947

The gravitational field about aspinning particle

D.G.C. McKeon

Abstract: We consider the action forN = 1 andN = 2 spinning particles in the presence of abackground gravitational field. The action for the gravitational field induced by one-loop effectsis examined to lowest order in the metric. This is the one-dimensional analogue of calculationsperformed in two-dimensional nonlinear sigma models. The inherent infrared divergences arequite severe, and it is found that the effective action depends crucially on how they are treated,as is the case in two-dimensional nonlinear sigma models. Using one approach, an intractableinfrared divergence arises, while with another technique, the effective action vanishes. Acalculational technique introduced by Onofri is employed. A novelN = 4 spinning particle isconsidered briefly.

PACS No.: 12.60Jv

Résumé: Nous étudions l’actionN = 1 etN = 2 pour des particules avec spin en présenced’un champ gravitationnel. Nous examinons l’action du champ gravitationnel induit par deseffets à une boucle à l’ordre le plus bas dans la métrique. Ceci est l’analogue 1-D de calculsfaits dans le modèle non linéaire sigma en 2-D. Les divergences infrarouges inhérentes sontimportantes et nous trouvons que l’action efficace dépend fortement de la façon dont ellessont traitées, comme c’est d’ailleurs le cas dans les modèles non linéaires sigma en 2-D. Dansune première approche, une divergence infrarouge incontrôlable apparaît, alors qu’une autretechnique donne une action efficace identiquement nulle. Nous utilisons une méthode de calculintroduite par Onofri. Nous étudions brièvement un nouveau casN = 4.

[Traduit par la rédaction]

1. Introduction

The classical action for a spinning particle [1, 2] is well known to be the analogue of supergravityin 0+ 1 dimensions. It is also similar in form to the action for spinning string [3]. The spinning particlecan interact with an external gravitational field in much the same way as a spinning string does. Weare led to consider the effective action for the gravitational field induced by quantum effects in thepropagation of a spinning particle in analogy with the corresponding calculation for a spinning string.This is essentially a nonlinear sigma model calculation in 0+ 1 dimensions.

In the next section we consider the action of spinning particles in the presence of a backgroundgravitational field in the case of bothN = 1 andN = 2 supersymmetry. The possibility of having anaction for a spinning particle withN = 4 supersymmetry is also briefly examined.

The third section deals with the actual calculation of this effective action for the external gravitational

Received April 27, 1999. Accepted September 24, 2000. Published on the NRC Research Press Web site onNovember 15, 2000.

D.G.C. McKeon.Department of Applied Mathematics, University of Western Ontario, London, ON N6A 5B7,Canada.Telephone: (519) 679-2111, ext. 8789; FAX: (519) 661-3523; e-mail: TMLEAFS@APMATHS.UWO.CA

Can. J. Phys.78: 947–957 (2000) © 2000 NRC Canada

948 Can. J. Phys. Vol. 78, 2000

field to one-loop order and to lowest order in the curvature. This is done by combining a normalcoordinate expansion for the background metric with a calculational technique for determining the heatkernel introduced by Onofri [4]. This technique provides a way of computing transition amplitudesentirely in terms of classical quantities without resorting to the functional path integral of Feynman;one has to merely compute functional derivatives of a quantity dependent on the classical Hamiltonian.Alternate ways of handling severe infrared problems lead to contradictory results.

2. The spinning particle

As is outlined in refs. 1 and 2, the action for a spinning particle withN = 1 supersymmetry is givenby the “supergravity” action

S1 = −i2

∫dτ dθ E(DaX

µ)(DαXµ) (1)

whereEMA is the “vierbein” field,DA = EMA ∂M andE = s detEAM . (We use the notation of ref. 1.) Inthe gauge in which

EαM = 3Eα

M, EaM = 31/2Ea

M (2)

(Eα

µ = 1,Ea

µ = 0,Eα

m = −iθm, Ea

m = δam,3-scalar superfield)we make the expansion

Xµ = φµ(τ)+ iθψµ(τ)√e(τ ), 3 = e(τ )+ iθχ(τ)

