Prequantisation from path integral viewpoint

PREQUANTISAFION FROM PATH INTEGRAL VIEWPOINT P.A. Horv~thy CNRS, Centre de Physique Th#orique, F-13288 M a r s e i l l e , Cedex 2. and I s t . Fis. Mat. d e l l ' U n i v e r s i t ~ Via Carlo Alberto I0, 10123 Torino, I t a l y .

ABSTRACT The quantum mechanically admissible d e f i n i t i o n s o f the f a c t o r

exp [ i / ~

S(~)]

needed in Feynman's i n t e g r a l - are put in b i j e c t i o n with the prequantisations o f Kostant and Souriau.

The d i f f e r e n t allowed expressions o f t h i s f a c t o r - the inequiva-

l e n t prequantisations - are c l a s s i f i e d in terms o f a l g e b r a i c topology.

1.

Introduction

In [ 1 ] a f i r s t [2,3]

attempt was made to use the gmometric tec~iques o f Kostant and Souriau

("K-S t h e o r y " ) in studying path inmegrals.

The method was applied to Dirac's

monopole and the Bohm - Aharonov experiment. Here we intend to develop a more general theory.

We show t h a t a general sjnnplectic

system is quant~n mechanically admissible (Q.M.A.S.) i f f i t t r a n s i t i o n functions depending on space-time v a r i a b l e s . is not simply connected, the d i f f e r e n t prequantisations.

A classification

is prequantisable with

I f the c o n f i g u r a t i o n space

physical s i t u a t i o n s correspond to d i f f e r e n t

scheme [ 6 ] - i m p l i c i t l y

recognized already by

Kostant [ 2 ] and D o w k e r [ 5 ] - is presented. The basic o b j e c t o f our considerations below is the f a c t o r exp [

Present address

i

S(T) ]

(I)

198 where

S(~)

is the c l a s s i c a l

action along the path ~ .

Our r e s u l t s c o n t r i b u t e to the physical interpretation of prequantisation, and are hoped to provide p h y s i c i s t s with a kind of introduction to t h i s theory.

2.

Quantum Mechanically Admissible Systems (Q.M.A.S.)

Let us r e s t r i c t

ourselves to c l a s s i c a l

E = T ~ Qx ~

(Q is a c o n f i g u r a t i o n space) and presymplectic s t r u c t u r e of the form e

d0 o

=

de

+ eF

~

- where g o is the r e s t r i c t i o n

canonical 1-form of T * ( Q x ~ )

systems (E, ~ ) with e v o l u t i o n space

(2)

to the energy surface H = H~ ( q , p , t )

- describes a free system; F

of the

- a closed 2-form on

space-time X = Q x n'< - represents the external f i e l d coupled to our system by the constant e (cf. [ 3 ] ) . I f the system admits a Lagrangian f u n c t i o n , then 6 = d @ and i t is e x a c t l y t h i s "Cartan 1-form" resp. f i n a l

0 which has to be integrated along paths in phase space (whose i n i t i a l

points project to the same x = ( q , t ) resp. x' = ( q ' , t ' )

~

X) when comput-

ing a path integral in phase space:

s(s

s

(3)

= ] 0 T

Now, by Poincar~'s lemma, for any point there e x i s t s a c o n t r a c t i b l e neighbourhood U. 3 ej defined here such t h a t ~ Uj = d @ j , Thus we would be tempted

and a l - f o r m to define -

exists.

Sj(~)

by (3) even i f no global Lagrangian - and consequently no global e

I t was pointed out in [ i ] ,

may be completely d i f f e r e n t . Definition

that the d i f f e r e n t

The f o l l o w i n g notion w i l l

expressions S j ( ~ )

and S k ( ~ )

be u s e f u l :

(E, ~ ) is a quantum mechanically admissible system (Q.M.A.S.) i f f

exists a collection

{Uj,

~j}

of pairs of open c o n t r a c t i b l e subsets

Uj

there

and 1-forms

~ j defined there - c a l l e d local system in what follows - such t h a t they are compatible,

i.e.

for any exp [ ~

i

~ J

c

Uj n Uk we have

~j]

Cjk ( x , x ' ) .

exp [ ~

i

~ Ok]

(4)

where the u n i t a r y factors Cjk depend only on the projections to space-time of the i n i t i a l resp. end point o f ~ , but not on ~ i t s e l f .

199 Clearly,in such situations the Feynman propagators corresponding to e j resp. be related by unobservable phase factors.

k will

In [ I ] we have shown that this happens i f f

2~;K.

~

E[

(5)

.~

S for any 2-cycle I emma)

S in space.

Expressed in fiber bundle language we have (Weil's

Theorem [ i ] A.) (E, 8 ) is Q.M.A.- iffprequantisable with t r a n s i t i o n Then for any ~ with end points in Uj we can define "

exp [ i~

S(~)]

functions depending on X.

