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.

Fluid dynamics

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

\nabla_t \mathbf{v}:= \partial_t \mathbf{v} + (\mathbf{v} \cdot \nabla) \mathbf{v}

where $\mathbf{v}$ 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

D_\mu := \partial_\mu - i e A_\mu

where $A_\mu$ 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 Gauge covariance requirement

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

\phi(x) \rightarrow U(x) \phi(x) \equiv e^{i\alpha(x)} \phi(x),

\phi^\dagger(x) \rightarrow \phi^\dagger(x) U(x)^\dagger \equiv \phi^\dagger(x) e^{-i\alpha(x)}, \qquad U^\dagger = U^{-1}.

where $\alpha(x)$ is an element of the Lie algebra associated with the Lie group of transformations, and can be expressed in terms of the generators as $\alpha(x) = \alpha^a(x) t^a$ .

The partial derivative $\partial_\mu$ transforms accordingly as

\partial_\mu \phi(x) \rightarrow U(x) \partial_\mu \phi(x) + (\partial_\mu U) \phi(x) \equiv e^{i\alpha(x)} \partial_\mu \phi(x) + i (\partial_\mu \alpha) e^{i\alpha(x)} \phi(x)

and a kinetic term of the form $\phi^\dagger \partial_\mu \phi$ is thus not invariant under this transformation.

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

D_\mu \phi(x) \rightarrow D'_\mu \phi'(x) = U(x) D_\mu \phi(x),

which in operatorial form takes the form

D'_\mu = U(x) D_\mu U^\dagger(x).

We thus compute (omitting the explicit $x$ dependences for brevity)

D_\mu \phi \rightarrow D'_\mu U \phi = UD_\mu \phi + (\delta D_\mu U + [D_\mu,U])\phi

where

D_\mu \rightarrow D'_\mu \equiv D_\mu + \delta D_\mu,

A_\mu \rightarrow A'_\mu = A_\mu + \delta A_\mu.

The requirement for $D_\mu$ to transform covariantly is now translated in the condition

(\delta D_\mu U + [D_\mu,U])\phi = 0.

To obtain an explicit expression we make the Ansatz

D_\mu = \partial_\mu - ig A_\mu,

from which it follows that

\delta D_\mu \equiv -ig \delta A_\mu

and

\delta A_\mu = [U,A_\mu]U^\dagger -\frac{i}{g} (\partial_\mu U)U^\dagger

which, using $U(x) = 1 + i \alpha(x) + \mathcal{O}(\alpha^2)$ , takes the form

\delta A_\mu = \frac{1}{g} ( \partial_\mu \alpha - ig [A_\mu,\alpha] ) + \mathcal{O}(\alpha^2) = \frac{1}{g} D_\mu \alpha + \mathcal{O}(\alpha^2)

We have thus found an object $D_\mu$ such that

\phi^\dagger(x) D_\mu \phi(x) \rightarrow \phi'^\dagger(x) D'_\mu \phi'(x) = \phi^\dagger(x) D_\mu \phi(x)

Quantum electrodynamics

If a gauge transformation is given by

\psi \mapsto e^{i\Lambda} \psi

and for the gauge potential

A_\mu \mapsto A_\mu + {1 \over e} (\partial_\mu \Lambda)

then $D_\mu$ transforms as

D_\mu \mapsto \partial_\mu - i e A_\mu - i (\partial_\mu \Lambda)

and $D_\mu \psi$ transforms as

D_\mu \psi \mapsto e^{i \Lambda} D_\mu \psi

and $\bar \psi := \psi^\dagger \gamma^0$ transforms as

\bar \psi \mapsto \bar \psi e^{-i \Lambda}

so that

\bar \psi D_\mu \psi \mapsto \bar \psi D_\mu \psi

and $\bar \psi D_\mu \psi$ 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 $\partial_\mu$ would not preserve the Lagrangian's gauge symmetry, since

\bar \psi \partial_\mu \psi \mapsto \bar \psi \partial_\mu \psi + i \bar \psi (\partial_\mu \Lambda) \psi

Quantum chromodynamics

In quantum chromodynamics, the gauge covariant derivative is^[1]

D_\mu := \partial_\mu - i g \, A_\mu^\alpha \, \lambda_\alpha

where $g$ is the coupling constant, $A$ is the gluon gauge field, for eight different gluons $\alpha=1 \dots 8$ , $\psi$ is a four-component Dirac spinor, and where $\lambda_\alpha$ is one of the eight Gell-Mann matrices, $\alpha=1 \dots 8$ .

Standard Model

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

D_\mu := \partial_\mu - i \frac{g_1}{2} \, Y \, B_\mu - i \frac{g_2}{2} \, \sigma_j \, W_\mu^j - i \frac{g_3}{2} \, \lambda_\alpha \, G_\mu^\alpha

References

↑ http://www.fuw.edu.pl/~dobaczew/maub-42w/node9.html
↑ 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.)