This part of the book is fully devoted to the physics of (1 + 1)-dimensional quantum or twodimensional classical models. As usual, I shall distinguish between classical and quantum systems only if it is necessary. Those models where the difference between quantum and classical is not relevant will be called ‘two-dimensional’; those where for some reason or other I want to emphasize the quantum aspect will be called ‘one-dimensional’.

The reason I spend so much time discussing two-dimensional physics is that here one finds a kind of paradise for strong interactions and nonperturbative effects. Those effects which are tricky or even impossible to achieve in higher dimensions appear in almost every two-dimensional model. To our greatest satisfaction there are theoretical tools at our disposal which allow us to solve many of the corresponding problems and describe these phenomena. The availability of these strong mathematical tools is a unique feature of two-dimensional physics. Even if it turns out to be impossible to generalize these tools for higher dimensions, I hope that the reader will be rewarded for the time spent by the pleasure obtained from contemplating their beauty.

The world we are about to enter has certain distinct features which are worth mentioning in the introduction. The first one concerns the second-order phase transitions. In (1 + 1)-dimensional systems such transitions may occur only at zero temperature.

In this chapter I continue to study the group of conformal transformations of the complex plane. The exposition of this and the following chapter is based on the pioneering paper by Belavin et al. (1984). Naturally, such studies are hardly necessary for the Gaussian model where one can calculate the correlation functions directly. However, these general considerations become indispensable in more complicated cases.

In two dimensions the group of conformal transformations is isomorphic to the group of analytic transformations of the complex plane. Since the number of analytic functions is infinite, this group is infinite-dimensional, i.e. has an infinite number of generators. An infinite number of generators generates an infinite number ofWard identities for correlation functions. These identities serve as differential equations on correlation functions. No matter how many operators you have in your correlation function, you always have enough Ward identities to specify it. Thus in conformally invariant theories (for brevity we shall call them ‘conformal’ theories) one can (at least in principle) calculate all multi-point correlation functions. We have already calculated multi-point correlation functions of the bosonic exponents for the Gaussian model; below we shall see less trivial examples.

Let us take in good faith that there are models besides the Gaussian one whose spectrum is linear, whose correlation functions factorize into products of analytic and antianalytic parts and which have operators transforming under analytic (antianalytic) transformations like (25.4). Such operators will be called primary fields.

Though it was quite beyond my original intentions to write a textbook, the book is often used to teach graduate students. To alleviate their misery I decided to extend the introductory chapters and spend more time discussing such topics as the equivalence of quantum mechanics and classical statistical mechanics. A separate chapter about Landau Fermi liquid theory is introduced. I still do not think that the book is fully suitable as a graduate textbook, but if people want to use it this way, I do not object.

Almost 10 years have passed since I began my work on the first edition. The use of field theoretical methods has extended enormously since then, making the task of rewriting the book very difficult. I no longer feel myself capable of presenting a brief course containing the ‘minimal body of knowledge necessary for any theoretician working in the field’. I strongly feel that such a body of knowledge should include not only general ideas, what is usually called ‘physics’, but also techniques, even technical tricks. Without this common background we shall not be able to maintain high standards of our profession and the fragmentation of our community will continue further. However, the best I can do is to include the material I can explain well and to mention briefly the material which I deem worthy of attention. In particular, I decided to include exact solutions and the Bethe ansatz. It was excluded from the first edition as being too esoteric, but now the astonishing new progress in calculations of correlation functions justifies its inclusion in the core text.

As mentioned at the beginning of Chapter 7, there have been two different attitudes in the experimenter's mind concerning the role of the μ+ in condensed matter, as shown schematically in Figure 8.1(a): at one extreme, the μ+ is treated as a gentle and passive probe to probe the condensed matter with minimal perturbation and to observe its intrinsic properties prior to the introduction of μ+; at the other extreme, the μ+ is treated as a violent and active probe introducing a perturbation in the host material so as to study new physics and chemistry created by the presence of the μ+. In this chapter, representative studies utilizing the second category, including similar studies of the μ−, are described.

The role of this type of muon spin rotation/relaxation/resonance (μSR) studies is quite significant in its contribution to the growth of our human daily life: (1) the localization and diffusion of the light hydrogen isotope Mu (μ+ e−) simulates a behavior of a dilute hydrogen atom in metals and other condensed matter which is quite difficult to monitor and important in various aspects of industrial constructions ; (2) a trace impurity hydrogen-like atom can “passivate” electrical activity of donors and acceptors or “hydrogenate” dangling bonds in semiconductors ; (3) the lightest hydrogen atom can explore the most fundamental mechanism of hydrogen chemical reaction in terms of mass dependence ; (4) the electron brought in by the energetic light hydrogen can be used to probe electron transport in conducting polymers and biological macromolecules.

Experimental studies in muon science can only be conducted when a reasonably intense muon beam of high quality is available. As described in Chapter 1, muons from various sources, including both accelerator-produced particles and those of cosmic-ray origin, are compared in terms of energy and intensity in Figure 1.7. It is easily seen that at very high energy (higher than 100 GeV), cosmic-ray muons are the only possibility, whereas at low energy accelerator-producing muons are almost exclusively used.

In this chapter, more detailed information is given on the types of muon that can be obtained using present production strategies. It should be understood, however, that future developments in both accelerator technology and in ideas for muon beam production are quite likely to change the situation drastically.

MeV accelerator muons

Muons can only be obtained through the decay of the pions which are produced in nuclear interactions between accelerated particles and nuclear targets. These days, the high-intensity proton accelerator is the most popular source of accelerated particles. Figure 2.1 gives a list of medium-energy proton accelerators currently in use for muon physics research. These days, most activities are based on the accelerators of the so-called meson factories such as the Tri-University Meson Facility (TRIUMF) and Paul Scherrer Institute (PSI) which have beams with an intensity of some 100 μA to mA. From the viewpoint of time structure of accelerators there are two types: continuous and pulsed.

Concept of muon catalysis of nuclear fusion

Of the two types of muons, only the μ− is involved in muon catalyzed fusion (hereafter designated μCF) processes. As depicted in Figure 5.1, nuclear fusion reactions take place when two nuclei such as d and t approach one another to within the range of the nuclear interaction rn(≅ a few times 10–13 cm). However, because of the Coulomb repulsion between positively charged nuclei which increases with decreasing distance, the realization of nuclear fusion is not at all easy.

In the concept of thermal nuclear fusion, the additional energy is given by thermal energy (kT) through the satisfaction of the condition kT ≥ e2/rn. By assuming rn ≅ 10–12 cm, the right-hand side of the inequality becomes 7 × 104 eV (note that the radius and binding energy for the ground state of a hydrogen-like atom are 0.53 × 10−8 cm and 13.6 eV), the required temperature is 7 × 108 K (while room temperature, 300 K, corresponds to 0.03 eV). In the μCF concept, the fusion reaction is mediated by the neutral small atom formed between μ− and a hydrogen isotope and the subsequent formation of a small muonic molecule, and the relevant energy is the appropriate overall formation energy.

Here, it might be relevant to mention significant features of fusion energy as a possible energy source in future centuries.