About symmetry

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KFC
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I read a introductory book on symmetry of physics system. The author says symmetry of quantum mechanical system is VERY different from that of classical system. I just wonder what's the main difference?
 

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I read a introductory book on symmetry of physics system. The author says symmetry of quantum mechanical system is VERY different from that of classical system. I just wonder what's the main difference?
I'd say the main difference is in something like angular momentum. Rotational invariance leads to conservation of angular momentum, but in QM you only know the angular momentum in one direction. There might also be a difference in time reversal, where spin 1/2 particles pick up a negative sign when performed twice or something, but I'm not sure about this.

Other than that, CM and QM are similar with regards to symmetries, and usually QM is constructed from analogy with Hamilton's CM. In CM, symmetry means the Hamiltonian is unchanged by the flow generated by a canonical generator of the symmetry - but there is a dual relationship that says that this implies that the generator is unchanged by the flow generated by time translation (which happens to be the Hamiltonian). So basically if the flow created by a generator does not change the generator of the flow of time, then the flow of time does not change the generator. Hence the generator is conserved. This is true in CM and QM.
 
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malawi_glenn
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which book?
 

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