Gauge covariant derivative ( $D_{μ}$ )

A gauge covariant derivative combines an ordinary derivative with a gauge field so that differentiating a field preserves its gauge-transformation law. In a matrix representation one common convention is

D_{μ} = \partial_{μ} - i g A_{μ}

Table of contents

Invariant integration by parts for a gauge covariant derivative

 0  0

For the Killing form or another invariant bilinear form on a Lie algebra,

\partial_{μ} κ (X, Y) = κ (D_{μ} X, Y) + κ (X, D_{μ} Y) .

(1)

The connection terms cancel by invariance. Thus, for compactly supported variations, a gauge covariant derivative can be moved to the other factor in an integral with a minus sign, just as an ordinary derivative.

// Target: quantum-field-theory.bigb

Conjugate gauge covariant derivative ( $(D_{μ} ϕ)^{} = (\partial_{μ} - i e A_{μ}) ϕ^{}$ )

 0  0

If a complex scalar field uses

D_{μ} ϕ = (\partial_{μ} + i e A_{μ}) ϕ

, its conjugate uses

(D_{μ} ϕ)^{*} = (\partial_{μ} - i e A_{μ}) ϕ^{*}

. Under

ϕ \mapsto e^{- i e α} ϕ

and

A \mapsto A + d α

, the two derivatives transform with opposite phases. Their product is gauge invariant. Applying the same plus-charge operator to both factors leaves an unwanted term proportional to

\partial_{μ} α

and fails gauge invariance.

Gauge covariance of a scalar covariant derivative ( $δ_{X} (D_{μ} ϕ) = ϵ R (X) D_{μ} ϕ$ )

 0  0

With

D_{μ} = \partial_{μ} + R (A_{μ})

and

δ_{X} A_{μ} = - ϵ \partial_{μ} X + ϵ [X, A_{μ}]

, the derivative terms in

δ_{X} (D_{μ} ϕ)

cancel. The remaining terms combine using

R ([X, A_{μ}]) = [R (X), R (A_{μ})]

, so the covariant derivative transforms in the same Lie algebra representation as the scalar. This convention absorbs the gauge coupling into the connection.

Adjoint covariant derivative

 0  0

On an adjoint-valued field

X

, the covariant derivative is

D_{μ} X = \partial_{μ} X + [A_{μ}, X]

. If

X \mapsto g X g^{- 1}

, then

D_{μ} X \mapsto g (D_{μ} X) g^{- 1}

Adjoint gauge variation for a positive-sign covariant derivative ( $δ A_{μ} = \partial_{μ} ω + i g [A_{μ}, ω]$ )

 0  0

Choose the gauge covariant derivative

D_{μ} = \partial_{μ} + i g A_{μ}

and matter transformation

ϕ^{'} = h ϕ

. Covariance

D_{μ}^{'} ϕ^{'} = h D_{μ} ϕ

forces

A_{μ}^{'} = h A_{μ} h^{- 1} + (i / g) (\partial_{μ} h) h^{- 1}

. For

h = 1 - i g ω + O (ω^{2})

, expansion gives the displayed adjoint covariant derivative variation. With an orthonormal Hermitian Lie-algebra basis,

δ A_{μ}^{a} = \partial_{μ} ω^{a} - g f^{ab c} A_{μ}^{b} ω^{c}

. Thus Lorenz-gauge variation is

δ (\partial \cdot A) = \partial \cdot D ω

. Choosing the Faddeev-Popov operator as

M = - \partial \cdot D

, with its field-independent sign absorbed in normalization, gives ghost action

- \overset{c}{ˉ} \partial \cdot D c

; consistent propagator and vertex signs follow from this choice.

Gauge-covariant integration by parts ( $\partial_{μ} κ (Y, Z) = κ (D_{μ} Y, Z) + κ (Y, D_{μ} Z)$ )

 0  0

For Lie-algebra-valued fields and a constant invariant bilinear form on a Lie algebra, the connection terms in the displayed identity cancel. Indeed, invariance gives