# Gauge covariant derivative in different books

It puzzles me that Zee uses throughout the book this definition of covariant derivative: $$D_{\mu} \phi=\partial_{\mu}\phi-ieA_{\mu}\phi$$ with a minus sign, despite of the use of the $(+---)$ convention.

But then I see that Srednicki, at least in the free preprint, uses too the same definition, with the same minus sign. The weird thing is that Srednicki uses $(-+++)$

I looked too into Peskin & Schröder, who stick to $(+---)$ (the same as Zee) and the covariant derivative there is:

$$D_{\mu} \phi=\partial_{\mu}\phi+ieA_{\mu}\phi$$

Now, can any of you tell Pocoyo what is happening here? Why can they consistently use different signs in that definition?

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We will work in units with $c=1=\hbar$. The $4$-potential $A^{\mu}$ with upper index is always defined as

$$A^{\mu}~=~(\Phi,{\bf A}).$$

1) Lowering the index of the $4$-potential depends on the sign convention

$$(+,-,-,-)\qquad \text{resp.} \qquad(-,+,+,+)$$

for the Minkowski metric $\eta_{\mu\nu}$. This Minkowski sign convention is used in

$$\text{Ref. 1 (p. xix) and Ref. 2 (p. xv)} \qquad \text{resp.} \qquad \text{Ref. 3 (eq. (1.9))}.$$

The $4$-potential $A_{\mu}$ with lower index is $$A_{\mu}~=~(\Phi,-{\bf A}) \qquad \text{resp.} \qquad A_{\mu}~=~(-\Phi,{\bf A}).$$

Maxwell's equations with sources are

$$d_{\mu}F^{\mu\nu}~=~j^{\nu} \qquad \text{resp.} \qquad d_{\mu}F^{\mu\nu}~=~-j^{\nu}.$$

The covariant derivative is

$$D_{\mu} ~=~d_{\mu}+iqA_{\mu}\qquad \text{resp.} \qquad D_{\mu} ~=~d_{\mu}-iqA_{\mu},$$

where $q=-|e|$ is the charge of the electron.

2) The sign convention for the elementary charge $e$ is

$$e~=~-|e| ~<~0 \qquad \text{resp.} \qquad e~=~|e|~>~0.$$

This charge sign convention is used in

$$\text{Ref. 1 (p. xxi) and Ref. 3 (below eq. (58.1))} \qquad \text{resp.} \qquad \text{Ref. 2.}$$

References:

1. M.E. Peskin and D.V Schroeder, An Introduction to QFT.

2. A. Zee, QFT in a nutshell.

3. M. Srednicki, QFT.

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thanks very much, it is a luxury to have that precise and quick answer! – Eduardo Guerras Valera Feb 24 at 1:04
@Eduardo Guerras Valera: Thanks. I updated the answer. – Qmechanic Feb 24 at 10:27
FYI: Srednicki mentions explicitly his convention below eq. (58.1). I'll try to pinpoint the others as well, and make an update at some point in the future. – Qmechanic Feb 24 at 23:17
FYI: W. Siegel, Fields, has Minkowski sign convention $(-,+,+,+)$ (p.55); has charge sign convention e=|e| (p.184,204); and covariant derivative $D_{\mu}=d_{\mu}+iqA_{\mu}$ (p.184,204), which is opposite. [Also note that Siegel's definition (p.169ff) of the action $S=\int\! dt ({\rm Pot.terms - Kin.terms})$ is opposite of the standard definition.] – Qmechanic Feb 25 at 21:25
FYI: (i) C. Itzykson and J.-B. Zuber, QFT, has Minkowski sign convention $(+,-,-,-)$ (eq.A-1); has charge sign convention e=-|e|; and covariant derivative $D_{\mu}=d_{\mu}+iqA_{\mu}$ (eq.4-77), like e.g. Ref. 1, and e.g. Bjorken and Drell. (ii) S. Weinberg, The Quantum Theory of Fields, has Minkowski sign convention $(-,+,+,+)$ (p.xxv); has charge sign convention e=|e| (p.xxvi); and covariant derivative $D_{\mu}=d_{\mu}-iqA_{\mu}$ (eq.8.1.21) – Qmechanic Feb 26 at 16:48