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Filozofia Nauki

Problem operacjonizmu z definicją czasu w mechanice kwantowej

Jakub Róg

Published Dec 12, 2023• 0 citations
Publisher

Abstract

Bridgman’s operationalism imposes a set of methodological requirements on its proponents, which become untenable when applied to the definition of time within the formalism of non-relativistic quantum mechanics. This paper explores the operational definition of time and investigates how far it can be aligned with the core principles of operationalism. To this end, Busch’s (2008) framework, which distinguishes three roles of time in the quantum mechanical formalism, is employed. One of these roles introduces demands that operationalism cannot meet without compromising its essential principles. As a result, it seems reasonable to conclude that operationalism and quantum mechanics are incompatible (at least in the context of the problem of time in quantum mechanics).

DOI
10.14394/filnau.2023.0006

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Bridgman’s operationalism imposes a set of methodological requirements on its proponents, which become untenable when applied to the definition of time within the formalism of non-relativistic quantum mechanics. This paper explores the operational definition of time and investigates how far it can be aligned with the core principles of operationalism. To this end, Busch’s (2008) framework, which distinguishes three roles of time in the quantum mechanical formalism, is employed. One of these roles introduces demands that operationalism cannot meet without compromising its essential principles. As a result, it seems reasonable to conclude that operationalism and quantum mechanics are incompatible (at least in the context of the problem of time in quantum mechanics).
I consider in this book a formulation of Quantum Mechanics. Usually QM is formulated based on the notion of time and space, both of which are thought a priori given quantities or notions. However, when we try to define the notion of velocity or momentum, we encounter a difficulty as we will see in chapter 1. The problem is that if the notion of time is given a priori, the velocity is definitely determined when given a position, which contradicts the uncertainty principle of Heisenberg. We then set the basis of QM on the notion of position and momentum operators as in chapter 2. Time of a local system then is defined approximately as a ratio ∣x∣/∣v∣|x|/|v|∣x∣/∣v∣ between the space coordinate xxx and the velocity vvv. In this formulation of QM, we can keep the uncertainty principle, and time is a quantity that does not have precise values unlike the usually supposed notion of time has. The feature of local time is that it is a time proper to each local system, which is defined as a finite set of quantum mechanical particles. We now have an infinite number of local times that are unique and proper to each local system. Based on the notion of local time, the motion inside a local system is described by the usual Schrödinger equation. We investigate such motion in a given local system in part II. This is a usual quantum mechanics. After some excursion of the investigation of local motion, we consider in part III the relative relation or motion between plural local systems. In the final part IV, we will prove that there is at least one Universe wave function ϕϕϕ in which all local systems have local motions and thus local times. This concludes our formulation of Quantum Mechanics.
I consider in this book a formulation of Quantum Mechanics. Usually QM is formulated based on the notion of time and space, both of which are thought a priori given quantities or notions. However, when we try to define the notion of velocity or momentum, we encounter a difficulty as we will see in chapter 1. The problem is that if the notion of time is given a priori, the velocity is definitely determined when given a position, which contradicts the uncertainty principle of Heisenberg. We then set the basis of QM on the notion of position and momentum operators as in chapter 2. Time of a local system then is defined approximately as a ratio ∣x∣/∣v∣|x|/|v|∣x∣/∣v∣ between the space coordinate xxx and the velocity vvv. In this formulation of QM, we can keep the uncertainty principle, and time is a quantity that does not have precise values unlike the usually supposed notion of time has. The feature of local time is that it is a time proper to each local system, which is defined as a finite set of quantum mechanical particles. We now have an infinite number of local times that are unique and proper to each local system. Based on the notion of local time, the motion inside a local system is described by the usual Schrodinger equation. We investigate such motion in a given local system in part II. This is a usual quantum mechanics. After some excursion of the investigation of local motion, we consider in part III the relative relation or motion between plural local systems. In the final part IV, we will prove that there is at least one Universe wave function ϕ\phiϕ in which all local systems have local motions and thus local times. This concludes our formulation of Quantum Mechanics.
W artykule omówiono ważniejsze argumenty przeciw operacjonizmowi Percy’ego W. Bridgmana sformułowane na przestrzeni ostatnich dekad oraz przedyskutowano operacyjne sposoby definiowania czasu w mechanice klasycznej oraz nierelatywistycznej mechanice kwantowej. O ile w mechanice klasycznej nie napotyka się większych trudności w sformułowaniu operacyjnej definicji czasu, o tyle w formalizmie teorii kwantowej zadanie to staje się niemożliwe. Co, niezależnie od dyskusyjnego statusu czasu w mechanice kwantowej, prowadzi do ogólniejszego pytania, czy można pogodzić mechanikę kwantową z podejściem operacyjnym.
Abstract The problem of time in quantum mechanics (QM) concerns the fact that in the Schrödinger equation time is a parameter, not an operator. Pauli's objection to a time–energy uncertainty relation analogue to the position–momentum one, conjectured by Heisenberg early on, seemed to exclude the existence of such an operator. However Dirac's formulation of an electron's relativistic QM does allow the introduction of a dynamical time operator that is self-adjoint. Consequently, it can be considered as the generator of a unitary transformation of the system, as well as an additional system observable subject to uncertainty. In the present paper these aspects are examined within the standard framework of relativistic QM.
There are good motivations for considering some type of quantum histories formalism. Several possible formalisms are known, defined by different definitions of event and by different selection criteria for sets of histories. These formalisms have a natural interpretation, according to which nature somehow chooses one set of histories from among those allowed, and then randomly chooses to realise one history from that set; other interpretations are possible, but their scientific implications are essentially the same. The selection criteria proposed to date are reasonably natural, and certainly raise new questions. For example, the validity of ordering inferences which we normally take for granted – such as that a particle in one region is necessarily in a larger region containing it – depends on whether or not our history respects the criterion of ordered consistency, or merely consistency. However, the known selection criteria, including consistency and medium decoherence, are very weak. It is not possible to derive the predictions of classical mechanics or Copenhagen quantum mechanics from the theories they define, even given observational data in an extended time interval. Attempts to refine the consistent histories approach so as to solve this problem by finding a definition of quasiclassicality have so far not succeeded. On the other hand, it is shown that dynamical collapse models, of the type originally proposed by Ghirardi-Rimini-Weber, can be re-interpreted as set selection criteria within a quantum histories framework, in which context they appear as candidate solutions to the set selection problem. This suggests a new route to relativistic generalisation of these models, since covariant definitions of a quantum event are known.