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I am studying quantum field theory and gauge theory, and I am confused about how to take the second-order gauge covariant derivative of a field.

(1) The first way is to write the second order gauge covariant derivative as:

$$ D_\mu D^\mu \phi = (\partial_\mu + i g A_\mu)(\partial^\mu - i g A^{* \mu}) \phi $$

where $D_\mu$ is the gauge covariant derivative, $\partial_\mu$ is the partial derivative, $g$ is the coupling constant, $A_\mu = A_\mu^a T^a$ is the gauge field, $T^a$ are the generators of the gauge group, and $\phi$ is the field. Expanding this expression using the distributive property of multiplication over addition, we get:

$$ D_\mu D^\mu \phi = \partial_\mu \partial^\mu \phi - i g A^{* \mu} \partial_\mu \phi + i g A_\mu \partial^\mu \phi + g^2 A_\mu A^{* \mu} \phi $$

(2) The second way is to write the second order gauge covariant derivative as:

$$ D_\mu D^\mu \phi = (\partial_\mu + i g A_\mu)(\partial^\mu \phi - i g A^{* \mu} \phi) $$

Expanding this expression we get:

$$ D_\mu D^\mu \phi = \partial_\mu \partial^\mu \phi - i g \partial_\mu A^{* \mu} \phi - i g A^{* \mu} \partial_\mu \phi + i g A_\mu \partial^\mu \phi + g^2 A_\mu A^{* \mu} \phi $$

The difference between the two ways is the order of the terms in the distribution. In the first way, we firstly define the operator $D_\mu D^\mu$, by which the partial derivative $\partial_\mu$ is written after the gauge field $A$, and it is understood to act on whatever function follows in the equation, not on $A$ itself. In the second way, the partial derivative is understood to act both on $\phi$ and $A$.

My questions are: (1) which of these two ways is the appropriate one for taking the second order gauge covariant derivative? (2) What is the physical or mathematical reason for choosing one over the other? (3) How does the order of the terms affect the outcome of the calculation or the interpretation of the result? (4) Are there any situations where one way is preferable or more convenient than the other?

I would appreciate any references or explanations that can help me understand this topic better. Thank you! 😊

Note: I'm sorry for any confusion, but I am not considering just electromagnetism (which is an Abelian gauge theory), but my question is general for both Abelian and Non-Abelian gauges. In Non-Abelian theories, the gauge field $A_\mu$ may be complex.

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  • $\begingroup$ If $D_\mu = \partial_\mu +i g A_\mu$, then $D^\mu = \partial^\mu +igA^\mu$ and not $\partial^\mu -igA^{\ast\mu}$. Apart from that, simply compute $D_\mu(D^\mu \phi)$ as it is taught in basic analysis. $\endgroup$
    – Hyperon
    Feb 23 at 14:08
  • $\begingroup$ @Hyperon in order to $D_\mu D^\mu$ be hermitian, shouldn't $D^\mu = (g^{\mu \nu} D_{\nu})^{\dagger} = (\partial^\mu + i g A^\mu)^{\dagger} = \partial^\mu - i g A^{* \mu}$? $\endgroup$
    – Ruan
    Feb 23 at 15:40
  • $\begingroup$ 1. The photon field $A_\mu$ is a real field. 2. Look at $\int d^4x (D_\mu \phi)^\ast D^\mu \phi$ and use partial integration. 3. $D^\mu$ is simply $g^{\mu \nu}D_\nu$. $\endgroup$
    – Hyperon
    Feb 23 at 16:01
  • $\begingroup$ Oh, yes. Sorry for any confusion, but in my question I am talking about both abelian and non-abelian gauges. Non-abelian gauges can have complex gauge field $A_\mu$. I'm going to add this to my question. $\endgroup$
    – Ruan
    Feb 23 at 16:16
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    $\begingroup$ Well, first of all you should understand the $U(1)$ case properly before you worry about the non-abelian case . $\endgroup$
    – Hyperon
    Feb 23 at 17:46

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