The gauge principle is used to understand the interactions between fundamental particles. According to this principle, the weak, electromagnetic, and strong forces are all described by the interactions of spin-1 gauge bosons with the quarks and leptons. Each of the gauge bosons is associated with an underlying symmetry. The electromagnetic force is mediated by the photon, the strong force by the gluons, and the weak forces by the charged W+ and W- and the neutral Z bosons.
A quantum mechanical state is described by a wave function ψ (x ) where x is the space and time coordinate. Then all physical observables are described by the interactions of operators O with the wave function of the system:
The only physical observable is the expectation value 〈O 〉 which is unchanged by changes in the phase of ψ (x ): where ϑ is a constant at every space and time pointx. The wave function itself cannot be measured; the only measurable quantity is the expectation value. The invariance of the expectation value under phase changes implies that the phase of the wave function has no physical significance and so also can never be measured in an experiment.
The set of all such global phase transformations (change of the wave function by a constant phase) forms a U(1) (Abelian) symmetry group.
Since ϑ has no physical importance, one would like to be able to choose ϑ to be different for different space and time locations x. If this were the case, the system would be invariant under phase changes that were different in different places: This is known as a local gauge transformation.
The interactions of particles in quantum mechanics (using the Dirac or Schrödinger equation, for example) always involve derivatives acting on the fields. Under a local phase change, the derivative operating on the wave function changes the wave function by a factor (∂μ = ∂/∂xμ):
In this equation, μ = 0, 1, 2, 3, with x0 being the time coordinate and x1, x2, x3 representing the spatial dimensions. The second term, proportional to ∂μθ(x ), destroys the invariance under the local gauge transformation. The local gauge invariance can be restored, however, if the derivative is replaced byDμ is called the gauge covariant derivative, whereas the field Aμ(x ) is called a gauge field and must change under local phase transformations as The parameter g describes the strength of the coupling of the gauge field to other particles, such as the electron.
Invariance of the laws of physics under local gauge transformations therefore requires the introduction of a massless gauge field Aμ(x ) and the replacement of all derivatives by gauge covariant derivatives. The simplest example of a gauge theory constructed according to this principle is quantum electrodynamics, describing the interactions of the photon with the electron.
Abelian Gauge Bosons
The electromagnetic field Aμ(x ) describing the interactions of the photon is an example of an Abelian gauge field. The interactions of the photon are described by a U(1) gauge symmetry. This symmetry requires that the interactions be invariant under local phase transformations that depend on the space-time point as explained in the previous section. The self-interactions of the photon are contained in the Lagrangian: where Fμν = ∂μAν(x ) - ∂νAμ(x ). This interaction is clearly unchanged by the shiftwhere e is the charge of the electron. A mass term for the photon would have the form
It is easy to see that this interaction violates the local gauge invariance, and so local gauge invariance requires that the photon be massless. Massless gauge bosons such as the photon have spin 1 and two transverse degrees of freedom, with the spin of a transverse photon being perpendicular to the photon's direction of motion.
The interactions of the photon with fermion fields ψ such as the electron are restricted by the requirements of local gauge invariance and described by the Dirac equation where me is the mass of the electron and γμ are 4 × 4 Dirac matrices. Since Dμ = ∂μ + ieAμ(x ), the Dirac equation represents a coupling between the photon and the fermion field with strength e. There are no free parameters in the Dirac theory since it depends only on the mass and charge of the electron, both of which are measured experimentally.
