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Quantization of space-time and the corresponding quantum mechanics

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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 Poincaré-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. Prugovečki'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 Prugovečki'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.

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  1. Central Research Institute for Physics, P.O.B. 49, H-1525, Budapest 114, Hungary

    M. Banai

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This paper is a revised version of the author's work, “Quantization of Space-time and the Corresponding Quantum Mechanics (Part I),” report KFKI-1981-48.

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Banai, M. Quantization of space-time and the corresponding quantum mechanics. Found Phys 15, 1203–1245 (1985). https://doi.org/10.1007/BF00735531

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