-brane propagator

PHYSICAL REVIEW D, VOLUME 64, 026003

Quenched, minisuperspace, bosonicp-brane propagator

Stefano Ansoldi*Dipartimento di Fisica Teorica, Universita` di Trieste, INFN, Sezione di Trieste, Trieste, Italy

Antonio Aurilia†

Department of Physics, California State Polytechnic University, Pomona, California 91768

Carlos Castro‡

Center for Theoretical Studies of Physical Systems, Clark Atlanta University, Atlanta, Georgia 30314

Euro Spallucci§

Dipartimento di Fisica Teorica, Universita` di Trieste, INFN, Sezione di Trieste, Trieste, Italy~Received 17 January 2001; published 25 June 2001!

We borrow the minisuperspace approximation from quantum cosmology and the quenching approximationfrom QCD in order to derive a new form of the bosonicp-brane propagator. In this new approximation weobtain an exact description of both the collective mode deformation of the brane and the center of massdynamics in the target space time. The collective mode dynamics is a generalization of string dynamics interms of area variables. The final result is that the evolution of ap-brane in the quenched-minisuperspaceapproximation is formally equivalent to the effective motion of a particle in a spacetime where points as wellas hypersurfaces are considered on the same footing as fundamental geometrical objects. This geometricequivalence leads us to define a new tension-shell condition that is a direct extension of the Klein-Gordoncondition for material particles to the case of a physicalp-brane.

DOI: 10.1103/PhysRevD.64.026003 PACS number~s!: 11.25.Mj

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I. INTRODUCTION

The purpose of this paper is to introduce a new appromation scheme to the quantum dynamics of extendedjects. Our approach differs from the more conventional onsuch as the normal mode expansion or higher dimensigravity, in that it is inspired by two different quantizatioschemes: one is the minisuperspace approach to quacosmology~QC!, the other is the quenching approximatioto QCD. Even though these schemes apply to different fitheories, they have a common rationale, that is, the idequantizing only a finite number of degrees of freedom whfreezing, or ‘‘quenching,’’ all the others. In QC this ideamounts, in practice, to quantizing a single scale fac~thereby selecting a class of cosmological models, forstance, the Friedmann-Robertson-Walker spacetime! whileneglecting the quantum fluctuations of the full metric. Teffect is to turn the exact, but intractable, Wheeler-DeWfunctional equation@1# into an ordinary quantum mechanicwave equation@2#. As a matter of fact, the various forms othe ‘‘wave function of the universe’’ that attempt to descrithe quantum birth of the cosmos are obtained throughkind of approximation@3#, or modern refinements of it@4#.On the other hand, in QCD, the dynamics of quarks agluons cannot be solved perturbatively outside the small cpling constant domain. The strong coupling regime is usu

*Email address: [email protected]†Email address: [email protected]‡Email address: [email protected]§Email address: [email protected]

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dealt with by studying the theory on a lattice. However, evin that case, the computation of the fermionic determinantMonte Carlo simulation is actually impossible. Thus, in t‘‘quenched approximation,’’ the quark determinant is sequal to unity, which amounts to neglecting the effectvirtual quark loops. In other words, this extreme approximtion in terms of heavy quarks with a vanishing numberflavors assumes that gauge fields affect quarks while quhave no dynamical effect on gauge fields@5#.

Let us compare the above two situations with the baproblem that one faces when dealing with the dynamics orelativistic extended object, orp-brane, for short. Ideally, onewould like to account for all local deformations of the objeconfiguration, i.e., those deformations that may take placa generic point on thep-brane. However, any attempt to provide a local description of this shape-shifting process leada functional differential equation similar to the WheeleDeWitt equation in QC. The conventional way of handlinthat functional equation, or the equivalent infinite set of odinary differential equations@6#, is through perturbationtheory. There, the idea is to quantize the small oscillatioabout a classical configuration and assign to them the rol‘‘particle states’’ @7#. Alternative to this approach is thquantization of a brane of preassigned geometry. T~minisuperspace! approach, pioneered in Ref.@8#, was usedin Refs.@9# and @10# in order to estimate the nucleation raof a spherical membrane. Presently, the minisuperspaceproximation is introduced for the purpose of providingexact algorithm for computing two specific componentsthe general dynamics of ap-brane: one is the brane collectivmode of oscillation in terms of global volume variations, tother is the evolution of the brane center of mass.

©2001 The American Physical Society03-1

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ANSOLDI, AURILIA, CASTRO, AND SPALLUCCI PHYSICAL REVIEW D64 026003

In broad terms, this paper is divided into two parts: Secdeals with classical dynamics; Sec. III deals with quantdynamics in terms of the path integral, or ‘‘sum over hisries.’’ Since the action for a classicalp-brane is not uniquewe start our discussion by providing the necessary baground with the intent of justifying our choice of action. Wthen take the first step in our approximation scheme in orto separate the center of mass motion from the bulkboundary dynamics. In Secs. II C and II D we derive anfective action for the bulk and boundary evolution, whileSec. II E we discuss the meaning of the ‘‘quenching appromation’’ at the classical level.

An approximation scheme for a dynamical problemtruly meaningful and useful only when the full theoryprecisely defined, so that the technical and logical steps leing to the approximate theory are clearly identified. ThusSec. III we first tackle the problem of computing the genequantum amplitude for ap-brane to evolve from an initiaconfiguration to a final one. The full quantum propagatorobtained as a sum over all possible histories of the womanifold of the relativistic extended object. In Sec. III A, wshow in detail what that ‘‘sum over histories’’ really meanboth mathematically and physically, in order to explain wthe bulk quantum dynamics cannot be solved exactly. Wcan be calculated, namely, the boundary and center of mpropagator, is discussed in Secs. III B and III C. The finexpression for the quantum propagator and the genera‘‘tension-shell’’ condition in the quenched-minisuperspaapproximation is given in Sec. III D. Finally, Sec. III Echecks the self-consistency of the result against some spcases of physical interest.

