Vol. 813 (1982)
ACTA PHYSICA POLONICA
No 1-2
INTRODUCTION
TO
THE
BACKGROUND
FIELD
METHOD*
By L. F. ABBoTT**
CERN, Geneva
(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-
-Mills
f3
function is computed.
PACS numbers: 11.10.Np, 11.10.Gh
1. Introduction
The background field method is a technique for quantizing gauge field theories without
losing explicit gauge
invariar~ce.
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 [3] and then discussed
in more detail by DeWitt [4], Boulware [5] and by myself [6].
It
is this extension, valid
to all orders ofperturbation theory, which I will present here. The background field method
is used extensively in gravity [7] and supergravity [8] theories. In addition, it has been
used to derive light-particle effective field theories from grand unified models [9], to com-
pute the Yang-Mills
fJ
fULction up to two loops [6] and to perform calculatioLs in lattice
gauge theories [10]. 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.
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