PHYSICAL REVIEW A JULY 1998VOLUME 58, NUMBER 1
Path integrals on a flux cone

E. S. Moreira, Jr.*
Instituto de Fı´sica Teo´rica, Universidade Estadual Paulista, Rua Pamplona 145, 01405-900 Sa˜o Paulo, Sa˜o Paulo, Brazil

~Received 1 December 1997!

This paper considers the Schro¨dinger propagator on a cone with the conical singularity carrying magnetic
flux ~‘‘flux cone’’ !. Starting from the operator formalism, and then combining techniques of path integration in
polar coordinates and in spaces with constraints, the propagator and its path integral representation are derived.
The approach shows that effective Lagrangian contains a quantum correction term and that configuration space
presents features of nontrivial connectivity.@S1050-2947~98!02707-3#

PACS number~s!: 03.65.Ca
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I. INTRODUCTION

Quantum mechanics on cones has been shown to
fruitful model for studying the interplay between quantu
mechanics and geometry. The nearly trivial geometry of
cone~curvature is concentrated at a single point, the con
singularity @1,2#, and the result is that the geometry is E
clidean everywhere except on a ray that starts at the sin
larity @3#! is responsible for Aharonov-Bohm~AB!-like ef-
fects that have been discovered throughout the years@3–7#.
Such findings can be used in the study of various~real! quan-
tum systems whose backgrounds can be regarded as b
conical with good approximation. Quantum matter arou
cosmic strings and black holes and statistical mechanic
identical particles in two dimensions are examples.

In this paper a path-integral representation for the pro
gator of the Schro¨dinger equation is derived from the oper
tor formalism on the cone. A magnetic flux is allowed to r
through the cone axis, so that one has an AB setup cou
with the conical geometry. The method contrasts with
one in the literature where path-integral representation
spaces with a singular point are obtained by angular dec
position of the Feynman prescription in Cartesian coor
nates, and by assuming a nonsimple connectivity of the c
figuration space@8–12#. In the present approach, instea
topological features arise naturally.

The paper is organized as follows. In Sec. II the ba
ground is briefly discussed~for more detailed accounts, se
Ref. @3#, and references therein!. In Sec. III, path-integral
prescription~and propagator! is derived by breaking the evo
lution operator up into an infinite product of short-time ev
lution operators, and then inserting completeness relat
for configuration-space eigenstates, whose orthonorma
relation is expressed in terms of stationary states.~Such a
procedure is straightforward in Euclidean space, but ra
elaborate in nontrivial backgrounds@12#.! Topological fea-
tures are identified in the resulting expression. The pa
closes with final remarks.

II. BACKGROUND

A cone is obtained from the Euclidean plane by remov
a wedge of angle 2p(12a) ~in fact, whena.1, a wedge is

*Electronic address: moreira@axp.ift.unesp.br
PRA 581050-2947/98/58~1!/91~5!/$15.00
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inserted!. Clearly, the line element is given by

dl25dr21r2dw2, ~1!

which is the line element of the Euclidean plane written
polar coordinates. The fact that there is ad function curva-
ture at the origin is encoded in the unusual identification

~r,w!;~r,w12pa!. ~2!

The behavior of a free particle with massM on a cone is
determined from the Lagrangian,

L5 1
2 M ~dl/dt!2

5 1
2 M ~ ṙ21r2ẇ2!. ~3!

Noting Eq. ~2!, it follows that orbits of particles~geodesic
motion on the cone! are simply broken straight lines with
uniform motion. As a constant magnetic fluxF running
through the cone axis does not affect classical motion o
particle~with chargee), then classical motion on a flux con
is nearly trivial. Quantum motion, on the other hand, reve
nontrivial features@13#.

III. THE PROPAGATOR AND ITS PATH-INTEGRAL
REPRESENTATION

Due to local flatness of the conical geometry the fr
Hamiltonian operator is just the free Hamiltonian operator
the plane,

H52
\2

2M

1

r

]

]rS r
]

]r D1
L2

2Mr2
, ~4!

whereLª2 i\]/]w. By choosing an appropriate gauge~the
one corresponding to a vector potential, which vanishes
erywhere, except on a ray! and observing Eq.~2!, it follows
that solutions of the Schro¨dinger equation satisfy@3#

c~r,w12pa!5exp$ i2ps%c~r,w!, ~5!

with sª2eF/ch. Boundary condition~5! carries all infor-
mation about the nontrivial geometry and magnetic field.

Consider the following effective Lagrangian:
91 © 1998 The American Physical Society



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92 PRA 58E. S. MOREIRA, JR.
Le f f5
M

2
~ ṙ21r2ẇ2!1

\2

8Mr2
, ~6!

which is obtained from Eq.~3! by adding a quantum correc
tion. The corresponding Hamiltonian is given by

He f f5
1

2M S pr
21

pw
2

r2
2

\2

4r2D . ~7!

