and p ( e J.L ) 'YJ.L , J..L = 1 , . . . , 2n, and extending it to a C-algebra homomorphism p : C2n End ( £F ) . Rela tions ( 1 . 14) show that the map p admits such an extension. Proposition 1 . 1 . The homomorphism p : C2 n End ( £F ) is a C-algebra isomorphism. =
.
.
. .
=
=
_,
->
1.1 that the representation p is irreducible: every operator in £F which commutes with all elements of the C-algebra p( C2n ) is a multiple of the identity operator. Then by Wedderburn's theorem p ( C2 n ) End ( £F ) , and since dim C2n = 2 2 n dim End ( £F ) , the map p is D an isomorphism.
Proof. It follows from Lemma
=
=
1 . Canonical anticommutation relations
313
Remark. The structure of a Clifford algebra with an odd number of gen erators is different. Thus the mapping p ( e k ) = (1k , where (1k , k = 1 , 2 , 3, are Pauli matrices ( see Section 1 of Chapter 4) , defines an irreducible represen tation of C3 in J'f'F = C 2 . However, in this case C3 � End ( C 2 ) 0 C [c-] , where c = ie 1 e e3 and satisfies c- 2 = 1 . 2
We define the chirality operator by r = e1r i N , where N = l::j=1 aj aj . Since the operator N has an integral spectrum, r 2 = I. Moreover, we have ( 1 . 15) [r, 111]+ = o , J..L = 1, . . . , 2n. Indeed, as follows from ( 1 .4 ) , N aj = aj ( N + I) and N aj = aj ( N - I ) , so that e1ri N aj aj e1ri ( N+ I) = -aj e1ri N and e1riN aj = aj e1ri( N -I) = - aj e 1ri N . Thus r anticommutes with all aj , aj , and hence with all 1w Since r 2 = I, the operators 1 P± = are orthogonal projection operators and we have a decomposition £'F = £'j EB £j; into the subspaces of positive and negative chirality spinors. It follows from ( 1 . 15) that =
2u ± r )
Also, smce . e 7ria*aj ;
=
I - 2 aj* aJ = -Z/2 . j - 1 /2j , we have r
( - i t'Yl . . . /2n · Remark. When n = 2 , 4 x 4 matrices /I . /2 , /3, /4 are celebrated Dirac gamma matrices ( for the Euclidean metric on �4) , and r /5 · =
=
Problem 1 . 2 . Show that the definition of a Clifford algebra C(V) is compatible with the field change: if k c K is a field extension and VK = K 0k vk , then =
K 0k C(Vk ) ·
C0 c C I c c en C ( V) be the natural filtration Clifford algebra C ( V) , where cr is spanned by the elements V I . . . Vs , s � r.
Problem 1 . 3 . Let
of a Let
C·
C(VK )
1
=
·
cgr (V)
=
·
·
=
n
EB c k ;c k - I
k=I be the associated graded algebra. Show that the skew-symmetrizer map VI
1\
···
1\
Vr r--->
� r.
L
a ESymr
( -l)" (a) Va(I)
· ·
·
Va ( r)
7. Fermion Systems
314
establishes a Z-graded algebra isomorphism A• (V) exterior algebra of V.
�
cgr (V) , where A• (v) is the
Problem 1 . 4 . Formulate and prove the analog of Proposition 1 . 1 for Clifford algebras with an odd number of generators. 2.
Grassmann algebras -
Grassmann algebras algebras with anticommuting generators - are nec essary for the semi-classical description of fermions. The corresponding math ematical definition is the following. Definition. A Grassmann algebra with n generators is a ((>algebra Grn with the generators 01 , . . . , On satisfying the relations eiej + ej ei = o , i , j 1 , . . . , n . =
In particular, these relations imply that generators of a Grassmann al gebra are nilpotent: e� = · · = e� = 0. Equivalently, Grn C ( 01 , . . , On) / J - a quotient of a free C-algebra C ( 01 , . . . , On ) , generated by 8 1 , . . . , On , by the two-sided ideal J generated by the elements eiej + ejei , i , j 1 , . ' n . Remark. It follows from ( 1 . 1 1 ) that in the semi-classical limit n 0 fermion operators Pk and Q k , k = 1 , , 2n, satisfy the defining relations of Grassmann algebra Gr2n · Comparison with the polynomial algebra C[x1 , . , Xn ] = C ( x 1 , . , Xn ) / 1 - a quotient of a free C-algebra C ( x 1 , . , Xn ) by the two-sided ideal generated by the elements XiXj - Xj Xi , i , j = 1 , , n shows that the Grassmann algebra Grn can also be considered as a polynomia l algebra in anticommut ing variables 8 1 , . . . , On . In what follows we will always use Roman letters for commuting variables and Greek letters for anticommuting variables, so that Grn = C [01 , . . . , On ] · Needless to say, the polynomial algebra C [x 1 . . . , xn ] is isomorphic to the symmetric algebra of the vector space spanned by x 1 , . , Xn , and the Grass mann algebra C [ 01 , . . . , On ] is isomorphic to the exterior algebra A• v of the vector space V C01 EB EB COn with the basis 81 , . . . , On . The Grassmann algebra Grn is a complex vector space of dimension 2n and is Z-graded: it admits a decomposition n (2. 1 ) Grn = ffi Gr� ·
=
.
=
. . .
. .
. .
.
.
. . .
-
,
=
·
·
·
k =O
.
.
. .
-+
2.
315
Grassmann algebras
into homogeneous components Gr� of degree k and dimension (�) , k = 0, . , n , where Gr� = C · l . Namely, denote by I I the degree of homogeneous elements in the Grassmann algebra, l ad = k for a E Gr� . Then multiplication in Grn satisfies Gr� Gr� c Gr�+ l , where Gr�+l 0 if k + l > n, and is graded-commutative:
..
·
·
=
( 2.2 )
for homogenous elements a, {3 E Grn . The elements of the Grassmann alge bra Grn of even degree are called even elements, and those of odd degree odd elements. 2 . 1 . Realization of canonical anticommutation relat ions. The Grass mann algebra provides us with the explicit representation of canonical anti commutation relations by multiplication and differentiation operators, which is analogous to the holomorphic representation for canonical commutation relations (see Section 2.7 of Chapter 2 ) . Namely, let Oi
=
f) (} f) i
:
Grn
-t
Grn
be left partial differentiation operators, defined on homogeneous monomials by
(}il . . . (}ik
where {}i t denotes the omission of the factor (}i t . The differentiation operators are of degree - 1 and satisfy the graded Leibniz rule,
Remark. One can also introduce right partial differentiation operators by
f)
�
( ei1 . . . eik ) & . = L..,. ( - 1 ) et =l
k-l
l which satisfy the following graded Leibniz rule:
(a{J) � = a f) (}i
({3�) f)(}i
�
8iiA1 . . . eiz . . . eik ,
+ ( - 1) 1131
(a� ) {3. f) ()i
To distinguish between the left and right partial derivatives of f E Grn , we f and f . will denote them, respectively, by
�
f) i
�
f) i
316
7. Fermion Systems
As a complex vector space, Grassmann algebra Grn carries a standard in ner product defined by the property that homogeneous monomials fh 1 ei" , for all 1 :s; i 1 < < ik :s; n, form an orthonormal basis, •
·
·
•
•
·
(2 . 3)
By checking on homogeneous monomials, it is elementary to verify that where Bi are left-multiplication by ei operators in Grn , so that {h = a; . It is also easy to verify that the operators Bi and Oi satisfy the anticommutation relations
where I is an identity operator in Grn . Thus we have the following result. Proposition 2 . 1 . The assignment
establishes an isomorphism Jlt'p ':::::' Grn between the fermion Hilbert space of identical particles, and the vector space of the Grassmann algebra with n generators. It preserves decompositions (1.9) and (2. 1 ) and has the property that
n
and
i
= 1, . . . , n .
Using Proposition 2 . 1 , it is also very easy to verify that the represen tation of canonical anticommutation relations in the fermion Hilbert space Jlt'p is irre �ucible. Indeed, suppose that B E End ( Grn ) commutes with all operators ()i and fA . Then 8i (B ( 1 ) )
=
B(8i ( 1 ) )
=
0,
i
=
1, . . . , n.
The only solution of the equations or f = = 8n f 0 is f c · 1, so that B ( 1 ) = c · 1 . Since B commutes with all creation operators i}i , we obtain B = cl. ·
·
·
=
=
Remark. We will show in Section 2 . 3 that by using the notion of Berezin
integral, the inner product (2.3) in Grn can be written in a form (2. 1 1 ) , which i s similar t o the definition of the inner product in the holomorphic representation, given by formula (2.52) in Section 2. 7 of Chapter 2 .
317
2. Grassmann algebras
2 . 2 . Differential forms. An algebra of differential forms in anticommut ing variables fh , . . . , On is a C-algebra n� with odd generators 01 , . . . , On and even generators d01 , . . . , dOn satisfying relations Oi · dOj
=
dOj · Oi ,
=
1 , . . . , n. Equivalently, n� is a symmetric tensor product over C of Grassmann alge bra Gr n and polynomial algebra C [d01 , . . . , dOn ] , and every w E n� can be uniquely written as (2 .4)
W
=
00
L fk (d01 ) k1 ·
k=O
•
i, j
· (dOn ) kn ,
fk E
Grn ,
where k ( k1 , . . . , kn ) is a multi-index and fk 0 for all k, except finitely many. By definition, the degree J wk l of a homogeneous component wk fk (d01 ) k 1 • · • (dOn ) kn E D� is l fk J , the degree of fk E Grn . Remark. It is instructive to compare the algebra n� with the algebra of polynomial differential forms on en . On the one hand, n� is an infinite dimensional algebra in commuting variables d(}1 , . . . , dOn with coefficients in a finite-dimensional algebra Grassmann algebra C [ 0 1 , . . . , On ] · On the other hand, the algebra of differential forms i n commuti ng variables x 1 , . . . , Xn is a finite-dimensional algebra in anticommuting variables dx1 , . . . , dx n with coefficients in an infinite-dimensional polynomial algebra C [x1 , . . . , x n ] · =
=
=
The analog of the exterior ( de Rham ) differential on the algebra n� is the mapping d : n� n� ' defined by -t
dw
=
k ft dOi (d01 ) k 1 �� i
k=O =l
t
·••
(dOn ) kn ,
fk
E Grn ,
where w is given by (2.4) . It can also be written as d = 2::::: � 1 dOiai , where it is understood that 8i (d0j ) = 0 for all i , j 1, . . . , n . =
Lemma 2 . 1 . The exterior differential d on n� satisfies the graded Leibniz
rule,
dw1 w2 + ( - 1 ) l w1 l w1 dw2 , and is nilpotent, d2 = 0 .
d(w1w2)
for homogeneous W I ,
=
Proof. The graded Leibniz rule for d follows from the corresponding prop erty of partial differentiation operators ai . The property d2 0 follows from the commutativity of "differentials" d(}i and anticommutativity of the partial derivatives ai . D =
The next result is an analog of the Poincare lemma for differential forms in anticommuting variables.
318
7. Fermion Systems
Lemma 2.2. Suppose that w is 'TJ E il� such that w d'TJ. =
D efinition. A 2-form
E n� is closed, dw = 0 . Then w is exact: there
w E n�,
(2.5) =
is called a symplectic form on a Grassmann algebra if it is closed, dw 0, and is non-degenerate: the n x n symmetric matrix {w ij }�j=l is invertible in Grn . In particular, the 2-form w with constant coefficients wij E C is al ways closed, and w is symplectic if and only if the matrix { wij}� =l is non j degenerate. Since every quadratic form over C can be written as a sum of squares, we can always assume that generators 01 , , On are chosen such that the symplectic form w with constant coefficients has a canonical form n (2.6) w = ! L dei dei . . . .
i =l
With every symplectic form (2.5) there is an associated Poisson bracket on Grassmann algebra, defined by
where { Wij }i, =l is the inverse matrix to { w ij }i, =l , and we are using no tation for thej left and right partial differentiationj operators, introduced in the previous section. The following result - an analog of Theorem 2.9 in Section 2.4 of Chapter 1 is fundamental for formulating Hamiltonian me chanics for the systems with anticommuting variables, which we will discuss in Chapter 8. -
Proposition 2 . 2 . Suppose that all coefficients
form w are even. Then the Poisson bracket map
wij of a closed symplectic
{ , } : Grn X Grn ---) Grn
satisfies the following properties. (i) (Graded skew-symmetry)
{!, g}
= -
( -1 ) 1f l l9l {g , !} .
(ii) (Graded Leibniz rule)
{ fg , h } = f{g , h } + ( - 1 ) 1 fl lglg{ f, h }.
2.
31 9
Grassmann algebras
( iii ) (Graded Jacobi identity) {!, {g, h } } + ( - 1 ) 1 / l ( lg l + l h l ) {g, {h, ! } }
+
( - 1 ) 1 hl ( l/ l +lgl ) {h, { f, g } }
for all J , g, h E Grn . Proof. Part ( i ) follows from (2.2) , the property
=
0
a - ( - ) 1 11 a ao/ 1 aoi ' j and the condition that coefficients wi are even. The graded Leibniz rule follows from the corresponding property of right partial differentiation op erators. The graded Jacobi identity for the Poisson bracket =
which corresponds to the canonical symplectic form (2.6) , can be verified by a direct computation. The proof of a general case is left to the reader. D Problem 2 . 1 . Complete the proof of Lemma 2. 1 . Problem 2 . 2 . Prove Lemma 2.2. Problem 2.3. Complete the proof of Proposition 2.2. 2.3. Berezin integral. There is a principal difference between differential forms in commuting and in anticommuting variables. The former can be differentiated and integrated, and the differential and integral are related by the Stokes ' formula. However, the latter can only be differentiated. Still, there is an analog of the integration over anticommuting variables. Definition. The integral on a Grassmann algebra Grn with an ordered set of generators 0 1 , . . . , On (Berezin integral) is a linear functional B : Grn .1 01 02 +···+>.m02m-l02m d01 . . . d02
m = A I . . . Am ,
so that Pf(A) 2 = det A. This relation holds for complex-valued A, since both sides are polynomials in variables aij , 1 ::; i < j ::; n, which coincide for real 0 Uij .
For a Grassmann algebra Gr2 n = C[01 , . . . , On , 01 , . . . , On ] with 2n gen erators denote by J dOdiJ J d0 1 d01 . . . dOn dOn the corresponding Berezin integral, =
J J (O, O-) dOdO- = 8f)0n 80f)n . . . 8f)01 80f)1
J,
Lemma 2.4. For any n x n matrix A = {aij } r,j = 1 ,
Proof. It follows from the definition of a matrix determinant that
( 2.10 )
0
Definition. An involution on a Grassmann algebra Grn over C is a complex
anti-linear mapping Grn g* f* for all J, g E Grn .
3
f
f-.+
f*
E
Grn satisfying (!* ) * = f and (fg)* =
The Grassmann algebra C[01 , . . . , On , 01 , . . . , Bn ] has a natural involution defined on generators by (BI ) * = 01 , ( 01 ) * 01 , . . . ' (On )* = On , ( On ) * = On . In particular, for n = (O f ) L =
we have n
f( O ) * = f ( O ) = L The next lemma expresses the inner product on the Grassmann algebra Grn , introduced in Section 2. 1 , in terms of the Berezin integral.
323
2. Grassmann algebras
Lemma 2 . 5 . The standard inner product (2 . 3 ) on the Grassmann alge
bra Grn C[B1 , . . . , Bn] is given by the following Berezin integral over the Grassmann algebra Gr 2 n C [ B1 , . . , Bn, iJ1 , . . . , iJn ] : =
=
.
(2 . 1 1 ) Proof. Put /I (O) = Bi1 Oi"' and /2 (0) = Bj 1 Bjz · It is clear that the integral ( 2 . 1 1 ) is 0, unless k = l and it JI , . . . , i k = Jk , in which case we •
•
•
.
•
.
=
have
IJ . ( Utl
•
.
