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Gauge covariant derivative in different books

+ 8 like - 0 dislike
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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 in Theoretical Physics by Eduardo Guerras (435 points) [ no revision ]

1 Answer

+ 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 Qmechanic (2,790 points) [ no revision ]
Most voted comments show all comments
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
Most recent comments show all comments
@EduardoGuerrasValera The classic tome on general relativity (Misner, Thorne and Wheeler) has on the inside front cover a massive table of sign conventions as used by different authors in GR (as of the publication date). Someone needs to put a table like that together for this now too.

This post imported from StackExchange Physics at 2014-06-14 12:53 (UCT), posted by SE-user Michael Brown
I'm surprised that Peskin and Srednicki take $e<0$. I've never seen that before. Is this common in QFT?

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

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