Causal localization observables and modular theory

This talk is about quantum-mechanical observables for spatial and spacetime localization 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 Poincaré covariant localization observables exist. Some features of QFT emerge automatically and naturally connect to modular theory, which defines an essentially unique spacetime localization observable via the Brunetti-Guido-Longo map. 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.