.
ll .
Ll .
Utk ' Utl
..
.
Ll .
Utk
)_
=
=
e J J !11 . . . BniJn . . . jjl d(JdiJ j Btjjl . . . Bn iJn dOdiJ ll .
Ut l
•
•
.
IJ . i'i. Utk Utk
•
•
•
il.
Ut l
-(01fh +··+OnOn) d(} d(}-
=
0
1.
The following result was already stated in Section 2 . 1 . Lemma 2 . 5 allows us to prove it in a way that is reminiscent of a holomorphic representation (see Section 2.7 of Chapter 2 ) .
8i and ei , i = 1 , . . . , n, are adjoint with respect to the inner product on Grn . Proof. Using Lemma 2.5, formulas 8d (0) 0, Oi e - 8 8 iJie-09, and the integration by parts formula, we obtain Corollary 2 . 1 . The operators
=
( 8dt , h )
= =
=
J 8d1 ( 0) /2 (O)e- 88dOdiJ ( j JI (O)h[ii) iJi e- 8odOdiJ j fi (O)Bd2 (0)e- 89dOdiJ - - l ) l h l + lh l
= - ( - l ) l h l + l fz l =
(/I , &i /2 ) ,
since the last integral and (/I , &i h ) are both 0 unless I !I I + I h i is odd. Problem 2.4. Evaluate the Berezin integral
where
ry1 ,
..
j exp { � t,Jt= l aiJ fMJi + kt= l rJkfh } del . . . dOn ,
.
, 'r/n
are Grassmann variables.
Problem 2 . 5 . Prove formula (2. 10) .
0
7.
3 24 Problem 2 . 6 . Prove that
J f (O)e- 69 d8d0
Fermion Systems
f (O)
=
- the constant term of expansion of f (O) into the sum of monomials in C [B1 , . . . , Bn] ·
3 . Graded linear algebra 3. 1 . Graded vector spaces and superalgebras. The notions of gra
ded vector spaces and superalgebras, introduced in this section, allow us to consider commuting and anticommuting variables on the same footing.
7l/ 27l graded vector space
(graded vector space for brevity, or super vector space) over C is a vector space W with a decomposition Definition. A
W = W0 EB W1
into even and odd subspaces. The elements in W0 U W1 \ {0} are called homogeneous and the parity is a map I · I : W0 U W 1 \ {0} {0, 1 } such that l w l = 0 for w E W0 and l w l = 1 for w E W1 . -t
We reserve the notation V for ordinary ( even ) vector spaces, denoting graded vector spaces by W. If W is finite-dimensional, we define graded dimension as a pair ( dim W0 , dim W1 ) , usually denoted by no ln1 , where i ni = dim w ' i 0, 1 . When W0 CP and W1 (:Q ' the corresponding graded vector space W is denoted by CP i q . A fermion Hilbert space £p is a graded vector space with the even and odd subspaces given by decomposition ( 1 .9) : £; = EB £k , £) = EB .Yfk . =
=
=
k even
k odd
The graded dimension of £p is 2 n-l 12 n-l.
Direct sums and tensor products of graded vector spaces are defined in the same way as for ordinary vector spaces. For the direct sums the homogeneous subspaces are defined by ( W1
EB
W2 ) k
=
Wf EB W�,
and for the tensor products they are defined as (W1 0 W2 ) k =
E9
i +j= k
wt 0 w4 ,
k
=
0, 1 ,
k = o, 1 .
The difference between ordinary and graded vector spaces becomes trans parent in the definition of the corresponding tensor categories. Namely, the associativity morphism C wl w2 w : 3
wl 0 ( W2 0 W3 )
-t
(Wl 0 W2 ) 0 w3
3. Graded linear algebra
325
for graded vector spaces is defined by the same formula Cw1 w2 w3 (W I 0 ( w2 0 w3) ) = ( wi 0 w2 ) 0 W3
as in the case of ordinary vector spaces, whereas the commutativity mor phism O'wl w2 : WI 0 w2 - w2 0 WI
is defined for the homogeneous elements by
2 O'w1 w2 ( W I 0 W2 ) = ( - l ) lwl l l w 1 w2 0 W I . algebra T(W) of a graded vector space W is
The tensor defined using the associativity morphism. However the exterior algebra A •w and symmetric algebra Sym ( W ) of W are defined as quotient algebras of T(W) by using the commutativity morphism. Namely, Sym ( W ) = T(W)/I, where I is the two-sided ideal in T(W) generated by W I 0 w2 - O' (w2 0 W I ) , WI , W2 E W , and A• (w) = T(W) / J, where J is the two-sided ideal in T(W) generated by W I 0 w2 + O' (w2 0 WI ) , O'W,W is the commutativity morphism. w 1 , w2 E W . Here Definition. Let W = W0 EB W I be a graded vector space. A parity-reversed vector space IIW is a graded vector space with ( IIW) 0 = W I and (IIW) I wo . 0' =
=
It immediately follows from the definitions that for even vector space V Sym ( IIV ) = A• (v) and A • ( rr V ) = Sym ( V ) . (3. 1 ) Definition. A superalgebra over C is a graded vector space A = A 0 EB A I with a C-algebra structure such that 1 E A 0 and A0 · A0 c A 0 , A0 · Ai c A 1 , A 1 · A1 c A0 •
A
superalgebra A is a commutative superalgebra if a · b = ( - l ) lal lbl b · a for homogeneous elements a, b E A. An even superalgebra A is just an ordinary C-algebra. Definition. A left ( super ) module for a superalgebra A is a graded vector space M with the linear map A 0 M 3 a 0 m t-t a · m E M such that I a m l = ( l a l + lm l ) mod 2 for homogeneous a E A and m E M, and a b · m = a · ( b · m ) for all a , b E A and m E M. ·
7.
326
Fermion Systems
Problem 3 . 1 . Let u E Symn - Show that the isomorphism W1 0 0 Wn � W.,.- 1 {1) 0 · · · 0 Wo- - l ( n) , · ·
·
induced by the commutativity morphism of graded vector spaces, is well defined does not depend on the representation of u as a product of transpositions in Symn .
Problem 3 . 2 . Show that for a graded vector space W the algebras Sym (W) and A • (W) are superalgebras. 3.2. Examples of superalgebras. Example 3 . 1 (Tensor algebra) . The tensor algebra 00
E£) v 0 k , V 0 = c · 1 , k=O of an even vector space V is a superalgebra. The multiplication is given by the tensor product, and even and odd subspaces - by EB v®k, T(V) l = EB v®k . T ( V) o k k odd Example 3.2 (Symmetric algebra). The symmetric algebra Sym(V) of an even vector space V is a commutative algebra. A choice of a basis x 1 , . . . , X n of V establishes the isomorphism Sym(V) C [x1 , . . . , X n ] - a polynomial algebra in commuting variables x 1 , . . . , Xn · Example 3.3 (Exterior algebra) . The exterior algebra A• v of an even vec tor space V is a commutative superalgebra, with multiplication given by the wedge product; the even and odd subspaces are images of the corresponding subspaces of T(V) under the surjective mapping T(V) A • v = T(V)j J, where J is the two-sided ideal in T ( V) , generated by the elements uQS>v + vQS>u, u, v E V. According to (3. 1 ) , A • (V) = Sym(IIV) . The choice of a basis fh , . . . , On of the odd vector space IIV establishes the isomorphism A• (v) c:::: C [01 , . . . , Bn l · Here C [01 , . . . , On ] is the Grassmann algebra - a polynomial algebra in anticommuting variables 0 1 , . . . , On . Example 3.4 (Algebra of differential forms) . Let M be an n-dimensional manifold. The graded algebra A• (M) of smooth differential forms on M is a commutative superalgebra. Example 3.5 (Clifford algebra) . The Clifford algebra C(V) of a quadratic vector space (V, Q) is a superalgebra, with the multiplication and grading descending from the tensor algebra T(V) under the surjective mapping T(V) A · v T(V) 1 J, T ( V)
=
=
even
c::::
-t
-t
=
3.
327
Graded linear algebra
where now J is the two-sided ideal in T ( V ) , generated by the elements E V. The natural map V C ( V ) is injective, and V is identified with its image in C ( V ) The elements of V are odd in C(V) . The fermion Hilbert space J"t'F is a left supermodule for C2n , and the ((>algebra isomorphism p : C2 n � End ( Jlt'p ) ( see Proposition 1 . 1 ) is an isomorphism of superalgebras. Example 3 . 6 ( Graded matrix algebra ) . Let W be a graded vector space. The vector space End ( W ) of all endomorphisms of W is a graded vector space: the even subspace consists of all endomorphisms which preserve the grading in W, and the odd subspace consists of those which reverse the grading. The vector space End ( W ) is a superalgebra with the product given by the composition of endomorphisms, and W is a module for End ( W ) . When W = CPiq, the superalgebra End ( W ) is usually denoted by Mat (p l q ) . Its elements can be conveniently represented by 2 x 2 block matrices
u 0 v + v 0 u - 2(u, v ) · 1 , u, v
(
.
'----t
)
An A12 A= A 21 A 2 2 , where An , A 12 , A 21 , and A22 are, respectively, matrices of orders p x p, p x q,
q x p, and q x q . The even and odd elements of Mat (p l q ) are, respectively, block-diagonal and anti-diagonal matrices ( A0 1 A� 2 ) and ( A� 1 A0 2 ) Example 3 . 7 ( Lie superalgebra ) . A graded vector space g is called a Lie superalgebra ( or, super Lie algebra) if it carries a Lie superbracket - a linear mapping [ , ] g 0 g � g satisfying the following properties. ( i ) ( Super skew-symmetry ) •
:
[x , y] - ( - 1)1x ii Y I [y , x] for homogeneous x, y E g . =
( ii ) ( Super Jacobi identity )
[x, [y, z]] + ( - 1) 1xi( I YI+Izl) [y, [z, x]] + ( - 1) 1zl(lxi+IYD [z, [x, y]] = 0 for homogeneous x, y, z E g . According to Proposition 2.2, a Grassmann algebra Grn with a Poisson bracket (2. 7) is a Lie superalgebra. Corresponding to classical simple Lie algebras there are associated Lie superalgebras. Problem 3.3. Verify all the statements in this section.
Problem 3.4. Show that a superalgebra A carries with a Lie superbracket defined by for homogeneous
a,
bE
A.
[a, b] = ab - ( - l ) l a l l bl ba
a
Lie superalgebra structure
7. Fermion Systems
328
3.3. Supertrace and Berezinian. Let A = A0 EB A 1 be a commutative superalgebra. We have the following general notion of a graded matrix alge bra. Definition. A graded matrix algebra with coefficients in A is a superalgebra Mat A (P i q) of 2 x 2 block-matrices
A=
(AA2n1
A12
A22
)
,
where A n A1 2 , A 2 1 , and A 22 are, respectively, matrices of orders p x p, p x q, q x p, and q x q with elements in A. The element A E M atA (P iq) is even if corresponding matrices A n and A22 consist of even elements of A, and matrices A1 2 and A 2 1 consist of odd elements of A. The element A E Mat A (P i q) is odd if matrices An and A 22 consist of odd elements of A, and A1 2 and A21 consist of even elements of A. The graded vector space MatA(P iq) is a superalgebra with the product given by the matrix multiplication. The algebra Mat (plq) in Example 3.6 corresponds to the case A = C; another interesting example is when A is a Grassmann algebra. It is quite remarkable that such basic notions of linear algebra as trace and determinant admit non-trivial generalizations for graded matrix alge bras.
Definition. A supertrace on a graded matrix algebra is a linear mapping Trs : Mat A (P i q) --+ A, defined by Trs A = Tr An - ( - l) IA I Tr A22 ,
A = (��� ���) E MatA(P iq) ,
where Tr is the ordinary matrix trace. Proposition 3 . 1 (The cyclic property of the supertrace) . We
Tr8 A B = ( - l) IA!IBI Trs B A ,
A, B E M at A (P i q).
check only the case when both 2 x 2 block-matrices A and B are even elements of Mat A (P i q) . Other cases are treated similarly and are left to the reader. We have, using the cyclic property of the trace, Proof.
Trs AB = Tr(A n B n + A 1 2 B2 1 ) - Tr(A 2 1 B 1 2 + A 22 B22 ) = Tr (Bn An - B21 A 1 2 ) - Tr( -B1 2 A 21 + B2 2 A 2 2 ) =
=
Trs B A. 0
The superalgebra End(£F) corresponds to the case A C and is iso morphic to Mat(2 n -l 1 2 n - 1 ) . The supertace on End(Yf'F) is given by Trs A = Tr An - Tr A22 = Tr Af , A E End(Yf'F ) , (3.2)
329
3. Graded linear algebra
where r is the chirality operator (see Section 1 .2 ) . It has the following 2 X 2 block-matrix form:
r=
(� ��) ,
where stands for the identity operator in ..Yt'J and £) . It is a non-trivial problem to define a superdeterminant a natural analog of the determinant for graded matrix algebras, which is multiplicative and generalizes the rule det e A = e Tr A . The corresponding notion, which is defined only for invertible A E MatA (Pi q ) , was introduced by F.A. Berezin. It is now commonly called Berezinian and is denoted by Ber( A) . A Consider first the case of even diagonal A ( J 1 A� 2 ) The relation
I
-
Ber( e A)
=
eTr•
A
=
•
defines the Berezinian by Ber( A)
=
det Au det A 2i .
Thus in this case the even q x q matrix A22 is necessarily invertible, i.e. , det A 2 2 is an invertible element of the commutative algebra A0, so that the inverse matrix A2i exists . Now consider the general even A E MatA (P i q) , and suppose that A 22 is invertible. It is easy to verify the following analog of the Gauss decomposition: 0 Au A 1 2 = A 1 2A2:f An - A12A2:f A2 1 0 0 0 A2 1 A2 2 A22 A2:f A2 1 where and Iq are, respectively p x p and q x q identity matrices. This justifies the following definition.
(
lp
) (lp lq ) (
) ( lp lq) '
Definition. Let A E MatA (Piq) be even and such that the corresponding even q x q matrix A 2 2 is invertible. Then the Berezinian (superdeterminant)
of A is given by
Assuming that the p x p matrix Au is invertible, we get the decompo sition A1 A i 12 = A2 A 1 l A A A A � 1 i 1 22 ' i1 which suggests that al so Ber( A) = det Au det (A 22 - A 21 A i/ A 1 2) - 1 .
(1�� 1��) ( : �J (
It i s
�
a fundamental fact that for even invertible A formulas for the Berezinian coincide.
J (� �: )
E
MatA(Pi q) these two
7. Fermion Systems
330
Theorem 3 . 1 . Let
A=
.
(An AA2212 ) A2 1
be an even element of MatA (P i q ) Then
(i) A is invertible if and only if the matrices An and A22 are invertible. (ii) For invertible A, Ber( A ) is an invertible element of A0 satisfying Ber(A) = det(An - A1 2 A2l A 2 1 ) det A 2l = det An det(A 22 - A21 A ii1 A12 ) - 1 and
(iii) If even A, B
E
Ber(A- 1 ) = Ber(A) -1 . MatA(Pi q ) are invertible, then Ber( A B) = Ber(A)Ber(B) .
Problem 3.5. Complete the proof of Proposition 3 . 1 . Problem 3.6. Prove Theorem 3 . 1 . 4.
Path integrals for anticommuting variables
4 . 1 . Wick and matrix symbols. Here we describe the calculus of Wick
and matrix symbols of operators in the fermion Hilbert space Jf'p , which is analogous to the treatment of Wick and matrix symbols in the holomorphic representation in Section 2. 7 of Chapter 2. As well as in the bosonic case, it is a tradition to work in anti-holomorphic representation by using Jf'p = C [01 , , On] as the fermion Hilbert space with the annihilation and creation operators
...
0
and a *k = {)Ok ' The inner product (2. 1 1 ) takes the form
(4 . 1 )
k = 1,
...
, n.
