Things happen. That may seem obvious, but it has also been maintained that all the ‘happening’ does not signify change and that the way of the world is periodic and repetitious:

I won't discuss the profound aspects of this passage, but I will do post-industrial age nitpicking. Only in the past century has humanity understood a distinction that exists among these cyclic behaviors. For the 'rising and setting of the sun, there is indeed little that is happening. To a good approximation this is non-dissipative. But as to the coming and going of generations, with the benefit of wisdom gained in building steam engines, we recognize that birth and death can occur only so long as there is a source of negative entropy.

I could continue in this vein and discuss how the failure to distinguish between free and frictional motion confused humanity's greatest minds as they grappled with elementary mechanics. But I wish to begin a technical discussion of irreversibility and only want to draw a lesson of humility from the historical perspective. Until the past few centuries, humanity failed to appreciate the most manifest of time's arrows, the second law of thermodynamics. Unless one realizes that Nature's dynamics are mostly time symmetric one does not know that there is a problem.

In this chapter we return to the discussion of quantum gravity which we began in Chapter 4. In the first section we describe some of the technical problems that are encountered in constructing a theory of quantum gravity and some of the ideas that may go into their resolution. We then give a definition of simplicial gravity in arbitrary dimensions and describe a representative sample of the numerical results that have been obtained. It is often convenient to consider the theory in a fixed dimension larger than two. We shall discuss the four-dimensional case since it is physically the most relevant, and will only occasionally consider three-dimensional gravity.

Basic problems in quantum gravity

Formulating a theory of quantum gravity in dimensions higher than two leads to a number of basic questions, some of which go beyond those encountered in dimension two. Among these are the following:

1. (i) What are the implications of the unboundedness from below of the Einstein–Hilbert action?

2. (ii) Is the non-renormalizability of the gravitational coupling a genuine obstacle to making sense of quantum gravity?

3. (iii) What is the relation between Euclidean and Lorentzian signatures and do there exist analogues of the Osterwalder–Schrader axioms allowing analytic continuation from Euclidean space to Lorentzian space-time?

4. (iv) What is the role of topology in view, for instance, of the fact that higher-dimensional topologies cannot be classified?

We do not have answers to these questions and our inability to deal with them may be an indication that there exists no theory of Euclidean quantum gravity in four dimensions or, possibly, that quantum gravity only makes sense when embedded in a larger theory such as string theory.

The idea of describing the physical world entirely in terms of geometry has a history dating back to Einstein and Klein in the early decades of the century. This approach to physics had early success in general relativity but the appearance of quantum mechanics guided the development of theoretical physics in a different direction for a long time. During the past quarter of a century the programme of Einstein and Klein has seen a renaissance embodied in gauge theories and, more recently, superstring theory. During this time we have also witnessed the happy marriage of statistical mechanics and quantum field theory in the subject of Euclidean quantum field theory, a development which could hardly have taken place without Feynman's path integral formulation of quantization. In this book we shall work almost exclusively in the Euclidean framework.

The unifying theme of the present work is the study of quantum field theories which have a natural representation as functional integrals or, if one prefers, statistical sums, over geometric objects: paths, surfaces and higher-dimensional manifolds. Our philosophy is to work directly with the geometry as far as possible and avoid parametrization and discretizations that break the natural invariances. In this introductory chapter we give an overview of the subject, put it in perspective and discuss its main ideas briefly.

Lagrangian field theories whose action can be expressed entirely in terms of geometrical quantities such as volume and curvature have a special beauty and simplicity.

The role of random walks as a theoretical tool in physics dates back at least to the explanation of the origin of Brownian motion at the turn of the century. The universal features of the large-scale phenomena associated with random walks are already transparent in Einstein's derivation of the diffusion equation in his famous 1905 paper on Brownian motion. Ever since, theories of random walks have played an increasingly important role in virtually every branch of physics and now form the basis of statistical theory in general in the subject of stochastic processes. The appearance of random walks in elementary particle physics was mediated by Feynman's path integral formulation of quantum theory, and a mathematically rigorous approach to the subject was made possible by the introduction of Euclidean quantum field theory in the 1960s building on Wieners's earlier work on random walks and the diffusion equation.

We start in the next three sections by introducing various discrete random walk models describing the propagation of scalar particles in space-time. We pay particular attention to critical behaviour in these models and their universality properties: many different discrete models give rise to the same continuum limit. As explained in the Introduction we are primarily interested in viewing random paths as geometric objects. Hence, we focus on such aspects as reparametrization invariance which are usually not stressed in the more standard treatments.

When quantizing gravity in the functional integral formalism, the metric on the base manifold M is a dynamical variable that is integrated out in the partition function. This was discussed at length in Chapters 4 and 6 in particular cases. Because of general covariance one actually integrates over equivalence classes of metrics related by diffeomorphisms of M, and expectation values of physical observables are therefore diffeomorphism invariants of M. The central problem of quantum gravity in this formalism is to attribute a mathematical meaning to, as well as a physical interpretation of, such expectation values. It is fair to say that substantial results have been obtained in only two dimensions so far.

