Calibration transformation

A calibration transformation changes the calibration fields of a physical theory (e.g. the electromagnetic potentials or the potential energy ) in such a way that the physically effective fields (e.g. the electromagnetic field or a force field ) and thus all observable processes remain the same. This is referred to as calibration freedom .

One differentiates:

Global calibration transformations: they are carried out at every location with the same value, e.g. B. the shift of the zero point of the potential energy, the choice of the reference potential when measuring electrical voltages, a constant phase factor on the complex wave function of quantum mechanics .
Local calibration transformations: the changes to a parameter are not determined by a single value, but with the help of a spatially and / or temporally varying function.

A physical effect that is invariant under local gauge transformations is called a gauge invariant effect. A theory that derives the physical equations of motion from a gauge-invariant effect according to the principle of the smallest effect is called a calibration theory . All fundamental interactions - gravitation , electromagnetism , weak interaction ( beta decay of the neutron) and the strong interaction (nuclear forces) - are described by such gauge theories.

According to Noether's theorem , the symmetry on which a gauge transformation is based indicates the existence of a conservation quantity.

Electrodynamics

The electrodynamics is invariant under the gauge transformation

{\ displaystyle \ phi '({\ vec {r}}, t) = \ phi ({\ vec {r}}, t) - {\ frac {\ partial} {\ partial t}} \ Lambda ({\ vec {r}}, t) \,}

{\ displaystyle {\ vec {A}} '({\ vec {r}}, t) = {\ vec {A}} ({\ vec {r}}, t) + \ mathrm {grad} \, \ Lambda ({\ vec {r}}, t) \,}

which changes the electrical potential and the magnetic potential by the partial derivatives of a freely selectable function . ${\ displaystyle \ phi}$ ${\ displaystyle {\ vec {A}}}$ ${\ displaystyle \ Lambda}$

This transformation does not change the magnetic field

{\ displaystyle {\ vec {B}} = \ mathrm {red} ~ {\ vec {A}}}

nor the electric field

{\ displaystyle {\ vec {E}} = - \ mathrm {grad} \, \ phi - {\ frac {\ partial} {\ partial t}} {\ vec {A}}}

For the definition of and see Gradient and Rotation . ${\ displaystyle \ mathrm {grad}}$ ${\ displaystyle \ mathrm {red}}$

Examples

Calibration transformations can be used to simplify calculations.

The examples also use the measurement system . ${\ displaystyle c = 1}$

Lorenz calibration

Through the calibration transformation named after Ludvig Lorenz with a calibration function that ${\ displaystyle \ psi \}$

{\ displaystyle \ nabla ^ {2} \ psi - {\ frac {\ partial ^ {2}} {\ partial t ^ {2}}} \ psi = - (\ nabla \ cdot {\ vec {A}} + {\ frac {\ partial} {\ partial t}} \ phi)}

fulfilled, the inhomogeneous Maxwell equations become two independent wave equations of and . ${\ displaystyle \ phi \}$ ${\ displaystyle {\ vec {A}}}$

Coulomb calibration

Meets the calibration function , however, ${\ displaystyle \ psi}$

{\ displaystyle \ nabla ^ {2} \ psi = - \ nabla \ cdot {\ vec {A}} \,}

so the transformation helps to transform the scalar field straight to the Coulomb potential of the charges; then satisfies the electrostatic Poisson equation . ${\ displaystyle \ phi}$ ${\ displaystyle \ phi}$

general theory of relativity

The general theory of relativity is also a gauge theory, the gauge transformation of which defines new coordinates as freely selectable functions of the previous coordinates:

{\ displaystyle x '^ {\ mu} = x' ^ {\ mu} (x ^ {\ mu}) \.}

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

literature

↑ Robert G. Brown: Gauge Transformations ( English ). Retrieved January 17, 2013: "A gauge transformation can be broadly defined as any formal, systematic transformation of the potentials that leaves the fields invariant."

[1] Robert G. Brown: Gauge Transformations ( English ). Retrieved January 17, 2013: "A gauge transformation can be broadly defined as any formal, systematic transformation of the potentials that leaves the fields invariant."