(4.2) and the monomials parametrized by subsets I = {i1 , . . . , i k } � { 1 , . . . , n } , form an orthonormal basis in Jf'p .
4.
331
Path integrals for anticommuting variables
Definition. A matrix symbol of an operator A : Y't} A(O, 9) E qol, . . . ' Bn, el, . . . ' BnJ , defined by
A(O, 9)
__.
£F is an element
= :l: )A !J , !I)fr(O)!J(O) I,J
= 'l.'.) A !J , !I) oil I,J
. . . oikeJl . . . ej 1 ·
Here summation goes over all subsets I { i 1 , . . . , i k } and J = {j1 , . . . , jz } of the set { 1 , . . . , n } , and as in Section 2.3, we denote by f (O ) a natural involution on the Grassmann algebra qo� , . . . , Bn, 81 , . . . , Bn J , =
According to Proposition 1 . 1, C2 n � End ( £F ) , so that every operator : £F __. £F can be uniquely represented in a Wick normal form as follows: A
A=
L Ku a;1 • • • a;"' aj1 • • • aj1 • I, J
Definition. A Wick symbol of an operator A A(O , 9 ) E qol , . . . ' Bn , el , . . . ' BnJ , defined by A( 0 , 9)
=
: £F
__.
£F is an element
'L: Ku Bi1 . . . BikeJ1 . . . Bj1 • I,J
Remark. The definition of matrix and Wick symbols in the fermion case
repeats verbatim the corresponding definition for the bosonic case in Sec tion 2.7 of Chapter 2. Note, however, that in the fermion case the product ejl . . . e)l in the definition of the matrix symbol has reverse ordering, as is required by the inner product (4.2) in C[B1 , . . . , On ] · To matrix and Wick symbols A(O, 9 ) and A(O , 9 ) of an operator A one canonically associates elements A(O , o ) , A(O , o ) and A(a , 9) , A ( a, 9 ) in the larger Grassmann algebra
C [o , a , 9, OJ
=
C [o: 1 , . . . , o:n , & 1 , . . . , &n , 8 1 , . . . , BnBI , . . . , BnJ ,
by replacing, correspondingly, Bi by O:i and Bi by &i . The incomplete Berezin integral J dada on qo , a, 9, OJ is defined by 8 8 8 8 � � j, f E C [o , a , 9, 0] , f d a da = UO:n UO:n UO:l U0:1 and has the property
J
(4 .3)
!:) -
•
•
•
j h (9 , O)g(a, a) dada
!:) -
=
h(9, 0)
j g(a,
a ) dada .
7. Fermion Systems
332
We will also use the incomplete Berezin integral J d0d9, defined by
-
8 8 8 8 8(}n 8Bn . . . 8(} 1 801 f , f E C[o:, a , 9, 8] . As follows from the proof of Corollary 2.1,
J
(4.4)
j
f d9d9 =
(aich)he-9 0 d0d9
=
- ( - 1 ) 1fi i + Ihl
j
h (a d2)e-90d0d9 ,
where operators a'k and ak are given by (4. 1 ) , and JI , h E C [o:, a, 9 , 0] . The next result shows that the matrix symb ol of an operator A in .Y6, whi ch is just a 2 n x 2 n matrix, can be considered as an integral kernel in anticommuting variables! Lemma 4 . 1 . Let A( O , 9) be the matrix symbol of an operator A in ,Yep .
Then for every f ( 0)
( 4.5 )
E
,Yep ,
( A f) (O) =
j
A ( 8 , o:) f (a) e - c.: a do: da .
Proof. It is sufficient to verify ( 4.5 ) fo r f = fK , where K { 1 , . . . , n}. Using ( 4.3 ) and Lemma 2.5, we get
j A (O, o:) fK (a) e - c.:ado:da
= =
L ( A !J, !I )fi (O) I, J
L (A fK , !I ) JI ( O )
= { k1 , . . . , km}
�
j fJ (a)fK (a) e -c.:ado:da =
(A fK ) (0) .
I
Next, we introduce Grassmann analogs of the coherent states ( see Sec tion 2.7 of C hapter 2 ) . Put 4>a(O) = e0a = L fi (O) fi (a) and a( 9 ) e - Ba L fi (a) fi (O) . =
I
=
I
As in the bosonic case, elements 4>a , a E C[ o:, a , 9, OJ satisfy
( 4.6 )
ak4>a
=
ak4>a and aka = -aka,
k = 1, . . . , n.
It is very easy to express the matrix symbol of an operator in terms of the coherent states. Lemma 4.2. Let A( a, o:) be the matrix symbol of an operator
Then
A in .Yep .
0
4.
333
Path integrals for anticommuting variables
Proof. The proof is a straightforward computation (cf. the proof of Lemma
2.4 in Section 2.7 of Chapter 2) :
A ( a, a )
=
L (A!J, fi) fi ( a)!J ( a) I,J
=
L (A !J !J ( a) , fi ( a)fi) = (Aa, �a ) · I,J
2.
0
The next result is an exact analog of Lemma 2.4 in Section 2. 7 of Chapter
Lemma 4.3. The matrix and Wick symbols of an operator A in &p are
related by
A( a, a ) = e6a A( a , a) . Moreover, A( a, 0) = e 68 A( a, 0) and A(O, a) = e 8a A( 8 a ) . ,
Proof. As follows from the definition of coherent states, ( a , � a ) = e6a .
Now representing the operator A in a Wick normal form and using properties (4.4) and (4.6) , we obtain
(Aa, �a) = L Ku(a;1 =
•
•
•
a;kaj1
•
•
I,J
•
1
aj a , � a)
aj1 a, ai k . ai1 �a) I,J "" ( - 1 ) kl Ku(ah . . aj1a, ai k ai 1 a) = L.,. =
L( - l ) kl+kKu(aj1
•
·
•
.
I,J
L (-l)k1Kuaj1 I,J
•
•
•
•
·
. . .
aj1 Ci:i 1 . Ci:ik (a, �a) •
•
=
= A( a , a ) ( a �a) e 6a A( a , a ) . The same computation, after replacing a by o , or � a by �o , proves the 0 remaining two formulas. Thus we have shown that 2 n x 2 n matrices - operators in the fermion Hilbert space &p can be considered as integral operators in anticom muting variables with the integral kernels given by matrix or Wick symbols. The next result is an exact analog of Theorem 2.2 in Section 2. 7 of Chapter 2, and establishes the calculus of symbols for operators in &p . ,
-
Theorem 4. 1 . Let A 1 and A2 be operators in &p with matrix symbols
A1 (0, 0) and A2 (0, 0) and Wick symbols A 1 (0, 0 ) and A 2 ( 0, 0 ) . Then the following formulas hold.
7.
334
Fermion Systems
(i) The matrix and Wick symbols of the operator A = A 1 A2 are given by A(O, 8)
=
A ( O, 8) =
j j
A 1 (0, a)A 2 (a, 8)e-aa d a da, A 1 (0, a )A2 (a , 8)e- (ii- a)(fJ- a ) da da .
( ii ) The trace and supertrace of an operator A in £F are given by Th A
=
Th8 A =
j 8)e-288diid8, j J A(iJ, 8)e-88d8diJ J A(iJ, 8)d8diJ. A (ii , 8) e-88dO d8
=
A (ii,
=
Proof. Part ( i ) for matrix symbols is proved by the following straightfor
ward computation:
=
j
A 1 (0, a )A 2 (a , 8)e-aada da
L L ( A dJ , fi) ( A2 !L , fK ) I,J K,L
j
fi (ii) !J (a) fK (a) h ( O ) e -aadada
= L (AdJ , Jr) (A2 !L , h ) !I ( O) h (ii ) I , J, L
= =
L
I, J,L
( A2 h , h) (!J , A i fr ) JI ( O ) h ( O )
L ( A2 fL , Ai !I ) !I (O) fL ( O ) I,L
=
L ( AlA2h , fi ) !I ( ii) h ( O ) I,L
= A(ii , 8) .
The corresponding formula for the Wick symbols now follows from Lemma 4.3. The proof of part ( ii ) is also straightforward. We have
j
A ( 0 , 8) e - 88dOd8
=
=
L ( A!J , !I ) I,J
A ( O , 8)e -88d8diJ = =
fr ( O ) !J (O) e -88diid8
j
fi (ii) JJ (ii) e -98d8diJ
L ( AJI , fr ) = Th A . I
Similarly,
j
j
L ( A!J , fr ) I,J
L ( - I ) I I I ( A fr , JI ) = 'frs A , I
4.
335
Path integrals for anticommuting variables
where I I I denotes the cardinality of the subset I � { 1 , . . . , n} . Corresponding formulas for the Wick symbol follow from Lemma 4.3. 0
Problem 4 . 1 . Verify directly all results in this section for the simplest case of one degree of freedom, when £F C 2 . =
Problem 4.2. Show that the Wick symbol of the product
A = At . . . A 1
is given
by !-1
A (O,
0) = I · · · I At (O, O:t - 1 ) . . . A 1 ( 6: 1 , 0) exp { L O:k (ak - 1 - o:k ) k= 1
+
O(o: t - 1 -
0)}
do: 1 d6: 1 . . . do: t - 1 d6: t - 1
where o: 0
0 and A k ( O, 0) are the Wick symbols of the operators Ak · Problem 4.3. Prove that the Wick symb ol r (O , 0) of the chirality operator r is - 88
e
2
=
.
4.2. Path integral for the evolution operator. Let H be
a Hamilton ian of a system of n fermions - an operator in J't'p with the Wick symbol H(iJ, 0) . Here we express the Wick symbol U(iJ, 0; T) of the evolution op erator U(T) = e - iTH by using the path integral over Grassmann variables. Our exposition will be parallel to that in Section 2.4 of Chapter 5, with ob vious simplification due to the fact that fermion Hilbert space Jlt'p is finite dimensional. Namely, the following elementary result replaces assumption (2. 1 1 ) , made in Section 2.4 of Chapter 5.
Lemma 4.4. Let U(fl t ) be the operator with the Wick symbol
Then U(T)
=
lim U(flt)N,
N -+oo
where flt
=
e- i H(ii,e)t:.t.
T . N
= I - iii flt - U (tlt) is a polynomial in flt with Grassmann algebra coefficients, which starts with the term (flt) 2 . It is easy to see that I I R(flt) ll � c(tltf for some c > 0, and Proof. The Wick symbol of the operator R(tlt)
U(T) = lim (I - iH tlt) N = lim (U(tlt) + R(flt) ) N = lim U(flt) N. N-+oo
N-+oo
N-+oo
0
Using the formula for the composition of Wick symbols ( see Theorem 4.1 and Problem 4.2 ) , we can represent the Wick symbol UN ( iJ, 0; T ) of the operator U(b. t ) N as an (N - 1 ) -fold Berezin integral. Namely, consider the anticommuting variables ak { a l , . . . , a k } , iik = { al , . . . , ak } , k = 1 , . . . , N - 1 - generators of the Grassmann algebra with involution - and =
7.
336
Fermion Systems
denote a k o. k = 2:: ?=1 a i a i , etc. Then N exp :�:::) a k (o.k- 1 - O.k ) + iJ (o. N - 9) UN (iJ, 9 ; T) =
J J { k=l ·
·
·
-iH(a.k , o.k-d�t)
N-1
} II do.kdo.k , k= 1
where o. o = 9, and we put O N = iJ . It follows from Lemma 4.4 that U( iJ, 9 ; T) = lim UN ( 8 , 9 ; T)
(4.7)
N-+oo
N
exp
=
lim N-+oo
J J ·
·
·
N-1
{ L:) o. k ( o.k - 1 - o.k ) + B ( o.N - 9) - iH(ak , o.k - 1 ) � t ) } II do. k da k . k=l
k=l
This formula looks exactly the same as the corresponding formula (2. 13) for the Wick symbol of the evolution operator in Section 2.4 of Chapter 5! Accordingly, we interpret the limit N oo as the following Feynman path integral for Grassmann variables ( or Grassmann path integral) : -
U ( iJ, 9 ; T)
(4.8)
=
{ &(a(0T)J) ==ii8 }
ei J� (i&a- H (& ,a))dt+ ii ( a ( T) - 8) � o. �O. .
Here the "integration" goes over all functions o.(t) , a (t ) with anticommut ing values2 on the interval [0, T] , satisfying boundary conditions o.(O) = 9, a ( T ) 8 , and =
As in Section 2.4 of Chapter 5, for 0 < t < T variables a ( t ) are conjugated to o. ( t ) with respect to the Grassmann algebra involution, while 0. (0) and o.(T) - also variables of integration - are not conjugated to the boundary values o. (O) = 9, a(T) = iJ . Remark. It should be emphasized that the only rigorous meaning of the Grassmann path integral ( 4.8) is the limit of multiple Berezin integrals in
(4.7) . However, as we have seen already in Chapter 5, it is very useful to pretend that the path integral has an independent definition, and formally work with it as if it was an actual integral. 2 For every 0 � t � T t here is a 1 (t) , . . . ' a n (t) , a 1 (t) , . . . ' a n (t) .
a
copy of the independent
Grassmann algebra
with generators
4.
337
Path integrals for anticommuting variables
Using Theorem 4.1 , it is easy to express the supertrace of the evolution operator U(T ) - an operator in a finite-dimensional Hilbert space Jf'F as a Grassmann path integral. We have
j
J J
Tr8 e - iTH = N� lim uN ( 8 , 8; T ) d8d8 = lim · · · exp { iJ ( a ( T ) - 8 ) N� oo oo N N- 1 + Z:::: Ui: k ( o: k - 1 - a k ) - iH(a k, a k - 1)Llt) } IT da k dii k d8d8 k =1 k =1 ei for (iaa- H (a , o:)) dt � a � a { O:o:(O)=o:( ( O) = a(T) } T)
J
- a Grassmann path integral with periodic boundary conditions. Here pe riodic boundary conditions emerge from ao = 8 and ii N = 8 in the same way as in Section 2.4 of Chapter 5. Namely, since 8 and 8 are now variables of integration, the identity (2 . 1 7) in Section 2.4 of Chapter 5, which is valid in the case of anticommuting variables as well, ensures that ON = 8 and iio = 8. Denoting by A the corresponding "Grassmann loop space" - the space of all functions with anticommuting values a(t) and a(t) conjugated with respect to the Grassmann algebra involution and satisfying periodic boundary conditions a(O) = a (T) and a(O) = a (T) - we can rewrite the previous formula as Trs
=
e-iTH
J ei ft (iaa-H(a,o:))dt �a �ii . A
.
Replacing the physical time t by the Euclidean time - it and T by -iT, we get the Grassmann integral representation for the Wick symbol of the operator U( -iT) e -TH , =
U(8, 8; -iT) =
and for the supertrace, (4.9)
Trs
e - TH
f { O:(o:(0)=8 T)=iJ }
=
i e - foT (aa+H (a,o:))dt �a �ii .
Problem 4.4. Express the matrix symbol of the evolution operator as the Grass mann path integral. Problem 4.5. Show that
I {a(O)=-a(T)} a(O)=-a(T)
7.
338
Fermion Systems
- the Grassmann path integral with anti-periodic boundary conditions.
4.3. Gaussian path integrals over Grassmann variables. For sim plicity, here we consider only the case n 1 . As in Section 5.3 of Chapter 5, for u(t) E C1 ( [0 , T] , JR) we put d 1 T uo = J{ u(t) dt and D = dt ' T o and consider on the interval [0, T] the first-order differential operator D + u (t) with periodic boundary conditions. The following result evaluates the simplest Gaussian path integral for Grassmann variables. =
Theorem 4 . 2 . We have
i e - f[(iicHu(t)O:a:)dt g&a!?&a
=
det ( D + u(t) )
=
1 - e - uo T.