II. CLASSICAL DYNAMICS

A. Background

The action of a classicalp-brane is not unique. The firs~mem!brane action dates back to 1962 and was introducedDirac in an attempt to resolve the electron-muon puzzle@11#.The Dirac action was reconsidered in Ref.@8# and quantizedfollowing the pioneering path traced by Nambu-Goto in tlower dimensional string case. The Dirac-Nambu-Gototion represents the world volume of the membrane trajecin spacetime. Thus, it can be generalized to higher dimsionalp-branes as follows:

SDNG@Y#52mp11ESp11

dp11sA2g,

g[det~]mYm]nYm!, ~2.1!

wheremp11 represents the ‘‘p-tension’’~we denote with ‘‘p’’the spatial dimensionality of the brane! and the coordinatessm, m50,1, . . . ,p, span the (p11)-dimensional worldmanifold S in parameter space. On the other hand, the ebedding functionsYm(s), m50,1, . . . ,D21, represent thebrane coordinates in the target spacetime.

An alternative description that preserves the reparamzation invariance of the world manifold is achieved by intr

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ducing an auxiliary metricgmn(s) in parameter space together with a ‘‘cosmological constant’’ on the worlmanifold @12,13#:

SHTP@Y,g#52mp11

2 ESdp11sA2g@gmn]mYm]nYm

2~p21!#, ~2.2!

whereg[detgmn . The two actions~2.1! and ~2.2! are clas-sically equivalent in the sense that the ‘‘field equationdS/dgmn(s)50 require the auxiliary world metric to matcthe induced metric, i.e.,gmn5gmn5]mYm]nYm . The two ac-tions are also complementary:SDNG provides an ‘‘extrinsic’’geometrical description in terms of the embedding functioYm(s) and the induced metricgmn , while SHTP assigns an‘‘intrinsic’’ geometry to the world manifoldS in terms of themetric gmn and the ‘‘cosmological constant’’mp11, with theYm(s) functions interpreted as a ‘‘multiplet of scalar fieldsthat propagate on a curved (p11)-dimensional manifold.

Note that in both functionals~2.1! and ~2.2!, the branetensionmp11 is a preassigned parameter. More recently, naction functionals have been proposed that bridge thebetween relativistic extended objects and gauge fields@14–16#. The brane tension itself, or world-manifold cosmologcal constant, has been lifted from ana priori assigned pa-rameter to a dynamically generated quantity that may atboth positive and vanishing values. Either a Kaluza-Kletype mechanism@17# or a modified integration measure havbeen proposed as dynamical processes for producing tenat the classical@18# and semi-classical level@19#.

For our present purposes, the form~2.2! of the p-braneaction is the more appropriate starting point. There aresentially two reasons for this choice:

~1! Unlike the Nambu-Goto-Dirac action, or the Schiaction @20#, Eq. ~2.2! is quadratic in the variables]mXm. Aswe shall see in the following subsections, this property,gether with the choice of an appropriate coordinate syson Sp11, facilitates the factorization of the center of mamotion from the deformations of the brane.

~2! Equation ~2.2! can be interpreted as a scalar fietheory in curved spacetime. From this point of view, tminisuperspace quantization approach is equivalent tquantum field theory in a fixed background geometry, at leas far as the auxiliary metric is concerned.

B. Center of mass dynamics

The dynamics of an extended body can be formulatedgeneral as the composition of the center of mass motionthe motion relative to the center of mass. Ap-brane is bydefinition a spatially extended object. Thus we expect toable to separate the motion of its center of mass fromshape shifting about the center of mass. However, givenpointlike nature of the center of mass, its spacetime coonates depend on one parameter only, say, the proper timt.Thus, the factorization of the center of mass motion aumatically breaks the general covariance of the action inrameter space since it breaks the symmetry between the

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QUENCHED, MINISUPERSPACE, BOSONICp-BRANE . . . PHYSICAL REVIEW D 64 026003

poral parametert and the spatial coordinatessi . We can turn‘‘needs into virtue’’ by choosing a coordinate mesh onSp11that reflects the breakdown of general covariance in pareter space. Indeed, we can choose the model manifoldSp11of the form

Sp115I ^ Sp , ]I 5$P0 ,P%, ]Sp5B, ~2.3!

where I is an open interval of the real axis, which has twpoints, sayP0 and P, as its boundary andSp is a finitevolume, p-dimensional manifold, without boundary. Thu]Sp115P0^ SpøP^ Sp and the spacetime image o]Sp11 under the embeddingY represents the initial and finabrane configuration in target spacetime.

In terms of coordinates, the above factorization ofSp11amounts to definingt as the center of mass proper time athesi ’s as spatial coordinates ofSp . Accordingly, the invari-ant line element reads

dl25gmndsmdsn52e2~t!dt21hi j ~sW !dsidsj ~2.4!

wheret plays the role of ‘‘cosmological time,’’ that is, alclocks onSp are synchronized with the center of mass cloNow, we are in a position to introduce the center of macoordinatesxm(t) and the relative coordinatesYm(t,si):

Xm~t,sW ![xm~t!11

Amp11

Ym~t,sW !, ~2.5!

xm~t![1

VpE

Sp

dpsAh~sW !Xm~t,sW ! ~2.6!

Vp[ESp

dpsAh~sW !, h~sW ![det~hi j !. ~2.7!

Using the above definitions in the action~2.2! and replac-ing gmn with gmn as indicated in Eq.~2.4!, we find

S521

2ESdp11sAg$mp11g00xm~t!xm~t!

1@ gmn]mYm]nYm2mp11~p21!#%, p>1

521

2mp11VpE

0

T

dtF2xm~t!xm~t!

e~t!1e~t!G

21

2ESp11

dp11sAg@ gmn]mYm]nYm2mp11p#.

~2.8!

