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In mathematics, and especially differential geometry and mathematical physics, gauge theory is the general study of connections on vector bundles, principal bundles, and fibre bundles. Gauge theory in mathematics should not be confused with the closely related concept of a gauge theory in physics, which is a field theory which admits gauge ...
These contributions to mathematics from gauge theory have led to a renewed interest in this area. The importance of gauge theories in physics is exemplified in the success of the mathematical formalism in providing a unified framework to describe the quantum field theories of electromagnetism, the weak force and the strong force.
A gauge group is a group of gauge symmetries of the Yang–Mills gauge theory of principal connections on a principal bundle. Given a principal bundle P → X {\displaystyle P\to X} with a structure Lie group G {\displaystyle G} , a gauge group is defined to be a group of its vertical automorphisms.
In mathematics, any Lagrangian system generally admits gauge symmetries, though it may happen that they are trivial. In theoretical physics , the notion of gauge symmetries depending on parameter functions is a cornerstone of contemporary field theory .
A gauge theory is a type of theory in physics.The word gauge means a measurement, a thickness, an in-between distance (as in railroad tracks), or a resulting number of units per certain parameter (a number of loops in an inch of fabric or a number of lead balls in a pound of ammunition). [1]
The mathematical framework for traditional gauge theory places the gauge potential as a 1-form on a principal bundle over spacetime. Higher gauge theories provide geometric and category-theoretic , especially higher category theoretic , frameworks for field theories that involve multiple higher differential forms.
Electromagnetic theory possess the simplest kind of local gauge symmetry called () (see unitary group). A theory that displays local gauge invariance is called a gauge theory. In order to formulate other gauge theories we turn the above reasoning inside out. This is the subject of the next section.
Gauge covariant derivative; Gauge fixing; Gauge gravitation theory; Gauge group (mathematics) Gauge symmetry (mathematics) Gauge principle; Gauge theory; Gauge theory gravity; Gaugino condensation; General covariant transformations; Gluon field; Gluon field strength tensor; Gribov ambiguity; Gupta–Bleuler formalism