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?

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

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.

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

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

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@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

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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

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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

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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

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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)

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

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thanks. If it is of any use, some books in my personal library follow: Schutz 2009(-+++); Chetaev 1989 (---+); Einstein 1921(---+); Wald 1984(-+++); Dirac 1967(+---); Susskind&Lindesay 2005(+---); Choudhuri 2010(-+++); Carroll&Ostlie 2007(+---); Tong 2007 classnotes on QFT (+---); Tong 2009 classnotes on ST (-+++...+), 't Hooft 2009 notes on BHs (-+++); Schneider&Ehlers&Falco 1989 (+---); Zee 2010 (+---). I write the signature below the title, so that I don't need to find it out again and again every time I consult something, I guess I'll have to add the electron charge now in QFT books.

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

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Why the h*** don't they stick to the original convention in the very first paper for everything? It is really so painful, or do they have some need to appear as original? The Susskind lectures have now the + sign, and the David Tong notes have God knows what convention in the Faraday tensor. Everytime I try to cross some details between books and specially the first time I use a new document or internet page, I have to spend the hell of a time figuring out which is the arbitrary convention of that author. The first ten times it was even funny, now it is a pain in the neck. Damn them all!

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

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@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

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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