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[edit]

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[edit]

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 signature (-, +, +, +), which is used in this article. For (+, -, -, -) the minus becomes a plus.)

Construction of the covariant derivative through the comparator[edit]

Construction of the covariant derivative through Gauge covariance requirement[edit]

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^\dagger (x) \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 dependencies for brevity)

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


 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


 \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[edit]

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[edit]

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[edit]

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

See also[edit]


  1. ^
  2. ^ See e.g. eq. 3.116 in C. Tully, Elementary Particle Physics in a Nutshell, 2011, Princeton University Press.