Yoshiaki Ohkuwa

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.

Third quantization of

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) = R2. 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.

f(R)

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) = R2. 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.

-type gravity II - General

In this paper, we analyse the Wheeler-DeWitt equation in the third quantized formalism. We will demonstrate that for certain operator ordering, the early stages of the universe are dominated by quantum fluctuations, and the universe becomes classical at later stages during the cosmic expansion. This is physically expected, if the universe is formed from quantum fluctuations in the third quantized formalism. So, we will argue that this physical requirement can be used to constrain the form of the operator ordering chosen. We will explicitly demonstrate this to be the case for two different cosmological models.

f(R)

We propose a canonical formalism of f(R)-type gravity using a set of phase variables which is partly different from that used in the formalism of Buchbinder and Lyakhovich (BL). The new coordinates corresponding to the time derivatives of the metric are taken to be the Lie derivatives of the metric, which is the same as in BL. The momenta canonically conjugate to the new coordinates and Hamiltonian density are defined similarly to the formalism of Ostrogradski. It is shown that, in our formalism, the Hamiltonian is invariant under the change of the original coordinates, which is not the case in the formalism of BL.A canonical formalism of f(R)-type gravity is proposed, resolving the problem in the formalism of Buchbinder and Lyakhovich(BL). The new coordinates corresponding to the time derivatives of the metric are taken to be its Lie derivatives which is the same as in BL. The momenta canonically conjugate to them and Hamiltonian density are defined similarly to the formalism of Ostrogradski. It is shown that our method surely resolves the problem of BL.

case

We present a canonical formalism of the f(R)-type gravity using the Lie derivatives instead of the time derivatives by refining the formalism of our group. The previous formalism is a direct generalization of the Ostrogradski’s formalism. However the use of the Lie derivatives was not sufficient, in that Lie derivatives and time derivatives are used in a mixed way, so that the expressions are somewhat complicated. In this paper, we use the Lie derivatives and foliation structure of the spacetime thoroughly, which makes the procedure and the expressions far more concise.

Third quantization of

f(R)-type gravity in the first order formalism is interpreted as Einstein gravity with non-minimal coupling arising from the use of unphysical frame. Identification of the corresponding second order higher-curvature gravity in the physical frame is proposed by requiring that the action is the same.f(R)-type gravity in the first order formalism is interpreted as Einstein gravity with non-minimal coupling arising from the use of unphysical frame. Identification of the corresponding second order higher-curvature gravity in the physical frame is proposed by requiring that the action is the same.

f(R)

In this paper, we will analyse virtual black holes using the third quantization formalism. As the virtual black hole model depends critically on the assumption that the quantum fluctuations dominate the geometry of spacetime at Planck scale, we will analyse the quantum fluctuations for a black hole using third quantization. We will demonstrate that these quantum fluctuations depend on the factor ordering chosen. So, we will show that only certain values of the factor ordering parameter are consistent with virtual black holes model of spacetime foam.

-type gravity II

We first review the equivalence theorem of the f(R)-type gravity to Einstein gravity with a scalar field by deriving it in a self-contained and pedagogical way. Then we describe the problem of to what extent the equivalence holds. Main problems are (i) Is the surface term given by Gibbons and Hawking which is necessary in Einstein gravity also necessary in the f(R)-type gravity? (ii) Does the equivalence hold also in quantum theory? (iii) Which metric is physical, i.e., which metric should be identified with the observed one? In this work, we clarify the problem (i) and review the problem (ii) in a canonical formalism which is the generalization of the Ostrogradski one. We briefly comment on the problem (iii). Some discussions are given on one of the results of (ii) concerning the general relativity in non-commutative spacetime.