How might one interpret Feynman's comment about how all fundamental forces are conservative?

Question

I know that there are several questions related to the topic of conservative forces. But I am trying to understand where Feynman might have been coming from when he said that "We shall take a deeper view of this than is usual, and state that there are no nonconservative forces! As a matter of fact, all the fundamental forces in nature appear to be conservative." (link to lecture text: https://www.feynmanlectures.caltech.edu/I_14.html). I have seen this statement made in other classical mechanics courses too. While I understand where Feynman is coming from when it comes to forces like friction, I do not understand how this argument might apply on the electromagnetic force. In particular, the magnetic force is not conservative. So is there a way to think of the electromagnetic force as being conservative that I am missing somehow?

I would mention that I have also been through the previous discussion in this link: Are fundamental forces conservative?. In fact, someone links Feynman's lecture there as well.

The main answer explains that the notion of "conservative force" might not be important outside of a limited context. But I still do not understand how Feynman (and others) might have been thinking about this.

Answer 1

All the known fundamental forces of nature are conservative in the sense that all of them conserve energy. The version of energy is a bit wider though, Feynman puts it somewhere in his lecture that the only rule is you can't have a category called "unaccounted" or something like that. All forms of energy must have a rigorous definition.

As you say, friction (fundamentally electrostatic) is conservative in that the rise in internal energy accounts for all lost macroscopic mechanical energy.

The magnetic force is an interesting one. One way to answer its apparent nonconservative nature is to note that it never does any work, so the issue does not matter. This is arguably valid for a lot of circumstances. A bit deeper of an answer would be to note that with a widened definition of energy to include the energy in the field itself, electromagnetism does not violate energy conservation. There are certainly circumstances where the energy is more useful and less useful, but the total energy is never changed.

This is why Feynman is saying it is "a deeper view of this than is usual" for his purposes. When people talk about conservative forces, they usually are thinking that the energy is still available to the bodies under consideration. From a fundamental perspective, the whole universe is the "bodies under consideration," so if some objects have less energy available to them and there is now more electromagnetic radiation in the universe, then that is perfectly consistent with a conservative force.

That is my take on his statement.

Answer 2

You are right, strictly speaking, Feynman Lectures are wrong there, since e.g. magnetic force or induced electric force is not conservative in the usual sense of the word.

Perhaps what the authors had in mind is that all fundamental forces, including non-conservative forces, are believed to obey some variant of law of local conservation of energy, where energy can't be created or destroyed, only locally transported and transformed. For example, net EM force, including magnetic force and induced electric force, obeys such law and can thus be considered as "force compatible with the principle of conservation of energy". This is not the same as being "conservative", but it is easy to use the term confusingly in this sense.

The familiar friction laws (Coulomb's law for solids, Newton's law of viscous force for fluids) do not obey the principle of conservation of energy in pure mechanics or field theory (free of thermodynamics ideas, like internal energy), they violate it.(*) The belief expressed in the Feynman Lectures is that it is possible and more fundamental to describe these macroscopic friction forces as a result of many microscopic forces that obey some law of conservation of energy, and only on the macro-level, we do not see how energy is conserved, even though it is.

(*) It is possible, in special cases, to define a function of position and momentum, with dimensions of energy, which even friction force, like the Stokes viscous friction force on a small droplet falling in fluid, $-bv$, conserves. However, value of this new function is not that of energy in the established sense (kinetic plus potential energy). It would be non-standard and confusing to change the definition of energy just to include that kind of friction force into the family of "nice forces" obeying the principle of conservation of energy.