The momentum operators associated withpr and withpw are
given by

pr→2 i\S ]r1
1

2r D pw→L, ~8!

where the presence of the term2 i\/2r ensures self-
adjointness ofpr ~if the wave functions do not diverge ver
rapidly at r50 @3#!, without spoiling the usual canonica
commutation relations@12,14#. It turns out that by perform-
ing the substitutions~8! in ~7!, the Hamiltonian operator~4!
is reproduced, which obviously would not be the case if
quantum correction was not present in Eq.~7! @15#. The ef-
fective Lagrangian~6! will be considered again below.

One seeks stationary states that span a space of w
functions where conservation of probability holds. This im
plies that the singularity at the origin must not be a source
a sink,

lim
r→0

E
0

2pa

dwrJr50, ~9!

whereJr is the usual expression for the radial componen
the probability current on the plane. Condition~9! is auto-
matically guaranteed if the stationary states are finite at
origin. ~Mildly divergent boundary conditions can be equa
compatible with conservation of probability and square in
grability of the wave function@16,17,3#. These possibilities
will not be considered here.! Functions

ck,m~r,w!5^r,wuk,m&5
1

A2pa
Jum1su/a~kr!ei ~m1s!w/a,

~10!

where 0<k,`, m is an integer andJn denotes a Besse
function of the first kind, are simultaneous eigenfunctions
H andL with eigenvalues\2k2/2M and (m1s)\/a, respec-
tively. Note that sinceJn(0) is finite for nonnegativen, these
stationary states are finite at the origin.

States ur,w& in Eq. ~10! are a complete set o
configuration-space eigenstates

E
0

`

dr rE
0

2pa

dwur,w&^r,wu51. ~11!

Since their orthonormality relation is

^r,wur8,w8&5
1

r
d~r2r8!d~w2w8!,

it follows that
e

ve
-
r

f

e

-

f

^r,wur8,w8&5 (
m52`

` E
0

`

dk kck,m~r,w!ck,m* ~r8,w8!.

~12!

This expression is the completeness relation of the eig
functionsck,m ,

(
m52`

` E
0

`

dk kuk,m&^k,mu51,

which may be derived by using the completeness relation
the Bessel functions

E
0

`

dk kJn~kr!Jn~kr8!5
1

r
d~r2r8!, ~13!

with Poisson’s formula

(
m52`

`

d~f12pm!5
1

2p (
m52`

`

exp$ imf%. ~14!

Expression~12! corresponds to the usual one in Cartes
coordinates wherêxux8& is expressed in terms of plan
waves ^xux8&5*(dk/2p)exp$ik(x2x8)%. Recall that plane
waves are simultaneous eigenfunctions of the free Ham
tonian and linear momentum operators, whereasck,m(r,w)
are simultaneous eigenfunctions of the free Hamiltonian
angular momentum operators.

The orthonormality relation for the eigenfunctionsck,m ,

^k,muk8,m8&5E
0

`

dr rE
0

2pa

dwck,m* ~r,w!ck8,m8~r,w!

5
1

k
d~k2k8!dmm8, ~15!

follows from the orthonormality relation

E
0

2pa

dw exp$ iw~m2n!/a%52padmn, ~16!

and Eq.~13!.
For a complete set of configuration-space eigensta

ur,w& the propagator of the Schro¨dinger equation is given by
K(r,w;r8,w8;t)5^r,wuU(t)ur8,w8&, where U(t)5exp
$2iHt/\% is the evolution operator andt5t2t8 is the time
interval. Slicing t into N11 slices of width e5tn5tn
2tn215t/(N11), the propagator reads

K~r,w;r8,w8;t!5^r,wu )
n51

N11

U~tn!ur8,w8&, ~17!

where the composition law of the evolution operator w
used with the identificationst[tN11 andt8[t0. By inserting
into Eq. ~17! N completeness relations~11! between each
pair of evolution operators, one is led to



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PRA 58 93PATH INTEGRALS ON A FLUX CONE
K~r,w;r8,w8;t!5 )
n51

N F E
0

`

drn rnE
0

2pa

dwnG
3 )

n51

N11

@^rn ,wnuU~e!urn21 ,wn21&#.

~18!

The identifications ur,w&[urN11 ,wN11& and ur8,w8&
[ur0 ,w0& were also used.

The short-time amplitudes in Eq.~18! may be rewritten as

^rn ,wnuU~e!urn21 ,wn21&

5^rn ,wnurn21 ,wn21&2 i
e

\
H^rn ,wnurn21 ,wn21&

1O~e2!. ~19!

