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Concerning symmetry in fundamental physics, it is usually said that symmetry indicates that laws of physics are invariant independently of something

For example, time translational symmetry indicates that laws of physics are invariant in time, spatial translational symmetry indicates that they are invariant in space and not depending in spatial coordinates, Lorentz symmetry indicates that they are invariant independently of the frame of reference of the observer...

Also, these symmetries do hold locally but there may be some models and metrics where they cease to hold globally. One example is the case of some expanding universe models, where, although conservation of energy holds perfectly well locally, it may not hold globally and for the universe as a whole due to time translational symmetry being broken. But we can go further, as there can be even models without any global symmetry, as far as I know.

But, if that is the case, and the presence of symmetries means that laws of physics are invariant, then, if there are no symmetries (at least global symmetries) then the laws of physics could change (in space, in time, depending on the frame of reference...) at least for the universe as a whole and at very large cosmological scales (just as it happens with time translational symmetry and the law of energy conservation in an expanding universe), right?

And if that is true then, shouldn't our most fundamental theories (the theory of relativity, quantum mechanics...) also cease to hold for the universe as a whole (except locally, of course)? I mean, since they would be based on many principles that would not be invariant, these theories would also be non-invariant and could change in some contexts... If that makes sense...

That is my reasoning but I do not know if it is right. If someone could correct the mistakes that would be appreciated.

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  • $\begingroup$ The point at physics.stackexchange.com/q/683939 might reduce confusion. Namely that "quantum" and "relativistic" are properties of a theory in modern language. $\endgroup$ May 15, 2023 at 14:36
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    $\begingroup$ The invariance is the symmetry. The laws of physics being invariant under e.g. time translations is synonymous with them being time-translationally symmetric. $\endgroup$ May 16, 2023 at 13:00
  • $\begingroup$ A symmetry means nothing other than the absence of a symmetry breaking mechanism. In general, however, even a system with a microscopic symmetry will often break this symmetry at the macroscopic level. Sound waves, for instance, do not obey Lorentz symmetry, even though the microscopic constituents of the matter they propagate in do. Charge conservation, OHOH, survives the microscopic/macroscopic transition. In other words... it's complicated, which is why we are never going to run out of physical systems. $\endgroup$ May 16, 2023 at 20:19

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Do symmetries indicate that laws of physics are invariant?

It's not that simple, or rather: you have to be more specific as to what you are talking about, what symmetries and invariances for what laws or quantities?

The existence of a continuous symmetry of a system implies the existence of an invariant quantity (Noether's theorem). For example time translation symmetry of the action functional yields conservation of energy in the system you are describing. This does not necessarily yield invariance of the physical laws you are using.

The existence of a discrete symmetry may leave you disappointed as it gives you much less/a different kind of information than the continuous one.

And then there are invariances of physical laws that do not stem from symmetries of the system. For example physical laws can be invariant under a change of the observer's reference frame because you are using tensors, even when describing an inhomogeneous and anisotropic material. Cf. the principle of objectivity/frame indifference in continuum mechanics.

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