Quantum Structural Studies: Classical Emergence From The Quantum Level

Classical Emergence from the Quantum Level

Ruth E Kastner, Jasmina Jekni?-Dugi?;George Jaroszkiewicz;;

• Publisher: WSPC (EUROPE)
• Year: 2016
• ISBN:
  9781786341426
• Topic: Biological Sciences
• Subtopic: Science General
• Index: Biological Sciences

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The structural aspects of composite quantum systems in the foundation, interpretation and application of quantum theory is an increasingly prominent topic of physics research. As an emerging field, it seeks to understand the origins of the classical world of experience from the quantum level.

Quantum Structural Studies presents conceptual fundamentals and mathematical methods for investigating the structuring of quantum systems into subsystems. Split into four sections, the topics covered include the historical and philosophical aspects of quantum structures, specific interpretive approaches and ontologies, and alternative methodological approaches to quantum mechanics. Questions addressed are:

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• Can the classically relevant degrees of freedom (such as the center of mass) be considered physically realistic, and if so, in what sense?
• In what sense might various emergent structures be relevant for the transition from the quantum description to the classical?
• Do suggested new approaches describe phenomenology and proposals for new experiments?

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Specialists, graduate students and researchers seeking an introduction to the field of emergent structures and new directions for research and experimentation can use this book to find up-to-date representative texts and reviews.

-->0 Readership: Specialists, graduate students and researchers seeking an introduction to the field of emergent structures in physics.
Quantum Theory, Physical Structure, Historical Quantum Structure, Centre-of-Mass, Emergent Structures, Primitive Ontology, Stern-Gerlach Experiment

• This is the first book of its type — the first collected volume focusing on this topic

Chapter 1

Quantum Structures: An Introduction

R. E. Kastner^∗, J. Jeknić-Dugić^† and G. Jaroszkiewicz^‡

^∗ Foundations of Physics Group,
University of Maryland, College Park, USA

^† Department of Physics,
Faculty of Science and Mathematics, University of Niš, Serbia

^‡ School of Mathematical Sciences,
The University of Nottingham, UK

1.Introduction

Quantum mechanics offers a striking, genuinely novel observation: it is possible to obtain more information about a closed composite system than about the subsystems constituting that system. This is symptomatic of the highly nontrivial concept of a quantum subsystem: specifically, there is much more to the “subsystem” concept in quantum mechanics than there is in classical physics.

In classical physics, complete knowledge about the total state of a system is equivalent to a complete knowledge about the state of every constituent subsystem. This accounts for the primary role of subsystems in classical physics, and the fact that the lack of knowledge about subsystems is subjective, i.e. observer–dependent. There are no fundamental physical limits in this regard, since there is no classical information limit at the fundamental physical level. This view of the classical world is typically interpreted as follows: every single classical system exists in space independently of any other physical system at every instant of time. Physical subsystems (“constituting particles”) are as physically realistic as the physical objects they build.

Mathematically, this idea is represented by the Cartesian product of a set of ontologically fundamental degrees of freedom (and analogously for continuous fields). In addition, in classical physics, useful artificial degrees of freedom can be defined; for example, the center of mass of an extended composite system. Mathematical manipulations with such constructed degrees of freedom do not directly describe the behavior of a realistic physical object. Thus, a description based on such mathematical degrees of freedom is typically incomplete and approximate, but is assumed ultimately to be reducible to the dynamics of the fundamental degrees of freedom.

This reductionistic attitude is sometimes criticized even in the context of classical physics: there is a conceptual and formal gap between the fundamental and the apparently emergent degrees of freedom [6]. Nevertheless, the primary role of the fundamental degrees of freedom is rarely, if at all, challenged in the physics literature: the existence of fundamental degrees of freedom is assumed to be a necessary condition for the emergent behavior that is observed.

