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Gauge fixing

From Biocrawler, the free encyclopedia.

In the physics of gauge theories, gauge fixing (also called choosing a gauge) denotes the act of removing redundant field variables. There is an enormous amount of freedom involved in the choice of which variables to remove. Judicious gauge fixing can simplify problems immensely. However, this procedure removes manifest gauge symmetry. As a result, one needs to check the gauge invariance of every result obtained using a specific gauge fixing. Gauge transformations are the acts of going from one gauge fixing to another.

Contents

5 Maximum Abelian gauge

6 See also

7 References and external links

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Gauge freedom

In electrodynamics the electric field E and magnetic field H can be specified in terms of the scalar potential φ and the vector potential A through the relations

${\mathbf E} = -\nabla\phi - \frac{\partial{\mathbf A}}{\partial t}$ and ${\mathbf H} = \nabla\times{\mathbf A}.$

Clearly, one has the freedom of adding the gradient of any function on spacetime, ψ(x,t), to A, and the time derivative of the same function to φ without changing the observable quantities which are the fields. The existence of arbitrary numbers of gauge functions, ψ(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 later.

In other gauge theories, the field equations 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 transformations in general relativity correspond to the action of general coordinate invariance.

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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 was a stright 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.

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Coulomb gauge

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

$\nabla\cdot{\mathbf A}=0.$

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

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Lorenz gauge

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

$\nabla\cdot{\mathbf A} - \frac{\partial\phi}{\partial t}=0.$

This gauge is incomplete, in the sense that there is a residual gauge freedom. This can be seen by examining the constraint that this gauge puts on the gauge function ψ(x,t). 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.

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Weyl gauge

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

φ = 0.

It is named after Hermann Weyl.

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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

$\int d^Dx \left[(A_\mu^1)^2+(A_\mu^2)^2\right].$ where ${\mathbf A}_\mu = A_\mu^a \sigma_a$

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 &\lambda;₃ and &\lambda;₈, then the maximum Abelian gauge is that which maximizes the function