√e(τ ) (3)

which reduces (1) to

S1 = −i2

∫dτ dθ

(∂Xµ

∂τ

)(DXµ)3−1

(D ≡ ∂

∂τ+ iθ

∂

∂θ

)(4a)

= 1

2

∫dτ

(φ2

e− iψ · ψ − i

eχφ · ψ

)(4b)

A supersymmetric mass term for this particle can be taken to be

Sm = m2

2

∫dτ dθ

∫dτ ′ dθ ′ E(τ, θ)

[1(τ, θ; τ ′, θ ′)

]E(τ ′, θ ′) (5)

where1 is the Green’s function for

(EEMa ∂M)1(τ, θ; τ ′, θ ′) = δ(τ − τ ′)δ(θ − θ ′) (6)

SinceEMB EAM = δAB , we see that in the gauge of (2),

Eµa = i3−1/2θa Ema = 3−1/2δma

Eµα = 3−1δ

µα Emα = 0

E = 31/2(7)

and consequently (5) and (6) reduce to

Sm = m2

2

∫dτ dθ dτ ′ dθ ′31/2(τ, θ)10(τ, θ; τ ′, θ ′) 31/2(τ ′, θ ′) (8)

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McKeon 949

with

D10(τ, θ; τ ′, θ ′) = δ(τ − τ ′)δ(θ − θ ′) (9)

The solution to (9) is

10 = −i2ε(τ − τ ′)− δ(τ − τ ′)θθ ′ (10)

and hence by (3)

Sm = m2

2

[∫dτ e(τ )+ i

8

∫dτ dτ ′ χ(τ)ε(τ − τ ′)χ(τ ′)

](11)

To eliminate the nonlocality in (11) we can define [5] the Grassmann variable

ψ5(τ ) = 1

4m

∫dτ ′ε(τ − τ ′)χ(τ ′) (12)

so that (11) can be written

Sm = m2

2

∫dτ

[e(τ )+ i

(−1

m2 ψ5ψ5 + 1

mχψ5

) ](13)

thereby reproducing the mass contribution to the action in ref. 1. (The nonlocality in (8) can be eliminatedby using a superfield analogue of (12).)

We also note thatD2 = i∂

∂τand that the Green’s function appropriate for (4a) that satisfies

∂

∂τD1(τ, θ; τ ′, θ ′) = δ(τ − τ ′)δ(θ − θ ′) (14)

is given by

1(τ, θ; τ ′, θ ′) = −i2

| τ − τ ′ | −12ε(τ − τ ′)θθ ′ (15)

ForN = 2 supersymmetry, the action of (1) becomes [1]

S2 = 14

∫dτ dθ2 dθ1Eεab(E

Ma ∂MX

µ)(ENb ∂NXµ) (16)

which, using the analogue of (2), reduces to

S2 = 14

∫dτ dθ2 dθ1 εab(DaX

µ)(DbXµ)3−1 (17)

whereDa = ∂

∂θa+ iθa

∂

∂τ. If now we expandXµ and3 so that

Xµ = φµ + iθaψµa + iθθFµ (18a)

3 = e + iθmχm + iθθf (18b)

(with AB = 12 εabAaBb), then (17) reduces to [1]

S2 = 12

∫dτ

[1

eφ2 + iψa · ψa + eF 2 + i

eφ · ψaχa − iψ · ψf −1

e(ψaχa) · (ψbχb)+ 2iψχ · F

](19)

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950 Can. J. Phys. Vol. 78, 2000

provided we rescaleψ andχ by√e andF andf by e. There does not appear to be a suitableN = 2

generalization of the action of (5). For completeness, we observe that the Green’s function1 satisfying

