Introduction

What is more natural than constructing space from elementary geometric building blocks? It is not as easy as one might think, based on our intuition of playing with Lego blocks in three-dimensional space. Imagine the building blocks are d-dimensional flat simplices all of whose side lengths are a, and let d > 2. The problem is that if we glue such blocks together carelessly we will with probability one create a space of no extension, in which it is possible to get from one vertex to any other in a few steps, moving along the one-dimensional edges of the simplicial manifold we have created. We can also say that the space has an extension which remains at the “cut-off” scale a. Our intuition coming from playing with Lego blocks is misleading here because it presupposes that the building blocks are embedded geometrically faithfully in Euclidean ℝ3, which is not the case for the intrinsic geometric construction of a simplicial space.

By contrast, let us now be more careful in our construction work by assigning to a simplicial space T – which we will interpret as a (Euclidean) spacetime – the weight e−S(T), where S(T) denotes the Einstein action associated with the piecewise linear geometry uniquely defined by our construction. As long as the (bare) gravitational coupling constant GN is large, we have the same situation as before.

Introduction: what is DSR?

The definition of doubly special relativity (DSR) (see for review) is deceptively simple. Recall that Special Relativity is based on two postulates: the relativity principle for inertial observers and the existence of a single observer independent scale associated with the velocity of light. In this DSR replaces the second postulate by assuming the existence of two observer-independent scales: the old one of velocity plus the scale of mass (or of momentum, or of energy). That's it.

Adding a new postulate has consequences, however. The most immediate one is the question: what does the second observer-independent scale mean physically? Before trying to answer this question, let us recall the concept of an observer-independent scale. It can be easily understood, when contrasted with the notion of dimensionful coupling constant, like the Planck constant ħ or the gravitational constant G. What is their status in relativity? Do they transform under Lorentz transformation? Well, naively, one would think that they should because they are given by dimensional quantities. But of course they do not. The point is that there is a special operational definition of these quantities. Namely each observer, synchronized with all the other observers, by means of the standard Einstein synchronization procedure, measures their values in an identical quasi-static experiment in her own reference frame (like the Cavendish experiment).

Introduction

The modern version of canonical Quantum Gravity is called loop quantum gravity (LQG), see for textbooks and for recent reviews. At present, there is no other canonical approach to Quantum Gravity which is equally well developed. LQG is a Quantum Field Theory of geometry and matter which is background independent and takes fully into account the backreaction of (quantum) matter on (quantum) geometry. Background independence means that there is no preferred spacetime metric available, rather the metric is a dynamical entity which evolves in tandem with matter, classically according to the Einstein equations. These precisely encode the backreaction. This is therefore an entirely new type of QFT which is radically different from ordinary QFT. One could even say that the reason for the fact that today there is not yet an established theory of Quantum Gravity is rooted in the background dependence of ordinary QFT. Therefore ordinary QFT (quantum mechanics) violates the background independence of classical GR while classical GR violates the quantum principle of QFT. This is the point where the two fundamental principles of modern physics collide. LQG tries to overcome this obstacle by constructing a background independent QFT.

In order to see in more detail where the background metric finds its way into the very definition of an ordinary QFT, recall the fundamental locality axiom of the algebraic approach.

Introduction

The principles of General Relativity

The construction of a quantum theory of gravity is a major ambition of modern physics research. However, the absence of any direct experimental evidence implies that we do not have any empirical point of reference about the principles that will underlie this theory. We therefore have to proceed mainly by theoretical arguments, trying to uncover such principles from the structure of the theories we already possess.

Clearly, the most relevant theory for this purpose is General Relativity, which provides the classical description of the gravitational field. General Relativity is essentially based on two principles, uncovered by Einstein after the continuous effort of seven years. The first one asserts the importance of the spacetime description: all gravitational phenomena can be expressed in terms of a Lorentzian metric on a four-dimensional manifold. The second one is the principle of general covariance: the Lorentzian metric is a dynamical variable, its equations of motion preserve their form in all coordinate systems of the underlying manifold.

The first principle defines the kinematics of General Relativity. It identifies the basic variables that are employed in the theory's mathematical description, and determines their relation to physical quantities measured in experiments. This principle implies that General Relativity is a geometric theory. It refers primarily to the relations between spacetime events: the metric determines their distance and causal relation.

Introduction

The problems of perturbative Quantum Field Theory (QFT) in relation to the UV behaviour of gravity have led to widespread pessimism about the possibility of constructing a fundamental QFT of gravity. Instead, we have become accustomed to thinking of General Relativity (GR) as an effective field theory, which only gives an accurate description of gravitational physics at low energies. The formalism of effective field theories provides a coherent framework in which quantum calculations can be performed even if the theory is not renormalizable. For example, quantum corrections to the gravitational potential have been discussed by several authors; see and references therein. This continuum QFT description is widely expected to break down at very short distances and to be replaced by something dramatically different beyond the Planck scale. There is, however, no proof that continuum QFT will fail, and the current situation may just be the result of the lack of suitable technical tools. Weinberg described a generalized, nonperturbative notion of renormalizability called “asymptotic safety” and suggested that GR may satisfy this condition, making it a consistent QFT at all energies. The essential ingredient of this approach is the existence of a Fixed Point (FP) in the Renormalization Group (RG) flow of gravitational couplings. Several calculations were performed using the ∈-expansion around d = 2 dimensions, supporting the view that gravity is asymptotically safe.

