Gravity quantized: Loop quantum gravity with a scalar field

Abstract

but we do not have quantum gravity." This phrase is often used when analysis of a physical problem enters the regime in which quantum gravity effects should be taken into account. In fact, there are several models of the gravitational field coupled to (scalar) fields for which the quantization procedure can be completed using loop quantum gravity techniques. The model we present in this paper consist of the gravitational field coupled to a scalar field. The result has similar structure to the loop quantum cosmology models, except for that it involves all the local degrees of freedom of the gravitational field because no symmetry reduction has been performed at the classical level. PACS numbers: 4.60.Pp; 04.60.-m; 03.65.Ta; 04.62.+v I. INTRODUCTIONThe recent advances in loop quantum gravity (LQG) [1][2][3][4], strongly suggest that the goal of constructing a candidate for quantum theory of gravity and the Standard Model is within reach. Remarkably, that goal can be addressed within the canonical formulation of the original Einstein's general relativity in four dimensional spacetime. A way to define 'physical' dynamics in a background independent theory, where spacetime diffeomorphisms are treated as a gauge symmetry, is the framework of relational Dirac observables (often also called "partial" observables [5], [6,7],[8] section I.2 of [2]). The main idea is, that part of the fields adopt the role of a dynamically coupled observer, with respect to which the physics of the remaining degrees of freedom in the system is formulated. In this framework the emergence of the dynamics, time and space can be explained as an effect of the relations between the fields. As far as technical issues of a corresponding quantum theory are concerned, the most powerful example of the relational observables framework is the deparametrization technique [9][10][11][12]. This allows to map canonical General Relativity into a theory with a (true) non-vanishing Hamiltonian, that is independent of the (emergent) time provided by the observer fields. All this can be achieved at the classical level, the framework of Loop Quantum Gravity (LQG) itself, provides then the tools of the quantum theory like quantum states, the Hilbert spaces, quantum operators of the geometry and fields and well defined quantum operators for the classical constraints of General Relativity (see [2],[4] and references therein). The combination of LQG with the relational observables and deparametrization framework makes it possible to construct general relativistic quantum models. Applying LQG techniques to perform the quantization step has the consequence that the quantum fields of the Standard Model have to be reintroduced within the scheme of LQG. This is due to the reason that the standard quantum field theory (QFT) defined on the Minkowski (or even ADS) background is incompatible with quantization approach used in LQG. Therefore, the resulting quantum theory of gravity cannot be just coupled to the Standard Model in it's present form. The formulation of the full St...

Loop quantumf(R)theories

“…For reviews in this field, we refer to [1][2][3][4]. It is remarkable that, as a non-renormalizable theory, GR can be non-perturbatively quantized by the loop quantization procedure [5]. This background-independent quantization relies on the key observation that classical GR can be cast into the connection dynamical formalism with structure group of S U (2).…”

Curvature operator for loop quantum gravity

“…Our hope is that the curvature operator we built will allow us to construct a more analytically manageable Hamiltonian operator and will give the possibility to construct solutions in order to get more insights on the dynamical sector of LQG both in the full theory [43,44] and in simplified models [45][46][47][48]. This will be the subject of upcoming works.…”

Separable Hilbert space for loop quantization

“…Recent progress in this field, involving in particular the application of deparametrization techniques with respect to the matter frames [3][4][5], has brought LQG to a level which makes it possible to probe its dynamical predictions [5].…”