Restoring Time Dependence into Quantum Cosmology

Mini superspace cosmology treats the scale factor $a(t)$, the lapse function $n(t)$, and an optional dilation field $ϕ(t)$ as canonical variables. While pre-fixing $n(t)$ means losing the Hamiltonian constraint, pre-fixing $a(t)$ is serendipitously harmless at this level. This suggests an alternative to the Hartle-Hawking approach, where the pre-fixed $a(t)$ and its derivatives are treated as explicit functions of time, leaving $n(t)$ and a now mandatory $ϕ(t)$ to serve as canonical variables. The naive gauge pre-fix $a(t)=const$ is clearly forbidden, causing evolution to freeze altogether, so pre-fixing the scale factor, say $a(t)=t$, necessarily introduces explicit time dependence into the Lagrangian. Invoking Dirac's prescription for dealing with constraints, we construct the corresponding mini superspace time dependent total Hamiltonian, and calculate the Dirac brackets, characterized by $\{n,ϕ\}_D\neq 0$, which are promoted to commutation relations in the quantum theory.

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