II.1.1 Gauge Symmetry

As we have seen in section I.1.2, gauge symmetry can be considered the basis for electromagnetism and conservation of charge, and it was already included in Maxwell s equations even before the advent of Quantum Mechanics, but its importance remained unnoticed in the earliest formulations. Hilbert also derived Einstein s field equations by postulating a symmetry under changing the coordinates. After the development of Quantum Mechanics, gauge symmetry was applied successfully in electrodynamics, and it was generalized mathematically in 1954 by Yang and Mills, in an attempt to describe the nuclear forces. In general, a gauge symmetry is analogous to how we can often describe the same thing in different ways by equivalent synonyms.

The word gauge means a measurement, or a distance between two points. Unlike global symmetries, where observations are invariant under the transformations that include the observer and the physical apparatus, gauge symmetries are formulated only in terms of the laws, so the transformations only change our description of the same physical situation, without changing the situation itself. In Magnetism, for example, Maxwell s equations could be simplified by introducing a vector potential, in terms of which both the electric and magnetic fields,  and , could be expressed. Consequently, one could perform a gauge transformation without changing the values of the fields. For decades, this symmetry was regarded as artificial, but it has been recently understood that the vector potential has direct observable meaning.

Therefore, 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. But in Quantum Electrodynamics, for example, 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 gauge symmetry there exists an associated conserved current. Since particles are the excitations of fields, the particle associated with exciting the gauge field is the gauge boson, which is the photon in the case of Quantum Electrodynamics. The aforementioned symmetry of the wave-function under global phase changes implies the conservation of electric charge.

Consequently, modern theories describe all physical forces in terms of fields, such as the electromagnetic field and the gravitational field, that cannot be directly measured. Nevertheless, some associated quantities, such as charges, energies, and velocities, can be precisely measured and evaluated. In field theories, different configurations of the un-observable fields can result in identical observable quantities. A transformation from one such field configuration to another is called a gauge transformation, and the lack of change in the measurable quantities, despite the field being transformed, is called gauge invariance. Any theory that has the property of gauge invariance is considered a gauge theory. Gauge symmetry underlies dynamics and it forces the existence of gauge bosons, which are massless particles that are associated with the vector potential and mediate the forces, and they have integer spin.

Furthermore, the realization that gauge symmetry is based on a sophisticated geometrical concept, called the fiber bundle, which is a space that is locally a product space, but globally may have a different topological structure. This mathematical description provided a deep and beautiful geometrical foundation for gauge symmetry. The fiber bundle combines an internal space together with space-time, to form a unified geometrical object which exhibits the gauge symmetry.

Gauge theories constrain the laws of physics, because all the changes induced by a gauge transformation have to cancel each other out when written in terms of observable quantities. The fundamental interactions, or forces, arise from the constraints imposed by local gauge symmetries, in which case the transformations vary from point to point in space and time.

Therefore, gauge symmetry is extremely useful in describing the forces of nature, because we can redefine the fields, or their particles, in terms of each other in such a way that the laws remain the same, because the changes made to one field or particle are canceled by other changes. If the particles described by the laws do not interact then the gauge symmetries are global, which means that the re-definitions are the same everywhere and at all times. But if we require that the re-definitions vary with space or time, then some auxiliary field must be defined to describe the interaction between the original particles, which is subject to the local gauge symmetry.

The difference between global and gauge symmetries is that the latter are redundancies in our description of the physical system, so they are not real. In global symmetries, the quantum eigen-states can be divide in terms of representations of the symmetry group, and the physical operators go between different states. In contrast, in local symmetries, the Hilbert space has only gauge-invariant states, so the whole system must be neutral and the states that have nonzero net charges are un-physical. So gauge symmetries are redundant, because any operators that create or annihilate charges are un-physical.

Nevertheless, gauge symmetries are very useful in obtaining physically meaningful theories, especially in particle physics. In relativistic theories of massless particles, because of Lorentz invariance, where space and time have opposite signs in the metric, some states will have negative norm, which means that the probability of measuring something on these states would give negative answers, which makes no sense at all, because it violates unitarity. With gauge theory we can then declare these negative-norm states un-physical, so we need to have local symmetries in order to kill all the un-physical states. This works fine with spin-one bosons, because the number of such states, though it is infinite, is the same number of space-time points. With spin-two particles, there are much more un-physical states to kill, and no ordinary gauge symmetry, like  or is enough. In this case, we must resort to space-time diffeomorphism, which is also a gauge symmetry, and this is the modern quantum-field-theoretical view of Einstein s General Relativity.

In general, the gauge transformations of a theory may not be commutative, and they are combined into the framework of a gauge group, where infinitesimal gauge transformations are the gauge group generators. Thus, the number of gauge bosons is the group dimension, that is the number of generators forming the basis of the corresponding Lie algebra. Quantum Chromodynamics has the gauge group , and the gauge bosons are eight gluons. In the electroweak theory, the 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.

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 summary: there are symmetrical transformations in real space and others in hypothetical fields that can be attached to each point of the real space. Gauge symmetry is the freedom to transform the fields independently from any global symmetry of the real space, so it is merely changing the description of the given field. For example,  rotation is tied to electric charge conservation,  is tied to weak charge conservation, and  is tied to color charge conservation; so  is like attaching a little circle in which particles can undergo gauge transformations, so that they don t move in real space, but only in these attached spaces. These gauge transformations can change the type of particles, so for example while  is trivial and doesn t change the particle,  can take an up-quark to a down-quark, and  allows quarks to change color. Some gauge transformations are also associated with the emission of gauge bosons. For example, a photon is emitted by  rotation,  or  bosons are  rotations, and gluons are  rotations.