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By Claudio Simeone

During this publication, homogeneous cosmological types whose Hamilton-Jacobi equation is separable are deparametrized by way of turning their motion useful into that of a standard gauge method. Canonical gauges imposed at the gauge procedure are used to outline an international part time by way of the canonical variables of the minisuperspaces. The approach truly exhibits how the geometry of the constraint floor restricts the alternative of time. the results that this has for direction essential quantization are mentioned, and the transition amplitude is received for relativistic isotropic versions, relativistic anisotropic versions (Kantowski-Sachs and Taub) and isotropic string cosmologies. an entire bankruptcy in regards to the software of the deparametrization software to the standard canonical quantization scheme is additionally integrated.

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As we have remarked, the gauge choice is not only a way to avoid divergences in the path integral for a constrained system, but also a reduction procedure to physical degrees of freedom. e. that has only one degree of freedom and one constraint, we select only one point of the phase space at each r. 2, t — r = 0, means that the paths in the phase space can only go from *i = n at T± to £2 = T2 a t T2; there is no other possibility. Hence, 44 Deparametrization and path integral quantization the probability that the system evolves from t\ at T\ to £2 at T2 cannot be anything else but unity.

But while for these systems the time coordinate always refers to an external clock, this is clearly not the case in cosmology; for example, in the case of pure gravitational dynamics the coordinates are the elements of the metric gab over spatial slices, and in principle there is not necessarily a connection between gab and anything "external". Rather, such a relation can be thought to exist for the derivatives dgab/d,T of the metric, as they appear in the expression for the extrinsic curvature Kab which describes the evolution of spacelike three-dimensional hypersurfaces in four-dimensional spacetime.

16) the appropriate generator is / = QPTU,(T(r),Po = 0 , P - P ) . 17), means that Q and r define a hypersurface in the original configuration space. The explicit form of the physical degree of freedom and of the Hamiltonian for the reduced system described by (Q,P) is Q = d ±— / VP v2 ^+ 4Ae6fidft 9P JTIT) JT(r) h = ^ = T \AP2+4Ae6T^. 20) OLT OT with =F = sign{-Ko). 16) we have ft = T, then the paths go from Q\ = 4>2 to Q2 = <^2 (note that once we have fixed a gauge the coordinate Q is no more an observable, in the sense that Q\x=o — does no more conmute with the constraint).

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