From general relativity to quantum gravity

Abstract

In general relativity (GR), spacetime geometry is no longer just a background arena but a physical and dynamical entity with its own degrees of freedom. We present an overview of approaches to quantum gravity in which this central feature of GR is at the forefront. However, the short distance dynamics in the quantum theory are quite different from those of GR and classical spacetimes and gravitons emerge only in a suitable limit. Our emphasis is on communicating the key strategies, the main results and open issues. In the spirit of this volume, we focus on a few avenues that have led to the most significant advances over the past 2-3 decades. 1

Minisuperspaces: Observables and Quantization

“…This general strategy is of course rather old (see, e.g., articles by Kuchař and Rovelli in [23] and the references they contain) although its full power does not seem to be always appreciated. What we wish to show in the next two subsections is: i) this strategy can be implemented in detail in all DIMT models; and, ii) the implementation enables us to ask and answer a number of "dynamical" questions of physical interest, including the fate of singularities.…”

“…Let us begin by noting that not all mathematically equivalent representations in quantum theory are suitable for addressing the issue of time. (For details, see, e.g., the article by Ashtekar in [23].) For the free relativistic particle, for example, time is explicit in the position representation; the quantum constraint equation, η ab ∇ a ∇ b Φ(x) = 0 can be immediately interpreted as the evolution equation.…”

Regularization of Kepler problem in κ-spacetime

“…Now, if we have canonical transformation connecting (x, p) and (y, q), then using the realization given by eqns. (14) and (15) we can establish a canonical transformation between (ψ, φ) and (η, ζ). This in turn assures that the system described in terms of either (ψ, φ) or (η, ζ) refers to the same physical system.…”

“…But since we live in a quantum world, a complete understanding of gravity requires formulating a theory of gravity consistent with principles of quantum mechanics. In fact, quantum theory of gravity is a holy grail of theoretical physics and people have been attacking the problem of quantizing gravity along numerous paths but not yet have been succeeded [12,13,14,15]. The reasoning that spacetime loses the operational meaning as one goes to energy scale of order of Planck scale have lead to introduction of spacetime with non-commuting coordinates [16].…”

Equivalence between the Lovelock–Cartan action and a constrained gauge theory

“…Essentially, these theories share the same problems of pure metric and Palatini theories of gravity [13,14,15]. In particular, the perturbative renormalizability or unitarity problems remain [16], as well as background independence [17,18] and so on.…”