This book is an elementary introduction to hadronic properties at finite temperature (and chemical potential). Its contents may be divided into three parts.

The first part, comprising the first three chapters, develops the (vacuum) theory of hadronic interactions at low energy. As propagators play a major role in applications, we review in Chapter 1 in detail the vacuum propagators for fields of different spins. This provides the background in which to discuss later (in Chapter 4) the form of thermal propagators. The next two chapters are devoted to strong interaction dynamics. Chapter 2 describes the phenomenon of spontaneous symmetry breaking leading to Goldstone bosons. It prepares the ground for chiral perturbation theory in Chapter 3. We review this theory in some detail, including Goldstone as well as (heavy) non-Goldstone fields interacting with the Goldstone fields. Chapters 2 and 3 constitute an elementary yet general introduction to the effective theory of strong interactions at low energy.

The second part consists of the next two chapters and presents the equilibrium thermal field theory. Broadly speaking, there are two formulations of this theory, concerning so-called real and imaginary time. Though the imaginary time formulation is used more often than that of real time, we choose the latter in this book and try to show some of its advantages. In Chapter 4 we derive in elementary ways the thermal matrix propagators for fields of low spin and go on to obtain the spectral representation of complete propagators for fields of arbitrary spin. In Chapter 5 the thermal perturbation theory is developed in relation to the matrix structure of the propagators.

The third part consists of the last four chapters. Here we apply the methods developed to study different thermal one- and two-point functions in the hadronic phase using chiral perturbation theory. In Chapter 6 we obtain the temperature dependence of a number of hadronic parameters to one loop. Some two-loop results are presented in Chapter 7. In Chapter 8 we derive the dilepton cross-section measured in heavy ion collisions. Finally, Chapter 9 describes the (non-equilibrium) transport phenomenon, whose discussion can be reduced to fluctuations in thermal equilibrium. Clearly the range of applications of thermal field theory in particle physics is much wider. Many such applications are covered in books (mentioned in Chapter 4), though mostly in imaginary time formulation.

Methods of thermal field theory can broadly be put into two categories, according to their use of imaginary time or real time in the ensemble averages. The method of imaginary time was first proposed by Matsubara [1]. The real time method was introduced later, originally by Schwinger [2] and Keldysh [3] and followed by others [4], who applied it to study non-equilibrium processes. It was also applied to equilibrium situations by Mills [5]. The real time method was reformulated in the so-called thermofield dynamics by Umezawa and collaborators [6], who suggested a doubling of field variables in the statistical context. Niemi and Semenoff [7] clarified this apparent doubling in terms of the time contour needed in the real time formulation. A survey of early literature is available in [8]. Physical applications including later works are presented also in books [9, 10].

For systems in thermal equilibrium, one can develop perturbative expansion in both methods in parallel to the conventional method in vacuum. The difference shows up in the form of the propagators. Because of the finite interval of imaginary time, the frequencies are discrete, giving the propagator as a sum over these frequencies. On the other hand, the real time method has two infinite intervals of time, giving rise to a 2 × 2 matrix for the propagator. The majority of calculations reported in the literature are done in the imaginary time method, avoiding matrix propagators in the real time method.

For systems out of thermal equilibrium, the real time method suggests itself as the appropriate framework to calculate physical quantities. However, for quantities such as the response functions (we address this problem in the last chapter) the end results relate to thermal equilibrium and so can be evaluated by either method. But, to follow the time dependence of quantities during non-equilibrium phases, such as in the early universe [11–13] – a topic too distant to discuss in this book – the real time method is the only option.

In this book we adopt the real time method to calculate thermal quantities in equilibrium. This chapter is devoted to deriving the matrix form of propagators arising in this method.

It is interesting to note that much of the effective theory of strong interactions was developed well before the existence of QCD [1]. The isotopic spin symmetry of strong interactions was established long ago by works in nuclear physics. Modern developments began in 1960, when Nambu recognised the existence of nearly massless pions as a symptom of a spontaneously broken symmetry [2]. (This topic is reviewed in the previous chapter.) He and co-workers also showed how to calculate amplitudes involving a single pion [3]. In 1964 Gell-Mann introduced current algebra [4], which allowed the treating of processes involving more than a single pion. In particular, exact sum rules could be written for pion–hadron amplitudes, taking the pion to be massless [5, 6]. But for processes involving three or more pions, the calculations became complicated. Further, there was no way to treat approximate symmetries.

The numerical success of the sum rules did show that strong interactions must possess an underlying SU(2) × SU(2) symmetry broken to the SU(2) symmetry of isospin. In an attempt to find a simpler and more physical method of calculation, Weinberg [7] introduced in 1967 the effective Lagrangian incorporating this symmetry. Originally it was justified on the basis of current algebra: an effective Lagrangian with the symmetry of the underlying theory would produce conserved Noether currents in terms of the effective fields. These currents in turn would yield the equal-time commutators of current algebra. So if one calculates low energy amplitudes directly with this Lagrangian, it must yield the same results as from current algebra. The essential uniqueness of the effective Lagrangian was also established [8].

The QCD theory was proposed in 1972 by Fritzsch and Gell-Mann [1] as a theory of quarks and gluons. Soon after, Leutwyler [9] found that not only the mass difference of u and d quarks, but also the masses themselves, were small compared to the scale of strong interactions, which immediately explained the SU(2) × SU(2) symmetry of strong interactions. (Including the heavier s quark leads to a less accurate SU(3) × SU(3) symmetry.) The mass terms provide a definite pattern of symmetry breaking.