15 Computational methods for lattice field theories
15.1 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.Renormalisationis 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, under-
standing 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. We try to explain the main concepts and ideas before entering
into computational details, but unfortunately we cannot give proofs and deriva-
tions, as a thorough introduction to the field would require a book on its own. For
details,thereaderisreferredtoRefs.[ 1 – 5 ].Inthenextsectionweshallbriefly
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