3.5.14 Gauge Freedom
A gauge theory is a theory that admits a symmetry with a local parameter. In every quantum theory, the global phase of the wave-function is arbitrary, and does not represent something physical. Consequently, the theory is invariant under a global change of phases, such as adding a constant to the phase of all wave-functions everywhere. This is called global symmetry.
In Quantum Electrodynamics, the theory is also invariant under a local change of phase, so one may shift the phase of all wave-functions so that the shift may be different at every point in space-time. This is called local symmetry.
However, in order for a well-defined derivative operator to exist, one must introduce a new field, called the gauge field, which also transforms in order for the local change of variables not to affect the derivative. In Quantum Electrodynamics, this gauge field is the electromagnetic field.
According to Noether’s theorem, for every such symmetry there exists an associated conserved current. The aforementioned symmetry of the wave-function under global phase changes implies the conservation of electric charge. Since the excitations of fields represent particles, the particle associated with excitations of the gauge field is the gauge boson, such the photon in the case of Quantum Electrodynamics.
The degrees of freedom in Quantum Field Theory are local fluctuations of the fields, and the existence of a gauge symmetry reduces the number of degrees of freedom, simply because some fluctuations of the fields can be transformed to zero by gauge transformations, so they are equivalent to having no fluctuations at all, and they, therefore, have no physical meaning. Such fluctuations are usually called non-physical degrees of freedom or gauge artifacts. Therefore, if a classical field theory has a gauge symmetry, then its quantized version will have this symmetry as well. In other words, a gauge symmetry cannot have a quantum anomaly.
In general, the gauge transformations of a theory consist of several different transformations, which may not be commutative. These transformations are combined into the framework of a gauge group; infinitesimal gauge transformations are the gauge group generators. Thus, the number of gauge bosons is the group dimension, i.e., the number of generators forming the basis of the corresponding Lie algebra.
All the known fundamental interactions in nature are described by gauge theories, including:
•Quantum Chromodynamics, whose gauge group is
, and the gauge bosons are eight gluons. •The electroweak theory, whose gauge group is
, and the gauge bosons are the photon and the massive
and
bosons. •Gravity, whose classical theory is General Relativity, relies on the equivalence principle, which is essentially a form of gauge symmetry, and its action may also be written as a gauge theory of the Lorentz group on tangent space.
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