Discrete Lorentzian quantum gravity

Abstract

Just as for non-abelian gauge theories at strong coupling, discrete lattice methods are a natural tool in the study of non-perturbative quantum gravity. They have to reflect the fact that the geometric degrees of freedom are dynamical, and that therefore also the lattice theory must be formulated in a background-independent way. After summarizing the status quo of discrete covariant lattice models for four-dimensional quantum gravity, I describe a new class of discrete gravity models whose starting point is a path integral over Lorentzian (rather than Euclidean) space-time geometries. A number of interesting and unexpected results that have been obtained for these dynamically triangulated models in two and three dimensions make discrete Lorentzian gravity a promising candidate for a non-trivial theory of quantum gravity. QUANTUM GRAVITYFor the purposes of this presentation, quantum gravity will be defined as the non-perturbative quantization of the classical theory of general relativity, with and without the inclusion of other matter fields and interactions. Judging from our current knowledge of the fundamental laws of physics, it seems highly likely that at sufficiently large energies also gravitational interactions should be governed by quantum rather than by classical equations of motion. Quantum gravity -whose theoretical formulation is still elusive -should include a consistent description of local quantum phenomena in the presence of strong gravitational fields.It has been known for a long time that perturbative quantum gravity, based on a decompositionof the Lorentzian space-time metric g µν into the flat Minkowski metric η µν and a linear perturbation h µν (representing the degrees of freedom of a massless spin-2 graviton), leads to a nonrenormalizable field theory. Although this does not preclude the use of the perturbation series as an effective description of quantum gravity in the presence of an energy cut-off, it cannot serve as the definition of a fundamental theory. The ensuing need to quantize gravity nonperturbatively is not confined to field theory, but persists in string-theoretic formulations (where it is an unsolved problem as well). Imagine trying to obtain a quantum state representing a 4d Schwarzschild black hole with metricby superposing gravitonic excitations within string theory. However, because of the proportionality G ∼ g 2 str for Newton's constant G [1], this involves arbitrary powers of the string coupling g str , and is therefore an intrinsically nonperturbative construction.1.1. How do we quantize gravity nonperturbatively? The great success of lattice models in describing non-perturbative properties of QCD has for a long time been a motivation for applying discrete methods also in quantum gravity. I will be reporting on the status quo of path-integral ("covariant") lattice models for quantum gravity, and on how to make them more Lorentzian. However, it should be pointed out that there are other ways of tackling the problem, most notably, in a canonical continuum approach based...

Quantum Gravity via Causal Dynamical Triangulations

“…Even if this is the case, it is in principle possible that by adding more terms to the bare lattice action and suitably tuning the associated new coupling constants, an interesting continuum theory may emerge after all. This possibility has been investigated in the past [19], as well as more recently [20], but there is no conclusive evidence at this point that these modified Euclidean models can reproduce the physical properties of quantum gravity from CDT, the Lorentzian lattice gravity theory to which we will turn next (see also [21] for a variety of reviews of the subject). Fig.…”

Black hole evaporation within a momentum-dependent metric

“…Lorentz-breaking observable effects appear in Grand-Unification Theories [4], in String Theories [5], in Quantum Gravity [6], in foam-like quantum spacetimes [7]; in spacetimes endowed with a nontrivial topology or with a discrete structure at the Planck length [8,9], or with a (canonical or noncanonical) noncommutative geometry [10,11,12]; in the so-called "extensions" of the Standard Model incorporating breaking of Lorentz and CPT symmetries [13]; in theories with a variable speed of light or variable physical constants. In particular, the M-Theory [5], the Loop Quantum Gravity [8,9,14] and the Causal Dynamical Triangulation [15] lead to postulate an essentially discrete and quantized spacetime, where a fundamental mass-energy scale naturally arises, in addition to and c. An intrinsic length is directly correlated to the existence of a "cut-off" in the transferred momentum necessary to avoid the occurrence of "UV catastrophes" in Quantum Field Theories. Moreover, some authors suspect that the Lorentz symmetry breaking may play a role in extreme astrophysical phenomena as, e.g., the observation of ultra-high energy cosmic rays with energies [16] beyond the GreisenZatsepin-Kuzmin [17] cut-off, and of gamma rays bursts with energies beyond 20 TeV originated in distant galactic sources [18].…”

A Discrete History of the Lorentzian Path Integral

“…At an intermediate stage of the construction, we use a regularization in terms of simplicial "Regge geometries", that is, piecewise linear manifolds. In this approach, "computing the path integral" amounts to a conceptually simple and geometrically transparent 2 One should not get confused here by the fact that in gauge formulations of gravity which work with vierbeins e a µ instead of the metric tensor g µν , one has an additional local invariance under SO(3,1)-frame rotations, ie. elements of the Lorentz group, in addition to diffeomorphism invariance.…”

“…If one works in a continuum metric formulation of gravity, the symmetry group of the Einstein action is instead the group Diff(M) of diffeomorphisms on M , which in terms of local charts are simply the smooth invertible coordinate transformations x µ → y µ (x µ ). 2 I will in the following describe a particular path integral approach to quantum gravity, which is non-perturbative from the outset in the sense of being defined on the "space of all geometries" (to be defined later), without distinguishing any background metric structure (see also [1,2] for related reviews). This is closely related in spirit with the canonical approach of loop quantum gravity [3] and its more recent incarnations using so-called spin networks [4,5], although there are significant differences in methodology and attitude.…”