Unitary in quantum field theory

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Discussion Overview

The discussion revolves around the concept of unitarity in quantum field theory (QFT), particularly in the context of infinite degrees of freedom and phase transitions. Participants explore whether unitarity can be considered broken in QFT and how this relates to the evolution of pure states into mixed states, especially in the context of black hole physics.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested

Main Points Raised

  • Some participants propose that in QFT, there exist infinite unitary nonequivalent representations, particularly after phase transitions.
  • One participant explains that while finite quantum mechanics has the Stone-von Neumann theorem ensuring physical equivalence of representations, no such theorem exists for QFT, leading to different phases that are physically distinct.
  • Another participant draws a parallel between phases in statistical thermodynamics and the idea of phase transitions in QFT, questioning if this analogy applies to changes in phase with evolution.
  • A later reply references a talk by Wald regarding the information paradox, suggesting that a pure state evolving into a mixed state does not necessarily imply a breaking of unitarity, and posits that this concept may be more comprehensible within QFT.
  • One participant suggests that the formation of a black hole can be viewed as a phase transition from a pure state to a mixed state.

Areas of Agreement / Disagreement

Participants express differing views on the implications of unitarity in QFT, particularly regarding the evolution of states and the nature of phase transitions. The discussion remains unresolved, with multiple competing perspectives presented.

Contextual Notes

Participants note the absence of a theorem analogous to the Stone-von Neumann theorem for QFT, which may limit the understanding of unitary representations in this context. The discussion also highlights the complexity of relating concepts from statistical mechanics to quantum field theory.

shuijing
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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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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 Schrödinger'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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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.
 

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