Quantization of Generally Covariant Systems with Extrinsic Time

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 ∗Electronic address: [email protected] †Electronic address: [email protected] 1 General relativity and quantum mechanics are the most important achievements of physics in this century. It seems essential to find a quantum theory of gravity by embracing both theories in a consistent one. However, despite the many efforts that have been made, that program has not been sucessfully completed [1]. One of the most difficult 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 significant quantities are independent of those labels: they are invariant under diffeomorfisms. 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 efforts directed to quantize general relativity (or some minisuperspace models) emphasize the analogy with the relativistic particle [3,4]. Actually, both systems have Hamiltonian constraints H that are hyperbolic on the momenta. If the role of the squared mass is played by a positive definite potential, then the analogy is complete in the sense that time is hidden in configuration space. In fact, the positive definite potential guarantees that the temporal component of the momentum is never null on the constraint hypersurface. Thus the Poisson bracket {q,H} is also never null, telling us that q 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) formalism. Unfortunately that analogy cannot be considered too seriously because the potential in general relativity is the (non positive definite) spatial curvature. This means that the time 2 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 2 gpμpν + v(q ) and definite positive metric g . The dynamics of the system will not change if a function of t, namely − t 2 2 is added to the Hamiltonian. So we write the action