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Is Time a Cyclic Dimension? How to Second-Quantize Fields by Means of Periodic Boundary Conditions. https://doi.org/10.1016/j.aop.2022.169182

Dolce, Donatello

Published Jan 1, 2025• 0 citations
Publisher

Abstract

If "quantization is an art", it can be greatly refined through the formalism of cyclic time. In past papers, we have demonstrated the effectiveness of a unified formulation of quantum and classical physics based on cyclic relativistic time. In this paper, we show, using theorems from Geometric Quantization, that the Poisson brackets of intrinsically cyclic time dynamics, classical in essence, lead naturally to the canonical commutation relations and the other quantization rules of standard Quantum Mechanics. Our results suggest that canonical quantization can be interpreted as an implicit imposition of closed time paths on classical dynamics, without explicitly stating that time is a cyclic dimension.

DOI
10.6084/m9.figshare.30788567

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If "quantization is an art", it can be greatly refined through the formalism of cyclic time. In past papers, we have demonstrated the effectiveness of a unified formulation of quantum and classical physics based on cyclic relativistic time. In this paper, we show, using theorems from Geometric Quantization, that the Poisson brackets of intrinsically cyclic time dynamics, classical in essence, lead naturally to the canonical commutation relations and the other quantization rules of standard Quantum Mechanics. Our results suggest that canonical quantization can be interpreted as an implicit imposition of closed time paths on classical dynamics, without explicitly stating that time is a cyclic dimension.
If “quantization is an art”, it can be greatly refined through the formalism of cyclic time. In past papers, we have demonstrated the effectiveness of a unified formulation of quantum and classical physics based on cyclic relativistic time. In this paper, we show, using theorems from Geometric Quantization, that the Poisson brackets of intrinsically cyclic time dynamics, classical in essence, lead naturally to the canonical commutation relations and the other quantization rules of standard Quantum Mechanics. Our results suggest that canonical quantization can be interpreted as an implicit imposition of closed time paths on classical dynamics, without explicitly stating that time is a cyclic dimension.
Wave-particle duality, together with the concept of elementary particles, was introduced by de Broglie in terms of intrinsically periodic phenomena. However, after nearly 90 years, the physical origin of such undulatory mechanics remains unrevealed. We propose a natural realization of the de Broglie periodic phenomenon in terms of harmonic vibrational modes associated to space-time periodicities. In this way we find that, similarly to a vibrating string or a particle in a box, the intrinsic recurrence imposed as a constraint to elementary particles represents a fully consistent quantization condition. The resulting cyclic dynamics formally match ordinary relativistic Quantum Mechanics in both the canonical and Feynman formulations. Interactions are introduced in a geometrodynamical way, similarly to general relativity, by simply considering that variations of kinematical state can be equivalently described in terms of modulations of space-time recurrences, as known from undulatory mechanics. We present this novel quantization prescription from an historical prospective.
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We interpret the relativistic and quantum behavior of elementary particles in terms of elementary cycles. This represents a generalization of de Broglie hypothesis of intrinsically "periodic phenomenon". Similarly to a "particle in a box" or to a "vibrating string", the constraint of intrinsic periodicity can be used as semi-classical quantization condition, with remarkable matchings to ordinary relativistic quantum mechanics. In this formalism the retarded and local variations of four-momentum characterizing relativistic interactions can be equivalently expressed in terms of retarded and local variations of "de Broglie internal clock" space-time periodicity.
Besides the purely digital or analog interpretation of reality there is a third possibility which incorporates important aspects of both. This is the cyclic formulation of elementary systems, in which elementary particles are represented as classical strings vibrating in compact space-time dimensions with periodic boundary conditions. We will address these cyclic solutions as "de Broglie internal clocks". They constitute the deterministic gears of a consistent semi-classical description of quantum relativistic physics, providing in addition an appealing formulation of the notion of time.