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If is a subset of a real or complex vector space, then the Minkowski functional or gauge of is defined to be the function: [,], valued in the extended real numbers, defined by ():= {: >}, where the infimum of the empty set is defined to be positive infinity (which is not a real number so that () would then not be real-valued).
In the physical literature on gauge theory, a structure group of a principal bundle often is called the gauge group. In quantum gauge theory, one considers a normal subgroup () of a gauge group () which is the stabilizer = {() (): = ~} of some point ~ of a group bundle ~. It is called the pointed gauge group. This group acts freely on a space ...
QED is generally regarded as the first, and simplest, physical gauge theory. The set of possible gauge transformations of the entire configuration of a given gauge theory also forms a group, the gauge group of the theory. An element of the gauge group can be parameterized by a smoothly varying function from the points of spacetime to the ...
Gauge theory in mathematics should not be confused with the closely related concept of a gauge theory in physics, which is a field theory that admits gauge symmetry. In mathematics theory means a mathematical theory , encapsulating the general study of a collection of concepts or phenomena, whereas in the physical sense a gauge theory is a ...
the gauge as used in the definition of the Henstock-Kurzweil integral, also known as the gauge integral; in fractal geometry, a synonym for dimension function; in control theory and dynamical systems, a synonym for Lyapunov candidate function; in gauge theory, a synonym for gauge symmetry. a type of Minkowski functional
A particular choice of the scalar and vector potentials is a gauge (more precisely, gauge potential) and a scalar function ψ used to change the gauge is called a gauge function. [citation needed] The existence of arbitrary numbers of gauge functions ψ(r, t) corresponds to the U(1) gauge freedom of this theory. Gauge fixing can be done in many ...
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.
For the leptons, the gauge group can be written SU(2) l × U(1) L × U(1) R. The two U(1) factors can be combined into U(1) Y × U(1) l, where l is the lepton number. Gauging of the lepton number is ruled out by experiment, leaving only the possible gauge group SU(2) L × U(1) Y. A similar argument in the quark sector also gives the same result ...