How does one show using QED that same/opposite electric charges repel/attract each other, respectively?

Question

Why do same charges repel each other and opposite charges attract each other (please explain the phenomenon using real laws of nature (QED) not with the approximation model)?

Answer 1

A short answer, is that to estimate interaction energy (which says if same charges attract or repel), you use propagators. Propagators come from the expression of Lagrangians. Finally, the time derivative part for dynamical freedom degrees in the action must be positive, and this has a consequence on the sign of the Lagrangian.

Choose a metrics $(1,-1,-1,-1)$

For instance, for scalar field (spin-0), we have ($i=1,2,3$ representing the spatial coordinate) the : $$S = \int d^4x ~(\partial_0 \Phi\partial^0 \Phi+\partial_i \Phi\partial^i \Phi)$$

Here, the time derivative part of the action is positive (because $g_{00}=1$), so all is OK. When we calculate energy interaction for particles wich interact via a spin-0 field, one finds that same charges attract each other.

Now, take a spin-1 Lagrangian (electromagnetism):

$$S \sim \int d^4x ~(\partial_\mu A_\nu - \partial_\nu A_\mu) (\partial^\mu A^\nu - \partial^\nu A^\mu)$$

The dynamical degrees of freedom are (some of) the spatial components $A_i$, so the time derivative of the dynamical degrees of freedom is :

$$S \sim \int d^4x ~\partial_0 A_i \partial^0 A^i$$

Now, there is a problem, because this is negative (because $g_{ii} = -1$), so to have the correct action, you must add a minus sign :

$$S \sim -\int d^4x ~ (\partial_\mu A_\nu - \partial_\nu A_\mu) (\partial^\mu A^\nu - \partial^\nu A^\mu)$$

This sign has a direct consequence on the propagators, and it has a direct consequence on interaction energy, which is calculated from propagators.

This explains while same charges interacting via a spin-0 (or spin-2) field attract, while same charges interacting via a spin-1 field repel.

See Zee (Quantum Field Theory in a nutshell), Chapter 1.5, for a complete discussion.

Answer 2

The title and the content ask two different things:

Content:

Why does same charge repel each other and opposite charge attract each other (please explain the phenomenon using real laws of nature not with the approximation model)?

The only answer is : Because.

It is an observational fact data gathered over centuries that made us separate electricity into charges, like ones repelling each other , unlike ones attracting. It is an experimental fact organized into this duality.

Physics is about mathematical models that are used to fit existing data and predict future phenomena.

The title asks:

Explanation of electromagnetism using quantum field theory

that is, it is asking for a specific mathematical model. Unfortunately, even though QFT can be used to predict the behavior of charged elementary particles it cannot explain why there are positive and negative charges and like charges repel and unlike attract. It still is an experimental fact inputted to the postulates of the theory.