Quantization of space-time and the corresponding quantum mechanics

Abstract

An axiomatic framework for describing general space-time models is presented. Space-time models to which irreducible propositional systems belong as causal logics are quantum (q) theoretically interpretable and their event spaces are Hilbert spaces. Such aq space-time is proposed via a “canonical” quantization. As a basic assumption, the time t and the radial coordinate r of aq particle satisfy the canonical commutation relation [t,r]=±i $$h =$$ . The two cases will be considered simultaneously. In that case the event space is the Hilbert space L2(ℝ3). Unitary symmetries consist of Poincare-like symmetries (translations, rotations, and inversion) and of gauge-like symmetries. Space inversion implies time inversion. Thisq space-time reveals a confinement phenomenon: Theq particle is “confined” in an $$h =$$ size region of Minkowski space $$\mathbb{M}^4$$ at any time. One particle mechanics overq space-time provides mass eigenvalue equations for elementary particles. Prugovecki's stochasticq mechanics andq space-time offer a natural way for introducing and interpreting consistently such aq space-time andq particles existing in it. The mass eigenstates ofq particles generate Prugovecki's extended elementary particles. When $$h =$$ →0, these particles shrink to point particles and $$\mathbb{M}^4$$ is recovered as the classical (c) limit ofq space-time. Conceptual considerations favor the case [t,r]=+i $$h =$$ , and applications in hadron physics give the fit $$h =$$ ⋍2/5 fermi/GeV.