12 εabDaDb1(τ, θ; τ ′, θ ′) = δ(τ − τ ′)(θ − θ ′)(θ − θ ′)

is given by

1 = −(θθ)(θ ′θ ′)δ(τ − τ ′)+ i

2θmθ

′mε(τ − τ ′)− 1

2|τ − τ ′|

An N = 4 supersymmetric generalization of (1) and (16) is

S4 = 1

48

∫dτ d4θ εabcd

(EKa ∂K E

Lb ∂LX

u) (

EMc ∂MENd ∂NX

µ)

(20)

Again, specializing to the gauge of (2), this becomes

S4 = 1

48

∫dτ d4θ εabcd

(3−1/2Da3

−1/2DbXµ) (3−1/2Dc3

−1/2DdXµ)

(21)

If we take

Xµ = φµ + iψµa θa + σµabθaθb + i

6εabcdχ

µa θbθcθd + 1

24f µεabcdθaθbθcθd (22)

and then further specialize to the case in which3 = 1, we obtain

S4 = 12

∫dτ

[φ2 − iψa · ψa + 2

3(σab · σab − 2σab · σab) +iχa · χa + f 2

](23)

(We note that if3 6= 1, then derivatives of3 with respect toτ will occur, unlike what happens in theN = 1 andN = 2 cases.) Upon setting8µ = φµ and9µ = ψµ, (23) reduces to the standard formfor a spinning particle with an additional fieldσab arising. There are more exhaustive treatments of theaction for theN = 4 supersymmetric particles in terms of component fields in refs. 6–8. In ref. 6, themodel is supersymmetric only on-shell. The approach used is similar to that used in the two-dimensionalnonlinear sigma model [9].

We now consider coupling the spinning particle to an external gravitational field [1, 6]. In theN = 1case, (1) is modified to give

S′1 = −i

2

∫dτdθE Gµν(X

λ)(DaXµ)(DαX

ν) (24)

while in theN = 2 case, (16) becomes

S′2 = 1

4

∫dτd2θE εabGµν(X

λ)(EMa ∂MXµ)(ENb ∂NX

ν) (25)

If we now work in the gauge of (2), with the parameterization of (3), we find that

Gµν(Xλ) = Gµν(φ

λ + iθφλ) = Gµν(φλ)+ iθφσ

∂Gµν(φλ)

∂φσ(26)

so that upon using the connection

0αβγ = 12 G

αλ

[∂Gλγ

∂φβ+ ∂Gλβ

∂φγ− ∂Gβγ

∂φλ

](27)

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McKeon 951

S′1 becomes

S′1 = 1

2

∫dτGµν(φ)

[1

eφµφν + i(Dτψ

µ)ψν + i

eφµψνχ

](28)

whereDτψµ = ψµ + 0µαβφ

αψβ .

For S′2 in (25), the parameterization ofEMA andXµ given in (7) and (18), combined with the

expansions

Gµν(X) = Gµν(φλ + iθmψ

λ + iθθFλ)

= Gµν(φ)− iψλm∂Gµν(φ)

∂φλθm +

[iF λ

∂Gµν(φ)

∂φλ+ψλψσ ∂

2Gµν(φ)

∂φλ∂φσ

]θθ (29)

can be used to generate the action for anN = 2 spinning particle moving in an external gravitationalfield. If in this action we utilize the equation of motion for the fieldFµ,

Fµ = −i(

−0µαβψαψβ + 1

eψµχ

)(30)

we obtain the action

S′2 = −1

2

∫dτ

{Gµν

e

(−φµφν − iDτψ

µmψ

νm + i

ef ψµψν − i

e(φ)µψνmχm

+ 3

e2 ψµχ ψνχ

)− 2

3e2 Rµανβψµψν ψαψβ

}(31)

where

Dτψµm = ψµm + 0

µαβφ

αψβm (32)

and

Rµανβ = 12

(Gµν,αβ +Gαβ,αβ −Gµβ,αν −Gαν,µβ

) +Gλσ0λαβ0

σµν −Gλσ0

λµβ0

σαν (33)

We now consider the effective action for the gravitational field.