"

(6a)

such that there exists phase factors Cjk with " For

E~

]

Sj(~)

"

= Cjk ( x , x ' )

"

= exp [ ~

"

exp

~

Sk(T)

]

(6b)

~r c Uj fl Uk , we have "

B.)

exp

exp [ ~

Explicitely,

Sj(~)]

we have the t r a n s i t i o n (~j

(~k

-

(6c)

~y 0 j ]

function

Zjk

:

Uj h Uk

> U(1)

with

d Zjk i Zjk

(6d)

Zjk (x) Zjk ( x ' )

(6e)

yielding Cjk ( x , x ' )

Let ~ be any path in E joining y = ( x , . ) to y' = ( x ' , . ) . Denote (Y,co, J-c) a prequantisation [3] of (E, ~ ). L i f t y to Y horizontally through a ~ ~J't -1 (y) , denote ~J the end point of this horizontal l i f t 7 " I f y, y' ~ Uj , we can write locally ~ = (y, Zj) ' ~' = ( y ' , Z~ ) j The expression (6a) is then "

exp

[

Sj(g)

"

-

Z~ J

(7)

200

7

3.

Geometric Expression for the Integrand

Now we can give a completely coordinate free form to the integrand in Feynman's expression. Following a suggestion of Friedmann and Sorkin [ 8 ] let us consider any path ~ Y projecting to ~ . Write (o)

:

~ ~ (y, Zj)

Lemma exp [ i ~ S j ( ~ ) ]

.

~(I) Z~ Zjj

:

:

~'

-~ (y', Zj)

exp [ i ~

I

~ ]

(8)

The product of two coordinate-dependent quantities is thus coordinate-independent! Now all we need is to remember that the wave functions can be represented by complex functions on Y satisfying [3] qJr( Zy

(~))

:

Z -~(~)

(9)

(where Zy denotes the action of U(1) on Y ) rather than merely functions on Q; the usual wave functions are the local representants of these objects obtained as tip ( ~ )

= Zj 9 ~JHj

~ E J'C -I (Uj)

(10)

Thus, we get f i n a l l y the geometric formula for the time evolution (~t,_t~u)

(~')

:

dq ~ Q

D ~

Pxx'

exp [ ~

~

(ii)

201

where

~ : ~ (o) ,

Pxx' = { ~ C E exp [ ~

~(o) = (x,.)

~(i)

= (x',.) I.

Note that

co

i s , in f a c t , a function of ~ , independently of the choice of ~- supposing ~ ( I )

= ~ '

is held fixed. Remarks. 1.

We do not t r y to give a geometric d e f i n i t i o n

for " D ~ " .

An attempt in t h i s d i r e c -

tion was made by Simms [ 9 ] . 2.

The introduction of the bundle

( Y , ~ , J ~ ) allows for developing a generalized

v a r i a t i o n a l formalism [ 8 ] and makes i t easy to study conserved q u a n t i t i e s .

4.

A C l a s s i f i c a t i o n Scheme [ 6 ]

I f the underlying space is not simply connected, we may have more than one prequantisation and thus several inequivalent meanings of (1) be e q u i v a l e n t i f

(two local systems are said to

t h e i r union is again an admissible local system).

The general construction for a l l the prequantisations are found in Souriau [ 3 ] . (~' ~ i '

q) the universal covering of E; define

~ = q

& " J'~ 1' the f i r s t

Denote

homotopy

group of E, acts then on ~ by symplectomorphisms. Let us choose a "reference prequantisation" (Yo,CJo,I~o) of (E, ~ ) . As (~, ~ is simply connected, i t has a unique prequantisation (~ , ~ , ~ ) , which can be obtained from

(Yo,Wo,~o)

as (~,~,~)

If

% : J~6 1

(Y,~,~)

> U(1)

:

q~

( yo,~o,1"(o )

(12

is a character, then J~l admits an isomorphic lift to

of the form

G ~ (#, ~ ) g E J'~1 ~ ~Yo

=

(g(~) , ~ ( g ) y o ( ~ )

denoting the action of

Z c U(1)

)

(13

on Yo"

Now, Souriau has shown that ( Y%,CJ%,Yd )

=

(Y"~'~)

/

A

~1 ~

(14

202 is a p r e q u a n t i s a t i o n of (E, 6 ), and a l l prequantisations can be obtained in t h i s way. The i n e q u i v a l e n t prequantisations are thus in (1-1) correspondence with the characters o f the homotopy group. In [ 1 ] we rederived t h i s theorem from our p a t h - i n t e g r a l c o n s i d e r a t i o n noting t h a t we are always allowed to add a closed but not exact I-form cx to

~ o' which - due to

non simply connectedness - may change the propagator in an i n e q u i v a l e n t way.

The

corresponding character is then

%(g)

exp

L ~ [

(15)

=g

For instance, in the Bohm-Aharonov experiment [4]

311 = 7L and a l l the characters

have t h i s form. This i s , however, not the general s i t u a t i o n . is t h a t o f identical particles [ 3 ] ,

A p h y s i c a l l y i n t e r e s t i n g counter-example

~10].

