Wilson loop

The Wilson Loop (or Wilson Line ) , named after Kenneth Wilson , a pioneer of lattice scale theories , is an expectation value of an operator in gauge theories , which is used to distinguish between the different phases of the theory. ${\displaystyle W_{C}}$

definition

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

${\displaystyle W_{C}=\mathrm {Tr} \,\left\{{\mathcal {P}}\exp i\oint _{C}A_{\mu }dx^{\mu }\right\}}$

Here designated

• ${\displaystyle C}$ the closed way
• ${\displaystyle {\mathcal {P}}}$that the product of the operators is ordered along the path.
• ${\displaystyle Tr}$(English Trace ) the trace regarding the calibration group . Because of the cyclic invariance of this track, the operator is gauge invariant. ${\displaystyle G}$

application

Lattice theories

A main area of ​​application of the Wilson loops are lattice theories, where order parameters for different phase states are obtained from them .

In quantum chromodynamics, for example (considered as quantum field theory on a lattice at finite temperature), Wilson loops are used to distinguish between:

• Confinement phases when the expression in the exponent in the spatial dimension areally behaves (proportional to the enclosed area; area law ). This can be imagined as a consequence of the additive contributions of many color-electric confinement flow tubes. The associated potential increases linearly with distance, similar to the elastic behavior of a rubber band .
• Deconfinement phases when the expression in the exponent behaves linearly in the spatial dimension (proportional to the circumference of the loop ; circumferential law ). There are no flow contributions through the loop or they cancel each other out on average. The associated potential is inversely proportional to the distance, as in electrodynamics ( Coulomb phase ). ${\displaystyle C}$

The Wilson loops are formed using closed curves in space-time , whereby the time is assumed to be imaginary , so that a Euclidean formalism similar to statistical mechanics results, only in four dimensions . The corresponding temperature is inversely proportional to the time, and periodic boundary conditions are assumed. The closed curves are usually guided over a time and a spatial direction; then the Wilson loops in the continuum limit of the grid correspond to the calculation of the Quark -Antiquark potential. However, purely spatial loops are also considered.

Electrodynamics

In electrodynamics, it is identical to the magnetic flux through the loop if it is spatial, as can be seen by applying Stokes' theorem. ${\displaystyle \oint _{C}A_{\mu }dx^{\mu }}$ ${\displaystyle C}$

String theory

Wilson loops are also considered in string theory . Here there is the possibility of non-contractable (contractible) loops in the compactified extra dimensions, depending on their topology .

In the loop quantum gravity of Ashtekar Wilson loops play an important role as a fundamental basis states of a quantized gravity theory . There a parallel transport of a quadruped along a closed path is considered. This is in direct analogy to the Wilson loops in the gauge theories, the description of which is mathematically similar ( fiber bundles with associated forms of connection describing the parallel transport ("connections"), which in the case of gauge theories are identical to the gauge fields).

Since the 1990s, the formalism of spin networks has increasingly been used in quantum gravity instead of Wilson loops .