Gauge covariant derivative

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The gauge covariant derivative is 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.

Fluid dynamics

In fluid dynamics, the gauge covariant derivative of a fluid may be defined as

where is a velocity vector field of a fluid.

Gauge theory

In gauge theory, which studies a particular class of fields which are of importance in quantum field theory, the minimally-coupled gauge covariant derivative is defined as

where is the electromagnetic vector potential.

(Note that this is valid for a Minkowski metric of signature , which is used in this article. For the minus becomes a plus.)

Construction of the covariant derivative throught the comparator

Construction of the covariant derivative throught Gauge covariance requirement

Consider a generic, possibly non abelian, Gauge transformation given by

where is an element of the Lie algebra associated with the Lie group of transformations, and can be expressed in terms of the generators as .

The partial derivative transforms accordingly as

and a kinetic term of the form is thus not invariant under this transformation.

We can introduce the covariant derivative in this context as a generalization of the partial derivative which transforms covariantly under the Gauge transformation, i.e. an object satisfying

which in operatorial form takes the form

We thus compute (omitting the explicit dependences for brevity)

,

where

The requirement for to transform covariantly is now translated in the condition

To obtain an explicit expression we make the Ansaz

from which it follows that

and

which, using , takes the form

We have thus found an object such that

Quantum electrodynamics

If a gauge transformation is given by

and for the gauge potential

then transforms as

,

and transforms as

and transforms as

so that

and in the QED Lagrangian is therefore gauge invariant, and the gauge covariant derivative is thus named aptly.

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]

where is the coupling constant, is the gluon gauge field, for eight different gluons , is a four-component Dirac spinor, and where is one of the eight Gell-Mann matrices, .

Standard Model

The covariant derivative in the Standard Model can be expressed in the following form:[2]

See also

References

  1. http://www.fuw.edu.pl/~dobaczew/maub-42w/node9.html
  2. See e.g. eq. 3.116 in C. Tully, Elementary Particle Physics in a Nutshell, 2011, Princeton University Press.