How can we claim that electromagnetism has positive definite energy if the energy density is underdetermined?

Question

This post claims that the energy of any particle + EM field system is positive definite.

However, according to Feynman Lectures Part 2, Section 27-4, the energy density of the electromagnetic field is underdetermined. To give a direct quote from here, he says,

Before we take up some applications of the Poynting formulas [Eqs. (27.14) and (27.15)], we would like to say that we have not really “proved” them. All we did was to find a possible “u” and a possible “S.” How do we know that by juggling the terms around some more we couldn’t find another formula for “u” and another formula for “S”? The new S and the new u would be different, but they would still satisfy Eq. (27.6). It’s possible. It can be done, but the forms that have been found always involve various derivatives of the field (and always with second-order terms like a second derivative or the square of a first derivative). There are, in fact, an infinite number of different possibilities for u and S, and so far no one has thought of an experimental way to tell which one is right! People have guessed that the simplest one is probably the correct one, but we must say that we do not know for certain what is the actual location in space of the electromagnetic field energy. So we too will take the easy way out and say that the field energy is given by Eq. (27.14). Then the flow vector S must be given by Eq. (27.15).

Here is a post discussing this quote.

So in this context, how can we claim that the energy of any EM + particle system is positive definite? Is there a proof that for any choice of energy density, the energy of the EM field is bounded below? Are there any references that discusses this?

R. M. Wald's Advanced Classical Electromagnetism discusses a similar issue (the uniqueness thing). While the book will only be launched on March, there is a preview of the first chapter on the publisher's website. In short, the coupling of Electromagnetism to gravity is pretty much the only way of testing the usual formulae and cosmological observations are in accord with them. — Níckolas Alves, Dec 24 '21 at 08:10

score 1 · Accepted Answer · answered Dec 25 '21 at 12:09

The posts you link already contain the answer: AccidentalFourierTransform shows here that the energies $U,U'$ for different choices $u,S$ and $u',S'$ only differ by an overall constant. Therefore, the "positive-definiteness" of $U$ for the ordinary choice directly shows that $U$' is bounded from below for all possible choices, since adding a constant does not change whether a function is bounded or not.

How can we claim that electromagnetism has positive definite energy if the energy density is underdetermined?

1 Answers1

Linked