Introduction

Although there is enormous uncertainty about the nature of Quantum Gravity (QG), one thing is quite certain: the commonly used ideas of space and time should break down at or before the Planck length is reached. For example, elementary scattering processes with a Planck-sized center-of-mass energy create large enough quantum fluctuations in the gravitational field that space-time can no longer be treated as a classical continuum. It is then natural to question the exactness of the Lorentz invariance (LI) that is pervasive in all more macroscopic theories. Exact LI requires that an object can be arbitrarily boosted. Since the corresponding Lorentz contractions involve arbitrarily small distances, there is an obvious tension with the expected breakdown of classical space-time at the Planck length. Indeed, quite general arguments are made that lead to violations of LI within the two most popular approaches towards QG: string theory and loop quantum gravity

This has given added impetus to the established line of research dedicated to the investigation of ways in which fundamental symmetries, like LI or CPT, could be broken. It was realized that extremely precise tests could be made with a sensitivity appropriate to certain order of magnitude estimates of violations of LI.

Introduction

It is the opinion of this author that many theories of Quantum Gravity have already been discovered, but that the one which applies to the real world still remains a mystery. The theories I am referring to all go under the rubric of M/string-theory, and most practitioners of this discipline would claim that they are all “vacuum states of a single theory”. The model for such a claim is a quantum field theory whose effective potential has many degenerate minima, but I believe this analogy is profoundly misleading.

Among these theories are some which live in asymptotically flat space-times of dimensions between 11 and 4. The gauge invariant observables of these theories are encoded in a scattering matrix. All of these theories are exactly supersymmetric, a fact that I consider to be an important clue to the physics of the real world. In addition, they all have continuous families of deformations. These families are very close to being analogs of the moduli spaces of vacuum states of supersymmetric quantum field theory. They all have the same high energy behavior, and one can create excitations at one value of the moduli which imitate the physics at another value, over an arbitrarily large region of space. Except for the maximally supersymmetric case, there is no argument that all of these models are connected by varying moduli in this way. One other feature of these models is noteworthy.

How can we reach a theory of Quantum Gravity? Many answers to this question are proposed in the different chapters of this book. A more specific set of questions might be: what demands should we put on our framework, so that it is best able to meet all the challenges involved in creating a theory of Quantum Gravity? What choices are most likely to give the correct theory, according to the clues we have from known physics? Are there any problems with our initial assumptions that may lead to trouble further down the road? The latter seems to be one of the most important strategic questions when beginning to formulate a candidate theory. For example, can a canonical approach overcome the multifaceted problem of time? And how far can a theory based on a fixed background spacetime be pushed? On the one hand, these questions may only be answered in the very attempt to formulate the theory. On the other, many such attempts have been made, and now that Quantum Gravity research has built up some history, perhaps it is time to plough some of the experience gained back into a new approach, laying the groundwork for our theory in such a way as to avoid well-known problems. The causal set program represents such an attempt.

In this review, some answers to the above questions, as embodied by the causal set program, are set out and explained, and some of their consequences are given.

Introduction

In this chapter we present noncommutative geometry (NCG) not as a ‘theory of everything’ but as a bridge between any future, perhaps combinatorial, theory of Quantum Gravity and the classical continuum geometry that has to be obtained in some limit. We consider for the present that NCG is simply a more general notion of geometry that by its noncommutative nature should be the correct setting for the phenomenology and testing of first next-to-classical Quantum Gravity corrections. Beyond that, the mathematical constraints of NCG may give us constraints on the structure of Quantum Gravity itself in so far as this has to emerge in a natural way from the true theory.

Also in this chapter we focus on the role of quantum groups or Hopf algebras as the most accessible tool of NCG, along the lines first introduced for Planck scale physics by the author in the 1980s. We provide a full introduction to our theory of ‘bicrossproduct quantum groups’, which is one of the two main classes of quantum group to come out of physics (the other class, the q-deformation quantum groups, came out of integrable systems rather than Quantum Gravity). The full machinery of noncommutative differential geometry such as gauge theory, bundles, quantum Riemannian manifolds, and spinors (at least in principle) has also been developed over the past two decades; these topics are deferred to a third article.

Introduction

In the early days of the subject, string theory was understood only as a perturbative theory. The theory arose from the study of S-matrices and was conceived of as a new class of theory describing perturbative interactions of massless particles including the gravitational quanta, as well as an infinite family of massive particles associated with excited string states. In string theory, instead of the one-dimensional world line of a pointlike particle tracing out a path through space-time, a two-dimensional surface describes the trajectory of an oscillating loop of string, which appears pointlike only to an observer much larger than the string.