3. The effective action

Following the approach of refs. 10–18, we consider the effective action for the gravitational fieldresulting from propagation of the particle described by the action of either (28) or (31). The fieldφµ issplit into the sum of a background and quantum part using Riemann normal coordinates so that

φµ(τ) = φµB(τ)+ πµ(ξ(τ )) (34a)

and

gµν(φµ) = gµν(φ

µB)− 1

3 Rµανβ(φµB)ξ

αξβ + . . . (34b)

φµ = φµB +Dξµ + 1

3 Rµαβνξ

αξβφνB + . . . (34c)

(Dξµ ≡ ˙ξµ + 0µαβξ

αφβB)

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952 Can. J. Phys. Vol. 78, 2000

as in refs. 9–19. The fieldψµ is taken to be purely quantum and the field3 of (7) is taken to be one. Thismeans that we are generating a 0+ 1-dimensional sigma model. If, in addition, we were to quantize the“graviton” field e(τ ) and the “gravitino” fieldχ(τ), we would be dealing with the propagator for thequantized Dirac field [20, 21]. This involves a functional integration overξµ(τ) andψµ(τ) as well ase(τ ) andχ(τ) in the quantum mechanical path integral. Such a representation of the propagator is usefulin quantum field theory [22–24]. The analogous step of integrating over the metric in 1+ 1 dimensionsin the quantized string action (and the corresponding gravitino when the string is supersymmetric) isalso considered in refs. 25–29.

The one-loop effective action arises from those contributions to the classical action that are bilinearin the quantum fields. For the actions of (28) and (31), respectively, these are

S′(2)1 = 1

2

∫dτ

[(Dabξb

)2 + RµabνφµBφ

νBξ

aξb + i(Dabψb

)ψa

](35a)

and

S′(2)2 = 1

2

∫dτ

[(Dabξb

)2 + RµabνφµBφ

νBξ

aξb + i(Dabψbm

)ψam

](35b)

An “einbein” eaµ has been introduced so thatXµ = eµa X

a andDabXb = Xa + ωabµ φµBX

b whereωabµ isthe associated spin connection [11].

Performing a functional integral over the quantum fieldsξ andψ (ignoring the functional measuresas in refs. 10–18, unlike ref. 19), we obtain one-loop contributions to the effective action of the form

W(1) = ln

{det −1/2

[−

(d

dτδap + ωapµ φ

µB

) (d

dτδpb + ωpbν φ

νB

)+ Rµabνφ

µBφ

νB

]

× det ε/2[−

(d

dτδap + ωapµ φ

µB

) (d

dτδpb + ωpbν φ

νB

)]}(36)

(ε = 1,2 forN = 1,2 respectively).We will use the techniques of Onofri [4] in considering the functional determinants in (36). This

alternative to the usual Feynman techniques, those based on the Schwinger expansion [30, 31] or pathintegrals [22–24] is useful as it only involves an unambiguous evaluation of functional derivatives ofclassical quantities. In general we have to deal with quantities of the form

I = ln det[

12(p − A)2 + V

], p ≡ −i d

dτ(37)

which, upon usingζ -function regularization [32–34] becomes

= − d

ds

∣∣∣∣0

1

0(s)

∫ ∞

0dt t s−1tr e−[1/2(p−A)2+V ]t (38)

For the heat kernel

Mxy = 〈x| e−[1/2(p−A)2+V ]t |y〉 (39)

we use theN -dimensional representation [4]

= e−(x−y)2/2t

(2πt)N/2exp

{∫ 1

0dt1

∫ 1

0dt2

[t

2t<(1 − t>)

δ

δλ(t1)

δ

δλ(t2)+ 1

2t

δ

δσ (t1)

δ

δσ (t2)

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McKeon 953

+i (θ(t2 − t1)− t2)δ

δσ (t1)

δ

δλ(t2)

]}T exp

{− t

∫ 1

0dτ

[− (1+ σ(τ)) · A(y(τ)+ λ(τ))

+12 A

2(y(τ )+ λ(τ))+ V (y(τ)+ λ(τ))

]}∣∣∣∣λ=σ=0

(40)

wheret<= min(t1, t2); t>= max(t1, t2); 1 = i(x − y)

t; y(τ) = y + τ(x − y); θ(κ) = 1(κ >0),=

0(κ<0),= 12(κ = 0), andT denotes standard time-ordering whenA andV are matrices. Equation (40)

is useful as it expresses the matrix elementMxy unambiguously in terms of the functional derivativesof c-numbers. (Ambiguities arise when one uses a path integral representation of the matrix element in(39) [22].)

