Loop Quantum Gravity

The First 30 Years

Abhay Ashtekar, Jorge Pullin;;;

• Editore: WSPC
• Anno: 2017
• ISBN:
  9789813220010
• Argomento: Scienze fisiche
• Categoria: Cosmologia

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This volume presents a snapshot of the state-of-the-art in loop quantum gravity from the perspective of younger leading researchers. It takes the reader from the basics to recent advances, thereby bridging an important gap.

The aim is two-fold — to provide a contemporary introduction to the entire field for students and post-docs, and to present an overview of the current status for more senior researchers. The contributions include the latest developments that are not discussed in existing books, particularly recent advances in quantum dynamics both in the Hamiltonian and sum over histories approaches; and applications to cosmology of the early universe and to the quantum aspects of black holes.

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Part 1

Introduction

... a really new field of experience will always lead to crystallization of a new system of scientific concepts and laws ... when faced with essentially new intellectual challenges, we continually follow the example of Columbus who possessed the courage to leave the known world in almost insane hope of finding land beyond the sea.

– Werner Heisenberg (Changes in the Foundation of Exact Science)

An Overview

Abhay Ashtekar^∗ and Jorge Pullin^†

^∗Institute for Gravitation and the Cosmos and Department of Physics, Pennsylvania State University, University Park, PA 16802, USA ^†Department of Physics and Astronomy, Louisiana State University, Baton Rouge, LA 70803-4001, USA

The quote on the last page from Heisenberg’s essay Changes in the Foundation of Exact Science succinctly captures the spirit that drives Loop Quantum Gravity (LQG). One leaves behind the terra firma of a rigid spacetime continuum in the hope of finding a more supple and richer habitat for physics beyond Einstein. In Columbus’ case, while the vision and hope that led to the expedition were indeed almost ‘insane’, he was well aware of the risks. Therefore he embarked on the voyage well prepared, equipped with the most reliable navigational charts and tools then available. Similarly, in LQG one starts with principles of general relativity and quantum mechanics that are firmly rooted in observations, knowing fully well that one will encounter surprises along the way and habitats at the destination will not look anything like those on these charts. Yet the known charts are essential at the point of departure to ensure that the sails are properly aligned and one does not drift into a fantasy landscape with little relation to the physical world we inhabit.

The five chapters in Part 2 of this volume describe current status of this voyage and the new habitats it has already led to. One starts with well established general relativity coupled to matter and uses proven tools from quantum mechanics insisting, however, that there be no fields in the background, not even a space-time metric. This insistence leads one to a rigorous mathematical framework whose conceptual implications are deep [1–3]. Quantum spacetime does not look like a 4-dimensional continuum at all; fundamental excitations of geometry — and hence of gravity — are polymer like; geometric observables have purely discrete eigenvalues; local curvature in the classical theory is replaced by non-local holonomies of a spin connection; and, quantum dynamics inherits a natural, built-in ultraviolet cut-off. There is no ‘objective time’ to describe quantum dynamics; there are no rigid light cones to formulate causality. Nonetheless, time evolution can be described in detail through relational dynamics in the cosmological setting, where familiar causality emerges through qualitatively new effective descriptions that are valid all the way to the full Planck regime. Thus, the final landscape is very different from that of general relativity and quantum mechanics, although both provided guiding principles at the point of departure.

The three Chapters of Part 3 illustrate the Planck scale ‘flora and fauna’ that inhabits this new landscape. The ultraviolet properties of geometry naturally tames the most important singularities of general relativity — in particular, the big bang is replaced by a quantum bounce. In the very early universe, cosmological perturbations propagate on these regular, bouncing quantum geometries, giving rise to effects that are within observational reach. Quantum geometry has also opened a new window on the microscopic degrees of freedom of horizons. Singularity resolution gives rise to a quantum extension of classical spacetimes, creating new paradigms for black hole evaporation in which the evolution is unitary. Finally, several possibilities have been proposed to test LQG ideas in astrophysics and cosmology. They all involve additional assumptions/hypotheses beyond mainstream LQG. Nonetheless, the very fact that relation to observations can be contemplated, sometimes through detailed calculations, provides a measure of the maturity of the subject.