The first term describes the free motion of the bulk centemass. The absence of a mixed term, one that would cothe center of mass to the bulk oscillation modes, is due tovanishing of the metric componentg0i in the adopted coor-dinate system~2.4!. The last term represents the usual bumode free action for a covariant ‘‘scalar field theory’’parameter space. Finally, if we define the brane volu

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mass,M0[Vpmp11, representing the brane inertia undvolume variation, then, from the above expression, weread off the center of mass action and the correspondLagrangian

Scm52M0

2 E0

T

dtF2xmxm

e~t!1e~t!G[E

0

T

dtLcm„xm;e~t!…,

~2.9!

where theeinbein e(t) ensurest reparametrization invari-ance along the center of mass world line.

Summarizing, the final result of this subsection is that,the adopted coordinate frame where the center of masstion is separated from the bulk and boundary dynamics,can write the total action as the sum of two terms

S5Scm21

2 ESp11

dp11sAg@ gmn]mYm]nYm2mp11p#.

~2.10!

We emphasize that, in order to derive the express~2.10!, it was necessary to break the full invariance undgeneral coordinate transformations of the initial theory, pserving only the more restricted symmetry under indepdent time and spatial coordinate reparametrizations.

C. Induced bulk and boundary actions

In this subsection we wish to discuss those features ofbrane classical dynamics which are instrumental for the ssequent evaluation of the quantum path integral.

In agreement with the restricted reparametrization invaance of the action~2.10!, as discussed in the previous susection, we first set up a ‘‘canonical formulation’’ whicpreserves that same symmetry through all computatiosteps. This means that all the world indicesm,n, . . . areraised, lowered and contracted by means of the centemass metricgmn . From the brane action~2.10! we extractthe branerelative momentum Pmm and the correspondingHamiltonianH:

Pmm[

]L

]]mYm~s!52gmn]nYm~s!

H[Pmm]mYm2L52

1

2@ gmnP

mmPnm1mp11p#.

~2.11!

Thus, we can write the action in the following canonicform:

S5ESp11

dp11sAg~Pmm]mYm2H !

5ESp11

dp11sAgFPmm]mYm1

1

2~ gabP

amPbm1pmp11!G .

~2.12!

The first term in Eq.~2.12! can be rewritten as follows:

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ESp11

dp11sAgPmm]mYm

5ESp11

dp11s@]m~AgPmmYm!2Ym]m~AgPm

m!#.

~2.13!

According to Eq.~2.13! we can write the total action as thsum of a boundary term plus a bulk term:

S5SB@]Sp11#1SJ@Sp11#

5ESp

dpsAhNnpnmym2E

Sp11

dp11sAgYm~s!¹mPmm

2ESp11

dp11sAgH, ~2.14!

wherepnm andym are the momentum and coordinate of t

boundary,dpsAhNn represents the oriented surface elemof the boundary, and¹m stands for the covariant derivativwith respect to the metricgmn . The distinctive feature of thisrearrangement is that the bulk coordinatesYm(s) enter theaction as Lagrange multipliers enforcing the classical eqtion of motion:

dS

dYm~s!50⇒]m~AgPm

m!50. ~2.15!

The general solution of Eq.~2.15! may be expressed afollows:

Pmm5gmn]nfm1

1

p!emm2•••mp11

3~P0)mm2•••mp11

]m2Ym2

•••]mp11Ymp11

1] [m2Amm3•••mp11] !

5gmn]nfm11

p!emm2•••mp11@P0)

mm2•••mp11

1F~A!mm2•••mp11# ~2.16!

with

h gfm~s!50 ~2.17!

]mP0)mm2•••mp11

50. ~2.18!

Here, the componentsfm represent localharmonic modes

on the bulk, andemm2•••mp11[(g)21/2d [mm2•••mp11] standsfor the totally antisymmetric tensor. Moreover, the constantisymmetric tensorP0)

mm2•••mp11represents the volum

momentum zero-mode or collective-mode, that describesglobal volume variation of the brane.

In order to be able to treatf, P0), andA as independenoscillation modes, we demand that the following orthogonity relations are satisfied:

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CI[ESp11

dp11sAgemm2•••mp11]mfm]m2Ym2

•••]mp11

3Ymp1150, ~2.19!

CII[ESp11

dp11sAg Fm m2...mp11 Pm m2...mp11

0) 50 ,

~2.20!

CIII[ESp11

dp11sAgemm2•••mp11]mfm] [m2Am3•••mp11]

50. ~2.21!

The three orthogonality constraints on the bulk determinefield behavior on the boundary through Stokes’ theoremparticularCI gives

CI5ESp

fmdym2`•••`dymp1150⇒fm~sW !50,

~2.22!

which is a Dirichlet boundary condition, whereas inCIII twointegrations remain:

CIII 5ESp

dpsAh Nm2Fm

m2m3...mp11 ym]m3ym3...]mp11

ymp11

2ESp11

dp11sAg ~]m2Fm

m2m3...mp11!

3Ym]m3Ym3...]mp11

Ymp11. ~2.23!

Since the two integrations are carried over the boundand the bulk, respectively, the orthogonality condition cansatisfied only if each integral is identically vanishing,

CIII 50⇒Amm3•••mp11

~sW !

5] [m3Lm

m4•••mp11] and ]m2Fm

m2m3•••mp1150.

~2.24!

Thus,A must solve free Maxwell-type equations on the buand reduce to a pure gauge configuration on the boundNote that under these conditions forf andA, CII is satisfiedas well.

In summary, the classical solution for the brane momtum reduces the original field content of the model to:

~i! a multipletfm of world, harmonic, scalar fields~targetspacetime vector! which satisfy Dirichlet boundary conditions;

~ii ! a multipletAmm3•••mp11

of world, Kalb-Ramond fields~target spacetime vector! which reduce to a pure gauge cofiguration on the boundary;

~iii ! a world-manifold ~cosmological! constantP0)

mm2•••mp11~target spacetime constant tensor! correspond-

ing to a constant energy background along the brane wmanifold.

