This is a book about techniques, and the first few chapters are about the techniques you have to learn before you can learn the real techniques.

Your mastery of these crucial chapters will be presumed in subsequent ones. So go though them carefully, paying special attention to exercises whose answers are numbered equations.

I begin with a review of basic ideas from thermodynamics and statistical mechanics. Some books are suggested at the end of the chapter [1–4].

In the beginning there was thermodynamics. It was developed before it was known that matter was made of atoms. It is notorious for its multiple but equivalent formalisms and its orgy of partial derivatives. I will try to provide one way to navigate this mess that will suffice for this book.

Energy and Entropy in Thermodynamics

I will illustrate the basic ideas by taking as an example a cylinder of gas, with a piston on top. The piston exerts some pressure P and encloses a volume V of gas. We say the system (gas) is in equilibrium when nothing changes at the macroscopic level. In equilibrium the gas may be represented by a point in the (P,V) plane.

The gas has an internal energy U. This was known even before knowing the gas was made of atoms. The main point about U is that it is a state variable: it has a unique value associated with every state, i.e., every point (P,V). It returns to that value if the gas is taken on a closed loop in the (P,V) plane.

There are two ways to change U. One is to move the piston and do some mechanical work, in which case

dU = –PdV

by the law of conservation of energy. The other is to put the gas on a hot or cold plate. In this case some heat δQ can be added and we write

dU = δQ

which acknowledges the fact that heat is a form of energy as well. The first law of thermodynamics simply expresses energy conservation:

dU = δQ–PdV.

I use δQ and not dQ since Q is not a state variable.

We have thus far looked at the classical statistical mechanics of a collection of Ising spins in terms of their partition function. We have seen how to extract the free energy per site and correlation functions of the d = 1 Ising model.

We will now consider a second way to do this. While the results are, of course, going to be the same, the method exposes a deep connection between classical statistical mechanics in d = 1 and the quantum mechanics of a single spin-12 particle. This is the simplest way to learn about a connection that has a much wider range of applicability: the classical statistical mechanics of a problem in d dimensions may be mapped on to a quantum problem in d – 1 dimensions. The nature of the quantum variable will depend on the nature of the classical variable. In general, the allowed values of the classical variable (s=±1 in our example) will correspond to the maximal set of simultaneous eigenvalues of the operators in the quantum problem (σz in our example). Subsequently we will study a problem where the classical variable called x lies in the range –∞ < x < ∞ and will correspond to the eigenvalues of the familiar position operator X that appears in the quantum problem. The correlation functions will become the expectation values in the ground state of a certain transfer matrix.

The different sites in the d = 1 lattice will correspond to different, discrete, times in the life of the quantum degree of freedom.

By the same token, a quantum problem may be mapped into a statistical mechanics problem in one-higher dimensions, essentially by running the derivation backwards. In this case the resulting partition function is also called the path integral for the quantum problem. There is, however, no guarantee that the classical partition function generated by a legitimate quantum problem (with a Hermitian Hamiltonian) will always make physical sense: the corresponding Boltzmann weights may be negative, or even complex!

In the quantum mechanics we first learn, time t is a real parameter. For our purposes we also need to get familiar with quantum mechanics in which time takes on purely imaginary values t =–iτ, where τ is real. It turns out that it is possible to define such a Euclidean quantum mechanics.

“Bosonization” refers to the possibility of describing a theory of relativistic Dirac fermions obeying standard anticommutation rules by a boson field theory. While this may be possible in all dimensions, it has so far proved most useful in d = 1, where the bosonic version of the given fermionic theory is local and simple, and often simpler than the Fermi theory. This chapter should be viewed as a stepping stone toward a more thorough approach, for which references are given at the end.

In this chapter I will set up the bosonization machine, explaining its basic logic and the dictionary for transcribing a fermionic theory to a bosonic theory. The next chapter will be devoted to applications.

To my knowledge, bosonization, as described here, was first carried out by Lieb and Mattis [1] in their exact solution of the Luttinger model [2]. Later, Luther and Peschel [3] showed how to use it to find asymptotic (low momentum and energy) correlation functions for more generic interacting Fermi systems. It was independently discovered in particle physics by Coleman [4], and further developed by Mandelstam [5]. Much of what I know and use is inspired by the work of Luther and Peschel.

Preamble

Before getting into any details, I would first like to answer two questions. First, if bosonization applies only to relativistic Dirac fermions, why is it of any interest to condensed matter theory where relativity is not essential? Second, what is the magic by which bosonization helps us tame interacting field theories?

As for the first question, there are two ways in which Dirac fermions enter condensed matter physics. The first is in the study of two-dimensional Ising models, where we have already encountered them. Recall that if we use the transfer matrix approach and convert the classical problem on an N ×N lattice to a quantum problem in one dimension we end up with a 2N-dimensional Hilbert space, with a Pauli matrix at each of N sites. The two dimensions at each site represent the twofold choice of values open to the Ising spins. Consider now a spinless fermion degree of freedom at each site. Here too we have two choices: the fermion state is occupied or empty.