Proof. Using Lemma 2.4, we get
= Nlim -+oo
j j ...
i
e-
J[ ( O: a+u(t) O:a: )dt �a�a
e�f:= 1 ( O: k (a:k-1 - a: k) - u(tk) O: ka:k-l�t)
II dak da k N
k =l
det A N . = Nlim -+oo Here ao = a N , iio = li N , tk = k t:l t , and A N is the following N x N matrix: bN 1 0 0 0 bl 1 0 0 0 b 1 0 0 2 0 AN = 1 0 0 0 0 bN - 1 1 0 0 0 where bk = - 1 + u(tk ) t:lt. We have N N det A N = 1 - ( - 1 ) N II bk 1 - II ( 1 - u(tk ) tlt ) , k =l
so that
=
k =l
lim det AN = 1 - e - faT u(t)dt 1 - e - uoT . Using Proposition 5.3 in Section 5.3 of Chapter 5 completes the proof. =
0
Example 4. 1 ( The fermion harmonic oscillator ) . The fermion analog of the
harmonic oscillator is the Hamiltonian H = �w(a*a - aa* ) = w ( a*a - �I) = w(N - � 1 ) ,
4.
339
Path integrals for anticommuting variables
where a * and a are creation and annihilation operators in the one fermion Hilbert space Yt'p = C2 ( see Section 1 . 1 ) . The Wick symbol H is H(a, a) = w ( aa - ! ) . Now using ( 4.9 ) and Theorem 4.2, we obtain wT
Trs
i
e -TH =
e - J[ ( a( HH( a ,o:)) dt q& a q& fi
rT - · + - ) e - J o ( o:o: w o:o: dt q& a q& fi
wT wT e T (1 - e - wT ) = 2 sinh - . 2 A Of course, the same result can be obtained directly since e- TH is just a wT wT 2 x 2 matrix with eigenvalues e T and e - 2 which correspond, respectively, =
eT
J
to the eigenspaces £fl and £j . Thus wT
=
_ wT
. wT 2 smh - . 2 Remark. We have the following analog of formula (2.6) of Chapter 6 for Tr 5 e - T H
=
e 2 -e
2
=
path integrals over Grassmann variables: (4. 10)
where
i
f ( a , fi) q& a q& fi = a (t ) =
J (io
() + f3 (t ) and
)
f (/3, 0; iJ , O) q&{3 q&j3 dOdO,
lo
T
f3(t ) dt
=
0,
and Ao is the space of Grassmann loops {3 (t) , lJ (t) with zero constant term. Example 4 . 2 . Using formula (4. 10) and Theorem 4 .2, we get
=
J
det ( D + w ) e -w T99 d() d0
so that
=
i
e - f[ (a( Hw ao: )dt q& a q& fi
{ e - J[ CN3+w/3{3)dt q&{3q&j3 = wT { e - f[ ( /3i3+w/3{3) dt �f3 �iJ , lAo
lAo
{ e - for /3/3dt q&(3 q&j3 1Ao Equivalently,
=
lim
w-->0
det ( D + w ) wT
= lim w -->0
1 - e - wT wT
=
1.
f e - f[ /3/3dt �/3 �!3 _!_ det' D, T lAo where det ' D = T ( see Proposition 5.3 in Section 5 . 3 of Chapter 5). Intro ducing =
(4. 1 1 )
-
-
({3( t ) + {3 (t) ) , 02 (t) = i-/2 ( /3( t ) - /3(t) ) , J2 so that Bj (t) = Oj (t) , j = 1 , 2 , we can rewrite (4. 1 1 ) as f e - ! J[ (o181 +92B2 )dt �e1 q&e = _!_ det ' D 1 . 0 1 (t) =
lAo
1
1
2
T
=
7.
340
Fermion Systems
Since Pf ' ( D ) = v'det ' = VT, we have 1 ( e - � faT 80dt fi)() Pf ' ( D ) = 1 . ( 4 . 12) jAo (IR) yT Here the domain of integration Ao (IR) consists of all functions O ( t) with values in the Grassmann algebra over IR, which have zero constant term and satisfy periodic boundary conditions 0(0) = O(T) . We will use formulas (4. 10) and (4. 12) in Section 2.2 of Chapter 8.
D
=
_ _
Problem 4.6. Show that e J { 0:(0) =-&(T) }
- g (aa+u(t)&a)dt � a � ii
=
1 +
e - uoT
a(O) = - a( T )
- the regularized determinant of the operator D + u ( t) on [0, T] with anti-periodic boundary conditions.
P roblem 4. 7. For the fermion harmonic oscillator verify that Tr e - T H
directly, and also by using results of Problems 4.5 and 4.6. ()
=
2 cosh wi
f3N and L:; �= l f3k = 0.)
Problem 4.8. Prove formula (4. 10) . (Hint: Follow the proof of Theorem 4.2, and use the decomposition ak
=
+
f3k , where /3o
=
Problem 4.9. Give a direct proof of formula (4. 12) . 5.
Notes and references
Canonical anticommutation relations, discussed in Section 1, were introduced by P. Jordan and E. Wigner in 1928 [JW28] . We refer the reader to F.A . Berezin ' s clas sical monograph [Ber66] for a comprehensive mathematical treatment of canonical commutation and anticommutation relations. Though (Ber66] is mainly devoted to quantum systems with infinitely many degrees of freedom, which are studied by quantum field theory, it also discusses the simpler case of finitely many degrees of freedom. A self-contained mathematical introduction to Clifford algebras, their rep resentations, and other topics can be found in (Var04] and references therein. The fundamental idea that systems with Grassmann variables appear as a semi-classical limit of fermions was formulated by 1.1. Martin in 1 959 (Mar59b, Mar59a] . In dependently, F.A. Berezin in (Ber61] introduced Grassmann variables for rigorous mathematical description of the second quantization for fermion systems by using generating functionals for vectors and operators. Material in Section 2 - differen tial and integral calculus on the Grassmann algebra - belongs to F.A. Berezin, and our exposition follows (Ber66J and [Ber87] . The superalgebra - graded lin ear algebra - which we very briefly describe in Section 3, was also discovered and developed by F.A. Berezin [Ber87] . The references [Man97] , [Fre99J , and [DM99] provide the reader with a more abstract mathematical treatment, while lecture notes [Var04J supplement the category theory approach of [DM99] with
5. Notes and references
341
motivation from physics. We refer to [Ber61 , Ber66] for the general discussion of the path integral over Grassmann variables, and of the matrix and Wick symbols of operators in the fermion Hilbert space. In Section 4 we follow the elegant exposition in [FS91] ; as in Section 2.4 of Chapter 5, we carefully treat boundary conditions for Grassmann path integrals for the Wick symbols ( as opposed to the formulas [Ber71a] ) .
Chapter
8
S upersymmet ry
1.
Supermanifolds
The coordinate vector space V = JRn has a natural structure of a smooth manifold, which to every open subset U � JRn assigns a commutative JR algebra C00 (U) of all smooth functions on U. The assignment for all open U � JRn defines a sheaf of commutative JR-algebras ( commutative rings ) on a topological space JRn , and turns it into a ringed space. Every smooth n-dimensional manifold M is a ringed space - a topological space with a sheaf of commutative rings, which is locally isomorphic to the ringed space JRn . It is easy to see that this definition is equivalent to the standard one, defined by gluing coordinate charts. The notion of a supermanifold generalizes this idea by using local models associated with graded vector spaces. Namely, let W = JRPi q be the coordinate graded vector space of dimension p!q over R The following definition formalizes the intuitive idea that odd coordinates on W are anticommuting. Definition. A supermanifold JRP i q is a topological space JRP with a sheaf of commutative JR-superalgebras ( supercommutative rings ) over JR, called the
structure sheaf, defined by the assignment
for all open U � JRP , where C00 (U) [0 1 , . . . , Oq] is a Grassmann algebra with generators 01 , . . . , ()Q over the commutative ring C00 ( U) .
343
8.
344 Remark. Elements of
f=
c oo ( U ) [ 0 1 ,
L !Ifl' I
01
Supersymmetry
. . . , Oq]
= t9 i 1
.
.
•
have the form eik and !I E C00 ( U )
for I = { i 1 , . . . , ik} C { 1 , . . . , q}, and are called functions on a supermanifold JR?.Pi q over U. Thus JR?.Pi q is a coordinate space with even coordinates x = (x 1 , . . . , xP) and odd coordinates 8 = (0 1 , . . . , Oq ) , and we will write f E C00 (U) [Bl , . . . , B q ] as f ( x, 8) . logical space X with a sheaf 0x of supercommutative rings over the structure sheaf, which is locally isomorphic to JR?.P i q .
IR?.,
Definition. A supermanifold of dimension p l q is a pair (X, Ox ) - a topo
called
Supermanifolds form a category: a morphism between supermanifolds (X, Ox ) and (Y, Oy) is a continuous map r.p : X ----? Y, together with a sheaf map rp * : Oy Ox over rp, a collection of homomorphisms of supercom mutative rings over IR?., 'P v : Oy (V) ----? Ox (r.p - 1 (V) ) , V � Y open, which commute with restriction maps of the sheaves. To every vector bundle E of rank q over an ordinary p-dimensional man ifold M there is an associated supermanifold IIE of dimension plq, defined by reversing parity in the fibers of E. Namely, for every open U . X A ntt na X £l ux P
and we obtain dj3 ( r , 0)
=
=
•
u
u
_
a
•
u
u
,
1 1 gttv (x ( t ) ) (iJtt (t) + r�).. (x (t) ) xa (t) e).. (t) ) ev (t) dt
fo1
(\1 "y(t ) (J (t) , 9 (t) ) dt.
0
2.
351
Equivariant cohomology and localization
As in Chapter 6, one can develop the integration theory on C(M) with respect to the Wiener measure associated with a Riemannian metric on M by using the corresponding heat kernel4 . One can also integrate differen tial forms over C(M) , provided that C(M) is orientable. As in the finite dimensional case, the latter is defined by the condition that the structure group of an C(SO(n) )-bundle C(FM) C (M) , where FM M is the frame bundle, reduces to the connected component of the identity. It follows from the transgression homomorphism ----t
�
that the image of the second Stiefel-Whitney class of M is the obstruction to orientability of C(M) . In particular, if M is a spin manifold, then C (M) is orientable, and if M is simply connected, then this is also a necessary condition. In what follows we will not attempt to develop the integration theory on C(M) , and will formulate and "prove" the infinite-dimensional version of the general Berline-Vergne localization theorem at a heuristic level only. The key observation, by Lemma 2.2, is that the differential form e -8 on C (M) is equivariantly closed, therefore the functional integral f.c(M) e-8 localizes reduces to the finite-dimensional integral over C (M)"r = M, the zero locus of the vector field � . To give the quantitative formulation of this result, recall the notion of the A-genus. Namely, the A-genus of a Riemannian manifold ( M, g) is a differential form A(M) , defined as follows. Let R E A2 (M, s o (TM ) ) be the Riemannian curvature of M, and let eJ.L be a local orthonormal frame of TM. Put
det
( sin��/2 )
A ( M) = j (R) - � = det
( sin� �;2 )
and let j (R)
=
/ be an even degree differential form on M. It does not depend on the choice of a local orthonormal frame of T M, and the A-genus of the manifold M is defined by5 /
1
2
E A• (M) .
It is a closed differential form, whose cohomology class does not depend on the choice of a Riemannian metric on M. 4In Chapter 6 we considered the case of M JR_n with the standard Euclidean metric. 5This is the geometer's definition of the A-genus; in topology one replaces R by (27ri) - 1 R. =
352
8. Supersymmetry
M,
Remark. In local coordinates on 1 - 2 RJ.LVpq
where RJ.L
v
pa
RJ.Lv
X o- ,
1\
is the Riemann curvature tensor.
Theorem 2 . 3 . Let
loop space
X
dP d
M be a compact orientable manifold such that the free
.C(M) is orientable. Then
r e -S (2 7ri ) - � r A ( M ) . jL(M) jM Proof. The proof is at a heuristic level of rigor. The integral =
{
{
e -S e -S('"Y,B) � x �(} jL(M) lrrrL(M) localizes at the zero locus = M, so it is sufficient to integrate over a small tubular neighborhood of in For this aim, consider Riemann normal coordinates of the normal bundle N M) to M at a point E ITTx0 M , given by E Tx 0 M and = E satisfying =
.C(M),y ITTM ITT.C(M). (ITT ITT y(t) (y1 (t), . . . , yn(t)) (xo, Bo) rJ(t) (77 1 (t), . . . , 77n (t)) ITTx0M, 11 yJ.L(t)dt = 0 and 1 1 71J.L(t)dt 0, 1 , Coordinates (Ey(t),ErJ(t)), where real E: is sufficiently small, correspond to a point (expx 0 (Ey(t)), P(e, t ) (Bo+rJ(t))), in the normal bundle N(ITTM) , where expx0 is the exponential map, and P(e,t) is a parallel transport in ITTM from xo along the path expx0(sy(t)) as s changes from 0 to E:. Since we are integrating over an arbitrarily small tubular neighborhood of ITT M in ITT .C( M), it is sufficient to expand S (!, B) up to terms of second order in local coordinates (y( t ) rJ(t)). Since in Riemann normal coordinates around xo on M ar ea or �a 9J.Lv ( x ) 8/J-v + 0 ( I l x 11 2 ) and RJ.Lvpa ( xo) = uxJ.L uxv we readily obtain that at (xo, Bo) ITTx0 M S('y, B) So(xo, Bo; y, rJ) + higher order terms in yJ.L(t), 77J.L(t), =
=
J-L
=
. . .
, n.
,
� - �- ,
=
E
=
where (2. 12)
and
2.
353
Equivariant cohomology and localization
Now us ing property (2.6) in Section 2.2 of Chapter 6 (where m
=
n
=
1 and
T = 1 ) , formula ( 4. 1 0 ) in Section 4.3 of Chapter 7, and performing Gaussian
integration, we get
(
)
{ e -So(xo,Oo;y,Tf) !»y!»'f/ dx o d8o ( 2n ) - � { Jrr.TM jN(ITT M) = (27r ) - � r det 1 ( D 2 8J.Lll - RJ.Lv ( x o, Bo )D) - � dxod8o , lrrrM where we have also used formula (4. 12) in Section 4.3 of Chapter 7. Since R {RJ.Lv } �,11= 1 is a skew-symmetric matrix with even matrix elements, there is an orthogonal matrix C of determinant 1 such that
{ e- S Jc(M)
=
-
=
c"Rc- 1 =
0 r1
r1 0
0 0
0 0
-
0 0
0 0
m = 2. n
0 rm -rm 0
It is easy to see that second order matrix differential operators -D2 J2 +
(� �r) D
and
- D 2 J2 +
( -�r �) D, i
where h is the 2 x 2 identity matrix, have the same spectrum. Using formula (3.5) in Section 3.2 of Chapter 6 with w = ±i rk , we get
-
I
det ( -D2 8J.L11 - RJ.LvD)
=
m
g det ( -D2 h + (r0k I
m
=
= =
-Tk
O
)
D
)
IT det1 ( -D 2 - irk D ) det1 ( -D2 + irkD)
k=l
IT ( sin rk /2 ) 2
k=1
rk /2
=
det
( ) s i� R/ 2 R/2
j (iR) .
Therefore, using the identification C00 (IIT M) :::: A• (M) we finally obtain
e -8 = (2 n ) - � f j (iR(x , B) ) - � dxd8 lrrrM 0 = (2 n i ) - � A(M) . M Remark. Using formulas (2.6) in Section 2 . 2 of Chapter 6 and (4. 1 0 ) in Section 4.3 of Chapter 7, we see that the same result holds for the space f
Jc(M)
J
354
8. Supersymmetry
£1 (M)
of loops on M parametrized by the interval I = [0, T] , 2 e - � J[ ( ll"'r( t) 1 1 +(6(t) ,V'"r(t) 6 (t)) )dt�x �8
r
1rrT£:.1(M) Problem 2.3. Problem 2.4. P roblem 2.5.
= (27ri ) - � r A ( M) . JM
Derive formula (2.8) . Derive formulas (2 . 1 1 ) and (2 . 12 ) .
