What is it all about?

Harold: For quite some time now, you have been talking about ‘gravity being an emergent phenomenon’ and a ‘thermodynamic perspective on gravity’. This is quite different from the conventional point of view in which gravity is a fundamental interaction and spacetime thermodynamics of, say, black holes is a particular result which can be derived in a specific context. Honestly, while I find your papers fascinating I am not clear about the broad picture you are trying to convey. Maybe you could begin by clarifying what this is all about, before we plunge into the details? What is the roadmap, so to speak?

Me: To begin with, I will show you that the equations of motion describing gravity in any diffeomorphism-invariant theory can be given [2] a suggestive thermodynamic reinterpretation (Sections 2.2, 2.3).

General remarks

What follows is essentially an extended summary of my actual talk, which I hope adequately conveys the gist of what I did report at the Stellenbosch meeting. I hope, also, that it can serve as a small token of the great respect that I have for George Ellis – in the honouring of his 70th birthday – both as a person and for the enormous contributions that he has made to science and to the cause of humanity.

I briefly discuss three different topics, all of which have relevance to the nature of quantum space-time geometry. The first has to do with the very framework of quantum theory in relation to Einstein's foundational principle of equivalence, and provides a reason for anticipating a change in the rules of quantum mechanics when superpositions of significant displacements of mass are involved.

Harold: It seems to me that there has been an enormous amount of resources spent on the various quantum gravity programs. Is there an actual proof that gravity has to be quantized at all?

Steph: Well, that depends on what you think would constitute a proof. Does classical mechanics have to be quantized? Apparently. Do we have a proof to that effect? No. We have a theory, it makes predictions and seems to agree with nature so we accept it. By the nature of the whole enterprise though, if we encounter a prediction that is wrong then we have to give up the theory. So, does gravity need to be quantized? I don't know. What I do know is that it is a fundamental interaction. The other three fundamental interactions all seem to have consistent quantum descriptions and, personally, I find that three out of four having quantum descriptions and only one being completely classical is unappealing. But maybe this is just the way nature is.

Harold: Where does the need to quantize gravity come from except for a belief in unification which may or may not be satisfied in reality?

Steph: I would say that it comes from a belief that at sufficiently small scales all the interactions exhibit quantum behaviour.

Introduction

Gauge–string duality is arguably one of the most profound problems in modern physics [1–11]. This duality implies that the gauge-invariant observables (operators) in QCD are in one-to-one correspondence with the physical states (vertex operators) in string theory. The reason why extended objects (such as strings) appear in QCD is quite natural. If we recall standard electrodynamics, there are two ways of describing it: either in terms of local electric fields (Coulomb's approach), or in terms of the geometry of electric field lines (Faraday's approach). In the case of electromagnetic theory, Coulomb's approach turns out to be far more efficient. In the case of QCD, however, things are quite different. While the electric field lines created by the charged particles are spread over the entire space, the gluon field lines are confined to thin flux tubes. These flux tubes, connecting quarks, can be naturally interpreted as one-dimensional extended objects, known as QCD strings.

Introduction

The field of non-perturbative and background-independent quantum gravity has progressed considerably over the past few decades [78]. New research directions are being developed, new important developments are taking place in existing approaches, and some of these approaches are converging to one another. As a result, ideas and tools from one become relevant to another, and trigger further progress. The group field theory (GFT) formalism [39, 77, 79] nicely captures this convergence of approaches and ideas. It is a generalization of the much studied matrix models for 2D quantum gravity and string theory [28, 53]. At the same time, it generalizes it, as we are going to explain, by incorporating the insights coming from canonical loop quantum gravity and its covariant spin foam formulation of the dynamics, and so it became an important part of this approach to the quantization of 4D gravity [72, 74, 81, 85]. Furthermore, it is a point of convergence of the same loop quantum gravity approach and of simplicial quantum gravity approaches, like quantum Regge calculus [93] and dynamical triangulations [3, 79], in that the covariant dynamics of the first takes the form, as we are going to see, of simplicial path integrals.

Introduction

The present chapter is based on [1], and elaborates on several issues and arguments that were not fully spelled out there. In that work, a first step was taken towards quantization of the one-dimensional “geodesic” E10/K(E10) coset model which had been proposed in [2] as a model of M-theory. Here, E10 denotes the hyperbolic Kac-Moody group E10 which is an infinite-dimensional extension of the exceptional Lie group E8, and plays a similarly distinguished role among the infinite-dimensional Lie algebras as E8 does among the finite-dimensional ones. The proposal of [2] had its roots both in the appearance of so-called “hidden symmetries” of exceptional type in the dimensional reduction of maximal supergravity to lower dimensions [3], as well as in the celebrated analysis of Belinskii, Khalatnikov and Lifshitz (BKL) [4] of the gravitational field equations in the vicinity of a generic space-like (cosmological) singularity.

