**Are you an EPFL student looking for a semester project?**

Work with us on data science and visualisation projects, and deploy your project as an app on top of GraphSearch.

Concept# Wilson loop

Summary

In quantum field theory, Wilson loops are gauge invariant operators arising from the parallel transport of gauge variables around closed loops. They encode all gauge information of the theory, allowing for the construction of loop representations which fully describe gauge theories in terms of these loops. In pure gauge theory they play the role of order operators for confinement, where they satisfy what is known as the area law. Originally formulated by Kenneth G. Wilson in 1974, they were used to construct links and plaquettes which are the fundamental parameters in lattice gauge theory. Wilson loops fall into the broader class of loop operators, with some other notable examples being 't Hooft loops, which are magnetic duals to Wilson loops, and Polyakov loops, which are the thermal version of Wilson loops.
To properly define Wilson loops in gauge theory requires considering the fiber bundle formulation of gauge theories. Here for each point in the -dimensional spacetime there is a copy of the gauge group forming what's known as a fiber of the fibre bundle. These fiber bundles are called principal bundles. Locally the resulting space looks like although globally it can have some twisted structure depending on how different fibers are glued together.
The issue that Wilson lines resolve is how to compare points on fibers at two different spacetime points. This is analogous to parallel transport in general relativity which compares tangent vectors that live in the tangent spaces at different points. For principal bundles there is a natural way to compare different fiber points through the introduction of a connection, which is equivalent to introducing a gauge field. This is because a connection is a way to separate out the tangent space of the principal bundle into two subspaces known as the vertical and horizontal subspaces. The former consists of all vectors pointing along the fiber while the latter consists of vectors that are perpendicular to the fiber.

Official source

This page is automatically generated and may contain information that is not correct, complete, up-to-date, or relevant to your search query. The same applies to every other page on this website. Please make sure to verify the information with EPFL's official sources.

Related courses (2)

PHYS-432: Quantum field theory II

The goal of the course is to introduce relativistic quantum field theory as the conceptual and mathematical framework describing fundamental interactions.

PHYS-502: Interacting quantum matter

This course presents modern aspects of theoretical condensed matter physics with interfaces to statistical physics, quantum information theory, quantum field theory and quantum simulation.

Related lectures (18)

Related concepts (14)

Feynman Rules II: QED

Explores Feynman rules in QED, emphasizing normal ordered product and Wick's theorem, instantons, and relativistic amplitudes.

B-physics anomalies: data, EFT analyses, and model-building

Explores B-physics anomalies, EFT analyses, and model-building considerations in the context of Lepton Flavor Universality violations and semi-leptonic decays of the b quark.

Quantum Field Theory: Loop Corrections and Counter Terms

Explores loop corrections and counter terms in quantum field theory, showcasing how divergences are handled and predictions are extracted from loop calculations.

Gauge theory

In physics, a gauge theory is a field theory in which the Lagrangian is invariant under local transformations according to certain smooth families of operations (Lie groups). The term gauge refers to any specific mathematical formalism to regulate redundant degrees of freedom in the Lagrangian of a physical system. The transformations between possible gauges, called gauge transformations, form a Lie group—referred to as the symmetry group or the gauge group of the theory. Associated with any Lie group is the Lie algebra of group generators.

Gluon field strength tensor

In theoretical particle physics, the gluon field strength tensor is a second order tensor field characterizing the gluon interaction between quarks. The strong interaction is one of the fundamental interactions of nature, and the quantum field theory (QFT) to describe it is called quantum chromodynamics (QCD). Quarks interact with each other by the strong force due to their color charge, mediated by gluons. Gluons themselves possess color charge and can mutually interact.

Wilson loop

In quantum field theory, Wilson loops are gauge invariant operators arising from the parallel transport of gauge variables around closed loops. They encode all gauge information of the theory, allowing for the construction of loop representations which fully describe gauge theories in terms of these loops. In pure gauge theory they play the role of order operators for confinement, where they satisfy what is known as the area law. Originally formulated by Kenneth G.