Justify the arguments in the "proof" of Theorem 2.3. (Hint: See the references in Section 7. )
3 . Classical mechanics on supermanifolds
As was mentioned in Chapter 4, spin is a pure quantum notion which has no classical analog, and the same applies to fermion systems, considered in Chapter 7. However, one can formally consider particles with anticommuting coordinates and formulate classical mechanics on supermanifolds. Though such systems have no physical interpretation6 , their formal quantization yields fermion systems. This allows us to interpret particles with odd degrees of freedom as semi-classical limit of fermions. 3 . 1 . Functions with anticommuting values. Any smooth map f : M -t N of smooth manifolds M and N gives rise to a Frechet algebra homomor
phism
f * : C 00 (N)
-t
C00 (M) ,
where f* ( r.p ) r.p f for r.p homomorphism F : C00 (N) C00 (M) is of this form for some smooth map o
=
f:M
-t
N.
-t
E c oo ( N) . Conversely, any Frechet algebra
By definition, a map ( morphism ) between supermanifolds X and Y is a homomorphism of superalgebras where C00 (X) and C00 (Y) are commutative superalgebras of global sections of corresponding structure sheaves Ox and Oy ( see Section 1 ) . We will denote by Map ( X, Y ) the space of all maps between supermanifolds X and Y. The simplest cases are the following. 1 . X = JR0 1 1 - odd one-dimensional supermanifold, and Y = M an ordinary ( even ) manifold. 2. X = JR. ( or 8 1 and I = [t0 , t 1 ] ) - even one-dimensional manifold, and Y JR.01 n - odd n-dimensional coordinate vector space. =
6 Classical mechanics, which describes physical phenomena at the macroscopic level, neces sarily uses commuting coordinates.
3.
355
Classical mechanics on supermanifolds -+
In the first case every homomorphism of superalgebras F : C00 (M) JR[e] has the form F ( = ( 8 1 , 82 , 83 ) , which correspond to the anticommuting coordinates () , satisfy the following anticommutation relations: (5. 1 ) 9 Here we put n 1 . 1 0 Here it is convenient to lower all indices by the standard Euclidean metric on =
JR3 .
5. Quantum mechanics on supermanifolds
365
Since operators J2 8JL define a representation of a Clifford algebra C3 , the only irreducible realization of (5. 1 ) is given by (5.2)
e Jl
=
f-L. =
V2 a-Jl , 1
1 , 2, 3,
where a-Jl are Pauli matrices (see Section 1 . 1 o f Chapter 4) . Thus the Hilbert space of the system is .Yt' L2 (IR3 ) @ C2 - the Hilbert space of a quantum particle of spin � - and the Hamiltonian operator is p2 H = -. 2m Using (5.2) and the multiplication table of Pauli matrices, we obtain the following form for quantum Noether integrals of motion (4. 2) : =
J = M + S, where M is the angular momentum operator (see Section 3.1 of Chapter 3) , and S = � a . Thus J is the total angular momentum operator of a quantum spin � particle (see Section 1 . 2 of Chapter 4) . Example 5.2 (Quantum spin � particle in constant magnetic field) . The Hilbert space is the same as in the previous example, while the Hamiltonian operator which corresponds to (3. 7) takes the form P2 P2 B·a i - -- · = + ( x 2 2m 2 B E> ) E> 2m Thus operator H is the Pauli Hamiltonian with total magnetic moment f-L. = � (see Section 2 . 1 of Chapter 4) . Example 5.3 (Supersymmetric quantum particle on JRn ) . The phase space is a supermanifold IR2 n f n with even coordinates p = (P I , . . . , Pn ) and x = ( x 1 , . . . , X n ) , and odd coordinates 9 = Uh , . . . , Bn) , with canonical Poisson brackets
H=
{pJl , x v } = OJlv
and
{ BJl , Bv } = i OJlv ,
f-L., V = 1, . . . , n .
Quantum operators P = (P1 , . . . , Pn ) and Q = (Q1 , . . . , Q n ) , which corre spond to canonical coordinates p and x, satisfy Heisenberg commutation relations, and operators (8 1 , . . . , 8 n ) , which correspond to the anti E> commuting coordinates 9, satisfy the following anticommutation relations: =
(5 .3)
The operators V2 8 Jl define a representation of a Clifford algebra Cn . When n is even, the only irreducible realization of (5.3) is e Jl
=
J2"�Jl • 1
8. Supersymmetry
366
where /p are operators in Y't'p = (C2)® d , d = � , given by formulas ( 1 . 12) ( 1 . 13) in Section 1.2 of Chapter 7. For odd n , operators 1 act in (C 2 )®d, where d = [�] ( see Problem 1 . 4 in Section 1 . 2 of Chapter 7) . In both cases, the Hilbert space of the system is Y't' = L2 (IR.n) ® Y't'p = L2 (IR.n) ® C2d , the Hamiltonian operator is given by
p2
H=
2m ' and quantum N oether integrals of motion are
lpv = QpPv - Q vPp - i8p8v ·
Quantization of the supercharge gives the operator . i 1 8 1 Q = t8pPJ.L = rn /pPJ.L = rn /p � = rn fJ, ( 5 .4) v2 v2 v 2 uxJ.L where 8 iJ = !p 8xJ.L is the Dirac operator on IR_n . The Dirac operator is skew-adjoint in Y't', =
-<J. is even, decomposition Y't'p = £/ (j*
When n EB Y't'F- into the subspaces of positive and negative chirality spinors ( see Section 1 . 2 of Chapter 7) gives decomposition Y't'
=
Y't'+ EB Y'f'_ ,
where Y't'± = L2 ( IR.n )
® Y't'l ,
and using that lp ( Y't'/ ) = Y't'F- ( see Section 1 . 2 of Chapter 7) , we can represent the Dirac operator in the following 2 x 2 block-matrix form:
Here the operator [J + have
:
Y't'+
-----+
[Q, Q] +
£_ is called the chiral Dirac operator. We
=
2 Q2
=
i
=
-2mH,
so that 2mH is the Dirac Laplacian -<J 2 . In matrix form, 2m H =
((j�(j+ 0
0 (j+(j�
)
.
Example 5 . 4 ( Supersymmetric quantum particle on Riemannian mani fold ) . We have seen in Example 2.4 in Section 2.4 of Chapter 2 that though for a free quantum particle on a Riemannian manifold ( M, g) it is not pos sible to construct operators P and Q which correspond to standard local coordinates (p , x ) on T* M , the Hamiltonian operator H is well defined as the Laplace-Beltrami operator of the Riemannian metric. It is remarkable
5. Quantum mechanics on supermanifolds
367
that consistent quantization of a supersymmetric particle requires M to be a spin manifold, and for even n dim M the corresponding supercharge operator Q coincides with the Dirac operator! Namely, let Spin(n) be the spin group: connected, simply connected Lie group which is a double cover of SO (n) . The oriented Riemannian manifold ( M, g) of dimension n is said to have a spin structure (and is called a spin manifold) if the bundle SO ( M) of oriented orthonormal frames over M principal SO (n)-bundle - can be extended to the principal Spin(n)-bundle Spin(M) . This is equivalent to the condition that for some open covering M U a EA ua transition functions ta{3 : ua n u{3 -t S O ( n) of a tangent bun dle TM can be lifted to the transition functions Taf3 : Ua n u{3 -t Spin (n) ; taf3 p( Taf3 ) , where p : Spin(n) -t S O (n) is the canonical projection. The manifold M is a spin manifold if and only if its second Stieffel-Whitney class w2 E H2 (M, Z2) vanishes; in this case different spin structures are parametrized by H1 (M, Z2) c:= Hom(1r 1 (M) , Z2 ) , where 1r1 (M) is the fun damental group of M . For even n = 2d, the irreducible representation p of the Clifford algebra Cn in Yf'p c:= C2d (see Section 1 . 2 of Chapter 7) defines unitary representation R of the spin group Spin(n) in Yf'p , which commutes with the parity operator r. By definition, the spinor bundle S on an even dimensional spin manifold M is a Hermitian vector bundle associated with the principal Spin(n)-bundle Spin(M) by the unitary representation R. In other words, s is a complex vector bundle with the transition functions R( Ta(3) : Ua n u{3 -t U(2d) , where U(2d) is the group of unitary 2 d X 2d matrices. Decomposition of vector spaces Yf'p = Yf'/ EB Yf'j; defines the decomposition s = s+ EB S=
-
=
=
of the spinor bundle s into the bundles s+ and s_ of positive and negative chirality spinors. The Dirac operator � : C00 ( M, S) -t C00 (M, S) of the even-dimensional spin manifold M is defined as follows. Let '\75 be the connection on the spinor bundle S induced by the Levi-Civita connection in the tangent bundle T M . Then i n the coordinate chart c M with local coordinates x (x 1 , . . . , x n ) we have
U
=
(5.5)
a�J.L
where V'� is the covariant derivative with respect to the vector field over U, and 'YJ.L (x) are endomorphisms of the spinor bundle S over U satisfying (5.6) where I is the identity endomorphism. It is easy to show that there is an open covering M U a EA Ua such that 'YJ.L ( x) exist over each Ua, and that local
=
8. Supersymmetry
368
expressions (5.5) give rise to a globally defined operator (J : C00 (M, S) C00 (M, S) . Equivalently, if � E C00 (M, S) is given by � = {�a}aEA, �a Ua � £F , where ea = R( Taj3) ej3 on Ua n Uf3, then
�
(5.7)
where f/Ja is given by (5.5) for U = Ua . We also have f/J : C00 ( M S±) � C00 (M, S=F) .
,
The Hermitian metric II l is in the spinor bundle S and the Riemannian metric g on M allow us to define the Hilbert space £ of square integrable global sections of S, £= e
{
E
f (M, S)
: iiell 2
=
JM ll� (x) ll1 dJ-L (x ) }·
0, 2 Tr e- T� � �+ Tr e - T� + � � . ind $ + = Tr8 eT� -
=
On the other hand, we have seen in Example 5 .4 in the last section that 2 -$ = 2H, where H is the Hamiltonian operator of free supersymmetric particle of mass m = 1 on the spin manifold M. Thus for every T > 0, ind $+ Tr8 e - 2TH . =
In Section 2.2 of Chapter 6 and in Sections 4.2 and 4.3 of Chapter 7 we developed a formalism for expressing traces and supertraces of the evolution operator in Euclidean time by path integrals. Using these results, we can, at a physical level of rigor, represent the supertrace Tr 8 e- 2 TH by the path integral (6 . 1 ) Here
1 { 2T
SE ( r, O ) = 2 lo ( 111' 1 1 2 + (O (t) , \1-yO (t) ) )dt
is the Euclidean action of a supersymmetric particle on a Riemannian man ifold M, obtained from the Lagrangian function (4. 1 1 ) by replacing time t by the Euclidean time -it, and £1 (M) is the space of free loops on M parametrized by the interval [0, 2T] . When11 2T = 1 , the integral in (6. 1 ) coincides with integral f.c(M) e-8 , considered i n Section 2.2. Using Theorem 2.3, we obtain ind $+
=
Tr8 e- H
=
(2 7ri) - �
JM A(M) ,
which is the celebrated Atiyah-Singer formula for the index of the Dirac operator on a spin manifold! 1 1 According
to a remark in Section 2 . 2 , the same result holds for every T > 0 .
370
8. Supersymmetry
Remark. Integrating over Grassmann variables in (6. 1 ) , we obtain
(6.2)
Tr 8
e-H
=
{ Pf ( V'-y ) dJLw , j.C ( M )
where dp,w is the Wiener measure on the loop space C(M ) associated with the Riemannian metric g on M. It can be shown that when M is a spin manifold, the Pfaffian Pf (V' -y) of a covariant derivative operator along r E C ( M ) is a well-defined function on C(M) . Thus formula (6.2) , as opposed to the local computation in the derivation of Theorem 2.3, captures global properties of the manifold M. Problem 6.1. Prove the McKean-Singer theorem. Problem 6.2. "Derive" formula (6. 1 ) . (Hint: See the references in the next sec
tion. ) 7.
Notes and references
The goal of Section 1, besides giving a definition of a supermanifold, was to intro duce the isomorphism A• (M) :::: C00 (IITM) , emphasized by E. Witten [Wit82a, Wit82b] , and commonly used in the physics literature. For a systematic introduc tion to supermanifolds, we refer to the classic texts [Kos77, Ber87] , as well as to the modern sources [Man97, DM99, Var04] and references therein. A detailed exposition of equivariant cohomology and localization in the finite-dimensional case can be found in the monograph [BGV04] ; our proof of the Berline-Vergne local ization theorem in Section 2 . 1 follows [SzaOO] . Our presentation of the infinite dimensional case, based on [BT9 5 , SzaOO] , is at a physical level of rigor. Theorem 2 . 3 was originally formulated by E. Witten in his famous path integral derivation of the Atiyah-Singer formula for the index of the Dirac operator. Witten's original ap proach was lucidly presented by Atiyah [Ati85] . Our exposition follows [Wit99b] , with special attention to constant factors; see also [AG83, Alv95] , as well as [BT9 5 , SzaOO] . For a detailed explanation of the functor of points and a discussion of classical mechanics on supermanifolds, see [Fre99] ; our examples of classical systems are taken from [Alv95J . Supersymmetry introduced in Sections 4.2-4. 3 is called N � supersymmetry and is obtained as a reduction of N 1 supersymmetry; see lectures [Alv95 , DF99b, Fre99, Wit99a, Wit99b] and [CFKS08] for more details and references. In particular, [Fre99 , Wit99a] describe useful superfield formalism for producing supersymmetric Lagrangians, sketched in Problem 4.3 in Section 4.3. Material in Section 5 is based on [Alv95] ; we refer the reader to the monograph [BGV04] for the invariant definition of Dirac operators on spin manifolds and related topics. =
=
It should be emphasized that our derivation of the Atiyah-Singer formula in Sec tion 6 is purely heuristic. Rigorous justification of this approach using Ito-Malliavin
7.
Notes and references
371
stochastic calculus was made by J.-M. Bismut [Bis84a, Bis84b, Bis85) ; it turns out that formula (6.2) in Section 6 requires an extra factor: the exponential of the integral along the path of a multiple of a scalar curvature. When T --+ 0, this rig orous formula for the index coincides with the heuristic expression (6.2) . Still, the integration of a differential form of "top degree" over the loop space in [Bis85] remains formal, and is a challenging open problem. Finally, we refer the interested reader to [ASW90] for a path integral derivation of the H. Weyl character formula, and to [Wit99b] for the treatment of the Dirac operator on the loop space.
B ibliography
[AM78]
R. Abraham and J.E. Marsden, Foundations of mechanics, jamin/Cummings Publishing Co. Inc . , Reading, MA, 1978 .
[AG93]
N.l. Akhiezer and I.M. Glazman, Theory of linear Dover Publications Inc . , New York, 1 993.
[AHK76]
Ben
operators in Hilbert space,
S .A. Albeverio and R.J. H(llegh-Krohn, Mathematical theory of Feynman path Lecture Notes in Mathematics, vol. 523, Springer-Verlag, Berlin, 1976 .
integrals,
[AE05] [AG83] [ASW90]
S.T. Ali and M. Englis, Quantization methods : a guide analysts, Rev. Math. Phys. 17 (2005) , no. 4, 39 1-490.
for physicists and
L. Alvarez-Gaume, Supersymmetry and the A tiyah-Singer Comm. Math. Phys. 90 ( 1 983) , no. 2, 161-173 . 0.
index theorem,
Alvarez, I . M . Singer, and P. Windey, Quantum mechanics and the geometry Nuclear Phys. B 337 ( 1990) , no. 2 , 467-486 .
of the Weyl character formula,
[Alv95]
0 . Alvarez, Lectures on quantum mechanics and the index theorem, Geometry and quantum field theory (Park City, UT, 1 99 1 ) , lAS/Park City Math. Ser . , vol. 1 , Amer. Math. Soc . , Providence, Rl, 1995, pp . 271-322.
[Apo76]
T.M. Apostol, Introduction to analytic number theory, Undergraduate Texts in Mathematics, Springer-Verlag, New York, 1976 .
