Sunil
- 227
- 109
Fine that you accept this.vanhees71 said:There's no need for ##A^{\mu}## to be "causal", because it's not observable. What must be causal is the gauge-invariant electromagnetic field ##F_{\mu \nu}## which always is causal ("retarded solutions"), no matter in which gauge you calculate the potentials ##A^{\mu}##.
First of all, it follows that you accept that such non-observable things like potentials have a place in physics, even if they are non-observable. Field theory with ##A^{\mu}## is much much simpler than field theory with ##F_{\mu \nu}## given that there is not even a reasonable Lagrangian for the interaction with a Dirac field using the ##F_{\mu \nu}##.
The more problematic question is about their reality. The ##F_{\mu \nu}##, or whatever defines them, are real, given that they have observable consequences. But are the ##A^{\mu}## real? Positivists have problems with this. Realists not. They know and accept that realist theories are hypotheses and will remain hypotheses forever, that we cannot prove them, only falsify them. So, what is real is, as part of a realist theory, hypothetical too. Some criteria for preference are obvious: Simplicity, Ockham's razor. Giving reality to the ##A^{\mu}## gives much simpler realist theories than accepting only the ##F_{\mu \nu}## as real.
What are then the equations for the ##A^{\mu}##? They have to explain the equations for the ##F_{\mu \nu}## which we can test in a much better way given observations. So we have simplicity and explanatory power as unproblematic guiding principles. If the ##F_{\mu \nu}## follow well-defined evolution equations with Lorentz symmetry, the simplest explanation would be that the ##A^{\mu}## follow such equations too. This would be ##\square A^{\mu}=0##. Why would a realist consider some more complicate realist theory? Don't complicate things without necessity.
What is, therefore, behind the "no need for ##A^{\mu}## to be "causal"? It is good old empiricism - the wish to derive the theories of physics from observations. We obviously cannot derive the equation ##\square A^{\mu}=0## from observation, given that there are a lot of other imaginable equations for the ##A^{\mu}## which give the same Maxwell equations for ##F_{\mu \nu}##. So, ##\square A^{\mu}=0## cannot be part of a theory based on empiricism.