Gauge transformation

A calibration transformation changes the so-called calibration fields of a physical theory ( such as electromagnetic potentials or the potential energy ) such that the effective physical fields (e.g., the electromagnetic field or a force field ), and thus all operations observable The same. remain This is referred to as a calibration freedom.

We distinguish global and local gauge transformations. A global gauge transformation is performed at each location with the same value, such as the shift of the zero point of the potential energy, the choice of the reference potential for the measurement of electrical voltages, a constant phase factor of the complex wave function of quantum mechanics. At a local gauge transformation, these changes are not determined on a parameter by a single value, but with the help of a local and / or time-varying function.

A physical effect that is invariant under local gauge transformations is called a gauge-invariant action. One theory that is gaining the physical equations of motion according to the principle of least action of a gauge invariant action is called a gauge theory. All fundamental interactions - gravity, electromagnetism, weak interaction (beta- decay of the neutron ) and the strong interaction ( nuclear forces ) - are described by such gauge theories. According to the Noether theorem, the underlying symmetry of a gauge transformation to the existence of a conserved quantity.

Electrodynamics

The electrodynamics is invariant under the gauge transformation

Changing the potential function of the electric field and the magnetic field to the partial derivatives of a freely selectable function.

This transformation does not change the magnetic field

Nor the electric field

The example uses the system of units with, for defining and see gradient and rotation.

Examples

Gauge transformations can be exploited to simplify calculations mathematically:

Lorenz gauge

By named after Ludvig Lorenz gauge transformation with a calibration function that

Fulfilled, the inhomogeneous Maxwell equations to two independent wave equations and.

Coulomb gauge

Meets the calibration function, however,

Helping the transformation to transform the straight to the Coulomb potential scalar field of the charges; then satisfies the electrostatic Poisson equation.

General Theory of Relativity

Similarly, the general theory of relativity is a gauge theory whose gauge transformation defines new coordinates as arbitrary functions of the old coordinates:

The effect of general relativity does not change under this gauge transformation.