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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