Volume 127, number 8,9 PHYSICS LETTERS A 14 March 1988

ON A PATH INTEGRAL WITH A TOPOLOGICAL CONSTRAINT

D.C. KHANDEKAR l

Research Centre BiBoS, University of Bielefeld, D-4800, Bielefeld 1, FRG

and Theoretical Physics Division, Bhabha Atomic Research Centre, Bombay 400 085, India

K.V. BHAGWAT

Nuclear Physics Division, Bhabha Atomic Research Centre, Bombay 400 085, India

and

F.W. WIEGEL

Center for Theoretical Physics, Twente University, 7500 AE Enschede, The Netherlands

Received 22 October 1987; revised manuscript received 28 December 1987; accepted for publication 8 January 1988

Communicated by J.P. Vigier

We discuss a new method to evaluate a path integral with a topological constraint involving a point singularity in a plane. The

path integration is performed explicitly in the universal covering space. Our method is an alternative to an earlier method of

Inomata.

In various problems in theoretical physics one has to evaluate a path integral with a constraint which is

of a topological nature. This arises, for example, in the theory of polymers or in the analysis of the

Aharonov-Bohm effect (see ref. [ 1 ] and the literature quoted therein).

In order to elucidate some of these problems Inomata and Singh [2] considered path integrals over trajec-

tories in a plane (with polar coordinates r and 0, 0 < r< oo, 0 < 0 < 2~ ) where the constraint could be expressed

as

T

f O

dt-0=0, (1)

o

where 0 is a real constant. The trajectories are followed during the time interval (0, T). They introduce the

constraint through a Dirac 6-function with the left-hand side of (1) as the argument. Next the 6-function is

written as a Fourier integral which enables them to write the propagator

K~(r", TIr', o)-g~(r", 0", TIr', 0', O)

for a quantum mechanical free particle of mass m obeying the constraint (1) as

K~(r",Tlr',o)=lfd2e~Ka(r",Tir',O),

(2)

--ot~

Alexander von Humboldt Fellow.

0375-9601/88/$ 03.50 © Elsevier Science Publishers B.V.

(North-Holland Physics Publishing Division)

379

Volume 127, number 8,9 PHYSICS LETTERS A 14 March 1988

where

K~(r", Tit',

0)= fexp

[(i/h)S;.(r(t))]

d[r(t)] (3)

denotes the propagator associated with the action Sa given by

T T

0 0

Note that (3) represents an unconstrained path integral, the effect of the constraint being taken care of through

the introduction of an extra term in the action (4) namely

T

, hfo dt.

0

In refs. [2] and [3] the essential step in the evaluation of K~ and hence K is the observation that for an

infinitesimal time interval t the change A0 in the angular coordinates can be expressed as a cosine term

-at AO~cos(AO+ae)-cos

(A0)+½a2e2+O(e 3) . (5)

This enables one to take all contributions up to order E in the infinitesimal propagator into account. Once the

terms involving A0 in the infinitesimal propagator are modified using (5) the integration over intermediate

angular coordinates from 0 to 2n in the polygonal scheme finally enables the authors of refs. [ 2 ] and [ 3 ] to

obtain the exact propagator associated with the problem.

The present note is devoted to an alternative way to calculate such path integrals which might be easier to

generalize to more complicated geometries and constraints. We first note that the introduction of a singularity

at the origin makes the space multiple connected. The various trajectories connecting

r'

and

r"

can be classified

into several distinct homotopy classes each characterized by the number of turns (winding number) around

the singular point. The constrained propagator (2), therefore, can be expressed as a sum

K= ~ K~')(r ", TIr',O),

(6)

n= --oo

where

K~o">(r ", TIr', O)=K~")(r ", 0", TIr', 0', O)

represents the path integral over the set of trajectories characterized by the winding number n. Secondly, one

performs the path integration by going over to the universal covering space which essentially means that one

lets the angular coordinate 0 take all values between -oo and + oo. Further, since we want to evaluate K", we

must demand

T

IO dt=O"-0'

(7) +2rcn.

0

With these observations we proceed to evaluate the propagator for a free particle obeying the constraint (1).