However, this view face a serious challenge in the quantum mechanical context for at least the following two reasons. First, knowledge of a composite system’s state does not imply, or assume or require knowledge about, the subsystems’ states. If the total system is in a pure state

, the subsystems are statistically described by “reduced density matrices” (“reduced statistical operators”) that are sometimes referred to as “improper mixtures”. For a bipartite 1 + 2 system, a pure state,

, can be always represented by a Schmidt canonical form,

with the orthonormalized bases,

Provided

the reduced (subsystems) mixed states

are obtained via the tracing out operation such that

The point is that

i.e. that the αth subsystem cannot be assumed to be in the

state. Rather, the subsystems’ states are not well defined.

Second, there is no unique ensemble description of a mixed quantum state

The formalism of quantum mechanics allows a non-unique ensemble interpretation of

, which is a positive semidefinite Hermitian operator of trace one. Typically, there is a possibility of choosing different ensemble-representations pertaining to the same mixed state

while

and

Therefore, the basic interpretational tool of classical statistical physics as embodied in the concept of “Gibbs ensembles” is not necessarily useful in quantum theory.

On the other hand, a closed quantum system [subjected to the Schrödinger unitary dynamics] in a pure state

provides all the possible information (predictions) about the system and about its subsystems that is allowed by standard quantum theory.^a Hence, there is a kind of reversal of the above-described reductionist classical thinking that, in the view of the editors, merits further critical attention. This motivates the following questions, to which contributors to this volume have applied themselves:

1.Is there a preferred structure (i.e. decomposition into subsystems) of a composite quantum system? If so, how is it defined and what are the conditions that give rise to it?

2.How successful are extant accounts of quantum/classical correspondence, and/or emergence of classically recognizable structures from a quantum level of description?

3.Can classically familiar degrees of freedom (such as the ‘center-ofmass’) be considered quantum mechanically realistic, and in what sense — ontologically, epistemologically, operationally?

4.What kinds of alternative structures might be relevant for the transition from a quantum description to a classical one?

5.How, and to what extent, might the answers to the above describe phenomenology, including proposals for new experiments?

This introduction aims to set the stage for the answers offered by the contributors, who present diverse positions. The goal of the Editors of this volume is two-fold: (i) to collect the current perspectives of authors who have contributed to the topic in the past; (ii) to place side-by-side diverse views and approaches to these questions concerning the nature of fundamental quantum structures. Thus, our idea is not to ask or look for consensus, but rather to emphasize the wealth of ideas and approaches on the topic, as well as to initiate a discussion that could provide a fruitful foundation for progress in theory and experiment devoted to this topic. The contributors, in their independent research, have come to the point at which they realized that the question “How do [quantum] components relate to a composite?” cannot be ignored or treated as a secondary issue.

A significant aspect of this book’s mission is two-fold: (1) to present (as comprehensively as possible given space limitations) and extend the historical perspective and (2) to highlight the ubiquity of our topic of quantum structures in modern quantum physics research. Our presentation is general; further details can be found as indicated in the references list. Our choice of the relevant results is by no means exhaustive and is certainly not the only one possible. In addition, not all those invited have been willing or able (due to time or other constraints) to contribute. Nevertheless, we offer a beginning that, hopefully, will be extended and enriched in the near future.

2.Quantum Structures: A Brief Review

Quantum structural studies (QSS) cannot be given a definite chronological origin, nor can breakthrough results be non-arguably distinguished. Rather, a re-discovery of certain basic insights and their diverse (often implicit) understandings and interpretations are presented in this collection. The resurgence of interest in the issue of fundamental structure is not surprising, bearing in mind that the concept of “structure”, i.e. of “decomposition into parts”, is omnipresent in physical theory, in its mathematical formulation, and in its applications. To this end, arguably the most fundamental endeavor in science is investigation of the finite...

Table of contents

• Cover Page
• Title
• Copyright
• About the Editors
• Contents
• 1. Quantum Structures: An Introduction
• Historical Aspects
• Philosophical Aspects
• Specific Interpretive Approaches and Ontologies
• Methodological Approaches
• Index