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D. Effective bulk and boundary actions

By inserting Eq.~2.18! into the action~2.12!, and takinginto account the conditions~2.19!, ~2.20!, ~2.21!, ~2.22!,~2.23!, we can write an effective classical action for the thrtypes of oscillation modes,

Se f f[SB1SJ ~2.25!

SB51

~p11!!P0)

mm2•••mp11E

]Sdsmm2•••mp11~sW !

1ESp

dpsymNm~sW !]mfm ~2.26!

51

~p11!!P0)

mm2•••mp11smm2•••mp11

1ESp

dpsymNm~sW !]mfm ~2.27!

SJ521

2mp11E

Sp11

dp11sAgS gmn]mfm]nfm21

p!

3Fmm2•••mp11

Fmm2•••mp11D1

1

2mp11

3F 1

~p11!!P0)

mm2•••mp11P0) mm2•••mp112mp11

2 pG3Vp11 , ~2.28!

whereNm(sW) represents the normal to the boundary and

smm2•••mp115E]S

ymdym2`•••`dymp11

[E]S

dsmm2•••mp11~sW !, p>1 ~2.29!

stands for the volume tensor of the brane in target spacetwhile dsmm2•••mp11(sW) represents the oriented volume elment attached to the originalp-brane at the contact poinxm5ym(sW). Finally, by definition, we set

Vp11[ESp11

dp11sAg

5E0

T

dte~t!ESp

dpsAh[VpE0

T

dte~t!.

~2.30!

Expression~2.29! allows us to establish a relation between functional derivatives inp-loop space,

d

dym~sW !5ym2•••mp11~sW !

d

dsmm2•••mp11~sW !,

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ym2•••mp11~sW ![em2•••mp11]m2ym2

•••]mp11ymp11.

~2.31!

The above relationship can be used to describe the shdeformations, or local distortions of thep-brane in terms ofthe Jacobi equation inp-loop space@21#. The boundary ef-fective actionSB leads us to define the boundary momentudensity as the dynamical variable canonically conjugatedthe boundary coordinateym(sW):

dSB

dym~sW ![Pm~sW !

51

p!em2•••mp11P0)

mm2•••mp11]m2

ym2•••]mp11

ymp11

1Nm]mfm . ~2.32!

Here,Pm(sW) describes the overall response of thep-braneboundary to local volume deformations encoded indsm1•••mp11(sW), as well as to induced harmonic deformtions, orthogonal to the boundary, described by the norderivative offm . In a similar way, we can define the energdensity of the system as the dynamical variable canonicconjugated to thep-brane history volume variation

]SJ

]Vp115

1

2mp11~p11!!P0)

mm2•••mp11P0) mm2•••mp11

2mp11

2p. ~2.33!

Finally, from the antisymmetry ofP0)mm2•••mp11

under indexpermutations we deduce the following identity:

PmPm[1

~p11!!P0)

mm2•••mp11P0) mm2•••mp11. ~2.34!

Thus, we arrive at the main result of the classical formulatin the form of a reparametrization invariant, relativistic, efective Jacobi equation

1

2mp11VpE

Sp

dps

AhS dSe f f

dym~s!2Nm~s!]mfmD

3S dSe f f

dym~s!2Nj~s!] jfmD 2

mp11

2p

5]Se f f

]Vp11. ~2.35!

This Jacobi equation encodes the boundary dynamicthe p-brane with respect to an evolution parameter repsented by the world volume of thep-brane history. This is ageneralization of the areal string dynamics originally intrduced by Eguchi@22# via reparametrization of the Schilaction @20,23#.

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E. ‘‘Classical quenching’’ \ volume dynamics

The Jacobi equation derived in the previous subsec

takes into account both the intrinsic fluctuationsdym(sW) andthe normal boundary deformationsdNm]mfm induced by thebulk field fm. The problem is that, even neglecting thboundary fluctuations induced by the bulk harmonic mothe p-loop space Jacobi equation is difficult to handle@24–26#. In order to make some progress, it is necessary to fothe local fluctuations of the brane in favor of the simplglobal description in terms of hyper-volume variations, witout reference to any specific point on thep-brane where alocal fluctuation may actually occur. Thus, in our formulatiof p-brane dynamics, classical quenching means havinrelinquish the idea of describing the local deformatiofm(s) of the brane, and to focus instead on the collectmode of oscillation. In turn, by ‘‘collective dynamics,’’ wemean volume variations with no reference to the local fltuations that cause the volume to vary. In this approximatwe can write a ‘‘global,’’ i.e., non-functional,p-brane waveequation.

The effective action that encodes the volume dynamsayS0, is obtained fromSJ1SB by ‘‘freezing’’ both the har-monic and Kalb-Ramond bulk modes. The simplificationthat the general action reduces to the following form:

SJ1SB→S051

~p11!!smm2•••mp11P0)

mm2•••mp11

1Vp11F 1

2mp11~p11!!P0)

mm2•••mp11

3P0) mm2•••mp112mp11

2pG ~2.36!

so that the functional equation reduces to a partial differtial equation

1

2mp11

]S0

]sm1•••mp11

]S0

]sm1•••mp112

mp11

2p5

]S0

]Vp11.

~2.37!

The collective-mode dynamics is much simpler to handIn fact, the functional derivatives that describe the shavariation of the brane have been replaced by ‘‘ordinary’’ ptial derivatives that take into account only hyper-volumvariations rather than local distortions. In other words, whthe original equation~2.35! describes the shape dynamicthe global equation~2.37! accounts for the collective dynamics of the brane. The advantage of the Jacobi equation~2.37!is that the partial derivative is taken with respect to a macoordinatesm1•••mp11

instead of the usual position fourvector.

The similarity with the point particle case (p50) sug-gests the following ansatz forS0:

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S0~s;V![B

2Vp11

1

~p11!!~sm1•••mp112s0

m1•••mp11!2

2mp11

2pVp11 , ~2.38!

wheres0m1•••mp11 represents a constant~matrix! of integra-

tion to be determined by the ‘‘initial conditions,’’ while thevalue of theB factor is fixed by Eq.~2.37!. Indeed

]S0

]sm1•••mp115

B

Vp11~sm1•••mp11

2s0)m1•••mp11!

