scispace - formally typeset

Journal ArticleDOI

On a path integral with a topological constraint

14 Mar 1988-Physics Letters A (Elsevier)-Vol. 127, pp 379-382

Abstract: 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.

Summary (1 min read)

Jump to:  and [JoL mtm) 8mr2_J Jt,~]]

JoL mtm) 8mr2_J Jt,~]

  • Here the quantity EQUATION is still a functional of the "radial" path, and the radial path integration has still to be performed.
  • To perform the radial path integration the authors make use of the identity.
  • The propagator expression obtained in (16) coincides with the corresponding expressions of refs. [ 2 ] and [ 3 ] , which shows that this method does indeed yield the correct results.

Did you find this useful? Give us your feedback

Content maybe subject to copyright    Report

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 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
Citations
More filters


Journal ArticleDOI
P. Girard1, Richard MacKenzie1Institutions (1)
Abstract: We apply the formalism of path integrals in multiply connected spaces to the problem of two anyons.

4 citations


Journal ArticleDOI
TL;DR: Another method is presented where the quantum correction term, ∝ (ħ/8 Mr 2 ) d t , arises naturally in carrying out the constrained path integration.
Abstract: Remarks are given on several methods for evaluating path integrals in the presence of a singular point in two dimensions. We present another method where the quantum correction term, ∝ (ħ/8 Mr 2 ) d t , arises naturally in carrying out the constrained path integration.

4 citations


Journal ArticleDOI
P. Girard1, Richard MacKenzie1Institutions (1)
Abstract: We apply the formalism of path integrals in multiply connected spaces to the problem of two anyons.

3 citations


Journal ArticleDOI
Abstract: The statistical mechanical properties of plane polymer loops enclosing a constant area are investigated, using a continuous model from the start. For this purpose an analytic expression for the generating functional is obtained, which in turn is used to derive (1) the distribution function for the enclosed area, (2) the average squared distance of a given repeating unit from the origin, and (3) the entropic force on a repeating unit.

3 citations


References
More filters

Journal ArticleDOI
Abstract: In this paper we present a simplification of the path integral solution of the Schrodinger equation in terms of coordinates which need not be Cartesian. After presenting the existing formula, we discuss the relationship between the distance and time differentials. Making this relationship precise through the technique of stationary phase, we are able to simplify the path integral. The resulting expression can be used to obtain a Hamiltonian path integral. Finally, we comment on a similar phenomenon involving differentials in the Ito integral.

132 citations


Journal ArticleDOI
D.C. Khandekar1, Suresh V. Lawande1Institutions (1)
Abstract: The status of exactly solvable problems within the path integral formulation of non-relativistic quantum mechanics is reviewed. Some applications of these exact results are presented.

130 citations


Journal ArticleDOI
Abstract: By using Feynman’s definition of a path integral, exact propagators for a time−dependent harmonic oscillator with and without an inverse quadratic potential have been evaluated. It is shown that these propagators depend only on the solutions of the classical unperturbed oscillator. The relations between these propagators, the invariants, and the Schrodinger equation are also discussed.

110 citations


Journal ArticleDOI
Abstract: Path integrals with a periodic constraint ∫θ ds =Θ+2πn (n=integer) are studied. In particular, the path integral for a string entangled around a singular point in two dimensions is evaluated in polar coordinates. Applications are made for the entangled polymers with and without interactions, the Aharonov–Bohm effect, and the angular momentum projection of a spinning top.

50 citations


Journal ArticleDOI
Abstract: The Aharonov–Bohm effect is formulated in terms of a constrained path integral. The path integral is explicitly evaluated in the covering space of the physical background to express the propagator as a sum of partial propagators corresponding to homotopically different paths. The interference terms are also calculated for an infinitely thin solenoid, which are found to contain the usual flux dependent shift as the dominant observable effect and an additional topological shift unnoticeable in the two slit interference experiment.

42 citations


Performance
Metrics
No. of citations received by the Paper in previous years
YearCitations
19952
19931
19911
19892
19881