# Can a consistent theory of electric force exist where charges of equal sign attract each other and charges of opposite sign repel each other?

I am preparing tomorrows lesson about electrostatics and the Coulomb law and wondered the following:

By simple experiments we can show that the electric force can be attractive and repulsive, so we need a charge which can be positive or negative. Also, we can show experimentally that bodies whose charges are of the same type (sign) repel each other, while bodies whose charges are of different type (opposing sign) attract each other.

Naturally, one could ask whether a world is possible where it is the other way around: There is an interaction with a charge, which can be positive and negative, as well, but charges of equal sign attract each other, while charges of opposing sign repel each other.

As far as I understand, if we work with Coulomb's force law, it's a matter of adding an additional minus sign, and we would get a theory which describes exactly such a world. However, if we consider quantum electrodynamics, no matter the sign in front of the coupling constant in the gauge covariant derivative $$D_\mu = \partial_\mu\pm ieA_\mu$$, the sign in the Coulomb law will be the same (I'd not be able to do the calculation off the cuff, but since the conjugation flips the sign, I'd assume that).

So, is it correct to state that classically there could be a consistent theory of such an interaction (Coulomb force with flipped sign) but in quantum theory, as we understand it today, the only consistent theory with one signed charge is the one that we observe in reality?

• Positive and negative are just names for charges which attract each other. you have to make a difference between charges which attract an charges which repel each other, So one you call + the other - one could have named them blue and red or pim and pom. but if you rub the same things like plastic and wool you get the same kind of charge any time, so you call it the same. and they repel each other, ans you call it maybe plastic charge and wool charge. but in no way there is an experiment that the same kind attracts each other, this has nothing to do with quantum mechanics. Sep 24 at 12:31
• I'm aware of the fact that in our world things work the way they work. But my question is, whether this has to be this way for consistency reasons, or a world could exist where the rules are different. Sep 24 at 12:32
• Well, pair production via vacuum fluctuations would make the vacuum unstable. Sep 24 at 12:38
• @Qmechanic Could you elaborate? Like, particle and antiparticle with opposing charge would now repel each other and thus not annihilate, so we'd get more and more fermion-antifermion pairs which would never annihilate? Sep 24 at 12:41
• $\uparrow$ Yes. Sep 24 at 12:43