~2.39!

and

]S0

]Vp1152

B

2Vp11~p11!!~sm1•••mp11

2s0)m1•••mp11!2

2mp11

2pVp11 ~2.40!

so that

B52mp11 ~2.41!

S0~s;V!52mp11

2Vp11

1

~p11!!~sm1•••mp112s0)

m1•••mp11!2

2mp11

2pVp11 . ~2.42!

For the sake of simplicity, one may choose the integratconstant to vanish, i.e., one may sets0)

m1•••mp1150. Thus, inthe quenching approximation, we have obtained the classJacobi action for the hyper-volume dynamics of a frp-brane.

III. THE FULL QUANTUM PROPAGATOR

A. ‘‘Momentum space’’ propagator

Eventually, one is interested in computing the quantamplitude for the brane to evolve from an initial~vacuum!state to a final, finite volume, state. In general, thep-brane‘‘two-point’’ Green’s function represents the correlatiofunction between an initial brane configurationy0(sW) and afinal configurationy(sW). In the quantum theory ofp-branesthe Green function is obtained as a sum over all posshistories of the world manifoldSp11 in the correspondingphase space. For the sake of simplicity, one may ‘‘squeethe initial boundary of the brane history to a single point.other words, the physical process that we have in mind rresents the quantum nucleation of ap-brane so that thepropagator that we wish to determine connects an inibrane of zero size to a final object of proper volumeVp . Thecorresponding amplitudeG is represented by the path integral:

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-c

is

tin

n

a

unc-the

to

tobulk

rtthatehese

-ble


G[E @Dgmn#Ex0

x

@Dx~t!#E @Dp~t!#Ey

@DYm#@DPmm#

3expH i ESp11

dp11sAgFpmxm2g001

2mp11pmpm

1mp11

2 G2 i ESp11

dp11sAgFPmm¹mYm~s!

21

2mp11gmnP

mmPnm2p

mp11

2 G J . ~3.1!

Summing over ‘‘all’’ the brane histories in phase spameans summing over all the dynamical variables, that is,shapesYm(s) of the world manifoldSp11, over the rates ofshape change,Pm

m(s), and over the bulk intrinsic geometries,gmn(s), with the overall condition that the shapethe boundary is described byxm5ym(sW) and its intrinsic ge-ometry byhi j (sW). Thus,

@Dgmn#[@Dgmn#d @gmn2gmn#

5@Dg00#@De#d @g002e2~t!#@Dgik#d @gik2hik~sW !#,

~3.2!

where the only non-trivial integration is over the uncostrained fielde(t). Carrying out all three functional integrations would give us a boundary effective theory encodingthe information about bulk quantum dynamics, in agreemwith the holographic principle. However, there are severatechnical difficulties that need to be overcome before reaing that goal.

Let us begin with the shape variablesYm(s); they appearin the path integral as ‘‘Fourier integration variables’’ linearly conjugated to the classical equation of motion. Henthe Y integration gives a~functional! Dirac delta that con-finesPm

m on-shell:

E @DYm#expH 2 i ESp11

dp11sAgYm~s!¹mPmmJ

}d@¹mPmm#. ~3.3!

The proportionality constant in front of the Dirac deltaphysically irrelevant and can be set equal to unity.

Integrating out the brane coordinates is equivalent‘‘shifting from configuration space to momentum space’’a functional sense. However, the momentum integrationnot free, but is restricted by Eq.~3.3! to the family of clas-sical trajectories that are solutions of Eq.~2.15!. Then, wecan write the two-points Green’s function as follows:

G5NE @dP0)#E @Dgmn#E @Df#@DA#exp~ iSe f f!,

~3.4!

whereN is a normalization factor to be determined at the eof the calculations,Se f f is the effective action~2.25! and weintegrate over the zero mode components in the ordin

02600

e

llnt

h-

e,

o

is

d

ry

sense, that is, we integrate over numbers and not over ftions. It may be worth emphasizing that we have tradedoriginal set of scalar fieldsYm(s) with the ‘‘Fourier conju-gated’’ modesf, A, P0). Then

G5NE @Dg#expH 2 imp11

2pE

Sp11

dp11sA2detgmnJ3K@s;g#Zf,A@g#. ~3.5!

We recall that

Vp11[ESp11

dp11sA2detgmn

5ESp

dpsA2dethi j E0

T

dte~t!

[VpE0

T

dte~t! ~3.6!

and

K@s;e#5NE @dPm1•••mp11

0) #

3expH i

~p11!!Pmm2•••mp11

0) smm2•••mp11J3expH iVp

2mp11

~p11!! E0

T

dte~t!Pmm2•••mp11

0)

3P0)mm2•••mp11J . ~3.7!

At this point, we would like to factor out of the wholepath integral the boundary dynamics, i.e., we would likewrite K5K(boundary)3K(bulk). In order to achieve thissplitting between bulk and boundary dynamics, we needremove the dependence on the 00 component of themetric gmn in K@s;e#. In other words, we are looking for apropagator whereAusu plays the role of ‘‘Euclidean dis-tance’’ between the initial and final configuration. In suppoof this interpretation, we also need a suitable parameterplays the role of ‘‘proper time’’ along the history of thbranes connecting the initial and final configurations. Tobvious candidate for that role is the proper time lap*0

Tdte(t). However, thee-field is subject to quantum fluctuations, so that the proper time lapse is a quantum variaitself. Accordingly, ac numberVp11 can be defined only asa quantum average of the proper world volume operator

Vp115VpK E0

T

dte~t!L . ~3.8!

