Construction of a Hilbert space and operators on it

[Fredrik]

When quantizing a classical field theory, e.g. Klein-Gordon theory, the Hilbert space of one-particle states is taken to be a set of equivalence classes of (Lebesgue) measurable and square integrable solutions of the classical field equation, but how do you use the classical theory to construct operators on this Hilbert space?

I asked that question in a different way here (in the Topology & Geometry forum), but got no replies, so I'm trying again here.

 

[George Jones]

When quantizing a classical field theory, e.g. Klein-Gordon theory, the Hilbert space of one-particle states is taken to be a set of equivalence classes of (Lebesgue) measurable and square integrable solutions of the classical field equation, but how do you use the classical theory to construct operators on this Hilbert space?

I asked that question in a different way here (in the Topology & Geometry forum), but got no replies, so I'm trying again here.

I have replied in the math thread.

 

[schieghoven]

When quantizing a classical field theory, e.g. Klein-Gordon theory, the Hilbert space of one-particle states is taken to be a set of equivalence classes of (Lebesgue) measurable and square integrable solutions of the classical field equation, but how do you use the classical theory to construct operators on this Hilbert space?

I think that this is a terrific question to ask -- developing field theory in terms of L^2 functions (Lebesgue-measurable, square-integrable functions) is not just an appealing conceptual aim, but it might provide hooks into some powerful techniques of functional analysis. As far as I know, it hasn't been done. The sticking point is always whether the mathematical formulation can be made consistent with perturbative QFT and renormalization, and to date that has only been possible under the framework of distributions over the Schwartz space, in which case the literature often refers to BPHZ renormalization. Two influential books on this topic are
* Bogoliubov & Shirkov, Intro to the theory of quantized fields, Wiley 1980
* Bogolubov, Logunov, Oksak \& Todorov, General Principles of Quantum Field Theory, Kluwer Academic 1990
which discuss the complete development of field theory, from free to interacting to perturbative expansion by Feynman diagrams and renormalization, in terms of distributions over the Schwartz space. The first is more pedagogical, the second more mathematically technical. Additionally, you might look at papers by Hepp or Zimmerman from the early 70s.

Extending the theory from Schwartz distributions to L^2 operators was, I think, an outstanding question of mathematical physics in the 70s, and remains an open question. Does anybody have an impression or advice on this?

Cheers,

Dave

 

[0xDEADBEEF]

Read "Reed & Simon: Methods of Modern Mathematical Physics" for a classic and you could look for "Arthur Jaffe" and "constructive quantum field theory" on google. He has some nice lecture notes hidden somewhere. It looks like the most modern way is to work on Sobolev spaces.