Gauge covariant derivative in different books

Question

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 1

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.

Answer 2

A late answer, but important in my opinion.

There is another aspect: the sign in the covariant derivative also depends on the sign convention used in the gauge transformation!

This is something that is overlooked a lot.

If the Dirac field transforms as $$ \psi \rightarrow e^{ig\alpha} \psi, $$ then the covariant derivative is defined as $$ D_\mu = \partial_\mu - ig A_\mu. $$

But if the Dirac field transforms as $$ \psi \rightarrow e^{-ig\alpha} \psi, $$ then the covariant derivative is defined as $$ D_\mu = \partial_\mu + ig A_\mu. $$

It is interesting to see that in both cases, the gauge field transforms as $$ A_\mu \rightarrow A_\mu + \partial_\mu \alpha. $$

Peskin and Schroeder use the first convention, with coupling constant $g=-|e|$ (this is indeed a bit confusing, but makes sense from a physical point of view, as the electromagnetic coupling to an electron should be negative). They start using the more general definitions with a $g$ from the moment they go to non-Abelian theories in chapter 15, and keep using the first convention.

The second convention is used eg. in Collins' new book "Foundations on pQCD". This has somewhat become the de-facto standard for the TMD community (transverse momentum dependent PDFs), so people should realise that they cannot simply combine formulas from this book with eg. Peskin and Schroeders'.

Btw, in the non-Abelian case, the sign change also propagates into the definition of the gauge field tensor (in front of the interaction part)

In these examples I assumed $(+---)$ for all, as is standard for particle physics (while $(-+++)$ is standard for GR and string theory/susy).