Thus, we replace the quantum proper volumeVp11@g# withthe quantum averageVp11 in K@s;e# and write the bound-ary propagator in the form

3-7

e

inuow

lta

st

e

ath

usicsutetor

hat


K@s;Vp11#5NE @dP0)#

3expH i

~p11!!Pmm2•••mp11

0) smm2•••mp11

1 iVp11

2mp11~p11!!Pmm2•••mp11

0)

3P0)mm2•••mp11J , ~3.9!

while the amplitude becomes

G5NE0

`

dVp11 expH 2 imp11

2pVp11J K@s;Vp11#

3E @Dg#dFVp112VpK E0

T

dte~t!L GZfA@g#.

~3.10!

The ‘‘bulk’’ quantum physics is encoded now into thpath integral

Zf,A@g#5E @Df#@DA#expH i

2mp11~p11!!

3ESp11

dp11sAgS gmn]mfm]nfm

21

p!Fm

m1•••mp~A!Fm m1•••mp

~A! D J .

~3.11!

Equation ~3.10! presents a new problem: the constraover the metric integration, which allowed us to factor othe boundary dynamics, is highly nonlinear as it dependsthe vacuum average of the quantum volume. However,can get around this difficulty by replacing the Dirac dewith an exponential weight factor

dFVp112VpK E0

T

dte~t!L G→expH 2 iLS Vp112VpE

0

T

dte~t! D J ,

~3.12!

where L is a constant Lagrange multiplier. Thus, we firperform all calculations withVp11 as an arbitrary evolutionparameter and only at the end we impose the condition

]

]LG50⇒Vp115VpK E

0

T

dte~t!L . ~3.13!

In this way, we can write theL dependent amplitude in thform

02600

ttne

G5NE0

`

dVp11 expH 2 iVp11S L1mp11

2pD J

3K@s;Vp11#E @Dg#ZfA@g;L#, ~3.14!

where the bulk quantum physics is encoded into the pintegral

Zf,A@g;L#5expH iLVpE0

T

dte~t!J E @Df#@DA#

3expH i

2mp11~p11!!

3ESp11

dp11sAgS gmn]mfm]nfm

21

p!Fm

m1•••mp~A!Fm

m1•••mp~A! D J .

~3.15!

B. The boundary propagator

The main point of the whole discussion in the previosubsection is this: even though the bulk quantum dynamcannot be solved exactly, since there is no way to compthe bulk fluctuations in closed form, the boundary propagacan be evaluated exactly. This is because the integral~3.9! isGaussian inP0):

K@s;Vp11#5F mp11

ipVp11G (1/2)(

Dp11)

expH imp11

2~p11!!Vp11

3sm1•••mp11sm1•••mp11J . ~3.16!

Moreover, one can check through an explicit calculation tthe kernelK solves the~matrix! Schrodinger equation

F 1

2mp11~p11!!

]2

]sm1•••mp11]sm1•••mp11

GK@s2s0 ;Vp11#

52 i]

]Vp11K@s2s0 ;Vp11# ~3.17!

with the boundary condition

limV→0

K@s2s0 ;Vp11#5d@ us2s0u#. ~3.18!

Notice that the average proper volumeVp11 enters theexpression of the kernelK only through the combinationmp11 /Vp11. Thus, the limit~3.18! is physically equivalentto the infinite tension limit whereVp11 is kept fixed andmp11→`:

limmp11→`

K@s2s0 ;Vp11#5d@ us2s0u#. ~3.19!

3-8

to

dyth

n.

rs

er

nessq.

aoto

ym

ror-atedy-

su-ld

all

n ofics

ede

pirit

s

s is


In the limit ~3.19! the infinite tension shrinks the branea pointlike object.

From the above discussion we infer that the quantumnamics of the collective mode can be described either byzero mode propagator~3.16! or by the wavelike equation

F 1

2mp11~p11!!

]2

]sm1•••mp11]sm1•••mp11

GC0@s;Vp11#

52 i]

]Vp11C0@s;Vp11#

5E @ds0#K@s2s0 ;Vp11#f@s0 ;0#, ~3.20!

where f@s0 ;0# represents the initial state wave functioComparing the ‘‘Schro¨dinger equation’’~3.17! with the Ja-cobi equation~2.35! suggests the followingcorrespondenceprinciple among classical variables and quantum operato

Pm1•••mp11

0) → i]

]sm1•••mp11~p>1! ~3.21!

E→2 i]

]Vp11. ~3.22!

In summary, the main result of this section is that the genform of the quantum propagator for a closed bosonicp-branecan be written in the following form:

G5NE0

`

dVp11 expH iVp11FL1mp11

2pG J

3F mp11

ipVp11G (1/2)(

Dp11)

3expH imp11

2~p11!!Vp11sm1•••mp11sm1•••mp11J

3E @De#Kcm@x2x0 ;e~t!#Z@e;L#. ~3.23!

IV. ‘‘MINISUPERSPACE-QUENCHED PROPAGATOR’’

At this stage in our discussion, we can explicitly defiour approximation scheme. It consists of three main step

The first essential step, discussed in Sec. II B, consistsplitting the metric of the world manifold according to E~2.4!. In a broad sense, this is a ‘‘minisuperspace approximtion’’ to the extent that it restricts the general covariancethe action in parameter space. In other words, separatingcenter of mass proper time from the spatial coordinatesthe world manifoldSp11 effectively breaks the full invari-ance under general coordinate transformations into two smetry groups:

General Diffs→~ time!Rep ~spatial!Diffs, ~4.1!

02600

-e

:

al

.of

-fhen

-

so that the metric~2.4! shows a residual symmetry undeindependent time reparametrizations and spatial diffeomphisms. By virtue of this operation, we were able to separthe center of mass motion from the bulk and boundarynamics. This is encoded in the split form~2.10! of the totalaction for thep-brane.

The second and more strict interpretation of the miniperspace approximation is that we now ‘‘freeze’’ the wormetric into a background configurationgmn where the spaceis a p-sphere, whileg00, or its square roote(t), is free tofluctuate. In other words, we work in a minisuperspace ofpossible world geometries.

