False vacuum in axiomatic QFT

Question

There is an elegant way to define the concept of an unstable particle in axiomatic QFT (let's use the Haag-Kastler axioms for definiteness), namely as complex poles in scattering amplitudes. Stable particles are much simpler from this point of view, since they correspond to the discrete part of the Poincare group spectrum of the theory (of course they also correspond to real poles).

The concept of a vacuum state is rather straightforward to define in the axiomatic framework. But what about false (unstable) vacua?

What is the definition of "QFT false vacuum" in the Haag-Kastler axiomatric approach to QFT?

EDIT: I have a wild guess. Perhaps a false vacuum sector corresponds to an irreducible Poincare-invariant continuous representation of the observable algebra which is non-Hermitean, i.e. the representation space is a Banach, or maybe a Hilbertian Banach space (regarded as a topological vector space, without preferred norm or inner product) and no condition involving the *-structure is satisfied. This representation is supposed to have a unique Poincare invariant vector corresponding to the false vacuum itself. It should be possible to define "expectation value" in this setting if some kind of a spectral decomposition exists, and the energy-momentum tensor has expectation value $\epsilon \eta_{\mu\nu}$ where $\epsilon$ is a complex number, the imaginary part signifying the decay rate (as Lubos suggested below). Btw, is it possible to prove the existence of the energy-momentum tensor in Haag-Kastler? Anyway, this is a purely intuitive guess and I don't see how to connect it to the actual physics

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Comments

Dear @Squark, the first sentence of your text is loaded because it suggests that the elegance of the complex poles – or even the discovery itself – came from axiomatic QFT. It has nothing to do with this particular research program. This program only "borrowed" a well-known physics fact. I don't see why you're talking about a failed research direction at all. Why don't you ask the same question in the context of proper QFT rather than "axiomatic" QFT?

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Obviously, an unstable vacuum is at least formally a vacuum with a complex energy density where the imaginary part is equal to the decay probability density per spacetime. However, any such non-real Hamiltonian eigenvalue requires one to discuss a broader picture (like scattering amplitudes of stable particles in the first case) and it's true here, too. All unstable vacua (their Hilbert spaces) have to be embedded into lower-C.C. (AdS or Minkowski) stable vacua.

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I haven't intended to suggest that. The reason I'm using axiomatic QFT as a context is that I want an answer which can be made mathematically precise, at least in principle.

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