On nonlinear parametric quantisation of gravity and cosmological models

A generalisation of the recently formulated nonlinear quantisation of a parameterised theory is presented in the context of quantum gravity. The parametric quantisation of a Friedmann universe with a massless scalar field is then considered in terms of the analytic solutions of the resulting evolution equations. PACS numbers: 04.20.Fy, 04.60.Ds, 04.60.Kz, 11.10.Lm In a recent paper [1] the quantisation of a physical system with finite degrees of freedom subject to a Hamiltonian constraint was re-examined. A new scheme, called parametric quantisation, was introduced in that work that advocates the idea of treating a chosen time variable as a constrained classical variable coupled to other dynamical variables to be quantised. The consideration was motivated by the cosmological approach to quantum gravity [2] whose dynamical structure is analogous to that of the parameterised theory of a relativistic particle in an (n+1)-dimensional pseudo-Riemannian spacetime with a metric tensor γμν in coordinates q , μ, ν = 0, 1, · · · , n, described by a Lagrangian of the following generic form: L(q, q̇, N) = 1 2N γμν(q )q̇q̇ −NV (q). (1) Here q = q(t) described the particle trajectory parameterised by t, whose choice is related to the function N(t) > 0, and q̇ := dq̇ μ dt . Classically this generates the Lagrange’s equations of motion d dt (