From (36), it is apparent that to determine contributions toW(1) that are at most quadratic in thecurvature tensor, only one or two factors ofA andV in the expansion of the second exponential in (40)need to be considered. To compute the functional derivatives, it is easiest to consider a single Fouriermode forA andV so that

(2π)NV (x) = V eikx, (2π)NA(x) = Aeikx (41)

We are thus left to consider inN = 1 dimensions

Mxy ≈ e−(x−y)2/2t

(2πt)1/2exp

{∫ 1

0dt1

∫ 1

0dt2

t

2t<(1 − t>)

δ

δλ(t1)

δ

δλ(t2)

}

×[1 +

∫ 1

0dt1

∫ 1

0dt2

(1

2t

δ

δσ (t1)

δ

δσ (t2)+ i(θ(t2 − t1)− t2)

δ

δσ (t1)

δ

δλ(t2)

)

× 1

2!(∫ 1

0dt1

∫ 1

0dt2 i(θ(t2 − t1)− t2)

δ

δσ (t1)

δ

δλ(t2)

)2]

×{(−t)

∫ 1

0dτ

1

2π

[−(1+ σ(τ))A+ V

]eik(y(τ )+λ(τ)) + 1

2(2π)2A1A2 ei(k1+k2)(y(τ )+λ(τ))

+ (−t)2(2π)2

∫ 1

0dτ1

∫ τ2

0dτ2

([−(1+ σ(τ1))A1(1+ σ(τ2))A2 + V1V2

]

× ei[k1(y(τ1)+λ(τ1))+k2(y(τ2)+λ(τ2))]) }∣∣∣∣

λ=σ=0(42)

Evaluating the functional derivatives in (42) is straightforward and leads to

= e−(x−y)2/2t

(2πt)1/2

{−t

∫ 1

0dτ

[1

2π

(−1A+ (1

2 − τ)kA+ V

)eiky(τ )e− t

2τ(1−τ)k2

+ 1

2(2π)2A1A2 ei(k1+k2)y(τ ) e− t

2τ(1−τ)(k1−k2)2]

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954 Can. J. Phys. Vol. 78, 2000

+ t2

(2π)2

∫ 1

0dτ1

∫ τ1

0dτ2

[V1V2 + (1A1)(1A2)+ A1A2

t+ A1A2

(k2

1(1 − τ1)(1

2 − τ1)

+ k22(−τ2)

(12 − τ2

) + k1k2((1

2 − τ1)(1

2 − τ2) + (1 − τ1)(−τ2)

) −1A1A2(k1

(32 − 2τ1

)

+k2(1

2 − 2τ2))]

expi [k1y(τ1)+ k2y(τ2)]

× exp− t

2

[k2

1τ1(1 − τ1)+ k22τ2(1 − τ2)+ 2k1k2τ2(1 − τ1)

](43)

If in (38) we wish to evaluate the integral overt prior to taking the trace, then the contribution of(43) toI can be determined by using a standard representation of the Bessel function

Kν(az) = 1

2

(z

a

)ν ∫ ∞

0dt t−ν−1 e−(a2t+z2/4t)

However, it is much more convenient to compute the trace first by settingx = y and the integratingoverx; the latter operation leads to overall momentum conservation. The contribution of (43) toT rM

is given by (remembering thatT rA = 0)

T rM = 1

(2πt)1/2tr

{−t

∫ 1

0dτ

[V δ(k)+ 1

2

1

(2π)A1A2δ(k1 + k2)

