The Vacuum as a Lagrangian subspace

Abstract

We unify and generalize the notions of vacuum and amplitude in linear quantum field theory in curved spacetime. Crucially, the generalized notion admits a localization in spacetime regions and on hypersurfaces. The underlying concept is that of a Lagrangian subspace of the space of complexified germs of solutions of the equations of motion on hypersurfaces. Traditional vacua and traditional amplitudes correspond to the special cases of definite and real Lagrangian subspaces respectively. Further, we introduce both infinitesimal and asymptotic methods for vacuum selection that involve a localized version of Wick rotation. We provide examples from Klein-Gordon theory in settings involving different types of regions and hypersurfaces to showcase generalized vacua and the application of the proposed vacuum selection methods. A recurrent theme is the occurrence of mixed vacua, where propagating solutions yield definite Lagrangian subspaces and evanescent solutions yield real Lagrangian subspaces. The examples cover Minkowski space, Rindler space, Euclidean space and de Sitter space. A simple formula allows for the calculation of expectation values for observables in the generalized vacua. *

Locality and General Vacua in Quantum Field Theory

“…The standard vacuum yields a Kähler polarization on the propagating modes in L p ∂M and there is no problem in constructing the corresponding Hilbert space. However, on the evanescent modes in L e ∂M the polarization is real [8]. In the mentioned work [6,7] on the S-matrix the problem of not being able to construct a Hilbert space for the evanescent modes was noted, but did not affect the result, because the modes are absent asymptotically due to their exponential decay.…”

“…Here, Σ denotes a copy of Σ with opposite orientation. 8 We glue M and N together along Σ, as illustrated in Figure 4. Then, the amplitude of the composite region satisfies justified in the same way from the (formal) properties of the path integral.…”

“…As mentioned in Section 1.5, the construction of the Hilbert spaces of states on each hypersurface relies on a choice of Kähler polarization representing the vacuum. The problem with this is that the standard vacuum of QFT in Minkowski space on a nonspacelike hypersurface generically corresponds to a polarization that is not Kähler [8]. 11 With this, the standard quantization prescription to obtain a Hilbert space of states breaks down.…”

“…In order to use the GBQFT framework for a truly local description of QFT we absolutely need to be able to decompose spacetime into regions that are "small" and certainly compact (recall Figure 5). But a compact region in Minkowski space has a boundary on which the polarization corresponding to the standard vacuum is generically not of Kähler type [8]. If we want to describe physical processes in such a region with realistic boundary conditions, we need to be able to deal with non-Kähler polarizations.…”

“…While this notation deliberately suggests that W P is a state and that ρ F M is a function on the corresponding state space we do neither define such a state space nor such a map for the moment. Here, the boundary conditions are determined by a polarization P, i.e., a Lagrangian subspace L P ∂M ⊆ L C ∂M of the complexified space of germs of solutions on the boundary ∂M [8]. We make the choice of polarization explicit with our notation.…”