Gauge Fixing

Prerequisites: Gauge Symmetry. Full path (14 pages)

Why is it interesting?
Intuitive
Concrete
Abstract

Why is it interesting?

Intuitive

In a gauge theory we describe the physical situation with more variables than there are physical degrees of freedom. In electrodynamics, for example, many different vector potentials give exactly the same electric and magnetic fields, and nothing we can measure distinguishes between them. A gauge symmetry transformation moves from one of these descriptions to another.

A simple analogy is the height of a mountain. Only height differences matter physically, but to write down numbers we must decide where height zero is: sea level or the valley floor. Each choice gives different numbers for the same mountain. Gauge fixing means making such a choice once and for all, so that each physical situation is described by one set of variables instead of many. In a gauge theory the choice can be made separately at every point in space and time, so a gauge condition is a rule that applies everywhere, such as requiring the scalar potential to vanish (temporal gauge).

We need to do this whenever the redundancy gets in the way. To solve the equations of motion as an initial value problem the solution must be unique, and without a gauge condition it is not. In the quantum theory, summing over all field configurations would count each physical configuration infinitely many times. Different gauges make different features of the same physics easy to see, which is why several are in use. A gauge condition can also fail to pick exactly one description, either because some freedom is left over, as in the temporal gauge, or because it is satisfied by several equivalent configurations; the second problem is discussed under Gribov ambiguities.

Concrete

For a nice discussion of gauge conditions see section 4.5 "Cauchy problem and gauge conditions" in Rubakov's "Classical Theory of Gauge Fields".

Unitary gauge: Is singular.

Temporal gauge: Does not fix the gauge completely. Time independent gauge transformations are still allowed.

Abstract

The presence of redundant variables complicates the formulation of the canonical formalism and the quantization. Only for independent dynamical degrees of freedom canonically conjugate variables may be defined and corresponding commutation relations may be associated. In a first step, one has to choose by a “gauge condition” a set of variables which are independent. […] Gauge theories are formulated in terms of redundant variables. Only in this way, a covariant, local representation of the dynamics of gauge degrees of freedom is possible. For quantization of the theory both canonically or in the path integral, redundant variables have to be eliminated. This procedure is called gauge fixing. It is not unique and the implications of a particular choice are generally not well understood.

Topological Concepts in Gauge Theories by F. Lenz

Different gauges lead to different interpretations of the theory:

For example, when considering the QCD vacuum one usually uses the temporal gauge. This gauge makes it possible to derive the usual periodic picture of the QCD vacuum. This picture is convenient, because it allows us to understand quite pictorial what instantons are, namely tunneling processes between the vacua. However, in a different gauge like the axial gauge this picture does not emerge and there is just one non-degenerate vacuum (See section 10.3 in the book Classical Solutions in Quantum Field Theory by Erick Weinberg and Interpretation of pseudoparticles in physical gauges by Claude W. Bernard and Erick J. Weinberg. A remark in this later paper is what gave Frank Wilczek the idea for the axion mechanism.) Moreover, in the Coulomb gauge the situation is even different. There, one encounters the famous Gribov ambiguities. When one tries to fix the Coulomb gauge, one notices that this is not globally possible. Instead, at some points several gauge potentials get "picked out" by the gauge condition. The $\theta$ parameter appears as a relative phase between two ambiguous gauge potentials. In addition, in the Coulomb gauge there is only one vacuum. (See: Coulomb gauge description of large Yang-Mills fields by R. Jackiw et. al.)
Another example is the Higgs mechanism. In the usually used unitary gauge, we get the now famous picture with the "Mexican Hat Potential" and the marble that runs down from the top to the new minimum, which corresponds to a non-zero vacuum expectation value of the Higgs field. However, in the temporal gauge this picture does not emerge. In this gauge the vacuum expectation value of the Higgs field is zero. (See Higgs Mechanism in the Temporal Gauge by Michael Creutz, Thomas N. Tudron and Higgs phenomenon without symmetry breaking order parameter by J. Fröhlich et. al.)

For more on problems with the unitary gauge, see: Ambiguities and breakdown configurations of the unitary gauge by C. Montonen, G. ’t Hooft (1981) Nucl. Phys. B190 and M.N. Chernodub, L.D. Faddeev, A.J. Niemi (2008) JHEP 12, 014 (http://arxiv.org/abs/0804.1544).

Path to this page

Leads to: Gribov Ambiguities.