]+ t2

2π

∫ 1

0dτ1

∫ τ1

0dτ2δ(k1 + k2)

×[V1V2 + A1A2

t+ A1A2k

21

(12 − τ1 + τ2

)2]e− t

2k21(τ1−τ2)(1−τ1+τ2)

}(44)

In using (44) in conjunction with the integral overt in (38), we see that there is an infrared divergenceatt = ∞. This we regulate as in refs. 11–17 by inserting an infrared regulator e−m2t/2 into the integrand.The massm2 used here as a regulator is obviously distinct from the massm2 appearing in (5); it can begenerated by inserting a termm2Gµν(φ)φ

µφν into the original Lagrangian.Together (38) and (40) given, upon evaluating the integral overt ,

I = −1

(2π)1/2tr

{−

(V + 1

4πA1A2

) (m2

2π

)−1/2

+ 1

2π

∫ 1

0dτ1

∫ τ1

0dτ2

[V1V2

√π

2

(m2

2+ k2

2(τ1 − τ2)(1 − τ1 + τ2)

)−3/2

+ k2(12 − τ1 + τ2

)2√π

2A1A2

(m2

2+ k2

2(τ1 − τ2)(1 − τ1 + τ2)

)−3/2

+√πA1A2

(m2

2+ k2

2(τ1 − τ2)(1 − τ1 + τ2)

)−1/2]}

(45)

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McKeon 955

Using the identities∫ 1

0dτ1

∫ τ1

0dτ2 f (τ1 − τ2) =

∫ 1

0dx(1 − x)f (x) (46a)

∫ 1

0dx(1 − 2x)g(x(1 − x)) = 0 (46b)

and an integration by parts, we can see that all dependence onA in (45) drops out (as expected, as thereis no field strength in 0+ 1 dimensions) leaving us with

I = tr

{V

m− 1

4πV1V2

1

m(m2 + k2

4 )

}(47)

Both terms in (47) are clearly singular as the infrared regulatorm approaches zero. For the first term,this corresponds to the infrared divergence that would arise if we were to use the Feynman diagramapproach in the 0+ 1-dimensional Feynman integral∫

dp

p2 +m2 = π

m(48a)

the second term corresponds to the divergence atm = 0 in

∫dp

(p2 +m2)((p − k)2 +m2)(48b)

In ref. 18 it was argued that ifζ -function regularization is used in conjunction with integrals of the formof (48), then ifm2 = 0, this tadpole integral would disappear. For the second term in (45)ζ -functionregularization whenm2 = 0 yields

I = − d

ds

∣∣∣∣0

1

0(s)

trV1V2

2(2π)3/2

(k2

2

)−s−3/20(s + 3

2)02(−s − 1

2)

0(−2s − 1)= 0 (49)

The ambiguity between (47) and (49) is a result of differing ways of handling the infrared divergenceoccurring at one-loop order. The infrared divergences in 0+ 1 dimensions is inherent in the nature ofunregulated one-dimensional Feynman integrals. This same ambiguity afflicts the analogous computa-tion in the 1+ 1-dimensional sigma model calculations of refs. 11–17 and 26–29, although in 1+ 1dimensions, the resulting effective action depends only on the logarithm of the regulating mass and notits reciprocal. Elimination of this divergence in 1+ 1 dimensions by using operator regularization isdiscussed in refs. 28 and 29.

4. Discussion

The classical actions forN = 1 andN = 2 spinning particles in a curved background have beenconsidered. TheN = 4 spinning particle has also been introduced and briefly examined, using asuperfield formalism.

Computing the effective action for the gravitational field about the spinning particle to one-looporder to second order in the curvature tensor turns out to be affected with ambiguities arising fromthe severe infrared difficulties occurring in 0+ 1 dimensions. Inserting a massm to act as an infraredregulator results in a finite result (which diverges asm → 0) whileζ -function regularization implies thatthese contributions to the effective action vanish. In both cases we have used the functional techniquesof Onofri to expand the heat kernel in powers of the background field.

