PREQUANTISAFION FROM PATH INTEGRAL VIEWPOINT. P.A. Horv~thy. CNRS, Centre de Physique Th#orique,. F-13288 Marseille, Cedex 2. and. Ist. Fis.
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