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.

Constraints on operator ordering from third quantization

“…Thus, it seems to be the physically plausible result as we do expect the quantum fluctuations to dominate the universe at the very early stages. It may be noted that a similar result has been discussed in the context of f (R) gravity [24]- [25]. Here we have seen that the physics of the system might depend critically on operator ordering.…”

“…In this analysis, the distribution function for the universes has been obtained. The uncertainty principle has also been discussed in the context of the third quantization of f (R) gravity theories [24]- [25]. The distribution function for the universes in the third quantization Kaluza-Klein theories have also been obtained [26].…”

Deformation of the Wheeler–DeWitt equation

“…This is because all the 10 500 different string theory vacuum states [48], can be viewed as real vacuum states of different universes [49]. The multiverse is most naturally analysed using a third quantized formalism of quantum gravity [50]- [52]. This is because the Wheeler-DeWitt equation can be viewed as the Schroedinger's equation for a single universe.…”

Fourth quantization

“…In the Schroedinger's picture this diagonalized operator version of φ(a) is represented by a time independent field φ, and the operator version of p(a) is represented by −i∂/∂φ. Now we can write the third quantized Schroedinger equation for the multiverse as [20] …”

“…In doing so a solution to the problem of time has also been proposed. The uncertainty principle for third quantization of f (R) gravity has also been analysed [20]. If we go to fourth quantization we will go to a theory of multi-multiverses.…”