gauge theory

Also found in: Dictionary, Wikipedia.
Related to gauge theory: Lattice gauge theory

Gauge theory

The theoretical foundation of the four fundamental forces of nature, the electromagnetic, weak, strong, and gravitational interactions. Gauge symmetry lies at the heart of gauge theory. A gauge symmetry differs from an ordinary symmetry in two important respects:

1. Gauge symmetry is a local symmetry rather than a global symmetry. For a local symmetry, the element of the symmetry group (G) that acts on the fields of a theory at a space-time point ( x , t) depends on the position x and time t, whereas for a global symmetry a fixed group element acts on fields at different space-time points.

2. A gauge transformation leaves a physical state invariant. Gauge symmetry reflects a redundancy in the variables used to describe a physical state. By contrast, a global symmetry acting on a physical state in general produces a new, distinct physical state. See Fundamental interactions, Space-time

The simplest example of a gauge theory is electromagnetism. In classical electrodynamics, gauge invariance reflects the arbitrariness that exists in choosing the potentials A ( x , t) and &phgr;( x , t) to represent the electric and magnetic fields, E and B , according to Eqs. (1), where c is the speed of light. If &Lgr;( x , t) is an arbitrary scalar field,

Eqs. (2) define a gauge transformation. The potentials A and &phgr; may equally well be used to represent the electromagnetic fields E and B . See Maxwell's equations, Potentials, Relativistic electrodynamics

In nonrelativistic quantum mechanics, gauge invariance is realized as follows. The Schrödinger equation for a particle with an electromagnetic charge q and mass m is Eq. (3),

where is the hamiltonian operator, is the momentum operator, &PSgr; is the wave function of the particle, and ℏ is Planck's constant divided by 2&pgr;. Equation (3) is invariant under the gauge transformation given by Eqs. (4).
Electromagnetism is a U(1) gauge symmetry [where U(1) is the one-dimensional unitary group, which is represented by the complex numbers ei&phiv; with 0 ≤ &phiv; < 2&pgr;] because a gauge transformation rotates the phase of the wave function in a space-time–dependent manner and adjusts the potentials A and &phgr; accordingly. See Quantum mechanics, Schr&ouml;dinger's wave equation, Unitary symmetry

The concept of a gauge theory may be generalized to larger, nonabelian Lie groups, such as G = SU(2), SU(3), SU(5), or SO(10). In 1954 C. N. Yang and R. L. Mills suggested gauging the SU(2) isospin symmetry, thus developing the first nonabelian gauge theory, also known as Yang-Mills theory. In 1971 G. `t Hooft demonstrated the renormalizability of nonabelian gauge theory. Nonabelian gauge theory is the foundation of the electroweak and strong interactions. In the electroweak theory, formulated by S. Weinberg and A. Salam in 1967, the gauge group G = SU(2)L × U(1)Y is spontaneously broken to U(1)Q, the gauge group of ordinary electromagnetism, by the condensation of a Higgs field. There are four kinds of particles that mediate the gauge interactions, called gauge bosons: two massive charged weak gauge bosons, the W+ and W-, with a mass of 80.4 GeV; a neutral gauge boson, the Z, with a mass of 91.2 GeV; and finally a massless photon (γ). The W± and the Z are responsible for the charged and neutral weak interactions, respectively, and the photon is the gauge particle responsible for the electromagnetic interaction. In quantum chromodynamics (QCD) the symmetry group is SU(3)color, which remains unbroken. There are eight gauge bosons called gluons. Quarks, which come in three colors, carry a color charge, and the quark field forms an SU(3) triplet. See Electroweak interaction, Symmetry breaking

