system
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human
Is there a conservation law associated with CPT symmetry? Shouldn't it be conservation of probability?

I've been reading a lot about symmetry lately and I can't articulate why but I feel like for there to be predictable laws that comes from an ultimately statistical (random) microscopic world, that the conservation of probability must come into play. So my ultimate question is, is there a conservation law associated with CPT symmetry?
gpt
Per Noether's theorem, only *continuous* symmetries (e.g. rotations, translations, etc) give rise to conservation laws. CPT is a discrete symmetry, and so doesn't have a corresponding conservation law.

Conservation of probability is associated with the unitarity of quantum mechanics: quantum states stay normalized as a function of time. In some sense this is a 'deeper' fact than any symmetry; it's true in any quantum system regardless of how symmetric the Hamiltonian (or Lagrangian) is.