# What are the fundamental equations of quantum electrodynamics?

I hope this question hasn't already been asked, but I looked and couldn't find a question with a similar title.

It is my understanding that Maxwell's equations and the Lorentz force law form the foundation of classical electromagnetism. These are fundamental because (in the classical limit) they are always obeyed, in contrast to Ohm's law, which is still a valid empirical law, but is restricted to a specific scope.

As I understand them (correct me if I have made a mistake), these describe basic assumptions used to create a model of electromagnetism:

• the net outflow of electric field lines through a closed surface is proportional to the amount of and polarity of the charge inside

• the net outflow of magnetic field lines through a closed surface is zero

• a magnetic field that changes with time will create a circulating electric field

• an electric field that changes with time will create a circulating magnetic field, taking into account the polarization current

• the electromagnetic force on a charged particle depends both on the electric field and the magnetic field

I've encountered some equations pertaining to phenomena described by QED, but I was wondering if there is a set of fundamental equations like Maxwell's equations and the Lorentz force law that describe all of the core requirements for accurate predictions in its model.

In other words, can I summarize QED the same way as I have done above for classical electromagnetism?

Yes, but it is very much a personal choice. All the informations about QED are contained in the QED Lagrangian:

$$\mathcal{L}_{QED} = \frac{-1}{4}F^{\mu \nu}F_{\mu \nu} + \bar{\psi}(i\gamma^\mu D_\mu^{QED} - m)\psi$$

And into the renormalization equations for the 3 free parameters.

From the Lagrangian you can extract all the relevant propagators, the equations of motions for the fields, the currents and so on.

What are the "fundamental" sets among this long list of equations is very much a personal choice, but all the relevant quantities can be computed out of the Lagrangian and the Renormalization Group [RG] equations.

To me what really is fundamental are the Lagrangian and the RG equations, but for you it could include the equations of motion, or the S-matrix elements, or whatever you want really

• Do you mean to say that from those tensor equations, you can extract a bullet-point list the same way I have? In other words, one can make a 1:1 correspondence between short English sentences and short mathematical equations, the same way that I have done above for Maxwell's equations and the Lorentz force law? Commented Sep 14, 2023 at 8:12
• Short english senteces are not physically relevant quantities. From the tensorial equations you can extract physically measurable predictions which are what is relevant in physics. Nonetheless if you care about english sentences you could of course recover some of them. For example all the Maxwell equations are contained into the Lagrangian, and also are the currents of matter fields and the Dirac equation Commented Sep 14, 2023 at 8:15
• I know that short English sentences are not always possible, but I know that the Einstein field equations can be described as, "The manner in which we have to modify the Pythagorean theorem to calculate the distance between two close points in spacetime, and how that manner changes as we move away from those points, is determined by the density and flux of the energy and momentum near those points." The first part is an expression containing the metric tensor and its derivatives, and the second part is the stress–energy tensor. This is easily recoverable given their definitions. Commented Sep 14, 2023 at 8:56
• But, I am glad to hear that everything is contained in the Lagrangian. I'm new to tensor calculus, so I look forward to learning how they are contained in it. Commented Sep 14, 2023 at 8:57
• Have you taken a course in analytical mechanics? The Lagrangian of a system contains all the equations of motion via the Euler-Lagrange equations and it also contains all the symmetries of the system. Via Noether theorem then the symmetries gives you all the conserved currents of the system. This has nothing to do with the tensorial nature of the involved quantities, it works for every system, non-relativistic point particles mechanics up to the full Lagrangian of the Standard Model Commented Sep 14, 2023 at 9:01