Computational methods for lattice field theories

Abstract

Introduction Classical field theory enables us to calculate the behaviour of fields within the framework of classical mechanics. Examples of fields are elastic strings and sheets, and the electromagnetic field. Quantum field theory is an extension of ordinary quantum mechanics which not only describes extended media such as string and sheets, but which is also supposed to describe elementary particles. Furthermore, ordinary quantum many-particle systems in the grand canonical ensemble can be formulated as quantum field theories. Finally, classical statistical mechanics can be considered as a field theory, in particular when the classical statistical model is formulated on a lattice, such as the Ising model on a square lattice, discussed in Chapter 7. In this chapter we shall describe various computational techniques that are used to extract numerical data from field theories. Renormalisation is a procedure without which field theories cannot be formulated consistently in continuous space-time. In computational physics, we formulate field theories usually on a lattice, thereby avoiding the problems inherent to a continuum formulation. Nevertheless, understanding the renormalisation concept is essential in lattice field theories in order to make the link to the real world. In particular, we want to make predictions about physical quantities (particle masses, interaction constants) which are independent of the lattice structure, and this is precisely where we need the renormalisation concept. Quantum field theory is difficult. It does not belong to the standard repertoire of every physicist.