Bosons and Fermions in a rigorous QFT

DarMM

I'm not sure what the (unique) Fock representation is as A. Neumaier mentioned it, perhaps he can explain.

Since generalised free-fields span the space of all Fock representations and the interacting theory is unitarily inequivalent to all free-theories, it is then unitarily inqueivalent to all Fock spaces.

A. Neumaier

The one and only Fock representation of a scalar particle of mass m is the standard representation given in each textbook, the direct sum of all symmetrized tensor product of the one-particle space.

What should an interacting Fock space be?

DrDu

Isn't it possible to construct a Fock space from the asymptotic in and out states of an interacting theory? E.g. if there are bound states, I don't see how it could be unitarily equivalent to the Fock space constructed from the free particles.

A. Neumaier

That's precisely what is being done in S-matrix theory. You get in- and out- Fock spaces that are the direct sum of free Fock spaces, one for each stable bound state. These are the only physical Fock space that exists, as it contains the physical particles. The Fock space in which the Lagrangian is expressed has no physical meaning and is only a crutch to ensure a correct classical limit, as it is composed of bare particles with masses that diverge during the renormalization procedure.

The problem is that this gives a free particle description at t=-inf and another one at t=+inf, but no dynamics for in between times. To get the dynamics right, one needs a representation that, by Haag's theorem, cannot be a Fock representation. (There are additional problems in case of gauge theories or massless fields; the above is just the simplest version.)