Vol. 813 (1982)
ACTA PHYSICA POLONICA
By L. F. ABBoTT**
(Received July 20, 1981)
The background field approach to calculations in gauge field theories is presented.
Conventional functional techniques are reviewed and the background field method is intro-
duced. Feynman rules and renormalization are discussed and, as an example, the Yang-
function is computed.
PACS numbers: 11.10.Np, 11.10.Gh
The background field method is a technique for quantizing gauge field theories without
losing explicit gauge
It makes gauge theories easier to understand and greatly
simplifies computations. In this review I will present the formalism of this method and
show how it is applied to gauge theory calculations.
The background field method was introduced by DeWitt [I, 2] in a formalism which
was applicable to one-loop processes. The extension to multi-loop calculatior.s, which
involved a reformulation of the method, was first made by 't Hooft  and then discussed
in more detail by DeWitt , Boulware  and by myself .
is this extension, valid
to all orders ofperturbation theory, which I will present here. The background field method
is used extensively in gravity  and supergravity  theories. In addition, it has been
used to derive light-particle effective field theories from grand unified models , to com-
pute the Yang-Mills
fULction up to two loops  and to perform calculatioLs in lattice
gauge theories . In all of these applications, the great simplifications introduced by the
method playa key role.
Any formulation of a gauge field theory begins with a gauge invariant Lagrangian.
However, in order to quantize the theory a gauge must be chosen. In the conventional
formulation, this means that the Lagrangian you actually use to derive Feynman rules
and perform calculations, consisting of the classical Lagrangian plus gauge-fixing and
ghost terms, is not gauge invariant. Of course, any physical quantity calculated will be
Presented at the XXI Cracow School ofTheoretical Physics, Paszk6wka, May 29
June 9, 1981.
Address: Physics Department, Brandeis University, Waltham, MA 02254, USA.