It is well known [4,5] that while performing the path integration in two-dimensional polar coordinates one

must add an extra

7

f

h 1

8m r2(t ~ ) dt

0

380

Volume 127, number 8,9 PHYSICS LETTERS A 14 March 1988

to the action functional. This term arises essentially from the scalar curvature of the space. Thus the effective

action functional for a free particle in two-dimensional polar coordinates can be written as

2

~[-1 [dry2 h2 1+ (_~t) 1

S:JoL mt, ) +Fm,

½mr2 dt, (8)

and consequently the propagator

K~ ")

can be expressed as

K~') =~((~-O" +O'-2xn) f exp (~ S) d[r(t)] ,

(9)

where the integral is over trajectories of winding number n.

Next, the path integration over the angular coordinates

O(t)

is identical to an integration over the paths of

a free particle with a time-dependent moment of inertia

mrZ(t).

An expression for the propagator for similar

systems has been evaluated by Khandekar and Lawande [ 6 ]. Following a similar line of steps the path inte-

gration over angular coordinates can be done and the result can be expressed as

K(~") =~(O"-O' + 2nn-¢)

xJ'exp{ h ~Fl {dr'~2+h2 l__]dt~( M ~,,2 [iM ,, +2nn)Z)d[r(t)] (10)

JoL mtm)

8mr2_J

Jt,~]

exPt2-h(0-0'

Here the quantity

T 1

M=m(!-~dt)-

(11)

is still a functional of the "radial" path, and the radial path integration has still to be performed.

To perform the radial path integration we make use of the identity

(M) 1/2 2

1

]-~j exP(2~h + )=~ ; [ ih 2

_ exPt-~-~¢ +i~)d~, (12)

and rewrite (10) as

K~ n) =6(0"-0'

+2nn-~) i exp[i~(0"-0' +2nn)]Kc d~. (13)

--oo

Here K¢ is the path integral

K¢=~f exp(~ X~) dD,(t)]

(14)

associated with the "radial action"

T

i rfl'dr'~2dt_h~_(~2_

) 1

zm ~

f-7£dt.

(15)

0 0

But this is exactly equal to the action of a fictitious one dimensional particle of mass m which moves on the

half line 0< r< oo under the influence of the potential

h 2 1

~m (¢~-I) 7"

381

Volume 127, number 8,9 PHYSICS LETTERS A 14 March 1988

For this system the propagator has been evaluated by Khandekar and Lawande [ 7 ]. Using their result we obtain

K~ "~ =~(0"-0' + 2nn-0)

×~m--~,~exp(i(r'2+r"2) m) i

L~ln~ \ ~ exp[i~(O"-O' + 2nn)]Ii¢l(mr'r"/ihT)

d~, (16)

where I is the modified Bessel function. The propagator expression obtained in (16) coincides with the cor-

responding expressions of refs. [ 2 ] and [ 3 ], which shows that this method does indeed yield the correct results.

D.C. Khandekar would like to thank Professor L. Streit for his kind hospitality at the Research Center BiBoS

of the University of Bielefeld and the Alexander von Humboldt Foundation for financial support through a

research fellowship. F.W. Wiegel would like to recall his pleasant stay at the Tara Institute of Fundamental

Research and the Bhabha Atomic Research Centre during a visit to Bombay in December 1986 where this work

was initiated.

References

[ 1 ] F.W. Wiegel, Introduction to path integrals in physics and polymer science (World Scientific, Singapore, 1986).

[2] A. Inomata and V.A. Singh, J. Math. Phys. 19 (1978) 2318.

[3] C.C. Bernido and A. lnomata, J. Math. Phys. 22 (1981) 715.

[4] D.W. McLaughlin and L.S. Schulman, J. Math. Phys. 12 (1971) 2520.

[ 5 ] S.F. Edwards and Y.V. Gulyaev, Proc. R. S0c. A 279 ( 1969 ) 229.

[ 6 ] D.C. Khandekar and S.V. Lawande, Phys. Rep. 137 (1986) 115.

[ 7 ] D.C. Khandekar and S.V. Lawande, J. Math. Phys. 16 ( 1975 ) 384.

382