Gauge covariant derivative
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.
In fluid dynamics, the gauge covariant derivative of a fluid may be defined as
where is a velocity vector field of a fluid.
where is the electromagnetic four potential.
(Note that this is valid for a Minkowski metric signature , which is used in this article. For the minus becomes a plus.)
Construction of the covariant derivative through the comparator
Construction of the covariant derivative through Gauge covariance requirement
Consider a generic (possibly non-Abelian) Gauge transformation given by
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 dependencies for brevity)
The requirement for to transform covariantly is now translated in the condition
To obtain an explicit expression we make the Ansatz
from which it follows that
which, using , takes the form
We have thus found an object such that
If a gauge transformation is given by
and for the gauge potential
then transforms as
and transforms as
and transforms as
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
- See e.g. eq. 3.116 in C. Tully, Elementary Particle Physics in a Nutshell, 2011, Princeton University Press.
- Tsutomu Kambe, Gauge Principle For Ideal Fluids And Variational Principle. (PDF file.)