Introduction

This chapter is devoted to a discussion of certain examples of theories in which general covariance is realized in a simpler way than by averaging over metrics. For example, a theory is generally covariant if its action is a functional of a set of fields on the manifold M which does not involve a metric at all and for which the functional integration measure over the fields is also metric independent. Alternatively, it may occur that although a metric enters the expression defining the action, the energy–momentum tensor nevertheless vanishes on the physical state space. Perhaps the best-known example of the former is the three-dimensional Chern–Simons gauge theory [391]. Examples of the latter are cohomological field theories [393], which include the two-dimensional twisted N = 2 superconformal models [167].

The main topic of this book is the work that has been carried out during the last 15 years under the general heading of random surfaces. The original motivation for the study of random surfaces came from lattice gauge theory, where one can represent various quantities of interest as weighted sums over surfaces embedded in a hypercubic lattice. A few years later, with the resurrection of string theory, random surfaces were used as regularization of that theory and, most recently, random surface models have been applied to two-dimensional quantum gravity. There is also an impressive body of work on random surfaces that has been carried out by membrane physicists, as well as by condensed matter physicists, so one often finds mathematically identical problems being studied in different branches of physics. Random surfaces are therefore not a physical theory but, rather, a theoretical tool and a methodology that can be applied to various physical problems in the same way as random walks find applications in many branches of science. The formalism that has been developed to deal with random surfaces carries over to the study of higher-dimensional manifolds, which are important for quantizing gravity in higher dimensions.

We address this book primarily to advanced graduate students in theoretical physics but we hope that more experienced researchers in the field, as well as mathematicians, may find it useful.

Aspects of quantum limits for the detection of a classical force

The development of the theory of QND measurements was triggered by the problem of detecting a small classical force that acts on a quantum probe―e.g. on a free mass or an oscillator, see the review article and the monograph. This problem arose most especially in gravitational-wave experiments. Astrophysical estimates indicate that the gravitational waves from cosmic sources should be so strong that they behave almost perfectly classically. During the measuring time the number of gravitons that pass within one wavelength of the measuring device is enormously larger than unity; and correspondingly, as would be the case if the waves were electromagnetic, quantum effects in the radiation are utterly negligible.

A central feature of gravitational waves is their extremely weak interaction with matter. Because of this weakness, the response of the measuring device's probe object to the waves is so small that it may be comparable to the object's quantum mechanical uncertainties. At the same time, the weakness of the interaction guarantees that there is almost no back action of the probe on the gravitational-wave field. As a result, the gravitational wave acts on the probe as through it were a precisely classical force (i.e. a force that is independent of the probe's quantum state).

The wave function

Before beginning to describe the quantum theory of measurement, we shall discuss briefly the main concepts of quantum mechanics. Readers familiar with quantum mechanics may skip this section and go on to section 2.2.

Consider, as a pedagogical aid, a classical particle that can move only along the x direction. For example, it might be a bead on an infinite needle. If we know at any moment of time the values of two numbers, the particle's position and momentum, then we can compute from them all other features of the particle's motion: the forces acting on it, its resulting acceleration, etc. In other words, this pair of numbers, the position and momentum, completely define the state of the classical particle at the chosen moment of time. Similarly, for three-dimensional motion the state is defined by six numbers: the three spatial coordinates x, y, z and the three corresponding components of the momentum px, py, pz.

A quantum object, with its dual particle-like and wave-like properties, is much more complex than a classical particle. Interferometric experiments (chapter I) show that the object can be spread out over some region of space, or for the bead on a needle, over an interval of x.

In the standard, textbook treatment of quantum mechanics, the contact with experiment is described in terms of probabilities for obtaining this, that, or another result, when identical experiments are performed on a huge number (ensemble) of identical objects. Such an ensemble description was adequate during the first half century of quantum theory, because the technology of that era was incapable of making a measurement without destroying or severely changing the measured object (photon, atom, …). In recent years, however, advances in technology and technique have made possible repetitive quantum measurements on a single quantum object, with each measurement influencing the object only minimally. Such measurements cannot be analyzed using solely the ensemble theory. Additional theoretical concepts are needed. The purpose of this book is to describe the methods and theory of such measurements (as well as the more elementary theory of ensembles of measurements).

This book is written for two types of readers: those who have had only a little previous contact with quantum mechanics, and those who have had much.

Readers of the first type are presumed to understand atomic physics at a qualitative level. The unstarred sections of this book offer such readers an overview of the present state-of-the-art of quantum measurements on single objects and prospects for the future, as well as an elementary overview of the theory of such measurements.