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A quantity which is gauge-invariant (i.e., invariant under gauge transformations) is the Wilson loop, which is defined over any closed path, γ, as follows:
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 line is the equivalent of a gauge function; it need not be straight. Almost any line is a valid gauge fixing, i.e., there is a large gauge freedom. In summary, to tell whether the rod is twisted, the gauge must be known. Physical quantities, such as the energy of the torsion, do not depend on the gauge, i.e., they are gauge invariant.
This gauge transformation property is often used to directly introduce the Wilson line in the presence of matter fields () transforming in the fundamental representation of the gauge group, where the Wilson line is an operator that makes the combination () † [,] gauge invariant. [4]
Calculating the expectation value in a gauge invariant way always gives zero, in agreement with Elitzur's theorem. The Higgs mechanism can however be reformulated entirely in a gauge invariant way in what is known as the Fröhlich–Morchio–Strocchi mechanism which does not involve spontaneous symmetry breaking of any symmetry. [11]
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 condition is Lorentz invariant. The Lorenz gauge condition does not completely determine the gauge: one can still make a gauge transformation +, where is the four-gradient and is any harmonic scalar function: that is, a scalar function obeying =, the equation of a massless scalar field.
In contrast to the Berry connection, which is physical only after integrating around a closed path, the Berry curvature is a gauge-invariant local manifestation of the geometric properties of the wavefunctions in the parameter space, and has proven to be an essential physical ingredient for understanding a variety of electronic properties. [4] [5]