In order to obtain the action ofH on ^rn ,wnurn21 ,wn21&
one expresses the latter in terms of eigenfunctions of
former, i.e., Eq.~12! is considered. Then Eq.~19! is recast as

^rn ,wnuU~e!urn21 ,wn21&

5 (
m52`

` E
0

`

dk ke2 i eEk /\ck,m~rn ,wn!ck,m* ~rn21 ,wn21!

1O~e2!, ~20!

where Ek denotes the eigenvalue ofH, i.e., \2k2/2M . By
using Eq.~20! in Eq. ~18! and taking the limitN→` (e
→0), a partitioned expression for the propagator is obtain

K~r,w;r8,w8;t!

5 lim
N→`

)
n51

N F E
0

`

drn rnE
0

2pa

dwnG
3 )

n51

N11 F (
m52`

` E
0

`

dk ke2 i eEk /\ck,m~rn ,wn!

3ck,m* ~rn21 ,wn21!G . ~21!

The integral overk in Eq. ~21! may be evaluated by using th
formula @18#

E
0

`

dx xe2ax2
Jn~bx!Jn~cx!5~ 1

2 a!e2~b21c2!/4aI n~bc/2a!,

~22!

where Rea.0, Ren.21. This integral corresponds, i
Cartesian coordinates, to the Gaussian integral. Analytic c
tinuation of Eq.~22! gives
e

d,

n-

K~r,w;r8,w8;t!

5 lim
N→`

M

2pa i e\ )
n51

N F E
0

`

drn rn

3E
0

2pa dwn

2pa i e\/M G )
n51

N11 FeiM ~rn
2
1rn21

2
!/2\e

3 (
m52`

`

I um1su/a~Mrnrn21 / i\e!ei ~m1s!~wn2wn21!/aG .

~23!

Whens is an integer and the space is Euclidean, i.e.,a51,
Eq. ~23! reduces to Feynman’s prescription for the propa
tor of a free particle. Indeed, by considering the Fourier
pansion of a plane wave,

exp$ ia cosf%5 (
m52`

`

I umu~ ia !eimf, ~24!

one sees from Eq.~23! the familiar partitioned expression

K0~x,x8;t!

5 lim
N→`

M

2p i e\ )
n51

N F E d2xn

2p i e\/M G
3expH i

\ (
n51

N11

e
M

2 S xn2xn21

e D 2J , ~25!

which is symbolically written as

K0~x,x8;t!5E D 2x expH i

\Et8

t

dt
M

2
ẋ2J . ~26!

Before rewriting the path-integral representation~23! in a
symbolic form that is analogous to Eq.~26!, the expression
for the propagator on the flux cone that was obtained in R
@4#, using a complex contour method, will be reproduc
here from Eq.~21!. ~References@6,11# have also reproduced
this propagator, whens50, using other methods. Referenc
@11# in particular has used a path-integral approach that
generalization for the cone of the method used in the AB
up @9#. The propagator, whena51, has been well known in
the literature@19#.! Observing Eq.~15!, it is seen that only
one sum overm and one integration overk remain in Eq.
~21!,

K~r,w;r8,w8;t!

5
1

2paE0

`

dk ke2 i tEk /\

3 (
m52`

`

Jum1su/a~kr!Jum1su/a~kr8!ei ~m1s!~w2w8!/a.

~27!

Then, using Eq.~22! to evaluate the integration overk, re-
sults in



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sti-

an

94 PRA 58E. S. MOREIRA, JR.
K~r,w;r8,w8;t!

5
M

2pa i t\
eiM ~r21r82!/2\t

3 (
m52`

`

I um1su/a~Mrr8/ i\t!ei ~m1s!~w2w8!/a, ~28!

which could have been guessed from Eq.~23!. From Eq.~24!
it follows that whens is an integer anda51, Eq. ~28!
collapses into the free Schro¨dinger propagator on the Euclid
ean plane, viz.

K0~x,x8;t!5
M

2p i t\
eiM ~x2x8!2/2\t.

Noting that*2`
` dlI l(z)d(l2n)exp$ilf%5In(z)exp$inf%,

and using Eq.~14!, Eq. ~28! becomes

K~r,w;r8,w8;t!5 (
l 52`

`

e2 i2p lsK̃~r,w12pa l ;r8,w8;t!,

~29!

with

K̃~r,w;r8,w8;t!

ª

M

2p i t\
eiM ~r21r82!/2\t

3E
2`

`

dl I ulu~Mrr8/ i\t!eil~w2w8!. ~30!

Likewise, Eq.~23! may be rewritten as

K~r,w;r8,w8;t!5 lim
N→`

M

2p i e\

3 )
n51

N F E
0

`

drnrnE
0

2pa dwn

2p i e\/M G
3 )

n51

N11 F (
l 52`

`

e2 i2p lseiM ~rn
2
1rn21

2
!/2\e

3E
2`

`

dl I ulu~Mrnrn21 / i\e!

