Third quantization of $f(R)$-type gravity

TL;DR

This paper explores the third quantization of $f(R)$-type gravity within a flat FLRW universe, analyzing quantum effects and classical emergence over cosmic evolution, especially for the case $f(R)=R^2$.

Contribution

It introduces a method for third quantization of $f(R)$ gravity and investigates the quantum-to-classical transition of the universe.

Findings

Quantum effects dominate at early universe when scale factor is small.

Spacetime tends to become classical at late times when scale factor is large.

Uncertainty relations support the quantum-classical transition in this model.

Abstract

We examine the third quantization of $f(R)$-type gravity, based on its effective Lagrangian in the case of a flat Friedmann-Lemaitre-Robertson-Walker metric. Starting from the effective Lagrangian, we execute a suitable change of variable and the second quantization, and we obtain the Wheeler-DeWitt equation. The third quantization of this theory is considered. And the uncertainty relation of the universe is investigated in the example of $f(R)$-type gravity, where $f(R)=R^2$. It is shown, when the time is late namely the scale factor of the universe is large, the spacetime does not contradict to become classical, and, when the time is early namely the scale factor of the universe is small, the quantum effects are dominating.