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Probability conservation and symmetry 
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#1
Feb1507, 01:42 PM

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What symmetry gives probability conservation? Or, what symmetry does probability conservation give?
I have been trying (unsuccessfully) to find an answer to this question. I think the question makes sense. That is, I can't see how the situation is different from, for example, that of spatial translation giving conservation of momentum. 


#2
Feb1507, 03:18 PM

P: 223

Unfortunately I don't have time to flip out a complete answer, but I would tell you to look at the Lagrangian formulation of the Schrodinger equation and to look up Noether's theorem. I seem to recall actually working this out at some point, but those notes are probably buried in a box somewhere.



#3
Feb1607, 12:52 AM

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Actually the theorem of Wigner insures probability conservation for any symmetry of the system. By that theorem, symmetry transformations are implemented in (rigged) Hilbert space language by either unitary or antiunitary operators which are known to conserve scalar products, hence probabilities.



#4
Feb1707, 12:19 AM

P: 2,068

Probability conservation and symmetry



#5
Feb1907, 09:32 AM

P: 223




#6
Feb1907, 11:11 PM

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And from this condition follows mathematically that they must be unitary or antiunitary representations of their group (because symmetries also always form a group, for the same physical reasons). EDIT: we can, because of this, apply two kinds of approaches to setting up the quantum description of a system. We can make a list of symmetries, and try to find a good representation of them, which can then serve as a quantum description ; or we can have another way of finding the quantum description, and then go fishing for the different symmetry representations it contains. 


#7
Feb2007, 02:30 AM

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Yes, Vanesch, you're right, that's why the original question
"What symmetry gives probability conservation? Or, what symmetry does probability conservation give?" should be answered: "Any symmetry. All symmetries". 


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