[Arn89]
V.I. Arnol'd, Math ematical methods of classical mechanics, Graduate Texts in Mathematics , vol. 60, Springer-Verlag, New York, 1989.
[AG90]
V.I. Arnol'd and A.B. Givental' , Symp lectic geometry, Dynamical systems IV, Encyclopaedia of Mathematical Sciences , vol . 4, Springer-Verlag, Berlin, 1 990, pp. 1-136.
[AKN97]
V.I. Arnol'd, V.V. Kozlov, and A.l. Neishtadt, Mathematical aspects of clas sical and celestial mechanics, Encyclopaedia of Mathematical Sciences, vol. 3 , Springer-Verlag, Berlin, 1 997.
[Ati85]
M.F. Atiyah, Circular symmetry and stationary-phase approximation, Asterisque ( 1985 ) , no. 1 3 1 , 43-59, CoJloquium in honor of Laurent Schwartz, Vol. 1 (Palaiseau, 1983 ) . -
373
374 [BI66a] [BI66b]
Bibliography M . Bander and C. Itzykson, Group Mod. Phys. 38 ( 1 966) , 330-345.
___
theory and the hydrogen atom. Part I,
, Group theory a n d t h e hydrogen atom. Part II,
( 1 966) , 346-358.
[Bar61] [BR86] [BW97] [BFF + 78a]
V. Bargmann, On a Hilbert Math. 3 ( 1 96 1 ) , 2 1 5-228.
Rev.
Rev. Mod. Phys . 38
space of analytic functions,
Comm. Pure Appl.
A.O. Barut and R. R�czka, Theory of group representations and second ed . , World Scientific Publishing Co. , Singapore, 1986.
applications,
S. Bates and A. Weinstein, Lectures on the geometry of quantization, Berkeley Mathematics Lecture Notes, vol. 8, Amer. Math. Soc . , Providence, RI , 1997. F.
Bayen, M. Flato ,
C.
Fronsdal, A. Lichnerowicz , and D. Sternheimer,
Ann. Physics 1 1 1 ( 1 978) , no. 1 , 61-1 10. + [BFF 78b] ___ , Deformation theory and quantization. Physics 1 1 1 ( 1 9 78) , no. 1 , 1 1 1-15 1 .
De-
formation theory and quantization. I. D eformations of symplectic structures, II. Physical applications,
Ann.
[Bel87]
J .S. Bell, Speakable and unspeakable in quantum mechanics, Collected papers on quantum philosophy, Cambridge University Press, Cambridge, 1987.
[Ber6 1]
F.A. Berezin,
[Ber66]
___ ,
Soviet Physics Dokl. 6 ( 1 96 1 ) , 2 1 2-2 15 (in Russian) .
Canonical operator transformation i n repres entation of sec
ondary quantization,
The method of second quantization,
Pure and Applied Physics, vol. 24,
[Ber71a]
___ , Non- Wiener path integrals, Teoret . Mat. Fiz. 6 ( 1 971) , no. 2, 194212 ( in Russian) , English translation in Theoret . and Math. Phys. 6 ( 1971 ) ,
[Ber71b]
___
[Ber74]
__
[Ber87]
___
Academic Press , New York, 1966.
, Wick and anti- Wick symbols of operators, Mat . Sb. (N. S . ) 86 ( 1 28) (1971 ) , 578-6 10 ( in Russian) , English translation in Math. USSR Sb. 15 ( 1971 ) , 577-606.
141-155.
, Quantization, Izv. Akad. Nauk SSSR Ser. Mat . 38 ( 1974) , 1 1 16-1 175 (in Russian) , English translation in Math. USSR-Izv. 78 (1974) , 1 109-1 165.
, Introduction to superanalysis, Mathematical Physics and Applied Mathematics , vol. 9, D. Reidel Publishing Co. , Dordrecht , 1987.
[BS91]
F.A. Berezin and M.A. Shubin, The Schrodinger equation, Mathematics and its Applications (Soviet Series) , vol. 66, Kluwer Academic Publishers Group, Dordrecht, 1991 .
[ B GV04]
N. Berline, E. Getzler, and M. Vergne, Heat kernels and Dirac operators, Corrected reprint of the 1992 original, Grundlehren Text Editions, Springer Verlag, Berlin, 2004. M. Sh. Birman and M . Z . Solomj ak, Spectral theory of selfadjoint operators in Hilbert space, Mathematics and its Applications (Soviet Series) , D. Reidel Publishing Co. , Dordrecht, 1987.
[BS87]
[Bis84a]
J .-M. Bismut , index theorem,
[Bis84b]
___ ,
J. Funct . Anal. 57 ( 1 984) , no. 1 , 56-99.
The A tiyah-Singer theorems: a probabilistic approach. I. The
schetz fixed point formulas, J .
[Bis85]
Funct . Anal. 5 7 ( 1 984) , no. 3, 329-348.
The A tiyah-Singer theorems: a pro babilistic approach. II. The Lef
, Index theorem and equivariant cohomology Math. Phys. 98 ( 1 985 ) , no. 2, 213-237.
___
on the loop space,
Comm.
375
Bibliography
[BT95]
M. Blau and G. Thompson,
Localization and diagonalization: a review of func
Math. Phys. 36 ( 1995) , no. 5, 2 192-2236.
tional integral techniques for low- dimensional gauge theories and topological field theories, J.
[Bry95]
R.L. Bryant , A n introduction to Lie groups and symp lectic geometry, Geom etry and quantum field theory (Park City, UT, 1991 ) , lAS/Park City Math. Ser., vol. 1, Amer. Math. Soc., Providence, RI, 1995, pp. 5-181.
[BFK9 1]
D . Burghelea, L . Friedlander, and
T.
Math. Phys. 138 (1991) , no. 1, 1-18 .
Kappeler,
On the determinant of elliptic
differential and finite difference operators in vector bundles over 8 1 ,
[BFK95]
___
,
segment,
[BF60]
[Cam63 ] [Cra83]
Proc. Amer. Math. Soc. 123 ( 1995) , no. 10, 3027-3038.
On the determinant of elliptic boundary value problems on a line
V.S. Buslaev and L.D. Faddeev, Formulas for traces for a singular Sturm Liouville differential operator, Dokl. Akad. Nauk. SSSR 132 ( 1960 ) , 13-16 (in Russian) , English translation in Soviet Math. Dokl. 1 ( 1 960) , 45 1-454.
R.H. Cameron, The Ilstow ( 1962/1963 ) , 287-36 1 .
and Feynman integrals, J.
M. Crampin, Tangent bundle 16 ( 1983) , no. 16 3755-3772. ,
[CFKS08]
Comm.
Analyse Math. 1 0
geometry for Lagrangian dynamics, J.
H.L. Cycon, R.G. Froese, W. Kirsch, and B. Simon,
·
Phys . A
Schrodinger operators
with application to quantum mechanics and glo bal geometry, Co rrected
and extended 2nd printing, Texts and Monographs in Physics, Springer-Verlag, Bedin, 2008. [Dav76]
A.S. Davydov, Quantum mechanics, International Series in Natural Philoso phy, vol. 1 , Pergamon Press, Oxford, 1976.
[DEF + 99]
P. Deligne, P. Etingof, D . S . Freed, L.C. Jeffrey, D. Kazhdan, J .W. Morgan, D.R. Morrison, and E. Witten (eds.) , Quantum fields and strings: a course for mathematicians. Vol. 1 , 2, Amer. Math. Soc. , Providence, RI , 1999 , Material from the Special Year on Quantum Field Theory held at the Institute for Advanced Study, Princeton, NJ, 1996-1997.
[DF99a]
P. Deligne and D.S. Freed, Classical field theory, Quantum fields and strings: a course for mathematicians, Vol. 1 , 2 (Princeton, NJ, 1 996/ 1997) , Amer. Math. Soc. , Providence, RI, 1 999, pp. 137-225.
[DF99b]
[DM99]
[Dik58]
[Dir47] [DR0 1]
___ ,
Superso lutions, Quantum fields and strings: a course for mathemati cians , Vol. 1, 2 (Princeton, NJ , 1996/ 1 997) , Amer. Math. Soc . , Providence, RI, 1999, pp. 227-355.
P. Deligne and J.W. Morgan, Notes on supersymmetry (following J. Bern stein} , Quantum fields and strings: a course for mathematicians, Vol. 1, 2 (Princeton, NJ , 1996/ 1997) , Amer. Math. Soc . , Providence, Rl, 1999, pp . 4197.
L.A. Dikil, Trace formulas for Sturm-Liouville differential operators, U spekhi Mat. Nauk (N. S . ) 13 ( 1958 ) , no . 3(81 ) , 1 1 1-143 (in Russian) , English trans lation in Amer. Math. Soc. Transl. (2) 18 ( 1 96 1 ) , 8 1-1 15. P.A.M. Dirac, 1 947.
The principles of quantum mechanics,
Clarendon Press, Oxford ,
W. Dittrich and M. Reuter, Classical and quantum dynamics. From classical third ed. , Advanced Texts in Physics, Springer-Verlag, Berlin, 200 1 .
paths to path integrals,
Bibliography
376 [Dri83] [Dri86] [Dri87]
[DFN84] [DFN85] [Dyn98] [EnrO l] [Erd56] [EK96]
[Fad 57]
V.G. Drinfel'd, Constant quasiclassical solutions of the Yang-Baxter quantum equation, Dokl. Akad. Nauk SSSR 273 ( 1983 ) , no. 3, 531-535 (in Russian) , English translation in Soviet Math. Dokl. 28 ( 1983 ) , 667-671 .
, Quantum groups, Zap. Nauchn. Sem. Leningrad . Otdel. Mat . lnst . S t eklov . (LOMI) 155 ( 1 986) , 18-49 (in Russian) , English translation in J . Soviet Math. 4 1 ( 1 988) , no . 2, 898-9 15. , Quantum groups, Proceedings of the International Congress of Math ematicians , Vol. 1 , 2 (Berkeley, Calif. , 1986) (Providence, RI) , Amer . Math. Soc . , 1987, pp. 798-820 . ___
___
B.A. Dubrovin, A.T. Fomenko, and S.P. Novikov, Modern geometry methods and applications. Part I, Graduate Texts in Mathematics, vol . 93, Springer-Verlag, New York, 1984. , Modern geometry - methods and applications . Part II, Graduate Texts in Mathematics, vol. 104, Springer-Verlag, New York, 1985. A. Dynin, A rigorous path integral construction in any dim ension, Lett. Math. Phys . 44 ( 1998) , no. 4, 317-329. B . Enriquez , Quantization of Lie bialgebras a n d shuffie algebras of L i e alge bras, Selecta M ath . ( N . S . ) 7 (200 1 ) , no. 3, 32 1-407. ___
A. Erdelyi ,
A s ymptotic expansions,
P. Etingof and D . Kazhdan, (N. S.) 2 ( 1996) , no. 1, 1-4 1 . L.D. Faddeev,
Dover Publications Inc. , New York, 1956.
Quantization of Lie bialgebras . I,
A n expression for the trace of the difference between two singu
lar differential operators of the Stu rm - L iouvil l e type,
(N. S . ) 1 1 5 ( 1957) , 878-88 1 (in Russian) .
___
[Fad64]
___
[Fad74]
___
, Properties of the S -matrix of the one-dimensional Schrodinger equa tion, Trudy Mat . Inst . Steklov. 73 ( 1964) , 314-336 (in Russian) , English trans lation in Amer. Math. Soc. Transl. (2) , Vol. 65: Nine papers on partial differ ential equations and functional analysis (1967) , 139-166.
, The invers e problem in t h e quantum theory of s cattering. II, Current problems in mathematics, Vol. 3, Akad. Nauk SSSR Vsesojuz. Inst . Naucn. i Tehn. Informacii, Moscow , 1974, pp. 93-180 (in Russian) , English translation in J . Soviet Math. 5 (1976) , no. 3, 334-396.
, Course 1. Introduction to functional methods, Methodes en theorie des champs/Methods in field theory ( Ecole d' Ete Phys . Theor. , Session XXVIII, Les Houches, 1975) , North-Holland, Amsterdam, 1976 , pp. 1-40. , A mathematician's view of the development of physics, Les relations entre les mathematiques et la physique theorique, Inst . Hautes Etudes Sci . , Bures, 1 998, pp. 73-79. ___
[Fad98]
___
[Fad99]
___
[FY80]
Do kl. Akad. Nauk SSSR
, The inverse pro b lem in the quantum theory of scatte ri ng, Uspekhi Mat . Nauk 14 ( 1959) , no. 4 (88) , 57-1 19 (in Russian) , English translation in J. Math. Phys. 4 ( 1 963) , 72-104.
[Fad59]
[Fad76]
Selecta Math.
, Elementary introduction to quantum field theory, Quantum fields and strings : a course for mathematicians, Vol. 1 , 2 (Princeton, NJ , 1996/ 1997) , Amer. Math. Soc . , Providence, RI , 1999, pp . 5 13-550. L . D . Faddeev and O.A. Yakubovskii , Lectures on quantum mechanics for mathematics students, Leningrad University Publishers, Leningrad , 1980 (in Russian) .
377
Bibliography
[FS9 1]
[FT07]
[Fey48] [Fey5 1]
L.D. Faddeev and A.A. Slavnov Gauge fields . Introduction to quantum theory, Frontiers in Physi cs , vol. 83, Addison-Wesley Publishing Company, Redwood City, CA, 1991. L . D . Faddeev and L.A. Takhtajan, Hamiltonian methods i n the theory of solitons, Reprint of I987 English edition, Classics in Mathematics, Springer Verlag, New York, 2007. R.P. Feynman, Space-time approach to non-relativistic quantum mechanics, Rev. Modern Physics 20 (1948 ) , 367-387. ,
___ , A n operator calculus having applications in quantum electrodynamics, Physical Rev. (2) 84 ( 1 95 1 ) , 1 0 8 1 2 8 R.P. Feynman and A.R. Hibbs , Quantum mechanics and path integrals, Mc Graw Hill, New York, 1 965. -
[FH65] [Fla82] [Foc32]
[Foc78] [Fre9 9) [Fuj 80) [FH91) [GL53)
.
M. Flat o , Deformation view of physical theories, Czechoslovak J . Phys . B 3 2 ( 1 982), 472-475 . V.A. Fock, Konfigurationsraum und zweite Quantelung, Z. Phys 75 ( 1 932 ) , no . 9- 10, 622-647. .
___ , Fundamentals of quantum mechanics, Mir Publishers , Moscow, 1978. D.S. Freed, Five lectures on supersymmetry, Amer. Math. Soc., Providence, RI , 1999. D. Fuj iwara, Remarks on convergence of the Feynman path integrals, Duke Math. J. 47 ( 1 980 ) , no. 3, 559-600. W. Fulton and J. Harris, Repres entation theory, Graduate Texts in Mathe matics , vol. 129, Springer-Verlag, New York, 199 1 . I . M . Gel'fand and B . M . Levitan, O n a simple identity for the characteristic values of a differential operator of the second order, Dokl. Akad. Nauk SSSR (N.S . ) 88 ( 1953) , 593-596 (in Russian) .
[GY56)
I . M . Gel'fand and A.M. Yaglom, Integration in function spaces and its app li cation to quantum physics, Uspekhi Mat . Nauk ( N S . ) 1 1 ( 1 95 6 ) , no. 1 (67) , 77-1 14, English translation in J . Math. Phys. 1 ( 1960) , 48-69 . Ya.L. Geronimus , Teoriya ortogonat nyh mnogoclenov, Gosudarstv. lzdat . Tehn.-Teor. Lit . , Moscow-Leningrad , 1950 (in Russian) . P.B . G i lkey Invariance theory, the heat equation, and the A tiyah-Sing er index theorem, second ed. , CRC Press , Boca Raton, FL, 1995. C. Godbillon, Geometrie differentielle et mecanique analytique, Hermann, Paris , 1969. I . C . Gohberg and M.G. Krein, Introduction to the theory of linear nons elfad joint operators, Translations of Mathematical Monographs, Vol. 18, Amer. Math. Soc . , Providence, RI, 1969. H . Goldstein, Classical mechanics, Addison Wesley, 1 980 . R. Goodman and N .R. Wallach, Repres entations and invariants of the classical groups, Encyclopedia of Mathematics and its Applications , vol. 68, Cambridge University Press, Cambridge, 1998. V. Guillemin and S. Sternberg, Geometric asymptotics, Mathematical Surveys, No. 14, Amer. Math. Soc., Providence, RI, 1977. .