Introduction

In its different incarnations, quantum gravity must face a diverse set of fascinating problems and difficulties, a set of issues best seen as both challenges and opportunities. One of the main problems in canonical approaches, for instance, is the issue of anomalies in the gauge algebra underlying space-time covariance. Classically, the gauge generators, given by constraints, have weakly vanishing Poisson brackets with each other: they vanish when the constraints are satisfied. After quantization, the same behavior must be realized for commutators of quantum constraints (or for Poisson brackets of effective constraints), or else the theory becomes inconsistent due to gauge anomalies. If and how canonical quantum gravity can be obtained in an anomaly-free way is an important question, not yet convincingly addressed in full generality.

Introduction

Obtaining an acceptable quantum theory of gravity is the key remaining problem in fundamental theoretical physics. A basic difficulty in formulating such a theory was already recognized in the earliest approaches to this problem in the 1930s: the dimensional character of Newton's constant gives rise to ultraviolet divergent quantum correction integrals. Naïve power counting of the degree of divergence Δ of an L-loop diagram in D-dimensional gravity theories yields the result

which grows linearly with loop order, implying the requirement for higher and higher-dimensional counterterms to renormalize the divergences. In the 1970s, this was confirmed explicitly in the first Feynman diagram calculations of the radiative corrections to systems containing gravity plus matter [51]. The time lag between the general perception of the UV divergence problem and its first concrete demonstration was due to the complexity of Feynman diagram calculations involving gravity.

Introduction: seeing atoms with the naked eye

At present, one of the most important tasks in theoretical physics is to understand the nature of spacetime at the Planck scale. Various indications from our current most successful theories point to this scale: quantum effects are to be expected to invalidate the general theory of relativity here. What should replace our current best understanding of spacetime? This question remains controversial as no theory of quantum gravity can yet be claimed to be complete. For example, some researchers are convinced that the kinematical structure used to replace the continuous manifolds of GR should be discrete, but others do not adhere to this requirement. George Ellis' great contribution to our understanding of spacetime, and his interest in the issue of spacetime discreteness, make this a very appropriate topic for these proceedings.

Introduction

Loop quantum gravity is a non-perturbative approach to the quantum theory of gravity, in which no classical background metric is used. In particular, its starting point is not a linearized theory of gravity. As a consequence, while it still operates according to the rules of quantum field theory, the details are quite different from those for field theories that operate on a fixed classical background space-time. It has considerable successes to its credit, perhaps most notably a quantum theory of spatial geometry, in which quantities such as area and volume are quantized in units of the Planck length, and a calculation of black hole entropy for static and rotating, charged and neutral black holes. But there are also open questions, many of them surrounding the dynamics (“quantum Einstein equations”) of the theory.

In contrast to other approaches such as string theory, loop quantum gravity is rather modest in its aims. It is not attempting a grand unification, and hence is not based on an overarching symmetry principle, or some deep reformulation of the rules of quantum field theory. Rather, the goal is to quantize Einstein gravity in four dimensions. While, as we will explain, a certain amount of unification of the description of matter and gravity is achieved, in fact, the question of whether matter fields must have special properties to be consistently coupled to gravity in the framework of loop quantum gravity is one of the important open questions in loop quantum gravity.

Introduction

The problem of quantizing gravity has proved to be a difficult one. To solve this problem, it seems to be necessary to answer the question “What is spacetime?” This challenges the most basic assumptions we are used to making; a radical new idea may be needed. Further, the hope of any guidance from experiment seems to be out of the question. One might conclude that the situation is hopeless. Drawing on recent insights from the AdS/CFT correspondence, we are nonetheless, optimistic.

The AdS/CFT correspondence [1] claims an exact equality between N =4 super Yang-Mills theory in flat (3+1)-dimensional Minkowski spacetime and Type IIB string theory on an asymptotically AdS5×S5 background. Type IIB string theory is a theory of closed strings; at least within string perturbation theory, theories of closed strings provide a consistent UV completion of gravity. The fact that such an equality exists is highly unexpected and non-trivial, and (as we will try to convince the reader) can be used to gain insight into the nature of spacetime. George Ellis opened the Foundations of Space and Time workshop by holding up two fingers and asking “are there an infinite or a finite number of places a particle could occupy between my fingers?” We don't know the answer to George's question.

