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Quantization of generally covariant systems with extrinsic time

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Quantization of generally covariant systems with extrinsic time Rafael Ferraro ?Instituto de Astronom′?a y F′?sica del Espacio,Casilla de Correo 67-Sucursal 28,1428Buenos Aires,Argentina and Departamento de F′?sica,Facultad de Ciencias Exactas y Naturales,Universidad de Buenos Aires -Ciudad Universitaria,Pabell′o n I,1428Buenos Aires,Argentina Daniel M.Sforza ?Instituto de Astronom′?a y F′?sica del Espacio,Casilla de Correo 67-Sucursal 28,1428Buenos Aires,Argentina Abstract A generally covariant system can be deparametrized by means of an “extrinsic”time,provided that the metric has a conformal “temporal”Killing vector and the potential exhibits a suitable behavior with respect to it.The quantization of the system is performed by giving the well ordered constraint operators which satisfy the algebra.The searching of these operators is enlightened by the methods of the BRST formalism.PACS numbers:04.60.Ds,11.30.Ly

General relativity and quantum mechanics are the most important achievements of physics in this century.It seems essential to?nd a quantum theory of gravity by em-bracing both theories in a consistent one.However,despite the many e?orts that have been made,that program has not been sucessfully completed[1].

One of the most di?cult features is the problem of time[2].In quantum mechanics, time is an absolute parameter;it is not on an equal footing with the other coordinates that turn out to be operators and observables.Instead,in general relativity“time”is merely an arbitrary label of a spatial hypersurface,and physically signi?cant quantities are independent of those labels:they are invariant under di?eomor?sms.General relativity is an example of a parametrized system(a system whose action is invariant under change of the integrating parameter).One can obtain such a kind of system by starting from an action which does not possess reparametrization invariance,and raising the time to the rank of a dynamical variable.So the original degrees of freedom and the time are left as functions of some physically irrelevant parameter.Time can be varied independently of the other degrees of freedom when a constraint together with the respective Lagrange multiplier are added.In this process,one ends with a special feature:the Hamiltonian is constrained to vanish.

Most e?orts directed to quantize general relativity(or some minisuperspace models)em-phasize the analogy with the relativistic particle[3,4].Actually,both systems have Hamil-tonian constraints H that are hyperbolic on the momenta.If the role of the squared mass is played by a positive de?nite potential,then the analogy is complete in the sense that time is hidden in con?guration space.In fact,the positive de?nite potential guarantees that the temporal component of the momentum is never null on the constraint hypersurface.Thus the Poisson bracket{q o,H}is also never null,telling us that q o evolves monotonically on any dynamical trajectory;this is the essential property of time.In this case,Ref.[3]shows the consistent operator ordering obtained from the Becchi-Rouet-Stora-Tyutin(BRST)for-malism.

Unfortunately that analogy cannot be considered too seriously because the potential in general relativity is the(non positive de?nite)spatial curvature.This means that the time

in general relativity must be suggested by another mechanical simile.

In order to essay a better mechanical model for general relativity,let us start with a system of n genuine degrees of freedom with a Hamiltonian h=1

is added to the Hamiltonian.So we write the action

S= pμdqμ22 dt,μ=1,...,n(1) The system is parametrized by regarding the integration variable t as a canonical variable whose conjugated momentum is(minus)the hamiltonian.This last condition enters the action as a constraint H=p t+h?t

dτ+pμ

dqμ

2 dτ(2)

where N is the Lagrange multiplier.

So far the constraint is parabolic in the momenta.However one can perform the canonical transformation,

q0=p t,p0=?t(3)

that turns the constraint H into a hyperbolic function of the momenta:

H=q0+h?1

p20+

G rs p r p s+V(q r),(4)

with r,s=0,1,...,n.The metric components read G00=?1,G0ν=0,Gμν=gμνand the potential is V(q i)=v(qμ)+q0.Then G rs is a Lorentzian metric,as is the supermetric in the Arnowitt-Deser-Misner(ADM)formalism of general relativity.The constraint(4)describes a parametrized system with extrinsic(hidden in the phase space)time[5],whose po tential is not positive de?nite.

For a complete analogy with general relativity,the“supermomenta constraints”can be introduced by adding m degrees of freedom q a.Their spurious character is stated by m

linear and homogeneus constraints G a≡ξr a p r,where ξa are m vectors?elds tangent to the coordinate curves associated with the q a’s.These m constraints G a can still be linearly combined

G a→G a′=A a a′(q)G a,det A=0,(5) to get an equivalent set of linear and homogeneous supermomenta constraints.The set (H,G a)is?rst class.

Finally the dynamics of the system is obtained by varying the action

S[q i,p i,N,N a]= p i dq i

≈0(7)

?q0

ξ0

V=1.(10) Di?ering from those approaches where a hyperbolic constraint like Eq.(4)is compared with the one of a relativistic particle,and the parameter of the Killing vector is regarded as the time[6],in our treatment the time(t=?p0)is the dynamical variable conjugated to the parameter of the Killing vector.