[Ger50] [Gil 95 ]
[God69] [GK69]
[Gol80) [GW98]
[ GS77] [GS03)
,
S.J. Gustafson and I.M. Sigal, Mathematical Universitext , Springer-Verlag, Berlin, 2003.
concepts of quantum mechanics,
378
Bibliography
[HS96]
P.D . Hislop and I.M. Sigal, Introduction to spectral theory: With applications to Schrodinger operators, Applied Mathematical Sciences, vol. 1 13, Springer Verlag, New York, 1996.
[IM74]
K. Ito and H.P. McKean, Jr. , Diffusion processes and their sample paths, Sec ond printing, corrected, Die Grundlehren der mathematischen Wissenschaften, Band 125, Springer-Verlag, Berlin, 1974.
[IZ80]
C . Itzykson and J.B. Zuber, Quantum field theory, International Series in Pure and Applied Physics , McGraw-Hill International Book Co. , New York, 1980.
[Jim85]
M. Jimbo, A q -difference analogue of U(g) Lett. Math. Phys. 10 ( 1 985) , no. 1, 63-69.
and the Yang-Baxter equation,
Uber das Paulische A quivalenzverbot,
Z. Phys. 47
[JW28]
P. Jordan and E. Wigner, ( 1928) , 631-658.
[Kac59)
M. Kac, Probability and related topics in physical sciences, Lectures in Applied Mathematics. Proceedings of the Summer Seminar, Boulder, CO, 1957, vol. 1 , Interscience Publishers, London-New York, 1959.
[Kac80] [Kaz99]
[Kho07] [Kir76]
, Integration in function spaces and some of its Fermiane, Accademia Nazionale dei Lincei , Pisa, 1980.
___
applications,
Lezioni
D. Kazhdan, Introduction to QFT, Quantum fields and strings: a course for mathematicians, Vol. 1 , 2 (Princeton, NJ, 1996/ 1997) , Amer. Math. Soc. , Providence, RI , 1999 , pp. 377-418.
D. Khoshnevisan, Probability, Graduate Studies in Mathematics, vol. 80, Amer. Math. Soc., Providence, Rl , 2007.
A.A. Kirillov, Elements of the theory of representations, Grundlehren der Mathematischen Wissenschaften, Band 220 , Springer-Verlag, Berlin, 1976.
, Geometric quantization, Dynamical systems IV, Encyclopaedia of Mathematical Sciences, vol. 4, Springer-Verlag, Berlin, 1990, pp. 137-172.
[Kir90]
___
[Kir04]
___
, Lectures on the orbit method, Graduate Studies in Mathematics, vol. 64, Amer. Math. Soc . , Providence, RI, 2004.
[KSV02)
A.Yu. Kitaev, A.H. Shen, and M.N. Vyalyi, Classical and quantum computa tion, Graduate Studies in Mathematics, vol. 47, Amer. Math. Soc., Providence, Rl, 2002.
[Kon03]
M. Kontsevich, Deformation quantization Phys. 66 (2003) , no. 3, 157-2 16.
of Poisson manifolds,
Lett. Math.
[Kos77]
B . Kostant, Graded manifolds, graded L i e theory, and prequantization, Dif ferential geometrical methods in mathematical physics (Proc. Sympos. , Univ. Bonn, Bonn, 1 975) , Lecture Notes in Math., vol. 570, Springer-Verlag, Berlin, 1977.
[Kre62)
M.G. Krein,
On perturbation determinants and a trace formula for unitary
and self-adjoint operators,
Dokl. Akad. Nauk SSSR 144 (1962) , 268-271 (in Russian) , English translation in Soviet Math. Dokl. 3 (1962) , 707-710. [LL58]
L.D. Landau and E.M. Lifshitz,
Quantum mechanics: non- relativistic theory.
Course of Theoretical Physics, Vol. 3,
Pergamon Press Ltd . , London-Paris,
1958. [LL76]
L.D. Landau and E.M. Lifshitz, Mechanics. 1, Pergamon Press, Oxford, 1976.
[Lan87]
S . Lang,
Elliptic functions,
Course of theoretical physics. Vol.
second ed. , Springer-Verlag, New York, 1987.
379
Bibliography
[Ler81)
[LS91)
[LV80) [Loe77) [Loe78) [Mac04) [Man97) [Mar86) [Mar59a) [Mar59b)
J. Leray, Lagrangian analysis and quantum mechanics, A mathematical struc ture related to asymptotic expansions and the Maslov index, MIT Press, Cam bridge, MA, 198 1 . B . M . Levitan and I . S . Sargsjan, Sturm-Liouville and Dirac operators, Math ematics and its Applications (Soviet Series) , vol. 59, Kluwer Academic Pub lishers Group, Dordrecht, 199 1 . G . Lion and M . Vergne, The Weil repres entation, Maslov index and theta series, Progress in Mathematics, vol. 6, Birkhiiuser, Boston, MA, 1980. M. Loeve, Probability theory. I, fourth ed. , Graduate Texts in Mathematics, vol. 45, Springer-Verlag, New York, 1977. , Probability theory. II, fourth ed. , Graduate Texts in Mathematics, vol . 46, Springer-Verlag, New York, 1978. G.W. Mackey, Mathematical foundations of quantum mechanics, Reprint of the 1963 original, Dover Publications Inc . , Mineola, NY, 2004. Y.I. Manin, Gauge field theory and complex geometry, second ed. , Grundlehren der Mathematischen Wissenschaften, vol. 289, Springer-Verlag, Berlin, 1997. V.A. Marchenko, Sturm- Liouville operators and applications, Operator The ory: Advances and Applications, vol. 22, Birkhauser Verlag, Basel, 1986. J.L. Martin, The Feynman principle for a Fermi system, Proc. Roy. Soc. Ser. A 2 5 1 ( 1 959) , no. 1267, 543-549. ___
___ ,
Generalized classical dynamics, and the 'classical analogue ' of a Fermi
oscillator,
[MF81)
[Mes99) [MP49)
[Mon52) [Nel59) [Nel64) [New02)
[Olv97) [PW35)
[Rab95)
[RS71)
Proc . Roy. Soc. Ser. A 2 5 1 ( 1 959) , no. 1267, 536-542 . V.P. Maslov and M.V. Fedoriuk, Semiclassical approximation in quantum me chanics, Mathematical Physics and Applied Mathematics , vol. 7, D. Reidel Publishing Co. , Dordrecht , 198 1 . A. Messiah, Quantum mechanics, Dover Publications Inc . , Mineola, NY, 1999. S. Minakshisundaram and A. Pleijel, Some properties of the eigenfunctions of the L aplace- operator on Riemannian manifolds, Canadian J . Math. 1 ( 1 949) , 242-256. E.W. Montroll, Markoff chains, Wiener integrals, a n d quantum theory, Comm. Pure Appl. Math. 5 ( 1 952) , 41 5-453 . E . Nelson, Analytic vectors, Ann. o f Math. ( 2 ) 7 0 ( 1 959) , 572-615. , Feynman integrals and the Schrodinger equation, J . Math. Phys . 5 ( 1964) , 332-343 . R. G . Newton, Scattering th eory of waves and particles, Reprint of the 1982 second edition, with list of errata prepared by the author, Dover Publications Inc . , Mineola, NY, 2002. F.W.J. Olver, A symptotics and special functions, Reprint of the 1974 original, AKP Classics , A K Peters Ltd . , Wellesley, MA, 1997. L . Pauling and E.B. Wilson, Introduction to quantum mechanics. With appli cations to chemistry, McGraw-Hill Book Company, New York and London, 1935. J.M. Rabin, Introduction to quantum field theory for mathematicians, Geom etry and quantum field theory (Park City, UT, 199 1 ) , lAS/Park City Math. Ser. , vol. 1, Amer. Math. Soc . , Providence, RI , 1995, pp. 183-269. D . B . Ray and I.M. Singer, R-torsion and the Laplacian on Riemannian man ifolds, Advances in Math. 7 ( 1971 ) , 145-2 10. ___
380 [RS80] [RS75] [RS79] [RS78)
Bibliography
M. Reed and B . Simon, Methods Press, New York, 1980 .
of modern mathematical physics. I,
___ , Methods of modern mathematical physics. a dj o i ntn e ss , Academic Press , New York, 1975.
, Methods of modern mathematical physics. demic Press, New York, 1979. ___
, Methods of modern mathematical Academic Press , New York, 1 978 .
___
Academic
II. Fourier analysis, self
III. Scattering theory,
Aca
physics. IV. Analysis of operators,
[RTF89)
N.Yu. Reshetikhin, L.A. Takhtadzhyan, and L.D. Faddeev, Quantization of Lie groups and Lie algebras, Algebra i Analiz 1 ( 1 989 ) , no. 1 , 1 78-206 (in Russian) , English translation in Leningrad Math. J. 1 ( 1 990) , 1 93-225.
[Ros04]
J . Rosenberg, A s elective history of the Stone-von Neumann theorem, Operator algebras, quantization, and noncommutative geometry, Contemp . Math. , vol. 365, Amer. Math. Soc. , Providence, RI, 2004, pp. 331-353.
[RSS94)
M . Z . Solomyak, and M.A. Shubin, Spectral theory of dif Partial differential equations. VII, Encyclopaedia of Math ematical Sciences , vol. 64, Springer-Verlag, Berlin, 1994.
G.V. Rozenblyum,
ferential operators,
[Rud87]
W . Rudin, Real and York, 1 987.
[Sak94)
J.J. Sakurai , pany, 1 994.
[See67]
R.T. Seeley, Complex powers of an el liptic operator, Singular Integrals (Proc . Sympos. Pure Math. , Chicago, IL, 1966) , Amer. Math. Soc . , Providence, RI, 1967, pp. 288-307.
[STS85)
M.A. Semenov-Tian-Shansky, Dressing transformations and Poisson group ac tions, Pub!. Res . Inst . Math. Sci. 2 1 ( 1 985) , no. 6, 1 237-1260.
[SW76]
D.J. Simms and N . M . J . Woodhouse, Lectures in g eome tric ture Notes in Physics, 53, Springer-Verlag, Berlin, 1976.
[Ste83]
S . Sternberg, Lectures Co. , New York, 1983.
[Str05]
F.
Strocchi ,
complex analysis,
third ed. , McGraw-Hill Book Co . , New
Modern quantum mechanics,
Addison-Wesley Publishing Com
on differential geometry,
quantization,
Lec
second ed. , Chelsea Publishing
A n introduction to th e mathematical structure of quantum me
chanics. A short cours e for mathematicians,
Advanced Series in Mathematical Physics, vol. 27, World Sci. Publishing, London-Singapore , 2005. [SzaOO)
R.J. Szabo, Equivariant cohomology and localization of path integrals, Lecture Notes in Physics. New Series: Monographs, vol. 63, Springer-Verlag, Berlin, 2000.
[Sze75)
G. Szego,
[Tak90]
L. A . Takhtaj an, Lectures on quantum groups, Introduction to quantum group and integrable massive models of quantum field theory (Nankai , 1989) , Nankai Lectures Math. Phys . , World Sci. Publishing, River Edge, NJ , 1 990, pp . 69197.
[Tob56) [Var04)
Orthogonal polynomials,
W. Tobocman, Transition ( 1 0 ) 3 ( 1 956) , 1213- 1229.
Amer. Math. Soc . , Providence, RI, 1 975.
amplitudes as sums over histories,
Nuovo Cimento
V.S. Varadaraj an, Supersymmetry for mathematicians : a n introduction, Courant Lecture Notes in Mathematics, vol. 1 1 , New York University Courant Institute of Mathematical Sciences, New York, 2004.
381
Bibliography [Vil68] [vN3 1] [vN96] [Vor05]
[Wey 5 0]
[ Wig5 9]
[Wit82a]
N.Ja. Vilenkin, Special functions and the theory of group repres entations, Translations of Mathematical Monographs, Vol. 22, Amer. Math. Soc . , Prov idence, RI, 1968. J . von Neumann, D i e Eindeutigkeit d e r Schrodingershen Operatoren, Mathe matische Annalen 104 ( 1 931 ) , 570-578 . , Mathematical foundations of quantum mechanics, Princeton Land marks in Mathematics , Princeton University Press, Princeton, NJ, 1996. A.A. Voronov, Notes on universal algebra, Graphs and patterns in mathe matics and theoretical physics ( Stony Brook, NY, 200 1 ) , Proc. Sympos . Pure Math. , vol. 73 , Amer. Math. Soc . , Providence , RI , 2005, pp. 81-103. H. Weyl, Theory of groups and quantum mechanics, Dover Publications , New York, 1950. ___
E.P. Wigner,
Pure and Applied Physics , Vol. 5 , Academic Press, New York, 1959. E. Witten, Constraints on supersymmetry breaking, Nuclear Phys. B 202 ( 1 982) , no. 2 , 253-3 16. , Supersymmetry and Morse theory, J. Differential Geom. 17 ( 1 982) , no. 4, 66 1-692 ( 1 983) . , Homework, Quantum fields and strings: a course for mathematicians , Vol. 1 , 2 (Princeton, NJ, 1996/ 1997) , Amer. Math. Soc., Providence, RI , 1999, pp. 609-717. , Index of Dirac operators, Quantum fields and strings: a course for mathematicians, Vol. 1 , 2 ( Princeton, NJ, 1996/ 1 997) , Amer. Math. Soc. , Providence, RI, 1999, pp. 475-5 1 1 .
[Wit82b]
__
[Wit99a]
___
[Wit99b]
___
[Woo92]
[Yaf92] [YI73]
Group theory: A nd its application to the quantum mechanics of
atomic spectra,
N.M.J. Woodhouse , Geometric quantization, second ed. , Oxford Mathematical Monographs, The Clarendon Press Oxford University Press , New York, 1992. D . R. Yafaev, Mathematical s cattering theory. General theory, Translations of Mathematical Monographs, vol . 105, Amer. Math. Soc . , Providence, Rl, 1992. K. Yano and S. Ishihara, Tangent and cotangent bundles: differential geometry, Pure and Applied Mathematics, No. 16, Marcel Dekker Inc., New York, 1 973.
