Causal quantum-mechanical localization observables in lattices of real projections

Abstract

Quantum-mechanical observables for spatial and spacetime localization are considered from a lattice-theoretic perspective. It is shown that when replacing the lattice of all complex orthogonal projections underlying the Born rule by the lattice of real linear projections with symplectic complementation, the well-known No-Go theorems of Hegerfeldt and Malament no longer apply: Causal and Poincar\'e covariant localization observables exist. In this setting, several features of quantum field theory, such as Lorentz symmetry and modular localization, emerge automatically. In the case of a particle described by a massive positive energy representation of the Poincar\'e group, the Brunetti-Guido-Longo map defines a spacetime localization observable that is unique under some natural further assumptions. Regarding possible probabilistic interpretations of such a structure, a Gleason theorem and a cluster theorem for symplectic complements are established. These imply that evaluating such localization observables in states yields a fuzzy probability measure that fails to be a measure because it is not additive. However, for separation scales that are large in comparison to the Compton wavelength, the emerging modular localization picture is essentially additive and approximates the one of Newton-Wigner.