Third quantization of $f(R)$-type gravity II - General $f(R)$ case -
  In the previous paper we examined the third quantization of the $f(R)$-type gravity and studied the Heisenberg uncertainty relation of the universe in the example of $f(R)=R^2$. In this work the Heisenberg uncertainty relation of the universe is investigated in the general $f(R)$-type gravity where tachyonic states are avoided. It is shown that, at late times namely the scale factor of the universe is large, the spacetime becomes classical, and, at early times namely the scale factor of the universe is small, the quantum effects dominate. 
Yoshiaki Ohkuwa, Yasuo Ezawa
gr-qc