# gauge theory

(redirected from*Yang-mills equation*)

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## 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,

**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),

*e*

^{iϕ}with 0 ≤ ϕ < 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, Schrödinger'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 α_{QCD} 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 α_{QCD} 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