Introduction

Einstein's recognition early last century that gravity can be interpreted as the curvature of space and time represented an enormous step forward in the way we think about fundamental physics. Besides its obvious impact for understanding gravity over astrophysical distances – complete with resolutions of earlier puzzles (like the detailed properties of Mercury's orbit) and novel predictions for new phenomena (like the bending of light and the slowing of clocks by gravitational fields) – its implications for other branches of physics have been equally profound.

These implications include many ideas we nowadays take for granted. One such is the universal association of fundamental degrees of freedom with fields (first identified for electromagnetism, but then cemented with its extension to gravity, together with the universal relativistic rejection of action at a distance).

Foreword

More than 40 years after its original proposal, there is no experimental evidence for string theory or, else, a satisfactory – possibly holographic – description of the QCD string is not yet available.

Still, the Veneziano model predicts a massless vector boson in the open string spectrum and the Shapiro-Virasoro model a massless tensor boson in the closed string spectrum. These two particles can naturally be associated with the two best known forces in Nature: gravity and electromagnetism. After GSO projection, supersymmetry then guarantees the presence of massless fermions.

Moreover, string theory makes a definite – albeit incorrect – prediction for the number of space-time dimensions: 26 for bosonic strings, 10 for superstrings. This basic fact led to many so-far unsuccessful attempts to get rid of the undesired extra dimensions. Calabi-Yau and orbifold compactifications, non-geometric Gepner models are the most famous examples.

Introduction

Stephen Hawking and George Ellis prefaced their seminal book, The Large Scale Structure of Space-Time, with the explanation that their aim was to understand spacetime “on length-scales from 10−13 cm, the radius of an elementary particle, up to 1028 cm, the radius of the universe” [24]. While many deep questions remain, ranging from cosmic censorship to the actual topology of our universe, we now understand the basic structure of spacetime at these scales: to the best of our ability to measure such a thing, it behaves as a smooth (3+1)-dimensional Riemannian manifold.

At much smaller scales, on the other hand, the proper description is far less obvious. While clever experimentalists have managed to probe some features down to distances close to the Planck scale [43], for the most part we have neither direct observations nor a generally accepted theoretical framework for describing the very small-scale structure of spacetime. Indeed, it is not completely clear that “space” and “time” are even the appropriate categories for such a description.

Quantum gravity and quantal reality

Why, in our attempts to unify our theories of gravity and the quantum, has progress been so slow? One reason, no doubt, is that it's simply a very hard problem. Another is that we lack clear guidance from experiments or astronomical observations. But I believe that a third thing holding us back is that we haven't learned how to think clearly about the quantum world in itself, without reference to “observers” and other external agents.

Because of this we don't really know how to think about the Planckian regime where quantum gravity is expected to be most relevant. We don’t know how to think about the vacuum on small scales, or about the inside of a black hole, or about the early universe. Nor do we have a way to pose questions about relativistic causality in such situations.

After almost a century, the field of quantum gravity remains as difficult, frustrating, inspiring, and alluring as ever. Built on answering just one question – How can quantum mechanics be merged with gravity? – it has developed into the modern muse of theoretical physics.

Things were not always this way. Indeed, inspired by the monumental victory against the laws of Nature that was quantum electrodynamics (QED), the 1950s saw the frontiers of quantum physics push to the new and unchartered territory of gravity with a remarkable sense of optimism. After all, if nothing else, gravity is orders of magnitude weaker than the electromagnetic interaction; surely it would succumb more easily. Nature, it would seem, is not without a sense of irony. For an appreciation of how this optimism eroded over the next 30 years, there is perhaps no better account than Feynman's Lectures on Gravitation. Contemporary with his epic Feynman Lectures on Physics, these lectures document Feynman's program of quantizing gravity “like a field theorist.” In it he sets out to reverse-engineer a theory of gravity starting from the purely phenomenological observations that gravity is a long-range, static interaction that couples to the energy content of matter with universal attraction.

Introduction

Since time plays such a prominent role in ordinary flat space quantum field theory defined by a Hamiltonian, it is of interest to study the role of time even in toy models of quantum gravity where the role of time is much more enigmatic. The model we will describe in this chapter is the so-called causal dynamical triangulation (CDT) model of quantum gravity. It starts by providing an ultraviolet regularization in the form of a lattice theory, the lattice link length being the (diffeomorphism-invariant) UV cut-off. In addition, the lattice respects causality. It is formulated in the spirit of asymptotic safety, where it is assumed that quantum gravity is described entirely by “conventional” quantum field theory, in this case by approaching a non-trivial fixed point [1, 2]. It is formulated in space-times with Lorentzian signature, but the regularized space-times which are used in the path integral defining the theory allow a rotation to Euclidean space-time. The action used is the Regge action for the piecewise linear geometry.