quantum field theory - 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?

Answer

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.