The third and last step in our procedure is an adaptatioone of the most useful approximations to the exact dynamof interacting quarks and gluons, namely, ‘‘quenchQCD.’’ There, the contribution of the determinant of thquark kinetic operator is set equal to one. In the same swe ‘‘quench’’ all the bulk oscillations:

ZfA@e;L#→exp$2 iLVp11%. ~4.2!

Combining these three steps, we obtain from Eq.~3.23!the quenched-minisuperspace propagator:

G@x2x0 ,s#5NE0

`

dVp11 expH ipVp11

mp11

2 J3F mp11

ipVp11G (1/2)(

Dp11)

3expH imp11

2Vp11

s2

~p11!! J3E @De#Kcm@x2x0 ;e~t!#

3dF E0

T

te~t!2Vp11

VpG . ~4.3!

The center of mass propagatorKcm@x2x0 ;e(t)# can becomputed as follows@27#. From the LagrangianLcm we candefine the center of mass momentumpm as follows:

pm[]Lcm

] xm~t!5M0

xm

e~t!. ~4.4!

By Legendre transformingLcm we obtain the center of masHamiltonian:

Hcm[pmxm2Lcm5e~t!

2M0@pmpm1M0

2#. ~4.5!

Moreover, the canonical form ofScm reads

Scm5E0

T

dtFpmxm2e~t!

2M0~pmpm1M0

2!G ~4.6!

so that the quantum dynamics of the bulk center of masdescribed by the path integral

3-9

ra

a

na

he

p-e

n

atoroxi-


Kcm@x2x0 ;e~t!#[E @Dx#@Dp#eiScm[x,T;e(t)] . ~4.7!

This path integral can be reduced to an ordinary integover the constant four momentumqm of a point particle ofmassM0

Kcm@x2x0 ;e~t!#5E dDq

~2p!Deiqm(xm2x0m)

3expF2E0

T

dte~t!

2M0~qmqm1M0

2!G .~4.8!

In order to get the explicit form of the center of mass propgator we have to integrate, in the ordinary sense, overqm .However, before that we must integrate, in the functiosense, over the einbein fielde(t).

Using the properties of the Dirac–delta distribution, teinbein field can be integrated out:

E @De#dF E0

T

dte~t!2Vp11 /VpG3expF2 i E

0

T

dte~t!

2M0~qmqm1M0

2!G5expF2 i

Vp11

2M0Vp~qmqm1M0

2!G ~4.9!

and

f

n

dtio

02600

l

-

l

E dDq

~2p!D eiqm(xm2x0m)expF2 i

Vp11

2M0VpqmqmG

5S pM0Vp

Vp11D D/2

expF2 iM0Vp

2Vp11~x2x0!2G .

~4.10!

Using the above results, the ‘‘QCD-QC combined aproximation’’ leads to the following expression for thpropagator:

G@x2x0 ,s#5NE0

`

dVp11 expH iVp11Fmp11

2p1

M0

2VpG J

3F mp11

ipVp11G (1/2)(

Dp11)

3expH imp11

2Vp11

s2

~p11!! J3S pM0Vp

iVp11D D/2

expF iM0Vp

2Vp11~x2x0!2G .

~4.11!

The amplitude~4.11! can be cast in a more familiar form iterms of the Schwinger-Feynman parametrization

s[Vp11

4Vp, dVp1154Vpds. ~4.12!

Using the above parametrization, the quantum propagtakes its final form in the minisuperspace-quenched apprmation

G@x2x0 ,s;M0#5NVpE0

`

dsS pM0

is D D/2

expF iM0

2s~x2x0!2GexpH is

M0

2~p11!J F M0

ipVp2sG (1/2)(

Dp11)

expH iM 0

2sVp2

s2

~p11!! J5

i

2M0E

0

`

dsS pM0

is D D/2F M0

ipVp2sG (1/2)(

Dp11)

expH isM0

2~p11!J expH i

M0

2s F ~x2x0!211

Vp2

s2

~p11!! G J .

~4.13!

en,a-ce.eful

a-f

Here we have setN5 i /2M0Vp in order to match the form othe point particle propagator. The formula~4.13! representsthe main result of all previous calculations and holds for ap in any number of spacetime dimensions.

A. Green’s function equation\ tension-shell condition

For completeness of exposition, in this subsection werive the master equation satisfied by the Green’s func

y

e-n

~4.13! in the quenched-minisuperspace approximation. Thby formally inverting that equation we arrive at an alterntive expression for the Green’s function in momentum spaThe advantage of this procedure is that it provides a usinsight into the structure of thep-brane propagator.

To begin with, it seems useful to remark that the propgation kernel in Eq.~4.13! is the product of the center omass kernelKcm(x2x0 ;s) and the volume kernelK(s;s);

3-10

toe

weth

r

te

par-r


each term carries a weight given by the phase facexp(iM0s) and exp(ipM0s), respectively. Finally, we integratover all the values of the Feynman parameters:

G@x2x0 ;M0#5i

2M0E

0

`

dsexpH iM0

2s~p11!J

3Kcm~x2x0 ;s!K~s;s!. ~4.14!

As is customary in the Green’s function technique,may add an infinitesimal imaginary part to the mass inexponent, that is, (M0/2)→(M0/2)1 i e, so that the oscilla-tory phase turns into an exponentially damped factor enfoing convergence at the upper integration limit. The ‘‘i e ’’prescription in the exponent allows one to perform an ingration by parts leading to the following expression:

ar

ne

02600

r

e

c-

-

G@x2x0 ;M0#51

M02~p11!

FexpS i S M0

21 i e D s~p11! D

3Kcm~x2x0 ;s!K~s;s!G0

`

21

M02~p11!

3E0

`

dsexpS i S M0

21 i e D s~p11! D

3]

]s„Kcm~x2x0 ;s!K~s;s!…. ~4.15!

