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. The willingness to question previously held truths and search for new answers resulted in a period of major scientific advancements, now known as the Scientific Revolution. ... Gauge theories are a class of physical theories based on the idea that symmetry transformations can be performed locally as well as globally. ... In physics, a field is an assignment of a quantity to every point in space. ... Gauge theories are a class of physical theories based on the idea that symmetry transformations can be performed locally as well as globally. ... Gauge theories are a class of physical theories based on the idea that symmetry transformations can be performed locally as well as globally. ...

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 Electromagnetism is the physics of the electromagnetic field: a field, encompassing all of space, composed of the electric field and the magnetic field. ...

 and 

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. Gauge theories are a class of physical theories based on the idea that symmetry transformations can be performed locally as well as globally. ... The adjoint representation of a Lie group G is the linearized version of the action of G on itself by conjugation. ... Gauge theories are a class of physical theories based on the idea that symmetry transformations can be performed locally as well as globally. ...

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. In mathematics, a vector bundle is a geometrical construct where to every point of a topological space (or manifold, or algebraic variety) we attach a vector space in a compatible way, so that all those vector spaces, glued together, form another topological space (or manifold or variety). ... Gauge theories are a class of physical theories based on the idea that symmetry transformations can be performed locally as well as globally. ... Section can be: A cross section (in the common sense or the physics sense) In mathematics: A conic section A section of a fiber bundle or sheaf A Caesarean section In UK law, Section 28 In the fictional Star Trek universe, Section 31 A military unit A section (land) is... Two-dimensional visualisation of space-time distortion. ... Gauge theories are a class of physical theories based on the idea that symmetry transformations can be performed locally as well as globally. ...

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. In mathematics, the unitary group of degree n over the field F (which is either the field R of real numbers or the field C of complex numbers) is the group of n by n unitary matrices with entries from F, with the group operation that of matrix multiplication. ...

Coulomb gauge

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

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.

Lorenz gauge

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

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. A simple introduction to this subject is provided in Special relativity for beginners Special relativity(SR) or the special theory of relativity is the physical theory published in 1905 by Albert Einstein. ... In category theory, see covariant functor. ...

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

φ = 0.

It is named after Hermann Weyl. Hermann Weyl (November 9, 1885 - December 8, 1955) was a German mathematician and physicist, one of the first people to combine general relativity with the laws of electromagnetism. ...

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 In mathematics, an abelian group is a commutative group, i. ... Gauge theories are a class of physical theories based on the idea that symmetry transformations can be performed locally as well as globally. ...

• 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

 where 

• 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

 where 

In mathematics, the special unitary group of degree n is the group of n by n unitary matrices with determinant 1 and entries from the field C of complex numbers, with the group operation that of matrix multiplication. ... The Pauli matrices are a set of 2 × 2 complex Hermitian matrices developed by Pauli. ... In mathematics, the special unitary group of degree n is the group of n by n unitary matrices with determinant 1 and entries from the field C of complex numbers, with the group operation that of matrix multiplication. ... The Gell-Mann matrices, named after Murray Gell-Mann, are the infinitesimal generators of su(3). ...

See also

• Gauge theory and gauge transformation
• The Aharonov-Bohm effect
• Vector bundles and sections
• Fadeev-Popov ghosts
• Gribov problem and Gribov ambiguity

Gauge theories are a class of physical theories based on the idea that symmetry transformations can be performed locally as well as globally. ... Gauge theories are a class of physical theories based on the idea that symmetry transformations can be performed locally as well as globally. ... The Aharonov-Bohm effect is a quantum mechanical phenomenon by which a charged particle is affected by electromagnetic fields in regions from which the particle is excluded, proposed by Aharonov and Bohm in 1959. ... In mathematics, a vector bundle is a geometrical construct where to every point of a topological space (or manifold, or algebraic variety) we attach a vector space in a compatible way, so that all those vector spaces, glued together, form another topological space (or manifold or variety). ... Section can be: A cross section (in the common sense or the physics sense) In mathematics: A conic section A section of a fiber bundle or sheaf A Caesarean section In UK law, Section 28 In the fictional Star Trek universe, Section 31 A military unit A section (land) is...

References and external links

• Landau and Lifschitz, "The classical theory of fields"
• Jackson, "Classical Electrodynamics"