A Coherent quantization, the non-unitary case

[mitchell porter]

TL;DR Summary

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?

 

[A. Neumaier]

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.

 

[vanhees71]

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.

 

[A. Neumaier]

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...

 

[vanhees71]

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.

 

[A. Neumaier]

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.