Locality and General Vacua in Quantum Field Theory

Abstract

We extend the framework of general boundary quantum field theory (GBQFT) to achieve a fully local description of realistic quantum field theories. This requires the quantization of non-Kähler polarizations which occur generically on timelike hypersurfaces in Lorentzian spacetimes as has been shown recently. We achieve this in two ways: On the one hand we replace Hilbert space states by observables localized on hypersurfaces, in the spirit of algebraic quantum field theory. On the other hand we apply the GNS construction to twisted star-structures to obtain Hilbert spaces, motivated by the notion of reflection positivity of the Euclidean approach to quantum field theory. As one consequence, the well-known representation of a vacuum state in terms of a sea of particle pairs in the Hilbert space of another vacuum admits a vast generalization to non-Kähler vacua, particularly relevant on timelike hypersurfaces.

Evanescent Particles

“…In the meantime, a new quantization prescription has been developed precisely to address vacua corresponding to non-Kähler polarizations [14]. Based on this, in the present work we are finally able to present a fully satisfactory quantization of Klein-Gordon theory on the timelike hyperplane that includes the evanescent modes.…”

“…We proceed to discuss the representation of observables on the Hilbert space H t of states. We can think of observables here as functions on the instantaneous phase space L t , although they should really be thought of as arising from slice observables in spacetime [15,14]. What is more, we require them to extend to holomorphic functions on the complexified phase space L C t [14].…”

“…We can think of observables here as functions on the instantaneous phase space L t , although they should really be thought of as arising from slice observables in spacetime [15,14]. What is more, we require them to extend to holomorphic functions on the complexified phase space L C t [14]. We restrict our considerations to Weyl observables, which arise as follows.…”

“…The associated transition probabilities are worked out in Section 5. In Section 6 we take advantage of the framework developed in [14] to implement the reflecting boundary condition of the Casimir effect as a pseudo-state on the timelike hyperplane encoding a change of vacuum. Finally, we present conclusions and an outlook is Section 7.…”

“…Finally, we present conclusions and an outlook is Section 7. We use throughout tools and conventions from General Boundary Quantum Field Theory (GBQFT), see [4,12,15,14] and references therein, but try to keep this paper as self-contained as possible. When considering predictions and probabilities we resort to the positive formalism where necessary, see [7] and references therein.…”