# Unitary in quantum field theory

## Main Question or Discussion Point

In QFT, which have infinite degree of freedom, there exlst infinite unitary nonequvilent representation. Expecially after phase transition, the two representation are unitary nonequvilent. So can we say that unitary are broken in QFT? Or a pure state can evolved to a mixed state which is a puzzle in black hole physics?

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julian
Gold Member
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.

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julian
Gold Member
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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Thanks for the response. I listened to a talk given by Wald about information paradox. His idea is that a pure state envoled into a mixed state doesn't break the unitary. I think that maybe in QFT this idea is better understandable.
The formation of black hole is just a phase transition which evolve from a pure state to a mixed state.