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956 Can. J. Phys. Vol. 78, 2000

Computing the anomaly in the conformal symmetry inherent in the 1+1 sigma model in the presenceof a background gauge field in the target space and with a quantized world-sheet geometry serves tofix the dimension of the target space [25, 26] and to generate the Einstein equations [35, 36]. There isno analogue of this in the 0+ 1-dimensional models of (1) and (16), as these models do not possessconformal invariance on the world-line. This is introduced only at the expense of having a compensating“gauge field” [37], although it is possible to explicitly have conformal symmetry in the target space [38,39]. Representations of the superconformal algebra in 0+ 1 dimensions for specific models are givenin refs. 40–42.

The peculiarities encountered in regulating the infrared divergences in 0+ 1 dimensions are remi-niscent of the analysis of chiral symmetry breaking in 1+ 1 dimensions [43].

If one were to compute the matrix element of (19) by evaluating a quantum mechanical path integralfor a spinning particle in curved space, rather than by using the approach of ref. 4, then the infraredproblem we have encountered might be ascribed to the quantum fluctuations of the particle in this space.However, the treatment of the quantum mechanical path integral for a particle in curved space requiresparticular care, as the effective action in the path integral acquires curvature dependent correctionsto the classical action. These corrections are related to ambiguities in ordering the operators that oneencounters when deriving the quantum mechanical path integral in curved space. For early discussionsof this, see ref. 44; further analysis appears in refs. 45 and 46. A more complete description of thequantum mechanical path integral in curved space appears in ref. 47.

Acknowledgements

The Natural Sciences and Engineering Research Council of Canada provided financial help. R. andD. MacKenzie had helpful suggestions.

References

1. L. Brink, P. Di Vecchia, and P. Howe. Nucl. Phys.B118, 76 (1974).2. L. Brink, S. Deser, P. Di Vecchia, P. Howe, and B. Zumino. Phys. Lett.64B, 435 (1975); F. Berezin and M.

Marinov. Ann. Phys. (NY),104, 336 (1977); R. Casalbuoni. Nuovo Cimento A,33, 115 (1976).3. L. Brink, P. Di Vecchia, and P. Howe. Phys. Lett.65B, 471 (1976); S. Deser and B. Zumino. Phys. Lett.

65B, 369 (1976).4. E. Onofri. Am. J. Phys.46, 379 (1978); D.G.C. McKeon. Can. J. Phys.24, 614 (1996); L. Martin and

D.G.C. McKeon. Int. J. Mod. Phys. A,14, 1337 (1999).5. A.M. Polyakov. Gauge fields and strings. Harwood Academic, Chur. 1987. Sect. 9.11.6. P. Howe, S. Penati, M. Pernici, and P. Townsend. Phys. Lett.215B, 555 (1988).7. E. Ivanov, S. Krivonos, and A. Pashnev. Class. Quantum Grav.8, 19 (1991).8. E. Donets, A. Pashnev, J. Rosales, and M. Tsulaia. Dubna report No. E2-99-218.9. D.Z. Freedman and P.K. Townsend. Nucl. Phys.B177, 282 (1981).

10. J. Honerkamp. Nucl. Phys.B36, 130 (1972).11. L. Alvarez-Gaumé, D.Z. Freedman, and S. Mukhi. Ann. Phys.134, 85 (1981).12. B. Fridling and A. Van de Ven. Nucl. Phys.B268, (1986) 719.13. R. Matsaev and A. Tseytlin. Sov. J. Nucl. Phys.47, 1136 (1988).14. D. Freidan. Phys. Rev. Lett.51, 334 (1980).15. E. Braaten, T. Curtright, and C. Zachos. Nucl. Phys.B260, 630 (1985).16. C. Hull and P.K. Townsend. Phys. Lett.191B, 115 (1987).17. M. Leblanc, R.B. Mann, D.G.C. McKeon, and T.N. Sherry. Nucl. Phys.B349, 494 (1991).18. D.G.C. McKeon. Can. J. Phys.67, 837 (1989).19. F. Bastianelli. Nucl. Phys.B376, 113 (1992).20. V. Fainberg and A. Marshakov. Nucl. Phys.B306, 659 (1988).21. E. Fradkin and D. Gitman. Phys. Rev. D: Part. Fields,44, 3230 (1991).22. M. Strassler. Nucl. Phys.B385, 145 (1992).23. M.G. Schmidt and C. Schubert. Phys. LettB318, 438 (1993).

