# Charge and the Dirac field

In Zee's quantum field theory in a nutshell, 2nd edition, pg 550 he has

$Q=\int {d^3p \over (2\pi)^3(E_p/m)} \sum_s \{b^\dagger(p,s)b(p,s)-d^\dagger(p,s)d(p,s)\}$

showing clearly that $b$ annihilates a negative charge and $d$ a positive charge.

I would very much appreciate an explanation of why its not the other way round. As the $d^\dagger(p,s)d(p,s)$ term has the negative sign why doesn't that mean it is associated with the negative charge particle?

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This is a matter of convention.

You are totally right: the $Q$ operator you have written implies that $b$ annihilates a positive charge and vice versa.

The thing is that in QED one usually defines $Q$ in a slightly different way, namely:

$$Q=-\left| e\right| \int {d^3p \over (2\pi)^3(E_p/m)} \sum_s \{b^\dagger(p,s)b(p,s)-d^\dagger(p,s)d(p,s)\}$$

with $-\left| e\right|$ the electron's charge. Then, with this definition, $b$ annihilates states with negative charge $-\left| e\right|$.

The confusion comes from what it is arguably the worst convention in the history of physics (and possibly chemistry and engineering): the electron's charge was chosen negative! Since there are many more electrons than positrons in the universe and in the Earth (and for this reason the appropriate convention that the electron is the particle and the positron its antiparticle, and not in the other way around), the electron's charges should have been chosen as positive! —so the positron should have been named negatron or perhaps the Dirac or why not simply anti-electron and having negative charge ;-)

Anyway, it is just a convention.

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Well the sign-convention was in place long before Dirac and there's never been enough consenting will to overhaul the system. Blame Franklin. Also, it's not a physics point, but... "You are totally $\textit{write}$"? We should probably try and get the English right as well as the physics! – James Sep 9 '12 at 21:59
@James Regarding the English: Thank you for letting me know the typo. I try to right correctly, although I know that I make many mistakes. I am honestly grateful to everyone who corrects me and helps me to improve. So if you find any wording/spelling/grammar mistake, please let me know. – Diego Mazón Sep 9 '12 at 22:24
@James Concerning the physics: Have I really said that the convention is due to Dirac or that it was not in place long before Dirac? If I did write this, please let me know where I did it because it is an English mistake again. The only place where I mentioned "Dirac" was in connection with his prediction of the positron and a possible alternative name for the positron in case the convention had been the opposite. Was it not clear? – Diego Mazón Sep 9 '12 at 23:17
No you didn't say that the convention is due to Dirac, but how you stated it ("The electron's charge was chosen negative", "the positron should have been named the negatron") rather makes it sound like the sign convention was imposed when the electron was discovered, or the positron predicted. – James Sep 10 '12 at 17:12