0

I was reading Why the Principle of Least Action? and the top voted answer says

You can go further mathematically by learning the path integral formulation of nonrelativistic quantum mechanics and seeing how it leads to high probability for paths of stationary action. [emphasis added]

That seems to imply that, while paths of stationary action are most probable, it's theoretically possible for some other path to be taken. That reminds me of the second law of thermodynamics and how it's fundamentally a probabilistic law - it's not actually guaranteed that the total entropy of a closed system will approach a maximum in any finite amount of time, it's just that the probability of this happening quickly approaches 100% as the size of the system and/or the amount of time increases. Is something similar the case with the principle of stationary action in general?

That is, could we, at least in principle, explain all instances of the principle of stationary action holding (in classical mechanics, QM, and QFT) in terms of probability? Or is this something specific to the path integral formulation of QM and the probability interpretation wouldn't make sense in other models where we apply the principle?

Mikayla Eckel Cifrese
  • 2,297
  • 1
    Is this a question about the principle of least action in classical mechanics or in quantum mechanics? – ACuriousMind Apr 10 '23 at 00:58
  • @acuriousmind, I guess both. I sort of figured the version in classical mechanics would be a special case/approximation to the version in QM. – Mikayla Eckel Cifrese Apr 10 '23 at 01:31
  • 1
    Since the Feynman path integral is clearly probabilistic, the title question (v6) is only non-trivial in a purely classical setting. – Qmechanic Apr 10 '23 at 04:56
  • @qmechanic, thanks for pointing that out. I further edited the question to clarify that I'm asking about whether the principle of stationary action is probabilistic in general or only in the path integral formulation of QM. – Mikayla Eckel Cifrese Apr 10 '23 at 06:04
  • In a purely classical context, multiple solutions to the stationary action principle seem to be an incomplete description. – Qmechanic Apr 10 '23 at 06:25
  • 3
    In classical mechanics we have Newton's second law and the work-energy theorem, which are two ways to say the same thing. In addition we have: in cases where the work-energy theorem holds good Hamilton's stationary action holds good also. That is a deterministic connection; not a probabilistic connection. As we know: quantum mechanics is subject to the constraint: must reproduce classical mechanics in the macroscopic limit. Both reproduce F=ma, but the nature of the connection is fundamentally different. – Cleonis Apr 10 '23 at 06:42

0 Answers0