A metric of event space

Ivan Chajda, Helmut Länger

Dorje C. Brody, Lane P. Hughston

Abstract The phase space of a relativistic system can be identified with the future tube of complexified Minkowski space. As well as a complex structure and a symplectic structure, the future tube, seen as an eight-dimensional real manifold, is endowed with a natural positive-definite Riemannian metric that accommodates the underlying geometry of the indefinite Minkowski space metric, together with its symmetry group. A unitary representation of the 15-parameter group of conformal transformations can then be constructed that acts upon the Hilbert space of square-integrable holomorphic functions on the future tube. These structures are enough to allow one to put forward a quantum theory of phase-space events. In particular, a theory of quantum measurement can be formulated in a relativistic setting, based on the use of positive operator valued measures, for the detection of phase-space events, hence allowing one to assign probabilities to the outcomes of joint space-time and four-momentum measurements in a manifestly covariant framework. This leads to a localization theorem for phase-space events in relativistic quantum theory, determined by the associated Compton wavelength.

Esteban Castro-Ruiz, Flaminia Giacomini, Alessio Belenchia +1 more

The standard formulation of quantum theory relies on a fixed space-time metric determining the localisation and causal order of events. In general relativity, the metric is influenced by matter, and is expected to become indefinite when matter behaves quantum mechanically. Here, we develop a framework to operationally define events and their localisation with respect to a quantum clock reference frame, also in the presence of gravitating quantum systems. We find that, when clocks interact gravitationally, the time localisability of events becomes relative, depending on the reference frame. This relativity ia a signature of an indefinite metric, where events can occur in an indefinite causal order. Even if the metric is indefinite, for any event we can find a reference frame where local quantum operations take their standard unitary dilation form. This form is preserved when changing clock reference frames, yielding physics covariant with respect to quantum reference frame transformations.

Werner Karl Heisenberg

Ignazio Francesco Licata

Preface Space-Time Geometry & Quantum Events The Transaction as a Quantum Concept Stochastic Foundation of Quantum Mechanics & the Origin of Particle Spin Fractal Space-Time as an Underlying Structure of the Standard Model Relativity of Scales, Fractal Space & Quantum Potentials Three Possible Implications of Space-Time Discreteness Quantum Computing Space-Time Metric Gauge Fields in Deformed Special Relativity Non-Local Granular Space-Time Foam as an Ultimate Arena at the Planck Scale Structural Approach to the Elementary Particle Theory Forming Physical Fields & Pseudometric & Metric Manifolds. Noncommutativity & Discrete Structures in Classical & Quantum Physics Index.