Deparametrization and path integral quantization of by Claudio Simeone

By Claudio Simeone

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

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4) The constraint H is equivalent to % because they differ only in a positive definite factor. 6) with + for 7Tfi > 0 and - for TTQ < 0. 7) Isotropic 47 universes with r) — sign(-KQ). 2, in order to quantize the model we define F = Q V 0 + f(r) so that the reduced Hamiltonian and the new variables are The system described by Q° and Fo has a constraint which is linear and homogeneous in the momenta. Its action functional is then invariant under general gauge transformations, so that there is gauge freedom at the end points and canonical gauges are admissible.

30) with 77 = ± 1 and T(T) a monotonous function. Then we have the canonical variables of the gauge system in terms of those of the minisuperspace: e° - |. T(T)) , P0 = TTX-Ky + 1, P = 7TX. 31) There is no problem with P as a denominator because P = irx cannot be zero on the constraint surface. 32) and a canonical gauge condition of the form x = <3° — T(r) = 0 with T a monotonic function of r, when imposed on the gauge system described by Ql and Pi, defines a global phase time for the minisuperspace described by ,fi,ir,p, TTQ.

1 End point terms Let us consider a complete solution [Landau&Lifshitz (I960)] of the Hamilton-Jacobi equation W(ql,afi,E) *{<•%)-* where H is not necessarily the original Hamiltonian constraint but it can be a scaled Hamiltonian, that is H — T~XH with T a positive definite function of ql. HE and the integration constants aM are matched to the new momenta P0 and P M respectively, then W(q\Pi) turns to be the l generator function of a canonical transformation (q ,Pi) ->• {Q ,Pi) defined by the equations Gauge-invariant action for a parametrized system 23 where K is a new Hamiltonian.

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