3eil~wn2wn2112pa l !G . ~31!

Now, by using

(
k,l 52`

`

e~k1 l !zE
0

c

dx f~kc1x!g~ lc2x!

5 (
l 52`

`

elzE
2`

`

dx f~x!g~ lc2x!,
one may extend the range of integration ofw from (0,2pa)
to (2`,`). This leaves only one sum in Eq.~31!, leading to
Eq. ~29!, but nowK̃(r,w12pa l ;r8,w8;t) is given as a par-
titioned expression

K̃~r,w12pa l ;r8,w8;t!

5 lim
N→`

M

2p i e\ )
n51

N F E
0

`

drn rnE
2`

` dwn

2p i e\/M G
3 )

n51

N11 FeiM ~rn
2
1rn21

2
!/2\e

3E
2`

`

dl I ulu~Mrnrn21 / i\e!eil~wn2wn2112pa ldn,N11!G .
~32!

Now the asymptotic behavior ofI n(z) for large uzu can be
used to derive@9#

E
2`

`

dl I ulu~z!exp$ ilf%'exp$z11/8z2zf2/2%,

which when used in Eq.~32! finally gives

K̃~r,w12pa l ;r8,w8;t!

5 lim
N→`

M

2p i e\ )
n51

N F E
0

`

drn rnE
2`

` dwn

2p i e\/M G
3expH i

\ (
n51

N11

eS M

2 F S rn2rn21

e D 2

1rnrn21S wn12pa ldn,N112wn21

e D 2G
1

\2

8Mrnrn21
D J , ~33!

or symbolically,

K̃~r,w12pa l ;r8,w8;t!

5E
0

`

Dr rE
2`

`

Dw expH i

\Et8

t

dtFM

2
~ ṙ21r2ẇ2!

1
\2

8Mr2G J . ~34!

IV. FINAL REMARKS

Expressions~29! and ~34! are the path-integral prescrip
tions where the corresponding action is the one made u
the effective LagrangianLe f f , ~6!. Recall thatLe f f is the
appropriate Lagrangian for quantization through the ‘‘sub
tution principle’’ ~8!. It is important to note that a naı¨ve
change from Cartesian to polar coordinates in the Feynm
prescription~26! does not lead to Eq.~34!, since the ‘‘quan-



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he

the
gh

PRA 58 95PATH INTEGRALS ON A FLUX CONE
tum correction’’\2/8Mr2 would be missing.~Quantum cor-
rections such as this one are typical of path integrals in n
trivial backgrounds@14#!. This is a simple example showin
that coordinate transformations within path-integral rep
sentations raise subtle issues.

Examining expressions~29! and~34! leads to the follow-
ing interpretation of this path-integral representation. Sin
there is a conical singularity and/or a magnetic flux at
origin, the configuration space is not simply connected. T
propagator is given by a sum of modulated propagators, e
one of them giving the contribution of all paths belonging
a homotopy class labeled by the winding numberl . Then the
sum overl in Eq. ~29! takes into account all paths circlin
around the ‘‘hole’’ at the origin. The modulated factors are
unitary representation of the fundamental groupZ, and the
particle travels in the covering space ofR22$0%. The par-
ticle is not free, but interacts with the ‘‘nontrivial’’ topology
through the quantum correction in the effective Lagrang
Le f f .

Recalling a study of quantum flow in Ref.@3#, one sees
that this interpretation may be appropriate whena,1 and/or
s is a noninteger. But, strictly speaking, it is incorrect wh
a>1 ands is an integer. In particular, whena51 ands
50, Eqs.~29! and~34! are just polar coordinate path-integr
prescriptions for a free particle moving on the Euclide
an

a-
n-

-

e
e
e
ch

n

plane—the apparent nontrivial topology is imparted by t
use of polar coordinates that are singular at the origin.

In principle, the material in this paper may be recons
ered in the context of other possible boundary conditions
the singularity. The result of such an investigation mig
reveal different features from the ones seen here. Procee
as in Sec. III, the crucial point would be the use of ne
stationary states to obtain the new propagators and their
responding path-integral representations. This proced
seems to answer a question in Ref.@16#, namely, how differ-
ent boundary conditions at the singularity are related to
path-integral approach. The use of the present method in
context of other geometries is also worth investigating.

Using the proper time representation for the Green fu
tions, the extension of the method to second quantizatio
straightforward. It would be interesting to investigate t
connections between this paper and Ref.@20#, where path
integrals in a black-hole background are considered.

ACKNOWLEDGMENTS

The author is grateful to George Matsas for reviewing
manuscript. This work was supported by FAPESP throu
Grant No. 96/12259-1.
s,

,

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