Example Consider two i d e n t i c a l p a r t i c l e s moving in 3-space. is then [10]

Q = Q/~2

The appropriate c o n f i g u r a t i o n

where

which has the homotopy group 3~C1 = ~L/2 . E is then T *Q x ~ with %1 (z)

=

& =

i

d@o(1) + dG~ 2)

and

)62 (z)

Y[1

:

2 2 has two characters:

i

where z is the interchange of two c o n f i g u r a t i o n s . Thus we have two prequantum l i f t s

trivial,

of~

while the second is twisted.

fe~mions,

Now, i t

1 and two p r e q u a n t i s a t i o n s , one of which is The f i r s t

corresponds to bosons, the second to

is easy to see t h a t %2 is not o f the form (15):

Proposition I f the homotopy group is f i n i t e ,

I J~ll

< o~ , then

HI(E, ~ ) = 0 , i . e . every closed

1-form is exact. Proof.

Let c~ be a closed 1-form on E, define

~ = q~o~ ;

~ = d ~

for ~ is simply

203 connected; define 1

>-

g~f

is i n v a r i a n t under g E J~C1 hand ~ = dh = d q * h = q ~

and projects thus to a

, and thus

h

:

E ~

R.

On the other

oc = dh.

The general s i t u a t i o n can be treated by algebraic t o p o l o g i c a l means [ 1 1 ] .

Consider

the exact sequence of groups 0

~~

~

2~Z 1 ~ U(1)

(16)

~ 0

g i v i n g r i s e to the long exact sequence (17) ~HZ(E, T Z ) - ~

HI(E,R)

I H I ( E , U(Z))

closed / exact

characters

,H2(E,Z) Chern cl ass

>H2(E,IT~,) curv. class

We can make the f o l l o w i n g observations: 2 (E,~) 1) ~ d e f i n e s , by (5), an integer-valued element of HdR theorem, is j u s t H2 2

which, by de Rham's

(E, R ).

The bundle is t o p o l o g i c a l l y completely characterized by i t s Chern class which s i t s in H2(E,TZ),

Thus we have as many bundles as elements in the kernel of

As U(1) is commutative, a character of~'C 1 depends only on ~1/[J~1,3-C~ is known to be [ii]

H I ( E , ~ ).

4,~ . , which

On the other hand, the Theorem in Universal C o e f f i c i e n t s

p. 76, y i e l d s that Hom ( H I ( E , ~ ) ,

U(1))

=

HI(E, U(1))

(18)

Thus HI(E, U(1)) is just the set of all characters classifying the different prequantisations.

Under q u i t e general c o n d i t i o n s , we have Hi(E,TL)

~

7L bi

~

Tors Hi (19)

Hi(E, Z )

~

~ bi

~

Tors Hi

where Tors Hi and Tors Hi are groups whose elements are a l l of f i n i t e order;

2O4

5) The kernel of the map H i ( E , ~ ) is a basis in Hi(E, IR) ;

is just Tors Hi', the image of 7Z bi

Hi(E,~)

6) Again, by the Theorem on Universal C o e f f i c i e n t s , Tors H2(E,Tz~) ~

Tors H I ( E , ~ )

( -~ Tors J ~ l / [ J ~ l , ~ . c 1 ] )

(2o)

Thus 2), 5), 6) give us Proposition The t o p o l o g i c a l l y d i s t i n c t

prequantum bundles are labelled by the elements of (20).

7) According to 5), the image of H I ( E , ~ ) in HI(E, ~ ) under I ) is made up of integer m u l t i p l e s of a basis. Thus H l ( E , ~ ) / i m H I ( E , ~ ) ~ ( s l ) bl and we get the exact sequence 0 ~

(S 1) ~

bl

HI(E, U(1))

} Tors H I ( E , ~ )

such that i t s value on i 2~

(21)

HI(E, R) we can associate a closed 1-form

Now by de Rham's theorem, to any element of ~/2~

~ 0

g E HI(E, R) is ~

(22)

Y

where the homology class of ~ is g. Next, by (21}, the image of ($I) bl

~(g) As (S1) bl

in HI&U(1)) is composed of characters of the form

exp

~

u

(23)

is connected, and Tors H2(E,TL) is f i n i t e , we have

Proposition The characters of the form (23) make up the connected eomponen~ c o n t a i n i n g ~ ~ 1 of the group of c h a r a c t e r s ;

8) Let us choose a basis ~ 1 . . . . 3 ~ bI

1 in HdR(E,N) and pick up a ~ k ~ HI(E' U(1))

corresponding to each element of Tors H2 = Tors It 1. Proposition Any character can be w r i t t e n as i

205

b1 ~C(g)

where aj E

~

= exp [ ~

(mod 2 ~ ) .

~ j =1

aj

The ~k

I s

~_ c