Conformal Symmetry and Quantum Relativity

Abstract

The relativistic conception of space and time is challenged by the quantum nature of physical observables. It has been known for a long time that Poincar\'e symmetry of field theory can be extended to the larger conformal symmetry. We use these symmetries to define quantum observables associated with positions in space-time, in the spirit of Einstein theory of relativity. This conception of localisation may be applied to massive as well as massless fields. Localisation observables are defined as to obey Lorentz covariant commutations relations and in particular include a time observable conjugated to energy. Whilst position components do not commute in presence of a non-vanishing spin, they still satisfy quantum relations which generalise the differential laws of classical relativity. We also give of these observables a representation in terms of canonical spatial positions, canonical spin components and a proper time operator conjugated to mass. These results plead for a new representation not only of space-time localisation but also of motion.