Wilson-Loop

Wilson -loop ( or Wilson line ) is an expected value of an operator to gauge theories which serves to distinguish the different phases of the theory. He is named after the Nobel Laureate Kenneth Wilson, one of the pioneers of lattice gauge theories, named.

The Wilson loop is defined as a gauge invariant expectation value of a phase factor, in which the field variables of the gauge theory, the ( four - ) vector potentials with values ​​in the underlying Lie group G of gauge theory, along a closed path (English loop) are multiplied together:

This refers to the closed path, and that the product of the operators is arranged along the path. Tr (English trace) is with respect to the trace of the gauge group G. Because of the cyclic invariance of the trace of the operator is gauge invariant.

A major field of application of Wilson loops are lattice gauge theories, where the order parameter is obtained from them for different phase states. In quantum chromodynamics, for example ( considered as a quantum field theory on a lattice at finite temperature ) they serve to distinguish between confinement and deconfinement phases, depending on whether the term in the exponent in the spatial dimension areally ( proportional to the enclosed area, so-called "area law " confinement) or linearly related ( proportional to the circumference of the loop C, the so-called " circumferential law " ). The first case ( Area Law ), one can vividly imagine as a result of the additive contributions of many colored electrical confinement - flow tubes. It shows a behavior of the linear increase of the potential corresponding to the distance, similarly as in the elastic properties of a rubber band. In the second case there are no such flux contributions through the loop or they cancel in the middle, there is a distance behavior of the associated potential inversely proportional to the distance in front as in electrodynamics before ( Coulombphase ). The Wilson loops are completed via closed curves in space- time, where the time is assumed imaginary, so that a Euclidean formalism similar to statistical mechanics, only in 4 dimensions, results (the corresponding temperature is inversely proportional to the time be accepted and periodic boundary conditions ). The closed curves are usually performed on a time and one spatial direction, as well as purely spatial loops will be considered. In the first case, the Wilson loops correspond in the continuum limit of the lattice calculation of the quark-antiquark potential.

In the electrical response is the same as the magnetic flux through the loop C, if it is spaced, as indicated by use of the set of Stokes.

Wilson loops are also considered in string theory, where there is also the possibility of not kontraktibler ( contractible ) loops results in the compactified extra dimensions, depending on their topology. In the loop quantum gravity of Ashtekar they play a major role as a fundamental basis states of a quantized theory of gravity. There is a parallel transfer of a four- leg is viewed along a closed path. This is in direct analogy to the Wilson loops in gauge theories, whose description is mathematically similar ( fiber bundle with associated, the parallel transport descriptive context shapes ( "connections "), which are identical to the gauge fields in the case of gauge theories ). Since the 1990s, the formalism of spin networks in quantum gravity is used increasingly.

• String theory
• Quantum field theory
• General Theory of Relativity