Gauge Fixing

GAUGE FREEDOM

In Electrodynamics the Electric Field

\mathbf{E}

and Magnetic Flux Density

\mathbf{B}

can be specified in terms of the Scalar Potential

arphi

and the Vector Potential

\mathbf{A}

through the relations:

:

{\mathbf E} = -
ablaarphi - rac{\partial{\mathbf A}}{\partial t}

and

{\mathbf B} = 
abla	imes{\mathbf A}.

However, the

\mathbf{E}

and

\mathbf{B}

fields are unchanged if we take ''any'' function

\psi(\mathbf{x},t)

and transform

\mathbf{A}

and

arphi

via:

:

\mathbf{A} 
ightarrow \mathbf{A} + 
abla \psi

arphi 
ightarrow arphi - rac{\partial\psi}{\partial t}

A particular choice of the scalar and vector potentials is a gauge, and a

\psi

scalar function used to change of gauge is called a '''gauge function'''. The existence of arbitrary numbers of gauge functions

\psi(\mathbf{x},t)

, corresponds to the U(1) '''gauge freedom''' of this theory. Gauge fixing can be done in many ways, some of which we exhibit below.

In other Gauge Theories , the Field Equation s allow similar gauge freedom, and one can perform gauge fixing in very similar fashion. Since the gauge potentials belong to the Adjoint Representation of the Gauge Group , one needs to fix a function corresponding to each component of this representation.

In the language of Vector Bundles , appropriate to classical Gauge Theories , the choice of a gauge corresponds to choosing a Section on the bundle. The term gauge fixing is also applied to the choice of coordinates in General Relativity . Gauge Transformation s in general relativity correspond to the action of General Coordinate Invariance .

An illustration

By looking at a cylindrical rod can one tell whether it is twisted? If the rod is perfectly cylindrical, then the circular symmetry of the cross section makes it impossible to give an answer. However, if there were a straight line drawn along the length of the rod, then one could easily say whether or not there is a twist by looking at state of the line. Drawing a line is gauge fixing. Drawing the line spoils the gauge symmetry, ie, the circular symmetry U(1) of the cross section at each point of the rod. The line is the equivalent of a '''gauge function'''; it need not be straight. Almost any line is a valid gauge fixing, ie, there is a large '''gauge freedom'''. To tell whether the rod is twisted, you need to first know the gauge. Physical quantities, such as the energy of the torsion does not depend on the gauge, ie, they are '''gauge invariant'''.

COULOMB GAUGE

The Coulomb gauge (also known as '''radiation''' or '''transverse gauge''') corresponds to choosing the gauge function in such a way that

::

abla\cdot{\mathbf A}=0.

This has the drawback that in this gauge fixing, sometimes A and φ may propagate faster than the speed of light. However, this is harmless, since A and φ alone are not observable, and the observable fields behave properly.

In the Coulomb gauge, it can be seen from Gauss' Law that the scalar potential is determined simply by Poisson's Equation based on the total charge density ρ (including Bound Charge ):

:

-
abla^2 \phi = rac{
ho}{arepsilon_0}

LORENZ GAUGE

The Lorenz Gauge is obtained by the choice of the gauge function which gives

::

abla\cdot{\mathbf A} + rac{1}{c^2}rac{\partial\phi}{\partial t}=0.

We find easily that in this way, we may change the gauge without breaking the relation above if the gauge function verifies the Wave Equation

{ \partial^2 \psi \over \partial t^2 } = c^2 
abla^2\psi

.
The Lorenz gauge is incomplete, in the sense that there is this residual gauge freedom. However, the gauge degrees of freedom propagate at the speed of light. In Special Relativity this is a Covariant gauge.

Note that this gauge is known after the Danish physicist Ludwig Lorenz and not after H. Lorentz .

WEYL GAUGE

The Weyl gauge (also known as the '''temporal gauge''') is an ''incomplete'' gauge obtained by the choice

::

\phi=0.

It is named after Hermann Weyl .

MAXIMUM ABELIAN GAUGE

In any non- Abelian Gauge Theory , any maximum Abelian gauge is an ''incomplete'' gauge which fixes the gauge freedom outside of the Maximum Abelian Subgroup . Examples are

For SU(2) gauge theory in D dimensions, the maximum Abelian subgroup is a U(1) subgroup. If this is chosen to be the one generated by the Pauli Matrix σ₃, then the maximum Abelian gauge is that which maximizes the function

\int d^Dx \left[(A_\mu^1)^2+(A_\mu^2)^2
ight].

{\mathbf A}_\mu = A_\mu^a \sigma_a

For SU(3) gauge theory in D dimensions, the maximum Abelian subgroup is a U(1)XU(1) subgroup. If this is chosen to be the one generated by the Gell-Mann Matrices λ₃ and λ₈, then the maximum Abelian gauge is that which maximizes the function

\int d^Dx \left[(A_\mu^1)^2+(A_\mu^2)^2+(A_\mu^4)^2+(A_\mu^5)^2+(A_\mu^6)^2+(A_\mu^7)^2
ight].

{\mathbf A}_\mu = A_\mu^a \lambda_a

	CATEGORIES ABOUT GAUGE FIXING

	electromagnetism

	quantum field theory

	quantum electrodynamics

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Information About

Gauge Fixing