The purpose of this Introduction is twofold: (i) To provide a global overview to aid the beginning researcher navigate through Parts II and III, especially by comparing and contrasting ideas in individual Chapters; and, (ii) Supplement the detailed discussions in these Parts with a brief discussion of a few general, conceptually important points. To keep the bibliography to a manageable size, we will refer only to reviews and monographs (rather than research articles) where further details can be found. We urge the beginning readers to read this Introduction first, as it spells out the overall viewpoint and motivation that is often taken for granted in individual Chapters.

The Setting

In LQG one adopts the viewpoint that among fundamental forces of Nature, gravity is special: it is encoded in the very geometry of spacetime. This is a central feature of GR, a crystallization of the equivalence principle that lies at the heart of the theory. Therefore, one argues, it should be incorporated at a fundamental level in a viable quantum theory.

The perturbative treatments which dominated the field since the 1960s ignored this aspect of gravity. They assumed that the underlying spacetime can be taken to be a continuum, endowed with a smooth background geometry, and the quantum gravitational field can be treated as any other quantum field on this background. But the resulting perturbation theory around Minkowski spacetime turned out to be non-renormalizable; the strategy failed to achieve the initial goals. The new strategy is to free oneself of the background spacetime continuum that seemed indispensable for formulating and addressing physical questions. In particular, in contrast to approaches developed by particle physicists, one does not begin with quantum matter on a background geometry and then use perturbation theory to incorporate quantum effects of gravity. Matter and geometry are both quantum-mechanical at birth. There is often an underlying manifold but no metric, or indeed any other physical fields, in the background.^a

In classical gravity, Riemannian geometry provides the appropriate mathematical language to formulate the physical, kinematical notions as well as the final dynamical equations. This role is now taken by quantum Riemannian geometry. In the classical domain, general relativity stands out as the best available theory of gravity, some of whose predictions have been tested to an amazing degree of accuracy, surpassing even the legendary tests of quantum electrodynamics. Therefore, it is natural to ask: Does quantum general relativity, coupled to suitable matter exist as a consistent theory non-perturbatively? There is no implication that such a theory would be the final, complete description of Nature. Nonetheless, this is a fascinating and important open question in its own right.

In particle physics circles the answer to this question is often assumed to be in the negative, not because there is concrete evidence against non-perturbative quantum gravity, but because of the analogy to the theory of weak interactions. There, one first had a 4-point interaction model due to Fermi which works quite well at low energies but which fails to be renormalizable. Progress occurred not by looking for non-perturbative formulations of the Fermi model but by replacing the model by the Glashow-Salam-Weinberg renormalizable theory of electro-weak interactions, in which the 4-point interaction is replaced by W ^± and Z propagators. Therefore, it is often assumed that perturbative non-renormalizability of quantum general relativity points in a similar direction. However this argument overlooks the crucial fact that, in the case of general relativity, there is a qualitatively new element. Perturbative treatments pre-suppose that the spacetime can be assumed to be a continuum at all scales of interest to physics under consideration. This assumption is safe for weak interactions. In the gravitational case, on the other hand, the scale of interest is the Planck length ℓ_Pl and there is no physical basis to pre-suppose that the continuum picture should be valid down to that scale. The failure of the standard perturbative treatments may largely be due to this grossly incorrect assumption and a non-perturbative treatment which correctly incorporates the physical micro-structure of geometry may well be free of these inconsistencies.

