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

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.

9 pages, Arbitrary constants in (4.19) are changed to arbitrary functions of $\phi$. Conclusions are not changed. References are added. Typos are corrected

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• Fourth Quantization
• Constraints on Operator Ordering from Third Quantization
• On the canonical formalism of $f(R)$-type gravity using Lie derivatives
• The end of spacetime
• Can quantum gravity be both consistent and complete?
• Operator Ordering Ambiguity and Third Quantization
• Virtual Black Holes in a Third Quantized Formalism