Third quantization of f(R)-type gravity

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 make a suitable change of variable and the second quantization, and we obtain the Wheeler–DeWitt equation. The third quantization of this theory has been considered. The uncertainty relation of the universe has been investigated in the example of f(R)-type gravity, where f(R) = R2. It has been shown that, at late times namely the scale factor of the universe is large, the spacetime does not contradict to become classical, and, at early times namely the scale factor of the universe is small, the quantum effects dominate.