Are there any situations, outside loop quantum gravity, where such physical expectations are borne out in detail mathematically? The answer is in the affirmative. There exist quantum field theories (such as the Gross-Neveu model in three dimensions) in which the standard perturbation expansion is not renormalizable although the theory is exactly soluble [5]! Failure of the standard perturbation expansion can occur because one insists on perturbing around the trivial, Gaussian point rather than the more physical, non-trivial fixed point of the renormalization group (RG) flow. Interestingly, thanks to developments in the Asymptotic Safety program there is now growing evidence that situation may be similar in quantum general relativity [6]. Although there are some basic differences [7] between LQG and the Asymptotic Safety program, these results provide concrete support to the idea that non-perturbative treatments of quantum general relativity can lead to an ultraviolet regular theory.^b

However, even if the LQG program could be carried out to completion, there is no a priori reason to assume that the result would be the ‘final’ theory of all known physics. In particular, as is the case with classical general relativity, while requirements of background independence and general covariance do restrict the form of interactions between gravity and matter fields and among matter fields themselves, the theory would not have a built-in principle which determines these interactions. Put differently, such a theory may not be a satisfactory candidate for unification of all known forces. However, just as general relativity has had powerful implications in spite of this limitation in the classical domain, LQG should have qualitatively new predictions, pushing further the existing frontiers of physics. Indeed, unification does not appear to be an essential criterion for usefulness of a theory even in other interactions. QCD, for example, is a powerful theory even though it does not unify strong interactions with electro-weak ones. Furthermore, the fact that we do not yet have a viable candidate for grand unified theory does not make QCD any less useful. Finally, as the three Chapters in Part 3 illustrate, LQG has already made interesting predictions for quantum physics of black holes and the very early universe, some of which are detailed and make direct contact with observations.

Quantum Riemannian Geometry

Since the basic dynamical variable in general relativity is the spacetime metric, Wheeler advocated the view that we should regard it as geometrodynamics, a dynamical theory of 3-metrics q_ab that constitute the configuration variable. For the three other basic forces of Nature, on the other hand, the dynamical variable is a connection 1-form that takes values in the Lie algebra of the appropriate internal group. In QED the connection enables one to parallel transport electrons and positrons while in QCD it serves as the vehicle to parallel transport quarks. Now the configuration variable is a spatial connections

we have theories of connection-dynamics. Weinberg, in particular, has emphasized that this difference has driven a ‘wedge between general relativity and the theory of elementary particles’ [9].

As described in Chapter 1, the starting point in LQG is a reformulation of general relativity as a dynamical theory of spin connections [4]. We now know that the idea can be traced back to Einstein and Schrödinger who, among others, had recast general relativity as a theory of connections already in the fifties. (For a brief account of this fascinating history, see [10].) However, they used the ‘Levi-Civita connection’ that features in the parallel transport of vectors and found that the theory becomes rather complicated. The situation is very different with self-dual (or anti-self-dual) spin connections. For example, the dynamical evolution dictated by Einstein’s equations can now be visualized simply as a geodesic motion on the ‘superspace’ of spin-connections (with respect to a natural metric extracted from the constraint equations). Furthermore, the (anti-)self-dual connections have a direct physical interpretation: they are the vehicles used to parallel transport spinors with definite helicities of the standard model. With this formulation of general relativity, Weinberg’s ‘wedge’ disappears. In particular, phase-space of general relativity is now the same as that of gauge theories of the other three forces of Nature [4, 11]. However, in the Lorentzian signature, the (anti-)self-dual connections are complex-valued and, so far, this fact has been a road-block in the construction of a rigorous mathematical framework in the passage to the quantum theory.^c Therefore, the strategy is to pass to real connection variables by performing a canonical transformation [1–3]. The canonical transformation introduces a real, dimensionless constant γ, referred to as the Barbero-Immirzi parameter; the (anti-) self-dual Hamiltonian framework is recovered by formally setting γ = ±i.

For real connection variables the ‘internal gauge group’ reduces to SU(2), which is compact. Therefore, as explained in Chapter 1, it is possible to introduce integral and differential calculus on the infinite-dimensional space of (generalized) connections rigorously without having to introduce background geometrical fiel...