Non-Abelian Gauge Boson
A gauge theory described by a special unitary group SU(N) is termed a non-Abelian gauge theory. An SU(N) gauge theory has N2 - 1 gauge bosons that interact in a manner exactly specified by the gauge theory. The simplest example of a non-Abelian gauge theory is the SU(2) gauge theory describing the electroweak interactions. This theory was first written down by Chen Ning Yang and Robert Mills. In SU(2) gauge theory, the interactions are invariant under the local gauge transformations: where σi, i = 1, 2, 3 are the 2 × 2 Pauli matrices, and θi, i = 1, 2, 3 are three real parameters that can depend on the space-time point x. The Pauli matrices can be written as
An SU(2) gauge group has three massless gauge bosons, Wiμ, i=1, 2, 3. (Each gauge boson has four components, corresponding to the energy of the boson and the three spatial directions). In order to maintain the local gauge invariance, derivatives acting on ψ must be replaced by covariant derivatives:
The strength of the gauge coupling is represented by the parameter g, and the self-interactions of the gauge bosons are given by the square of the field strength tensor: where εijk changes sign under the exchange of any two of its indices. The non-Abelian gauge bosons have self-interactions between two and three gauge bosons, unlike the Abelian gauge bosons of quantum electrodynamics. Because of the self-interactions of the gauge bosons, the strength of the non-Abelian gauge boson self-interactions decreases at high energy (corresponding to short distances) and increases at low energy (large distances). This property is known as asymptotic freedom.
The strong interactions are described by an SU (3)c gauge theory called quantum chromodynamics. This theory contains eight massless gauge bosons termed gluons that provide the interactions between quarks. Since the theory is non-Abelian, the strength of the coupling between the quarks and gluons increases with large distances and so provides the force that confines quarks into hadrons such as the proton.
Spontaneously Broken Gauge Theories
An unbroken non-Abelian gauge theory contains only massless gauge bosons. The Standard Model of electroweak interactions consists of a product group, SU(2) × U(1), which contains a spontaneously broken gauge symmetry. A spontaneously broken gauge symmetry has at least one scalar field, termed a Higgs field. This scalar field is used to break the gauge symmetry, while maintaining the gauge invariance of the interactions. When the gauge symmetry of the SU(2) × U(1) electroweak theory is broken, three of the gauge bosons receive masses, while one remains as the massless photon of quantum electrodynamics. The massive bosons are linear combinations of the SU(2) gauge bosons Wiμ (i = 1, 2, 3) and the U(1) gauge boson Bμ: The angle θW is called the weak mixing angle and is experimentally measured to be sin2 θW = .23. The weak mixing angle is a measure of the mixing between the SU(2) gauge bosons and the U(1) gauge boson. The remaining combination of neutral gauge bosons remains massless after the spontaneous symmetry breaking and is identified with the photon of quantum electrodynamics.
The massive gauge bosons contain three degrees of freedom: two are the transverse polarizations described in the previous section for the photon, and the third is the longitudinal polarization in which the spin of the gauge boson is parallel to the direction of motion of the gauge boson.
Experimental Successes of Gauge Theories
The predictions of quantum electrodynamics have been spectacularly confirmed by atomic physics measurements, such as the Lamb shift, and by high-energy measurements, such as the anomalous magnetic moments of the electron and the muon. These measurements leave no doubt that quantum electrodynamics describes the interactions of the photon with fermions.
The SU(2) × U(1) gauge theory of electroweak interactions has also received substantial experimental confirmation. The masses of the electroweak gauge bosons, W± and Z , are predicted in terms of the weak mixing angle and the Fermi coupling of beta decay as MW = 81 GeV and MZ = 91 GeV. These masses were predicted before the experimental discoveries of the gauge bosons and have been verified by measurements at the Fermilab Tevatron and the European Laboratory for Particle Physics (CERN) LEP collider. The interactions of the quarks and leptons with the gauge bosons are completely specified in terms of the gauge coupling constants. Many of these interactions, particularly those of the quarks and leptons with the Z boson, have been precisely measured, with most measurements agreeing with the predictions to within a percent.
Abers, E., and Lee, B. "Gauge Theories." Physics Reports9 , 1–143 (1973).
Quigg, C. Gauge Theories of the Strong, Weak, and Electromagnetic Interactions (Benjamin-Cummings, Menlo Park, CA, 1983).
Yang, C., and Mills, R. "Conservation of Isotopic Spin and Isotopic Gauge Invariance." Physical Review96 , 191–195(1954).