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Banach Center Publications

Positive energy quantization of linear dynamics

Jan Dereziński, Christian Gérard

Published Jan 1, 2010• 4 citations
Publisher

Abstract

The abstract mathematical structure behind the positive energy quantization of linear classical systems is described. It is separated into three stages: the description of a classical system, the algebraic quantization and the Hilbert space quantizatio
DOI
10.4064/bc89-0-4

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The mathematical framework of quantum mechanics is conventionally founded upon a set of abstract axioms governing a Hilbert space. This work investigates whether this formalism can be derived as the effective description of a deeper, deterministic physical reality. We propose a mechanical model wherein a fundamental particle is treated as a 1-dimensional elastic defect (a string-like oscillator) topologically anchored to a background medium. By analyzing the dynamics of this system in the linear regime, we demonstrate that the core tenets of quantum theory emerge as necessary theorems of classical wave mechanicssubject to geometric constraints. Specifically, we show that: (1) the quantization of energy levels is a direct consequence of boundary conditions and resonance; (2) the Hilbert space structure is the natural mathematical arena for describing the superposition of stable vibrational modes; and (3) the momentum and energy operators, along with the Schrödinger equation, arise uniquely from the definition of observables and the conservation of energy. This approach grounds the quantum formalism in the mechanics of a continuous medium, suggesting that quantum behavior is not a break from classical physics, but the natural manifestation of wave dynamics in a confined topology.
The mass relationships associated with a system of chemical equations are studied in a general way as a linear algebraic structure; initially the linear space is represented in terms of the matrix relations of stoichiometry, and subsequently more abstractly in terms of linear operators working on a carrier space. Various mathematical structures arise naturally so that ultimately the framework might be extended to a Hilbert space. The determination of a consistent set of atomic weights from the directly accessible experimental information recorded in equivalent weights was a notorious indeterminate problem in the nineteenth century. The indeterminacy arises in the present formalism because of the freedom to represent the linear space in terms of any valid basis, while the physical interpretation requires the selection of a privileged representation. It is argued that at a fundamental level this should arise through quantization, and the paper represents a first step in that direction by formulating the stoichiometric equations as a classical linear operator theory.
We draw a distinction between 'positive' and 'non-positive' quantization schemes. Although the former are somewhat known to philosophers (in both canonical and path integral versions), the latter are almost entirely unstudied in the philosophy literature; moreover, physics writings on non-positive quantization often proceed in a manner which is not maximally systematic. As such, this paper aims both to introduce to philosophers the powerful tools of non-positive quantization schemes and their applications and to provide a systematic lay-of-the-land for such approaches which should be of value to philosophers and physicists alike.