Unitary in quantum field theory

In summary, there are two types of unitary transformations in QFT: one involving the representation of operators on a concrete Hilbert space, and the other involving the evolution of pure states to mixed states. In QFT, due to the infinite degrees of freedom, there can be different phases or sectors that are physically distinct from each other. The concept of phase transitions and the formation of black holes can help to understand the idea of pure states evolving into mixed states without breaking the unitary. This was discussed by Wald in his talk on the information paradox.
  • #1
shuijing
4
0
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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  • #2
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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  • #3
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.
 

What is a unitary in quantum field theory?

A unitary in quantum field theory refers to a mathematical operator that preserves the inner product of a quantum state. It is a key concept in quantum mechanics and is used to describe the evolution of quantum systems over time.

Why is unitarity important in quantum field theory?

Unitarity is important in quantum field theory because it ensures the conservation of probability and the preservation of physical observables. This means that the total probability of all possible outcomes of a quantum system must always be equal to one, and that measurable quantities such as energy and momentum must be conserved.

How does unitarity affect the behavior of quantum fields?

Unitarity imposes constraints on the behavior of quantum fields, particularly in terms of their interactions and transformations. It ensures that quantum fields can only evolve in ways that are consistent with the principles of quantum mechanics, and that they must always have a well-defined and consistent mathematical description.

Can unitarity be violated in quantum field theory?

In general, unitarity is a fundamental principle in quantum field theory and cannot be violated. However, there are certain situations, such as in black holes or during the early stages of the universe, where unitarity may appear to break down. These situations are still not fully understood and are the subject of ongoing research and debate.

What are some applications of unitarity in quantum field theory?

Unitarity is a crucial concept in many areas of quantum field theory, including particle physics, condensed matter physics, and quantum information theory. It is used to study and understand the behavior of quantum systems and to make predictions about their properties and interactions. Additionally, the preservation of unitarity is a key criterion in the development of new quantum technologies such as quantum computing and quantum cryptography.

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