Convergence of the integral enables us to express thetial derivative]/]s by means of the diffusion equations foKcm and K and to move the differential operators]m]m,]2/]s2 out of the integral:

G@x2x0 ;M0#51

M02~p11!

@d~x2x0!d~s;s!#21

M03~p11!

E0

`

dsexpH iM0

2s~p11!J

3F „K~s;s!]m]mKcm~x2x0 ;s!…S Kcm~x2x0 ;s!Vp

~p11!!

]2K~s;s!

]sm1•••mp11]sm1•••mp11

D G52

1

M02~p11!

d~x2x0!d~s;s!11

M02~p11!S ]m]m1

Vp

~p11!!

]2

]sm1•••mp11]sm1•••mp11D G~x2x0 ,s;s!,

~4.16!

thel-

-r-

di-ly--xtour

from which we deduce the desired result,

F ]m]m1Vp

~p11!!

]2

]sm1•••mp11]sm1•••mp11

2~p11!M02G

3G~x2x0 ,s!5dD~x2x0!d~s!. ~4.17!

This is the Green’s function equation for the non-standdifferential operator]m]m1Vp]2/]s2. Finally, we ‘‘Fouriertransform’’ the Green’s function by extending the mometum space to a larger space that includes the volume momtum as well:

G~x2x0 ,s!5E dDq

~2p!DE @dkm1•••mp11#

3expS iqm~x2x0!m1i

~p11!!

3km1•••mp11sm1•••mp11D

d

-n-

31

q21Vp

2

~p11!!km1•••mp11

2 1~p11!M02

.

~4.18!

The vanishing of the denominator in Eq.~4.18! defines anew tension-shell condition:

q21Vp

2

~p11!!km1•••mp11

2 1~p11!M0250. ~4.19!

Real branes, as opposed to virtual branes, must satisfycondition~4.19! which links together center of mass and voume momentum squared. Equation~4.19! represents an extension of the familiar Klein-Gordon condition for point paticles to relativistic extended objects.

Some physical consequences of the ‘‘tension-shell contion,’’ as well as the mathematical structure of the undering spacetime geometry@28#, are currently under investigation and will be reported in a forthcoming paper. In the nesubsection we limit ourselves to check the consistency ofresults against some familiar cases of physical interest.

3-11

wthur

th

rewhe

c-

rdre

at

imfo

ne

atda-

and

de-s to

on,

vesta-oxi-.

ng


B. Checks

1. Infinite tension limit

When probed at low energy~resolution! an extended ob-ject effectively looks like a point particle. In this case, ‘‘loenergy’’ means an energy that is small compared withenergy scale determined by the brane tension. In natunits, the tension of ap-brane has dimension:@Tp#5(energy)p11. Thus, when probing the brane at energyE!(Tp)1/p11 one cannot resolve the extended structure ofobject. From this perspective, the ‘‘point-like limit’’ of ap-brane is equivalent to the ‘‘infinite tension limit.’’ In eithecase, no higher vibration modes are excited and one expthe brane to appear concentrated, or ‘‘collapsed,’’ in its ocenter of mass. This critical limit can be obtained from tgeneral result~4.13! by setting p50 and performing thelimit Vp→0 using the familiar representation of the Diradelta distribution:

d~x![ lime→0

S 1

pe D d/2

exp~2x2/e!. ~4.20!

In our case, x2→s2/(p11)!, e→2 iM 0/4sVp2 , d

→(p11D ), and the whole dependence on the volume coo

nates of the brane reduces to a delta function that is diffefrom zero only whens50. In this case,G@x2x0 ;M0# re-duces to the familiar expression for the Feynman propagfor a point particle of massM0,

G@x2x0 ;M0#5d@s2#i

2M0E

0

`

dsS pM0

is D D/2

3expS 2 iM0

2sDexpH i

M0

2s~x2x0!2J .

~4.21!

2. Spherical membrane wave function

From the results of the previous subsections we canmediately extract the generalized Klein-Gordon equationa 2-brane in four spacetime dimensions:

F ]m]m1V2

3!

]2

]sm1m2m3]sm1m2m3

2~p11!M02GC~x,s!50.

~4.22!

In the following we show briefly how this wave equatiospecializes to the case of a gauge fixed, or spherical, m

,n

02600

eal

e

ctsn

i-nt

or

-r

m-

brane of fixed radius@8,9#. Moreover, since it is widely be-lieved that p-brane physics may be especially relevantPlanckian energy, we assume that our 2-brane is a funmental object characterized by Planck units of tensionlength. Thus, for a spherical 2-brane of radiusR we have

V254p l Pl2 , ~4.23!

sm1m2m35dx[m1dy

m2dzm3] 4p

3R3, ~4.24!

dsm1m2m35dx[m1dy

m2dzm3] 4pR2dR, ~4.25!

]

]sm1m2m35

1

4pR2 d [m1

x dm2

y dm3]z ]

]R. ~4.26!

Since there is no mixing between ordinary and volumerivative, we can use the method of separation of variablefactorize the dependence of the wave function onx ands

C~x,s!5f~x!c0~R!⇒@]m]m2M02#f~x!50.

~4.27!

Here, f0(x) represents the center of mass wave functiwhile the ‘‘relative motion wave function’’c(R) must sat-isfy the following equation:

F 1

4pR2

]

]R

1

R2

]

]R2

pM02

l Pl2 Gc~R!50. ~4.28!

Apart from some ordering ambiguities, Eq.~4.28! is thewave equation found in Refs.@8–10# for the zero energyeigenstate, provided we identify the membrane tensionrthrough the expression

r25pM0

2

4p l Pl2

. ~4.29!

It may be worth emphasizing that our approach presertime reparametrization invariance throughout all computional steps, while the conventional minisuperspace apprmation assumes a gauge choice from the very beginning

ACKNOWLEDGMENT

The authors are indebted to Dr. Kai Lam for proofreadithe manuscript.

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3-12

d,,

rav.

,’’ys.

rav.

c-

o-inrds,

cz


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