One remarkable property of quantum chromodynamics is that color charge is antiscreened rather than screened, a feature arising from the nonabelian character of the gauge symmetry. This situation is opposite to that in QED. Consequently, as shorter distance scales are probed, the coupling constant αQCD decreases, so that at very short distances QCD approaches a free (noninteracting) field theory. This feature, pointed out in 1973 by D. Gross and F. Wilczek and by H. D. Politzer, is known as asymptotic freedom. Although on large scales QCD is a strongly coupled theory, at very small scales, which may be probed by scattering at very high energies, the constituents of hadrons (quarks and gluons) behave almost as if they were free particles, originally called partons. Similarly, at larger distances scales αQCD becomes larger, and perturbation theory breaks down. This increase in the coupling constant at large distances leads to a phenomenon known as confinement, which prevents colored objects (such as quarks or gluons) from being isolated. See Gluons, Quantum chromodynamics, Quarks

gauge theory

[′gāj ‚thē·ə·rē]
Any field theory in which, as the result of the conservation of some quantity, it is possible to perform a transformation in which the phase of the fields is altered by a function of space and time without altering any measurable physical quantity, so that the fields obtained by any such transformation give a valid description of a given physical situation.
References in periodicals archive ?
Davide Gaiotto for far-reaching new insights about duality, gauge theory, and geometry, and especially for his work linking theories in different dimensions in most unexpected ways.
of Miami-Coral Gables, Florida) introduces second-year graduate students to the theory of three-dimensional manifolds and its role in the modern four-dimensional topology and gauge theory.
They introduce the basics, such as the interaction of physics and mathematics, and the characteristics of basic topological exotica, then explain the algebraic tools for topology, the geometry of smooth folds, the relations between bundles and geometry with gauge theory, the relations between gauge theory and moduli space, the classification of manifolds, early exotic manifolds, the first results in dimension four, the modern approach to the Seiberg-Witten theory, physical implications, and the move from differential structures to operator algebras and geometric structures.
He has published extensively in his areas of research interest, which include harmonic maps, variational problems in fiber bundles, and generalized gauge theory and its applications in mechanics and mathematical physics.
In recent years the deformation theory of geometric structures has received revived attention due to new developments, which involve in a deeper way the connections to Lie theory and gauge theory.
The 19 papers in the proceedings discuss such topics as quantum field theory and the volume conjecture, representations and the colored Jones polynomial of a torus knot, delta-groupoids and ideal triangulations, spin foam state sums and the Chern-Simons theory, Yang-Mills in two dimensions and Chern-Simons in three, fermionization and convergent perturbation expansions in Chern-Simons gauge theory, and analytic continuation of Chern-Simons theory.
In this collection of recent research in the field, contributors from Japan, the US, Russia, France, and Argentina describe work in areas such as the application of string theory to noncommutative gauge theory, the minimal supersymetric standard model string, classical and quantum aspects of non-critical string theory, and string theory with a continuous spin spectrum.
We provide a U(1) gauge theory of Quantum Electrodynamics (QED), from which we derive photon as a quantum Dirac-Wilson loop W(z, z) of this model.
The tantalizing predictions of lattice gauge theory hopefully will become transparent with the Cray XD1.
As an example, it is possible to construct the integrand of the complete, four-loop four-point scattering amplitude in maximally supersymmetric gauge theory in four dimensions using only planar data, and the solution of a linear system of Jacobi identities.
They also present some modern developments in the field, such as Bose operators, Fermi operators, coherent states, squeezed state, gauge theory, quantum groups, and super Lie algebras.
He explains the action approach, including its relation to Maxwell's equation and Dirac fields, then examines continuous symmetries and conservation laws, magnetostatics, multivalued fields in superfluids and superconductors, magnetic monopoles, electric charge confinement, multivalued mapping from ideal crystals to crystals with defects, defect melting, relativistic mechanics in curvilinear coordinates, torsion and curvature from defects and embedding, mutivalued mapping, field equations of gravitation, fields of integer spin, particles with half-integer spins, covariant conservation gravitation of spinning matter as a gauge theory, evanescent properties of torsion in gravity, the teleparallel theory of gravitation embedding, and emerging gravity.