In order to quantize the theory,we must?nd well ordered?rst class constraint operators satisfying the quantum constraint algebra,

[?H,?G a]=?c00a?H+?c b0a?G b(11)

[?G a,?G b]=?C c ab(q)?G c,(12) where the structure function c b0a is linear in the momenta,c b0a(q,p)=c bj0a(q)p j.

However,it is apparent that the potential V commute with the linear constraints G a(it is gauge invariant)and as a consecuence the structure function?c00a must vanish.The algebra (11),(12)with?c00a=0was already solved in Ref.[3].There,the Dirac constraint operators were obtained within the framework of the BRST formalism:

?H=12?p

i G ij f?p j f?1

f12+V(13)

and

=f12,(14) where the function f=f(q)satis?es

C b ab=f?1(fξi a),i=div?α ξa,(15) (?αis the volume?α≡f dq0∧...∧dq n+m).1

The Dirac constraint operators (13),(14)were obtained from the quantum BRST gen-erator,the central object of the method.The BRST generator is a fermionic real function in an extended phase space spanned by the original canonical pairs (q i ,p i )and by m +1fermionic canonical pairs (ηa ,P a )(one for each constraint).The quantum BRST generator is a nilpotent Hermitian operator which reads for the system under consideration:

??=?ηo ?H +?ηa ?G a +

12?ηa ?ηb ?C c ab ?P c =?ηo 12?p i G ij f ?p j f ?12f 12

+V +?ηa f 12+12c bj oa ?p j f ?12?p j c bj oa f 12?ηa ?ηb C c ab ?P c (16)

(in this η?P ordering,the Dirac constraint operators and the structure function operators can be directly read from it [7]).

In Ref.[3]we started with a pseudo-Riemannian metric and a constant potential (a relativistic particle in curved space).That system had the property c 00a =0,which facilitated the search for the nilpotent BRST generator.After that,a general (but positive de?nite)potential was introduced by means of a unitary transformation of the BRST generator,and c 00a turned to be non null.This procedure gave to the constraint operators the invariance under scaling of the superhamiltonian constraint (without having recourse to a curvature term).

In the present case the system is not a relativistic particle:the potential is not constant (and neither is it positive de?nite),and the time is not hidden among the coordinates.However c 00a is still null due to the gauge invariance of the potential.Once again,the

invariance under scaling of the superhamiltonian will be introduced by performing a unitary transformation in the extended space.

The scaling of the Hamiltonian constraint

H→H=F H,F>0(17) (then G ij→G ij=F G ij,V→V=F V)relaxes the geometrical properties of ξ0:

| ξ0|=1→| ξ0|=F?1

2| ξ0|?1f?12| ξ0|?1+

2c aj oa?p j f?1 2ξi a?p i f?1

2?ηo?ηa| ξ0|?1(f12+f?12)| ξ0|?1?P b+

2| ξ0|?1f?12| ξ0|?1+

2C aj oa?p j f?1

=| ξ0|f12| ξ0|?1,(25) with the corresponding set of structure functions,

?C o

=?2ξi a(ln| ξ0|),i,(26)

?C b oa =

2C bj oa?p j f?12?p j C bj oa f1

(?1,?2)=1

2π dq dq 0 m δ(χ) J ??1(q 0,q γ)| ξ0|?1 dq ′0e ?it 0(q 0?q ′0)| ξ

0|?1?2(q ′0,q γ).(33)It should be noticed that the integration in the coordinate q ′0is evaluated along the vector

?eld lines of ξ

0.After completing the quantization,one can further understand the obtained ordering,Eqs.

(24),(28).It ful?lls the invariance properties imposed to the theory:(i)coordinate changes,(ii)combinations of the supermomenta [Eq.(5)],and (iii)scaling of the super-Hamiltonian [Eq.(17)].The physical gauge-invariant inner product of the Dirac wave functions,Eq.(33),must be invariant under any of these transformations.On account of the change of the Faddeev-Popov determinant under (ii)and (iii),the inner product will remain invariant if the Dirac wave function changes according to

?→?′=(det A )

12,| ξ

0|±1in the constraint operators are just what are needed in order that ?G a ?,?H?,and ?C b oa

?transform as ?,so preserving the geometrical character of the Dirac wave function [3].

Concerning the extension of the here exposed treatment to general relativity,Kuchaˇr has shown in Ref.[6]that a conformal timelike Killing vector actually exits in the superspace of the ADM formalism.But,the question whether or not it satis?es property (20)remains open.

As a ?nal remark,it is worth mentioning that although the idea of an extrinsic time is not new in general relativity [8],its use in the quantization problems is rather scarce [9,10].

As it was shown,the mechanical model presented here can lead to a better understanding of its implementation.

ACKNOWLEDGMENTS

This research was supported by Universidad de Buenos Aires(Proy.TX064)and Consejo Nacional de Investigaciones Cient′??cas y T′e cnicas.

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