The Equivalence Theorem in the Generalized Gravity of f(R)-Type and Canonical Quantization

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.Comment: 23 pages. Ecept for the change of style from {book} to {article} and related changes, e.g., addition of abstract and the form of References, as well as the addition of Appendix B, the work has been published as one of the chapters in the book "Advances in Quantum Theory" (2012, ed. Ion I. Cotaescu; InTech Open Access Publisher

Time without time: a stochastic clock model

We study a classical reparametrization-invariant system, in which ``time'' is not a priori defined. It consists of a nonrelativistic particle moving in five dimensions, two of which are compactified to form a torus. There, assuming a suitable potential, the internal motion is ergodic or more strongly irregular. We consider quasi-local observables which measure the system's ``change'' in a coarse-grained way. Based on this, we construct a statistical timelike parameter, particularly with the help of maximum entropy method and Fisher-Rao information metric. The emergent reparametrization-invariant ``time'' does not run smoothly but is simply related to the proper time on the average. For sufficiently low energy, the external motion is then described by a unitary quantum mechanical evolution in accordance with the Schr\"odinger equation.Comment: 18 pages; LaTeX. 4 (.ps) plus 2 (.gif) figure file

Third quantization of f(R)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 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.Comment: 9 pages, Arbitrary constants in (4.19) are changed to arbitrary functions of $\varphi$. Conclusions are not changed. References are added. Typos are correcte

Classical and Quantum Solutions and the Problem of Time in R2R^2 Cosmology

We have studied various classical solutions in $R^2$ cosmology. Especially we have obtained general classical solutions in pure $R^2$\ cosmology. Even in the quantum theory, we can solve the Wheeler-DeWitt equation in pure $R^2$\ cosmology exactly. Comparing these classical and quantum solutions in $R^2$\ cosmology, we have studied the problem of time in general relativity.Comment: 17 pages, latex, no figure, one reference is correcte

Some Aspects of Virtual Black Holes

In this paper we shall consistently third quantize modified gravity. Then we shall analyse certain aspects of virtual black holes in this third quantized modified gravity. We will see how a statistical mechanical origin for the Bekenstein-Hawking entropy naturally arises in this model. Furthermore, in this model the area and thus the entropy of a real macroscopic black hole is quantized. Virtual black holes cause loss of quantum coherence and this gives an intrinsic entropy to all physical systems which can be used to define a direction of time and hence provide a solution to the problem of time.Comment: 11 pages, 0 figures, accepted for publication in JET

Born-Infeld Theory and Stringy Causality

Fluctuations around a non-trivial solution of Born-Infeld theory have a limiting speed given not by the Einstein metric but the Boillat metric. The Boillat metric is S-duality invariant and conformal to the open string metric. It also governs the propagation of scalars and spinors in Born-Infeld theory. We discuss the potential clash between causality determined by the closed string and open string light cones and find that the latter never lie outside the former. Both cones touch along the principal null directions of the background Born-Infeld field. We consider black hole solutions in situations in which the distinction between bulk and brane is not sharp such as space filling branes and find that the location of the event horizon and the thermodynamic properties do not depend on whether one uses the closed or open string metric. Analogous statements hold in the more general context of non-linear electrodynamics or effective quantum-corrected metrics. We show how Born-Infeld action to second order might be obtained from higher-curvature gravity in Kaluza-Klein theory. Finally we point out some intriguing analogies with Einstein-Schr\"odinger theory.Comment: 31 pages, 4 figures, LaTex; Some comments and references adde