As the theory developed further, the need for a nonperturbative description of the theory became clear. The M(atrix) model of M-theory, and the AdS/CFT correspondence, each of which is reviewed in another chapter of this volume, are nonperturbative descriptions of string theory in space-time backgrounds with fixed asymptotic forms. These approaches to string theory give true nonperturbative formulations of the theory, which fulfill in some sense one of the primary theoretical goals of string theory: the formulation of a nonperturbative theory of Quantum Gravity.

There are a number of questions, however, which cannot – even in principle – be answered using perturbative methods or the nonperturbative M(atrix) and AdS/CFT descriptions. Recent experimental evidence points strongly to the conclusion that the space-time in which we live has a small but nonzero positive cosmological constant.

Assuming that “quantum spacetime” is fundamentally discrete, how might this discreteness show itself? Some of its potential effects are more evident, others less so. The atomic and molecular structure of ordinary matter influences the propagation of both waves and particles in a material medium. Classically, particles can be deflected by collisions and also retarded in their motion, giving rise in particular to viscosity and Brownian motion. In the case of spatio-temporal discreteness, viscosity is excluded by Lorentz symmetry, but fluctuating deviations from rectilinear motion are still possible. Such “swerves” have been described in and. They depend (for a massive particle) on a single phenomenological parameter, essentially a diffusion constant in velocity space. As far as I know, the corresponding analysis for a quantal particle with mass has not been carried out yet, but for massless quanta such as photons the diffusion equation of can be adapted to say something, and it then describes fluctuations of both energy and polarization (but not of direction), as well as a secular “reddening” (or its opposite). A more complete quantal story, however, would require that particles be treated as wave packets, raising the general question of how spatiotemporal discreteness affects the propagation of waves. Here, the analogy with a material medium suggests effects such as scattering and extinction, as well as possible nonlinear effects. Further generalization to a “second-quantized field” might have more dramatic, if less obvious, consequences.

Introduction

This chapter wants to be two things. On the one hand it wants to review a number of approaches to the problem of Quantum Gravity that are new and have not been widely discussed so far. On the other hand it wants to offer a new look at the problem of Quantum Gravity. The different approaches can be organized according to how they answer the following questions: Is the concept of a spacetime fundamental? Is a background time used? Are Einstein's equations assumed or derived? (See figure 7.1.)

In string theory, loop quantum gravity, and most other approaches reviewed in this book spacetime plays a fundamental role. In string theory a given spacetime is used to formulate the theory, in loop quantum gravity one tries to make sense of quantum superpositions of spacetimes. It is these spacetimes in the fundamental formulation of the theory that are directly related to the spacetime we see around us. In this broad sense these approaches treat spacetime as something fundamental. Here we want to focus our attention on approaches that take a different view. In these approaches spacetime emerges from a more fundamental theory.

The next questions concern the role of time. The models that we will be looking at will all have some sort of given time variable. They differ though in the way they treat this time variable. One attitude is to use this time variable in the emergent theory.

Introduction

In recent years, loop quantum gravity (LQG) has become a promising approach to Quantum Gravity (see e.g. for reviews). It has produced concrete results such as a rigorous derivation of the kinematical Hilbert space with discrete spectra for areas and volumes, the resulting finite isolated horizon entropy counting and regularization of black hole singularities, a well-defined framework for a (loop) quantum cosmology, and so on. Nevertheless, the model still has to face several key issues: a well-defined dynamics with a semi-classical regime described by Newton's gravity law and General Relativity, the existence of a physical semi-classical state corresponding to an approximately flat space-time, a proof that the no-gravity limit of LQG coupled to matter is standard quantum field theory, the Immirzi ambiguity, etc. Here, we address a fundamental issue at the root of LQG, which is necessarily related to these questions: why the SU(2) gauge group of loop quantum gravity? Indeed, the compactness of the SU(2) gauge group is directly responsible for the discrete spectra of areas and volumes, and therefore is at the origin of most of the successes of LQG: what happens if we drop this assumption?

Let us start by reviewing the general structure of LQG and how the SU(2) gauge group arises. In a first order formalism, General Relativity (GR) is formulated in term of tetrad e which indicates the local Lorentz frame and a Lorentz connection ω which describes the parallel transport.

Introduction

Any of us who has used the Global Positioning System (GPS) in one of the gadgets of everyday life has also relied on the accuracy of the predictions of Einstein's theory of gravity, General Relativity (GR). GPS systems accurately provide your position relative to satellites positioned thousands of kilometres from the Earth, and their ability to do so requires being able to understand time and position measurements to better than 1 part in 1010. Such an accuracy is comparable to the predicted relativistic effects for such measurements in the Earth's gravitational field, which are of order GM⊕/R⊕c2 ~ 10−10, where G is Newton's constant, M⊕ and R⊕ are the Earth's mass and mean radius, and c is the speed of light. GR also does well when compared with other precise measurements within the solar system, as well as in some extra-solar settings.

So we live in an age when engineers must know about General Relativity in order to understand why some their instruments work so accurately. And yet we also are often told there is a crisis in reconciling GR with quantum mechanics, with the size of quantum effects being said to be infinite (or – what is the same – to be unpredictable) for gravitating systems.