The gauge covariant derivative (pronounced: [geɪdʒ koʊ'vɛriənt dɪ'rɪvətɪv]) is like a generalization of the covariant derivative used in general relativity. If a theory has gauge transformations, it means that some physical properties of certain equations are preserved under those transformations. Likewise, the gauge covariant derivative is the ordinary derivative modified in such a way as to make it behave like a true vector operator, so that equations written using the covariant derivative preserve their physical properties under gauge transformations. This is a concise version of the International Phonetic Alphabet for English sounds. ... Jump to: navigation, search In mathematics, the covariant derivative is a way of specifying a derivative along vector fields on a manifold. ... Jump to: navigation, search General relativity (GR) is the geometrical theory of gravity published by Albert Einstein in 1915. ... Gauge theories are a class of physical theories based on the idea that symmetry transformations can be performed locally as well as globally. ...

Fluid dynamics

In fluid dynamics, the gauge covariant derivative of a fluid may be defined as Fluid mechanics or fluid dynamics is the study of the macroscopic physical behaviour of fluids . ...

where v is a velocity vector field of a fluid. Vector field given by vectors of the form (-y, x) In mathematics a vector field is a construction in vector calculus which associates a vector to every point in a Euclidean space. ...

Quantum field theory

In quantum field theory, the gauge covariant derivative is defined as Quantum field theory (QFT) is the application of quantum mechanics to fields. ...

where A is the electromagnetic vector potential. In vector calculus, a vector potential is a vector field whose curl is a given vector field. ...

What happens to the covariant derivative under a gauge transformation

If a gauge transformation is given by

and

where Λ is a Lorentz transformation, then D_μ transforms as Jump to: navigation, search The Lorentz transformation (LT), named after its discoverer, the Dutch physicist and mathematician Hendrik Antoon Lorentz (1853-1928), forms the basis for the special theory of relativity, which has been introduced to remove contradictions between the theories of electromagnetism and classical mechanics. ...

,

also D_μψ transforms as

and transforms as

so that

and is therefore Lorentz covariant, so that the QED Lagrangian is gauge invariant, and the gauge covariant derivative is thus named aptly. Lorentz covariance is a term in physics for the property of space time, that in two different frames of reference, located at the same event in spacetime but moving relative to each other, all non-gravitational laws must make the same predictions for identical experiments. ... A Lagrangian of a dynamical system, named after Joseph Louis Lagrange, is a functional of the dynamical variables which concisely describes the equations of motion of the system. ...

On the other hand, the non-covariant derivative would not preserve the Lagrangian's gauge symmetry, since

.

Quantum chromodynamics

In quantum chromodynamics, the gauge covariant derivative is [1] Jump to: navigation, search Quantum chromodynamics (QCD) is the theory describing one of the fundamental forces, the strong interaction. ...

where g is the coupling constant, A is the gluon gauge field, for eight different gluons α=1...8, ψ is a four-component Dirac spinor, and where λ_α is one of the eight Gell-Mann matrices, α=1...8. In physics, a coupling constant, usually denoted g, is a number that determines the strength of an interaction. ... Gauge theories are a class of physical theories based on the idea that symmetry transformations can be performed locally as well as globally. ... In mathematics and physics, in particular in the theory of the orthogonal groups, spinors are certain kinds of mathematical objects (group representations of Spin(N), roughly speaking) similar to vectors, but which change sign under a rotation of radians. ... The Gell-Mann matrices, named after Murray Gell-Mann, are the infinitesimal generators of su(3). ...

General relativity

In general relativity, the gauge covariant derivative is defined as Jump to: navigation, search General relativity (GR) is the geometrical theory of gravity published by Albert Einstein in 1915. ...

where Γ is the Christoffel symbol. In mathematics and physics, the Christoffel symbols, named for Elwin Bruno Christoffel (1829-1900), are coordinate-space expressions for the Levi-Civita connection derived from the metric tensor. ...

See also

• Kinetic momentum

References

• Tsutomu Kambe, Gauge Principle For Ideal Fluids And Variational Principle. (PDF file.)

External links

• Question & Answer