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If I understand correctly, in many context in physics (quantum mechanics?), a physical system is specified by giving its Hamiltonian. I also hear that symmetries are rather essential.

As far as the most natural symmetries are concerned, it seems fairly easy to recognise them. They should apparently be linear self-maps of the underlying Hilbert space, which seems natural enough, and they should surely commute with the Hamiltonian (if you transform the system by symmetry and then use the Hamiltonian to figure out how it evolves, the result should be the same as when you first look at time evolution, and then apply symmetry).

For a while I thought this is just what a symmetry is supposed to be doing, but then I realised that any map that keeps the eigenspaces of the Hamiltonian invariant would do, and I suspect this is not quite right. Also, there seem also to be some other operators around, often used when defining the Hamiltonian --- I'm not sure if a symmetry is supposed to commute with these as well...

I would be extremely grateful if someone could explain to me what symmetries of a physical system (viewed from the point of view of the Hamiltonian) actually are, and why is it that they are defined like this and not differently. An explanation in layman's/mathematician's terms would be even more appreciated.

Disclaimer: I am a mathematician, not a physicist.

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For a good introduction to this rather technical subject, I urge you -- as one mathematician to another! -- to refer to Bojowald's recent book Canonical Gravity and Applications, specifically ch. 3. –  Alex Nelson Apr 9 '13 at 23:15
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2 Answers

up vote 4 down vote accepted

There is a fundamental result called Wigner's theorem on symmetries in quantum mechanics that I think will shed a great deal of light on this issue. The theorem characterizes symmetries in quantum mechanics given that they are defined generally as mappings on Hilbert space that preserve transition probabilities. There is no other general restriction on symmetries in quantum mechanics as far as I am aware. There is a nice (not completely rigorous) discussion of this theorem in Weinberg's QFT volume 1 appendix A.

On the other hand, there is a related notion, that of conserved quantities. In the context of quantum mechanics, observable quantities (self-adjoint operators) that commute with the Hamiltonian will be conserved by virtue of Heisenberg's equation for time-evolution.

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There are two big fields of physics: General Relativity and Quantum Fields theory.

General Relativity symmetries:

  • Lorentz invariant: Arises from speed of light is constant regardless of one's frame of reference. Michelson–Morley experiment

  • Gauge invariant: $U(1)$

Quantum Fields symmetries (Standard model, there are others models with others symmetries but standard is wherever accepted):

  • Lorentz invariant: Same reason.

  • Gauge invariant: $U(1)\times SU(2)\times SU(3)$

$U(1)$ arises from electromagnetic interaction.

$SU(2)$ arises from experimental weak interaction symmetries.

$SU(3)$ arises from experimental strong interaction symmetries.

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