Canonical formulation of curvature squared action in the presence of lapse function

TL;DR

This paper formulates the canonical structure of curvature squared gravity with a lapse function in minisuperspace models, deriving Hamiltonian constraints and quantum equations, and verifies the approach across different cosmological backgrounds.

Contribution

It provides a systematic canonical formulation of R^2 gravity with lapse function in minisuperspace, including Hamiltonian and quantum equations, extending previous methods to higher order gravity.

Findings

Canonical Hamiltonian form derived for R^2 gravity in minisuperspace.

Quantum Schrödinger-like equation obtained with favorable features.

Classical and semiclassical solutions presented for various models.

Abstract

Lapse function appears as Lagrange multiplier in Einstein-Hilbert action and its variation leads to the (0 0) equation of Einstein, which corresponds to the Hamiltonian constraint equation. In higher order theory of gravity the situation is not that simple. Here, we take up the curvature squared (R^2) action being supplemented by an appropriate boundary term in the background of Robertson-Walker minisuperspace metric, and show how to identify the constraint equation and formulate the Hamiltonian without detailed constraint analysis. The action is finally expressed in the canonical form $A = \int(\dot h_{ij} \pi^{ij} + \dot K_{ij}\Pi^{ij} - N{\mathcal H})dt \sim d^3 x$, where, the lapse function appears as Lagrange multiplier, once again. Canonical quantization yields Schr\"odinger like equation, with nice features. To show that our result is not an artifact of having reduced the theory to a measure zero subset of its configuration space, the role of the lapse function as Lagrangian multiplier has also been investigated in Bianchi-I, Kantowski-Sachs and Bianchi-III minisuperspace metrics. Classical and semiclassical solutions have finally been presented.