©2000 NRC Canada

McKeon 957

24. D.G.C. McKeon. Can. J. Phys.70, 455 (1992); D.G.C. McKeon and A. Rebhan. Phys. Rev. D: Part. Fields,48, 2891 (1993).

25. A.M. Polyakov. Phys. Lett.103B, 207 (1982);103B, 211 (1982).26. C. Hull and P.K. Townsend. Nucl. Phys.B274, 349 (1986).27. U. Kraemer and A. Rebhan. Phys. Lett.196B, 477 (1987).28. R. Knienider, U. Kraemer, and A. Rebhan. Phys. Rev. D: Part. Fields,39, 3625 (1989).29. M. Leblanc, R.B. Mann, and B. Shadwick. Phys. Rev. D: Part. Fields,37, 3548 (1988).30. J. Schwinger. Phys. Rev.82, 664 (1951).31. D.G.C. McKeon and T.N. Sherry. Phys. Rev. Lett.59, 532 (1987).32. A. Salam and J. Strathdee. Nucl. Phys.B90, 203 (1975).33. J. Dowker and R. Critchley. Phys. Rev. D: Part. Fields,13, 3224 (1976).34. S. Hawking. Comm. Math. Phys.55, 133 (1977).35. E. Fradkin and A. Tseytlin. Nucl. Phys.B261, 1 (1985).36. C.G. Callen, D. Friedan, E. Martinec, and M. Perry. Nucl. Phys.B262, 593 (1985).37. F. Dilkes and D.G.C. McKeon. Int. J. of Mod. Phys.A14, 761 (1999).38. R. Marnelius. Phys. Rev. D: Part. Fields,20, 2091 (1979); W. Siegel. Int. J. Mod. Phys. A,3, 2713 (1988).39. A. Pashnev. Dubna preprint No. EZ-99-42.40. S. Fubini and E. Rabinovici. Nucl. Phys.B245, 17 (1984).41. V.P. Akulov and A.I. Pashnev. T.M.P.56, 862 (1983).42. E. D’Hocker and L. Vinet. Phys. Lett.137B, 72 (1983).43. D.G.C. McKeon and T.N. Sherry. Ann. Phys. (NY),218, 325 (1992).44. W. Pauli. Pauli lectures in physics. Vol. 6.Edited byCharles P. Engz. MIT Press. 1973); B.S. DeWitt. Rev.

Mod. Phys.29, 377 (1957).45. K.S. Chen. J. Math. Phys.13, 1723 (1972); I. Ben-Abraham and A. Lonke. J. Math. Phys.14, 1935 (1973);

J. Dowker. J. Phys. A,7, 1256 (1974); G.A. Ringwood. J. Phys. A,9, 1253 (1976); M. Omote. Nucl. Phys.B120, 325 (1977); M.S. Marinov. Phys. Rep.60, 1 (1980); J. Beckenstein and L. Parker. Phys. Rev. D: Part.Fields,23, 2850 (1981); D.C. Khandakar, S.V. Lawande, and K.V. Bhagwat. Path integral methods and theirapplications. World Scientific, Singapore. 1993.

46. F. Bastianelli and P. Van Nieuwenhuizen. Nucl. Phys.B389, 53 (1993); J. de Boer, B. Peeters, K. Skenderis,and P. Van Nieuwenhuizen. Nucl. Phys.B459, 631 (1996);B446, 211 (1995).

47. F.A. Dilkes. Ph.D. thesis, University of Western Ontario. 1999.

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