Coherent quantization, the non-unitary case

In summary, @A. Neumaier says that the reason why he is not restricted to unitary representations is because they are not necessary and that dissipation in relativistic QFT requires nonunitary representations.
  • #1
mitchell porter
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How can quantization be non-unitary?
This is a question specifically for @A. Neumaier !

At Peter Woit's blog, Arnold commented about his formalism for quantum mechanics, coherent quantization. I left a question but Peter Woit doesn't always let comments through, so, here is the question:

Why aren’t you restricted to unitary representations for physical applications? Is it because you start in Euclidean space, without time evolution? Do you impose physical unitarity in an extra step, when you transform to physical space-time?
 
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  • #2
There are two reasons:

1. Nothing in the development requires unitarity, so why should I make this restriction?

2. Dissipative quantum mechanics requires nonunitary representations. The unitary case only gives conservative dynamics. For example, to model a single unstable particle in the relativistic case, one needs a nonunitary representation of the Poincare group.
 
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  • #3
I don't understand the latter statement. What is non-unitary in the usual treatment of decay processes in relativistic QFT?

If you are puristic, of course, you could argue that one should rather consider both the production and decay process of the decaying particle (or rather a resonance) with asymptotic in and out states, described by the unitary S-matrix.
 
  • #4
vanhees71 said:
I don't understand the latter statement. What is non-unitary in the usual treatment of decay processes in relativistic QFT?
If one includes the decay products, everything is unitary. But a reduced description may be much more economical if the system of interest is only the undecayed part.

In the nonrelativistic case, this is modeled by Gamov states (also called Siegert states). Unlike scattering states, Gamov states are not normalizable and satisfy different (namely outgoing) boundary conditions. They are are very useful computationally, and are much used in nuclear physics and quantum chemistry. There is a large literature on this topic...
 
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  • #5
Sure, but after all all this is based on the unitary time evolution. Of course, descriptions of "open quantum systems" as parts of closed systems are non-unitary.
 
  • #6
vanhees71 said:
Sure, but after all all this is based on the unitary time evolution. Of course, descriptions of "open quantum systems" as parts of closed systems are non-unitary.
Yes.

Therefore, to discuss open quantum systems from a group theoretic point of view, one needs nonunitary representations. That one doesn't need them on the fundamental unitary level does not make these representations less useful.
 
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FAQ: Coherent quantization, the non-unitary case

What is coherent quantization?

Coherent quantization is a mathematical process used in quantum mechanics to represent the continuous states of a system as discrete values. It is based on the idea that the information contained in a continuous state can be preserved by representing it as a superposition of discrete states.

What is the non-unitary case in coherent quantization?

The non-unitary case in coherent quantization refers to situations where the quantum system is not isolated and experiences interactions with its environment. This can result in a loss of information and a breakdown of the unitary evolution that is typically used in quantum mechanics.

How is coherent quantization different from other quantization methods?

Coherent quantization differs from other quantization methods, such as canonical quantization, in that it preserves the symmetries of the system and allows for a more complete representation of the quantum states. It also allows for a more intuitive understanding of the quantum system and its dynamics.

What are the applications of coherent quantization?

Coherent quantization has applications in a variety of fields, including quantum information theory, quantum computing, and quantum optics. It is also used in the study of quantum systems with continuous degrees of freedom, such as quantum fields and quantum fluids.

What are the challenges in implementing coherent quantization in experiments?

One of the main challenges in implementing coherent quantization in experiments is maintaining the coherence of the quantum system. This requires precise control and isolation of the system from its environment. Another challenge is the complexity of the calculations involved in coherent quantization, which can make it difficult to apply in practical situations.

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