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What is the difference between laws (e.g., Newton's Laws, Boyle's Law) and principles (e.g., Principle of Least Action , Heisenberg Uncertainty Principle)? As far as I know, both break down to some limit.

Qmechanic
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quanity
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On paper, a principle should be something that is postulated, while a law is proven within a given framework.

The problem is that some physics results that were principles in the past have been proven since then, but we keep calling them principles for historical reasons, out of habit:

  • Heisenberg's uncertainty principle (which is neither a principle nor about uncertainty)
  • Pauli exclusion principle (proven under the name spin-statistics theorem) and so on.

Also, the way "principle" is used may vary a bit from language to language. In my language (French), both laws of thermodynamics are called principles, even though it's a rather obsolete denomination for those too.

The principle of least action, as far as I'm aware, actually is a principle. It's the basic axiom we're using, coupled with Noether's theorem, to build all known theories.

Miyase
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  • So law is a model – quanity Aug 04 '22 at 07:16
  • Close enough. A model is a choice of mathematical objects, and relations between them, that can be used to successfully predict the outcome of some experiments. Laws are the relations, but the model also requires objects (vector for velocity in classical mechanics, scalar field for temperature in thermodynamics…) – Miyase Aug 04 '22 at 07:19
  • Yes So principles are frameworks and laws are models build from them – quanity Aug 04 '22 at 07:26
  • That would be "first principles". A simple principle can also be a law that hasn't been proven yet but is known to work well, typically because it's been infered from experiment. – Miyase Aug 04 '22 at 07:32
  • like e=mc*2 Not proven yet – quanity Aug 04 '22 at 07:37
  • Within special relativity, yes it is. – Miyase Aug 04 '22 at 07:39
  • Least Action also breaks down in quantum physics. It is a purely classical principle. – quanity Aug 04 '22 at 07:43
  • No it isn't. It's the basic tool that, used with Noether's theorem, allows to deduce field equations from Lagrangian in quantum field theory... – Miyase Aug 04 '22 at 07:58
  • https://physics.stackexchange.com/questions/698164/principle-of-least-action-breaking-down – quanity Aug 04 '22 at 08:08
  • The post you link doesn't have any precise answer, so I wouldn't accept it at face value... Anyway, please show me a proof of Dirac's equation in QED that doesn't rely on the principle of least action... – Miyase Aug 04 '22 at 08:10
  • Let us continue this discussion in chat. – quanity Aug 04 '22 at 08:20
  • Classical Mechanics: Path of a particle is that of least action and therefore normal to the surfaces of constant action Quantum Mechanics: Because of the waves associated with a particle, it does not know which path to take. It takes all possible paths with certain probability amplitude and phase and these probability amplitudes interfere. The phase depends on the action. – quanity Aug 04 '22 at 08:41
  • And how do you study those interferences that lead to the "real" path? You use the principal of least action to find a stationary action, which leads to (for example) Dirac's equation. – Miyase Aug 04 '22 at 08:48
  • Could you explain me how do you define the term "breaks down" .I am a bit confused... – quanity Aug 04 '22 at 09:09
  • https://physics.stackexchange.com/questions/717412/breaking-down-of-2nd-law-of-thermodynamics – quanity Aug 04 '22 at 09:10
  • This is completely off-topic for this question. Please open a new one if necessary. – Miyase Aug 04 '22 at 09:13
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As @agaminon mentions in his comment, the difference is rather arbitrary and subjective. To my knowledge, there has generally never been any adherence to a particular criteria for what laws or principles are supposed to be in typical physics literature.

This is of course, not the case in modern mathematical literature, even those that dabble in physics, however, the terminology is different. Math literature generally talks about these laws and principles via very precise definitions and propositions. These propositions could be axioms (these are assumed to be true and serve as starting points for deductive reasoning), lemmas (auxillary propositions that are used to prove theorems) or theorems (the big results, often named).

At the very least however, depending on the formulation of your theory you can determine using the above criteria, which physical laws or principles are axiomatic, which are definitions and which are theorems. There are of course, technicalities that I don't go into here. From the examples you mention, we have,

  • Boyle's Law - Theorem (macroscopic thermodynamic laws can essentially be derived from statistics),
  • Newton's Laws: 1st - Theorem (special case of the 2nd law); 2nd - Definition (of Force); 3rd - Theorem (provable using linear algebra),
  • Principle of Stationary Action - Axiom or Theorem (depending upon your formulation),
  • Heisenberg's Uncertainty Principle - Theorem (in most formulations).

And as @Miyase highlights, even the names can often be misleading due to historical reasons, let alone their status as a law or principle.

Song of Physics
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