Index *-product,
136
action classical, 35, 208, 260 functional, 5 abbreviated, 35 algebra of classical observables, 38, 46 angular momentum conservation of, 20 internal, 2 1 7 operator, 1 8 1 , 2 1 7 vector, 2 0 , 180 angular velocity, 13 anti-holomorphic representation, asymptotics semi -classical, 206, 280 short-wave, 206 Berezin integral, 3 1 9 incomplete, 331 Berezinian, 329 Born -von Neumann formula, bosons, 220
113
70
canonical anticommutation relations, 308 commutation relations, 104, 1 1 0, 307 coordinates, 44 transformation, 39 free, 42 generating function of, 41 chirality operator, 313 classical trajectory, 4 Clebsch-Gordan decomposition, 184 Clifford algebra, 3 1 1
completeness relation, 1 6 5 , 187 complex probability amplitude, configuration space, 4 extended, 1 8 conservation law, 1 5 coordinate operator, 8 1 , 86 for fermions, 310 representation, 86 coordinates even, 344 normal, 1 2 odd, 344 correspondence principle, 80 Coulomb potential, 193 problem, 193 Darboux' theorem, 44 deformation quantization, 143 degrees of freedom, 4 internal, 2 1 7 determinant characteristic, 266 Fredholm, 273 regularized, 262 van Vleck, 285 Dirac bra and ket vectors , 9 1 gamma matrices, 3 1 3 Laplacian, 366, 368 operator chiral, 366, 368 on !Rn , 366 on a spin manifold, 367 Dirac-von Neumann axioms , 73
240
-
383
384
Index
distribution function, 67
on Grassmann algebra, 321 measure
joint , 91
finite-dimensional, 289 energy, 15
infinite-dimensional, 2 9 1
centrifugal, 2 3
path integral, 2 6 1
conservation of, 1 6
for a free particle , 2 6 1
effective potential, 2 3
for Grassmann variables , 338
kinetic , 1 0 potential, 1 0 total , 1 7 equations
for the harmonic oscillator, 264 Wiener integral for Dirichlet boundary conditions, 301 for periodic boundary conditions, 302
geodesic, 13 of motion, 4, 6, 7 of a rigid body, 1 3
quantum, 7 5 Euclidean p a t h integral, 2 5 4
generalized accelerations , 4
coordinates, 4 cyclic, 22
forces, 1 9
Euler 's equations, 1 4
momenta, 1 9
Euler- Lagrange equations , 7 , 28
velocities , 4
evolution operator in classical mechanics, 52 in quantum mechanics, 76 expectation value of classical observables, 58 of quantum observables, 67 extremals, 5 central field of, 35
graded dimension, 324 matrix algebra, 327, 328 vector space, 324 Grassmann algebra, 3 1 4 inner product , 3 1 6 , 322 with involution, 322 path integral, 336
fermions , 220 Feynman path integral for Grassmann variables, 336 for t he pq-symbol , 250 for the qp-symbol, 2 5 1 for t h e harmonic oscillator, 257 for the Weyl symbol, 2 5 1
Hamilton's canonical equations , 29
equations
for classical observables, 38, 46 picture , 59 , 75 principle, 5
for t h e Wick symbol, 2 5 3
Hamilton- J acobi equation, 36 , 205, 207
in t h e configuration space , 246 , 248
Hamiltonian
in the phase space, 245, 247 Feynman-Kac formula, 297 first variation with fixed ends, 8 with free ends, 8
action , 48
function, 2 8 , 45 operator, 75 of N particles ,
99
of a complex atom, 1 00 of a Newtonian particle, 98
force, 1 0 conservative, 1 0 Fourier transform, 9 1 frame of reference, 8 inertial ,
9
phase flow, 3 1 , 45 . system, 45 vector field, 3 1 , 45 harmonic oscillator
Fresnel integral , 24 1 , 245, 247
classical , 1 2 , 2 1
fundamental solution, 240
quantum, 1 0 3 , 109 fermion, 338
Galilean group ,
Heisenberg
9
transformation , 9
algebra, 8 2 , 3 1 1 commutation relations, 8 1 , 82
Galileo's relativity principle , 9
equation of motion, 76
Gato derivat ive , 7
group, 83
Gaussian
picture, 75
integral, 255 in complex domain, 256
uncertainty relations , 74 Hilbert space, 63
385
Index
boson , 3 09 fermion , 309 holomorphic representation, 1 1 3 index of a chiral Dirac operator, 369 inertia principal axes of, 1 4 principal moments of, 14 tensor, 14 integral first, 1 5 Noether, 1 7 o f motion, 1 5 , 3 9 quantum, 78 Jacobi inversion formula, 26 5 operator, 1 5 , 259, 2 6 1 , 281 theta series, 265 Jost solutions, 1 5 6 Kepler's first law, 24 problem, 24 second law, 23 third law, 24 kinetic energy operator , 149 Lagrangian, 4 function, 4 submanifold, 33, 43 subspace, 86, 89 system, 4 closed, 9 non-degenerate, 26 Laplace-Runge-Lenz operator, 1 99 vector, 2 5 , 50, 199 Legendre transform, 28 Levi-Civita connection, 13 Liouville 's canonical 1-form, 28 equation , 6 0 picture , 60 , 77 theorem, 33 volume form, 33 Lippman-Schwinger equation, 187 localization theorem Berline-Vergne, 348 for the free loop space ; 352 Lorentz force, 12 mass, 9 reduced, 22 total , 22
Maupertuis' principle, 34 measurement in classical mechanics, 56 in quantum mechanics, 67 momentum, 1 9 conservation of, 19 operator, 81, 88 for fermions, 3 1 0 representation, 89 monodromy matrix, 275 , 2 78 Morse index, 2 08 , 259 motion finite, 2 1 infinite, 2 1 Newton's equations, 10 law of gravitation, 1 1 law o f inertia, 9 third law, 1 0 Newton-Laplace principle, 4 Noether t heorem with symmetries, 48 Noether's theorem , 17 observables classical, 38 quantum, 6 6 complete system of, 9 0 simultaneously measured, 73 operator adjoint , 63 annihilation, 109, 307 fermion , 309 closed, 63 compact, 6 4 creation, 1 0 9 , 307 fermion, 309 differential first-order, 278 essentially self-adjoint , 64 Hilbert- Schmidt , 65 matrix-valued Sturm-Liouville with Dirichlet boundary conditions, 273
with periodic boundary conditions, 277 of trace class , 64 self-adj oint , 64 Sturm-Liouville , 2 6 8 with Dirichlet boundary conditions, 268
with periodic boundary conditions , 2 74 symmetric, 6 3 , 64 operator symbol pq, 135 qp, 135 matrix, 116
386
Index
for fermions, 331 Weyl, 132 Wick, 1 14 for fermions, 331 orthogonality relation,
pq, qp,
rules of Bohr-Wilson-Sommerfeld,
79
102,
213 168, 187
particle( s), 4 charged, in electromagnetic field, free, 9 on Riemannian manifold, 1 2 in a potential field, 1 1 interacting, 1 0 quantum free, 93 Pauli exclusion principle, 227 Hamiltonian, 222, 365 matrices, 2 1 8 wave equation, 222 Pfaffian, 32 1 phase shift, 191 phase space, 28, 45 extended, 33 Planck constant, 76 Poincare-Cartan form, 33 Poisson action, 48 algebra, 39 bracket, 46, 52 canonical, 39 on Grassmann algebra, 318 manifold, 51 non-degenerate, 53 structure, 5 1 tensor, 5 2 theorem, 4 7 potential effective, 189 long-range, 190 repulsive, 191 short-range, 190 potential energy operator, 149 potential field, 11 central, 1 1 principle o f the least action in the configuration space, 5 in the phase space, 33 projection-valued measure, 68, 72 resolution of the identity, 68 propagator, 240, 245 of a free quantum particle, 242 propogator for the harmonic oscillator, 257 quantization,
135 135
11
Weyl, 132 quantum bracket, quantum number azimuthal, 192 magnetic, 192 principal, 192 radial, 192
76
reflection coefficient, 173 regular set, 63 regularized product, 263, 266 representation by occupation numbers, for fermions, 310 resolvent operator, 63 rigid body, 1 3 ringed space, 343 scattering amplitude, 187 matrix, 1 73 operator, 1 72 , 187 solutions, 1 72 theory non-stationary, 172 stationary, 172 Schrodinger equation of motion, 77 radial, 189 stationary, 79 time-dependent, 78 operator, 99, 149 of a charged particle, 101 of a complex atom, 100, 151 of a harmonic oscillator, 103 of a hydrogen atom, 100, 1 5 1 one-dimensional, 1 5 5 radial, 189 picture, 77 representation, 89 for n degrees of freedom, 91 Schur-Weyl duality, 232 Sommerfeld's radiation conditions, 1 86 space of states classical, 58 quantum, 66 spectral theorem, 68 spectrum, 63 absolutely continuous, 86, 93, 96, 153 essential, 152 joint, 73 point, 63, 1 05 singular, 153
112
387
Index
spin, 2 1 7 manifold, 367 operators, 218 singlet space, 228 structure, 367 total, 218 triplet space, 228 standard coordinates on r• M, 27 on TM , 6 state, 4 bound, 79 ground, 107, 308 , 309 stationary, 78 states coherent, 1 14 for fermions, 332 in classical mechanics, 58 mixed, 58 pure, 58 in quantum mechanics, 66 mixed, 67 pure, 66 superalgebra, 325 commutative, 325 Lie, 327 supercharge, 361 supermanifold, 344 functions on, 344 supersymmetry generator, 361 supersymmetry transformation, 361 supertrace, 328 symmetrization postulate, 226 symmetry, 17 group, 17, 48 infinitesimal, 17 symplectic form, 42 canonical, 32 on Grassmann algebra, 318 manifold, 42 vector field, 46 system closed, 4 quantum, 66 composite, 66 systems on supermanifolds classical, 356 quantum, 364 time Euclidean, 242 physical, 9, 242 slicing, 244 trace of operator, 65
transition coefficients, 159 transmission coefficient , 1 73 transport equation, 207 turning point, 2 1 , 210 variance, 74 variation infinitesimal, 5 with fixed ends, 5 virial theorem in classical mechanics, 1 1 i n quantum mechanics, 154 wave operators, 170 plane, 174 scattering, 173 stationary, 187 wave function, 88 total, 224 coordinate part of, 224 spin part of, 224 Weyl inversion formula, 129 module, 232 quantization, 132 relations, 84, 1 1 8 transform, 1 1 9, 1 24 Wick normal form, 1 1 4 for fermions, 3 3 1 theorem, 290 Wiener integral, 294 measure conditional, 296 on C ( [O, oo) , JRn ; O) , 294 on C ( [O, oo) , JRn ; qo ) , 294 on the free loop space, 298 WKB method, 209 wave function, 210 Young diagram, 229 symmetrizer, 230 tableau, 229 canonical, 229 zeta-function Hurwitz, 279 operator, 262 Riemann, 263
Titles in This Series 95 L e o n A . Takhtajan, Quantum mechanics for mathematicians, 2008 94 James E . Humphreys, Representations of semisimple Lie algebras in the BGG category 0, 2008 93 Peter W. Michor , Topics in differential geometry, 2008 92 I . Martin Isaacs, Finite group theory, 2008 91 Louis Halle Rowen, Graduate algebra: Noncommutative view , 2008 90 Larry J. Gerstein, Basic quadratic forms, 2008 89 Anthony Bonato , A course on t he web graph , 2008 88 87
Nathania! P. Brown and Narutaka Ozawa, c•-algebras and finite-dimensional
approximations, 2008 Srikanth B. Iyengar, Graham J. Leuschke, Anton Leykin, Claudia Miller, Ezra Miller, Anurag K . S ingh, and Uli Walther, Twenty-fm.1r hours of local cohomology,
2007 86 Yulij Ilyashenko and Sergei Yakovenko, Lectures on analytic differential equations, 2007 85 John M. Alongi and Gail S . Nelson, Recurrence and topology, 2007
84
83
82
81
C haralambos D . Aliprantis and Rabee Tourky, Cones and duality,
2007
Wolfgang Ebeling, Functions of several complex variables and their singularities
(translated by Philip
G. Spain) , 2007
theorem ( translated by Stephen S. Wilson ) , 2007
Serge Alinhac and Patrick Gerard , Pseudo-differential operators and the Nash-Moser
V . V. Prasolov, Elements of homology theory,
2007
79 William Stein, Modular forms, a computational approach ( with an appendix by Paul E. Gunnells ) , 2007
80
Davar Khoshnevisan, Probability,
2007
78 Harry Dym, Linear algebra in action , 2007
77 Bennett Chow, Peng Lu, and Lei Ni, Hamilton's Ricci flow , 2006 Pe t e r D. Miller, Applied asymptotic analysis,
76 Michael E . Taylor, Measure theory and integration, 2006 75
74
2006
V. V. Prasolov, Elements of combinatorial and differential topology ,
2006
73 Louis Halle Rowen, Graduate algebra: Commutative view, 2006 72 R. J. Williams, Introduction the the mathematics of finance, 2006 71
S . P. Novikov and I . A. Taimanov, Modern geometric structures and fields,
2006
70 Sean Dineen, Probability theory in finance, 2005 69 Sebast ian Mont iel and Antonio Ros, Curves and surfaces, 2005 68 Luis Caffarelli and Sandro Salsa, A geometric approach to free boundary problems, 2005 67 T.Y. Lam, Introduction to quadratic forms over fields, 2004 66 Yuli Eidelman , Vitali Milman, and Antonis Tsolomitis, Functional analysis, An introduction, 2004 65 S . Ramanan, Global calculus, 2004 64 A . A. Kirillov, Lectures on the orbit method, 2004 63 Steven Dale Cutkosky, Resolution of singularities, 2004 62 T . W. Korner, A companion to analysis: A second first and first second course in analysis, 2004 61
Thomas A . Ivey and J. M. Landsberg, Cartan for beginners: Differential geometry via
moving frames and exterior differential systems, 2003
60 Alb erto Candel and Lawrence Conlon, Foliations II, 2003
TITLES IN THIS SERIES
59 Steven H. Weintraub , Representation theory of finite groups: algebra and arithmetic, 2003 58 Cedric Villani, Topics in optimal transportation, 2003 57 Robert P lato, Concise numerical mathematics, 2003 56 E. B . Vinberg, A course in algebra, 2003 55
C. Herbert C lemens, A scrapbook of complex curve theory, second edition,
2003
54 Alexander Barvinok, A course in convexity, 2002 53 Henryk Iwaniec, Spectral methods of automorphic forms, 2002 52 Ilka Agricola and Thomas Friedrich, Global analysis: Differential forms in analysis, geometry and physics , 2002 5 1 Y . A. Abramovich and C . D . Aliprant is , Problems in operator theory, 2002 50 Y. A . Abramovich and C. D . Aliprantis, A n invitation to operator theory, 2002 49
John R. Harp er, Secondary cohomology operations,
2002
48 Y. Eliashberg and N . M ishachev, Introduction to the h-principle, 2002 47 A. Yu. Kitaev, A. H. Shen, and M. N . Vyalyi , Classical and quantum computation, 2002 46
Joseph L . Taylor, Several complex variables with connections to algebraic geometry and
Lie groups , 2002
45 lnder K . Rana, A n introduction to measure and integration , second edition , 2002 44
J i m Agler and John E. McC arthy , Pick interpolation and Hilbert function spaces,
43
N . V. Krylov, Introduction to the theory of random processes,
42
Jin Hong and Seok-Jin Kang, Introduction to quantum groups and crystal bases,
41
Georgi V . S m irnov, Introduction to the theory of differential inclusions ,
2002
2002 2002
2002
40 Robert E. Greene and Steven G . Krantz, Function theory of one complex variable, third edition , 2006 39 Larry C . Grove, Classical groups and geometric algebra, 2002 38
Elton P. Hsu, Stochastic analysis on manifolds,
2002
37 Hershel M. Farkas and Irwin Kra, Theta constant s , Riemann surfaces and the modular group, 2001 36 Mart in Schechter, Principles of functional analysis, second edition, 2002 35 James F . Davis and Paul Kirk, Lecture notes in algebraic topology, 200 1
34 Sigurdur Helgason , Differential geometry, Lie groups , and symmetric spaces , 2001 33 Dmitri B urago , Yuri B urago , and Sergei Ivanov, A course in metric geometry, 2001 32 Robert G . Bart le, A modern theory of integration, 2001 31
Ralf Korn and Elke Korn, Option pricing and portfolio optimization: Modern methods of financial mathematics,
2001
30 J. C. McConnell and J . C . Robson, Noncommutative Noetherian rings, 2001 29 Javier D uoandikoetxea, Fourier analysis , 2001 28 Liviu I. Nicolaescu , Notes on Seiberg-Witten theory, 2000
2 7 Thierry Aubin, A course in differential geometry, 2001 26
Rolf B erndt , An introduction to symplect ic geometry,
25
Thomas Friedrich,
200 1
Dirac operators in Riemannian geometry,
2000
24 Helmut Koch, Number theory: Algebraic numbers and functions, 2000 23
Alberto C andel and Lawrence Conlon, Foliations I ,
2000
For a complete list of titles in this series, visit the AMS Bookstore at www.ams.org/bookstore/ .
