Continuous time and discrete space

Another form of lattice-gauge theory that is very intuitively appealing is the theory's Hamiltonian form. In many problems it is very instructive to make energyestimates, look at wavefunctions, and estimate the masses, sizes, and shapes of bound states etc. that a quantum-mechanical formulation with a Hamiltonian gives. Sometimes these simple things are hard to extract from a path-integral formulation. Mechanisms of confinement, flux-tube dynamics, and chiral-symmetry breaking are also amenable to Hamiltonian analysis. Decon-finement at finite T and transitions as functions of chemical potential will also be discussed in this formulation below. Since there is much more pioneering physics to be done in QCD in extreme environments, it is important to have many approaches available.

In the Hamiltonian formulation one uses a spatial lattice and leaves the time variable continuous, as we have illustrated in several 1 + 1 examples above. The Hamiltonian is the generator of time evolution and acts on quantized states. The Hamiltonian could be obtained methodically from the path-integral formulation by calculating the system's transfer matrix and then taking the time continuum limit [9]. It is more instructive, however, to construct the Hamiltonian from scratch. In fact, we can think about just two spatial points, write the Hamiltonian for that system, and then consider a full three-dimensional spatial lattice.

As in the Euclidean formulation, the key to the construction is the requirement that local color-gauge invariance be an exact symmetry for any lattice spacing a. Let there be two sites with the link between them.

The phases of QCD have presented a challenge to theoretical physics for many years. Originally it was considered folly to think about topics such as confinement, chiral-symmetry breaking, and different states of matter in relativistic field theories. Anything beyond traditional perturbation theory was met with skepticism. However, the development of semi-classical methods in field theory, the lattice-gauge formulation of the subject, and the generalizations of analytic methods of analysis from two-dimensional systems, such as duality and transitions driven by topological excitations, have changed all that.

This book is a look at the fundamentals of QCD from this perspective. After introducing lattice-gauge theory, beginning with fundamentals and reaching important recent developments, it will emphasize the application of QCD to the study of matter in extreme environments. Effective Lagrangians, which incorporate the constraints of low-energy dynamics and their symmetry realizations, will also be developed to provide a complementary, and insightful, perspective. Application of perturbative methods will also be presented in the regime of their validity.

Why extreme environments? A major theme of this book is the idea that, to understand the dynamics of QCD in ordinary circumstances, one needs to master them in extreme environments.

For example, to appreciate confinement, one can heat the system until the thermal fluctuations prevent the formation of the thin flux tubes which give rise to a linearly confining potential. In doing so, the special ingredients in the theory's dynamics which favor a vacuum pressure that squeezes flux into thin continuous tubes are emphasized.

In this lecture we shall continue to consider the perturbation theory of quantum electrodynamics with a photon with a non-zero mass λ. The conventional QED with λ = 0 has its own specific features, which will be addressed later.

The reason for considering this theory and for accounting for all complications related to spin, is that in the field theory with vector mesons one may expect the appearance of Regge particles at small values of the coupling constant g. We note that although in the gφ3 theory there is a Regge pole, it is situated not near j = 0, which is required for reggeization of a scalar particle, but near j = − 1. From the point of view of Feynman diagrams this is related to the fact that in the t-channel with a fixed momentum transfer we had two particles instead of one in the Born pole diagram. Each additional line in the t-channel reduces the asymptotics of a diagram by one power of the large variable s. A compensation of this suppression is possible owing to the Azimov spin shifting, which is made manifest by the appearance of the factor sσ in the numerator of the Feynman amplitude, with σ the spin of the added particle. One can see from this that to make reggeization of a ‘nucleon’ possible in perturbation theory we should have a particle with σ = 1. Such a situation is realized in QED (see Fig. 9.1).

This book presents the lecture course given in 1969 at the Ioffe Physicotechnical Institute in Leningrad (now St Petersburg, Russia) by one of the key figures of twentieth century theoretical physics, Vladimir N. Gribov.

The original motivation, brought up by A. Aronov, was for Gribov to present a course on the theory of complex angular momenta that would be understandable for condensed matter physicists. (Those were the days when condensed matter and particle physicists worked hand in hand, shared thoughts, methods and seminars.) The audience consisted of young bright theorists (though some had little or no experience in this specific field). So, to appreciate the book fully and learn from it, the reader should be familiar with the basics of quantum field theory (relativistic invariance, quantum theory of scattering, Feynman diagrams).

The lecture notes were taken down by A. Anselm, Ya. Azimov, A. Bukhvostov, I. Dyatlov, G. Danilov, E. Levin, G. Frolov, V. Gorshkov, L. Lipatov, E. Malkov, V. Shekhter and I.M. Shmushkevich (then in charge of the Ioffe PTI theory department), and edited by L. Frankfurt. E. Kuraev initiated their publication in the form of two voluminous preprints (in Russian) issued by the Kharkov PTI. This informal publication, known as the ‘Kharkov preprint’, immediately became a rarity.

More than 25 years later the effort of preparing an English version of the course was undertaken by E. Kuraev, B. Shaikhatdenov and V. Bytev. The lectures were then edited by G. Korchemsky and L. Lipatov. The final English translation was carried out by V. von Schlippe; the authors of the following Introduction share responsibility for the final scientific editing.