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nLab gauge field

Context

Field Theory

algebraic quantum field theory (perturbative, on curved spacetimes, homotopical)

Introduction

Concepts

field theory:

Lagrangian field theory

quantization

quantum mechanical system, quantum probability

free field quantization

gauge theories

interacting field quantization

renormalization

Theorems

States and observables

Operator algebra

Local QFT

Perturbative QFT

Differential cohomology

Contents

Idea

In physics, specifically in field theory, a gauge field is a type of field in gauge theory. In the standard model of particle physics, gauge fields are the force fields whose quanta are bosons, in contrast to matter-fields whose quanta are fermions.

The eponymous property of gauge fields is that they are identified only up to equivalences called gauge transformations.

The archetypical example of a gauge field is the electromagnetic field in (classically) electromagnetism and (quantumly) in quantum electrodynamics.

The most widely considered kind of gauge field, often the one understood by default, is a non-abelian generalization of electromagnetism, generally called the Yang-Mills field, of which specific instances are the gluon-field in quantum chromodynamics carrying the strong nuclear force, and a similar field carrying the weak nuclear force.

Mathematically, field configurations of such “ordinary” gauge fields are:

The eponymous gauge transformations between gauge fields are thereby identified with isomorphisms between these connections.

The isomorphism class of the underlying principal bundles, hence its class in nonabelian cohomology, reflects the “topological charge” carried by the gauge field, attributed to solitonic field configurations such as Abrikosov vortices, magnetic monopoles and instantons.

(To some extent also the gravitational field behaves like a gauge field with structure/gauge group a Poincaré group, but a further soldering-constraint makes gravity be described more specifically by Cartan connections which are often not counted under “gauge fields” – see at first-order formulation of gravity for more on this.)

Much technology in variational calculus for Lagrangian field theory and subsequent quantization-procedures has been and is being developed already for handling gauge fields with their gauge transformations just locally – common machinery is known as “BRST-BV-formalism” or variants thereof.

Comparatively less attention has traditionally been paid to the global aspects of gauge fields (for instance available BRST-BV formalism does not really reflect or handle them). But realizing that connections on U ( 1 ) U(1) -principal bundles globally describing the electromagnetic field are equivalently 2-cocycles in ordinary differential cohomology, and that connections on general principal bundles may still be understood as cocycles in nonabelian differential cohomology, it is helpful to understand the global nature of gauge fields generally as reflected by notions of cohomology:

CohomologyGauge fields
-theoryflux-quantization law
cocyclefield configuration
coboundarygauge transformation
characterflux densities
ordinary-electromagnetic
differential-gauge potentials
twisted-background fields
equivariant-on orbifolds
Real-on orientifolds
nonabelian-non-linear Gauss law

Cohomology and gauge fields. While cohomology has of course many and diverse applications, in physics no less than in other fields, the role of cohomology specifically in the global description of (higher) gauge fields (“force fields”) is profound: In generalization of the seminal historical observation (“Dirac charge quantization”) that electromagnetic field configurations are globally to be identified with 2-cocycles in ordinary differential cohomology of spacetime, higher gauge field species are similarly to be identified with generalized cohomology theories whose further properties and attributes closely reflect the field’s physical nature, as indicated in the above table.

This cohomological perspective is particularly apt for generalizations of gauge fields such as to (hypothetical) higher gauge fields. For instance, the RR-fields of type II supergravity/type II string theory have famously been conjectured to be cocycles in (twisted equivariant differential) topological K-theory instead of ordinary cohomology.

References

For the time being see the references at gauge theory.

Last revised on November 16, 2024 at 09:13:46. See the history of this page for a list of all contributions to it.