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Primitive Ontology アプローチのGRW理論とBohm力学の問題点

紘平 森田, Morita Kohei

Published Aug 8, 2019• 0 citations
Publisher

Abstract

GRW theory and Bohmian mechanics are typical interpretations of quantum mechanics. One of the problems for these interpretations arises from the gap of the space-dimensionality. Quantum mechanics is high dimensional, but its experiences appear in our three dimensional space. Primitive ontology has been proposed, in order to fill the gap, but Ney recently criticized this approach. Admittedly, the primitive ontology approach is not sufficient at all. But her criticism remains ambiguous, or at least not enough. Instead of her criticism, this article reveals that the main drawbacks for primitive ontology lie in the relationships with classical mechanics.

DOI
10.18999/nagjp.14.9

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GRW theory and Bohmian mechanics are typical interpretations of quantum mechanics. One of the problems for these interpretations arises from the gap of the space-dimensionality. Quantum mechanics is high dimensional, but its experiences appear in our three dimensional space. Primitive ontology has been proposed, in order to fill the gap, but Ney recently criticized this approach. Admittedly, the primitive ontology approach is not sufficient at all. But her criticism remains ambiguous, or at least not enough. Instead of her criticism, this article reveals that the main drawbacks for primitive ontology lie in the relationships with classical mechanics.
A major disagreement between different views about the foundations of quantum mechanics concerns whether for a theory to be intelligible as a fundamental physical theory it must involve a ‘primitive ontology’ (PO), i.e. variables describing the distribution of matter in four-dimensional space–time. In this article, we illustrate the value of having a PO. We do so by focussing on the role that the PO plays for extracting predictions from a given theory and discuss valid and invalid derivations of predictions. To this end, we investigate a number of examples based on toy models built from the elements of familiar interpretations of quantum theory.11 Introduction2 The GRWm and GRWf Theories 2.1 The GRW process 2.2 GRWm 2.3 GRWf3 Predictions and Primitive Ontology 3.1 Calibration functions 3.2 Taking the PO seriously 3.3 Examples from the literature 3.4 The main theorem about operators in the GRW formalism 3.5 The GRW formalism4 A Set of Examples 4.1 Bohmian mechanics 4.2 Bohmian trajectories and GRW collapses 4.2.1 Bohm’s law and GRW’s law 4.2.2 Bohm’s law and a modified GRW law 4.2.3 Trajectories from the GRW wave function 4.2.4 Configuration jumps and GRW law 4.2.5 Another way of configuration jumps and GRW law 4.3 MBM: Bohm-like trajectories from the master equation 4.3.1 Empirical equivalence of MBM with GRWm and GRWf4.4 Master equation and matter density4.5 Master equation and flashes5 Conclusions
The paper points out that the modern formulation of Bohm's quantum theory known as Bohmian mechanics is committed only to particles' positions and a law of motion. We explain how this view can avoid the open questions that the traditional view faces according to which Bohm's theory is committed to a wave-function that is a physical entity over and above the particles, although it is defined on configuration space instead of three-dimensional space. We then enquire into the status of the law of motion, elaborating on how the main philosophical options to ground a law of motion, namely Humeanism and dispositionalism, can be applied to Bohmian mechanics. In conclusion, we sketch out how these options apply to primitive ontology approaches to quantum mechanics in general.
In recent years, there has been a controversial debate in the literature regarding the choice among many interpretations of quantum mechanics. Based on certain realist desiderata, this debate has been partly addressed in the context of the choice between two opposite views about the mathematical space in which the fundamental ontology of this theory lives. Indeed, some scholars advocate a worldview in which this ontology lives in the abstract high-dimensional space, whilst others advocate a three-dimensional worldview, according to which the ontology lives in the physical space of our everyday experience. In this contribution, I will critically evaluate the tenability of solving the resulting underdetermination between the three/high-dimensional worldviews in the context of one particular quantum theory: the Bohmian approach. In so doing, I will be following two strategies: firstly, I will undermine the assumption that a set of theoretical or metaphysical virtues blocks the underdetermination problem associated with the mathematical space in which the fundamental Bohmian ontology lives; and secondly, I will propose a structural Bohmian interpretation through the use of dynamical symmetries (symplectic Lie groups), according to which the three/high-dimensional space distinction is just apparent as both spaces are grounded by the same dynamical, group-theoretic structure.
Abstract The three major interpretations of quantum mechanics—Bohm, GRW, and many worlds—all treat the wave function as a physical entity. But this entity apparently resides in a high-dimensional configuration space, with three coordinates for each particle in the universe. Some conclude on this basis that the three-dimensionality of the world of experience is an illusion. Others respond by adding three-dimensional ontology to the theory or by reinterpreting the wave function either as a law or as an indirect representation of properties in three-dimensional space. It is argued that the latter possibility makes it unnecessary to postulate either additional ontology or a fundamentally high-dimensional world.
In my dissertation I analyze the structure of fundamental physical theories. I start with an analysis of what an adequate primitive ontology is, discussing the measurement problem in quantum mechanics and theirs solutions. It is commonly said that these theories have little in common. I argue instead that the moral of the measurement problem is that the wave function cannot represent physical objects and a common structure between these solutions can be recognized: each of them is about a clear three-dimensional primitive ontology that evolves according to a law determined by the wave function. The primitive ontology is what matter is made of while the wave function tells the matter how to move. One might think that what is important in the notion of primitive ontology is their three-dimensionality. If so, in a theory like classical electrodynamics electromagnetic fields would be part of the primitive ontology. I argue that, reflecting on what the purpose of a fundamental physical theory is, namely to explain the behavior of objects in three--dimensional space, one can recognize that a fundamental physical theory has a particular architecture. If so, electromagnetic fields play a different role in the theory than the particles and therefore should be considered, like the wave function, as part of the law.Therefore, we can characterize the general structure of a fundamental physical theory as a mathematical structure grounded on a primitive ontology. I explore this idea to better understand theories like classical mechanics and relativity, emphasizing that primitive ontology is crucial in the process of building new theories, being fundamental in identifying the symmetries. Finally, I analyze what it means to explain the word around us in terms of the notion of primitive ontology in the case of regularities of statistical character. Here is where the notion of typicality comes into play: we have explained a phenomenon if the typical histories of the primitive ontology give rise to the statistical regularities we observe.
Abstract It has been argued that in the context of Bohm’s approach to quantum mechanics, the postulation of a three-dimensional ontology (as opposed to a high-dimensional one) is presumed to be the only interpretation that may reliably support object-oriented realism by virtue of the fact that this ontology is approximately preserved through scientific change, at least in the classical–quantum transition. Based on an interpretative analysis of the Bohmian formulation, I shall critically evaluate the tenability of this argument. In so doing, I shall argue that this formulation admits the postulation of an alternative ontology that is properly defined in a high-dimensional space, and is approximately preserved among this transition. As a result, both the three-dimensional and the high-dimensional worldviews shall form the basis of an equally reliable endorsement of object-oriented realism in the context of the Bohmian framework. I shall conclude that this argument becomes a problem of metaphysical underdetermination between a three and a high-dimensional Bohmian interpretation.