# Gauge covariant derivative in different books

+ 8 like - 0 dislike
408 views

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?

This post imported from StackExchange Physics at 2014-06-14 12:53 (UCT), posted by SE-user Eduardo Guerras Valera
asked Feb 24, 2013

+ 9 like - 0 dislike

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.

This post imported from StackExchange Physics at 2014-06-14 12:53 (UCT), posted by SE-user Qmechanic
answered Feb 24, 2013 by (3,120 points)
thanks very much, it is a luxury to have that precise and quick answer!

This post imported from StackExchange Physics at 2014-06-14 12:53 (UCT), posted by SE-user Eduardo Guerras Valera
@Eduardo Guerras Valera: Thanks. I updated the answer.

This post imported from StackExchange Physics at 2014-06-14 12:53 (UCT), posted by SE-user Qmechanic
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.

This post imported from StackExchange Physics at 2014-06-14 12:53 (UCT), posted by SE-user Qmechanic
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.]

This post imported from StackExchange Physics at 2014-06-14 12:53 (UCT), posted by SE-user Qmechanic
thank you so much for the additional clarifications in today edit! It is a pity I cannot upvote more than once or give two green marks!

This post imported from StackExchange Physics at 2014-06-14 12:53 (UCT), posted by SE-user Eduardo Guerras Valera
I'm surprised that Peskin and Srednicki take $e<0$. I've never seen that before. Is this common in QFT?
 Please use answers only to (at least partly) answer questions. To comment, discuss, or ask for clarification, leave a comment instead. To mask links under text, please type your text, highlight it, and click the "link" button. You can then enter your link URL. Please consult the FAQ for as to how to format your post. This is the answer box; if you want to write a comment instead, please use the 'add comment' button. Live preview (may slow down editor)   Preview Your name to display (optional): Email me at this address if my answer is selected or commented on: Privacy: Your email address will only be used for sending these notifications. Anti-spam verification: If you are a human please identify the position of the character covered by the symbol $\varnothing$ in the following word:p$\hbar$ysicsOve$\varnothing$flowThen drag the red bullet below over the corresponding character of our banner. When you drop it there, the bullet changes to green (on slow internet connections after a few seconds). To avoid this verification in future, please log in or register.