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

Answer 1

The exponential decay dynamics of an unstable particle (defined in terms of a complex pole) is dissipative, since the decay products are ignored in the description. Therefore its symmetries are described by the Poincare semigroup only, where momenta are restricted to be timelike, resulting in a dynamics forward in time without reversibility.

Then the classification of irreducible subunitary representations (characterized by $U(g)^*U(g)\le 1$, which replaces unitarity in the dissipative case) allows further possibilities, among which one finds those for unstable particles. See Schulman, Annals of Physics 59 (1970), 201-218.

The space where the semigroup acts is not a Hilbert space, so this doesn't quite fit the $C^*$-algebra framework of algebraic QFT. Instead one needs the rigged Hilbert space extension of quantum mechanics to accomodate unstable particles. See, e.g., Bohm et al. hep-th/9911059. The rigged Hilbert space can accomodate a deformed inner product in which the continuous spectum is moved far enough into the nonphysical sheet that the pole there becomes visible.

See also my answer to https://physics.stackexchange.com/a/29765/7924

Edit: On the other hand, a false vacuum is given by a tachyonic state with negative mass square, not by an unstable particle, whose complex mass has a positive real part. Thus it corresponds to a unitary but unphysical representation of the Poincare group. It is known that these sectors cannot satisfy causal commutation rules, hence they are excluded in algebraic QFT.

Indeed, in the algebraic treatment of gauge theories by the Epstein-Glaser method (see Scharf's true ghost story), broken symmetries don't arise as tachyons but lead directly to a unitary representation with generated masses.