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- Thread starter shuijing
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julian

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To make it clear there are two different types of unitary transformations here:

a. Given abstract commutation relations between configuration and momentum variables (or rather their exponentiated versions) there can be different representations as concrete operators on an concrete Hilbert space. Two such representations are physically equivalent if they can be related by a unitary transformation. For finite QM the the Stone-von Neuman theorem guarantees that all representations are physically equivalent...this answered the question do Schrodinger's equation and Heisenberg's matrix mechanics give the same answers. There is no such theorem for quantum theories with infinite degrees of freedom - i.e. for QFT. This can be understood as there being different `phases' (or sectors) that are physically different from each other just as one gets in the thermodynamical limit (taking infinite degrees of freedom limit) of statistical mechanical systems.

b. There is unitary evolution which guarantees conservation of probability in time - or the evolution that preserves pure states.

a. Given abstract commutation relations between configuration and momentum variables (or rather their exponentiated versions) there can be different representations as concrete operators on an concrete Hilbert space. Two such representations are physically equivalent if they can be related by a unitary transformation. For finite QM the the Stone-von Neuman theorem guarantees that all representations are physically equivalent...this answered the question do Schrodinger's equation and Heisenberg's matrix mechanics give the same answers. There is no such theorem for quantum theories with infinite degrees of freedom - i.e. for QFT. This can be understood as there being different `phases' (or sectors) that are physically different from each other just as one gets in the thermodynamical limit (taking infinite degrees of freedom limit) of statistical mechanical systems.

b. There is unitary evolution which guarantees conservation of probability in time - or the evolution that preserves pure states.

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julian

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In statistical thermodynamics different phases are cut off from each other because it would take an infinite amount of energy to go from one to the other - but is this the kind of idea you are getting at - a change in phase with evolution?

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The formation of black hole is just a phase transition which evolve from a pure state to a mixed state.

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