Particle populations and number operators in quantum theory

The purpose of the present paper is to give a general theory of the quantum mechanical representation of particle populations.The first part of the paper, Sections 1 to 5, is devoted to a review of mathematical principles of quantum theory, with particular emphasis on the role played by probability concepts, using an approach adapted to the subsequent development of the theory of particle populations. This approach, which goes back in its essentials to von Neumann [20], leans heavily on the subsequent work of Wigner, Mackey, Jauch, Segal, Wightman and many others (see e.g., Mackey [15], Jauch [11], Streater and Wightman [26]). Sections 6 to 9 deal with the representation of finite particle populations: i.e., quantum systems where the total number of particles is an observable. In Section 10 a brief sketch is given of the generalization of the theory to infinite populations where the total number of particles is not an observable, as e.g., in the statistical theory of an infinitely extended gas (see Ruelle [22]). Finally, Section 11 treats some simple examples.

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On individuality in quantum theory

Facta universitatis - series Physics Chemistry and Technology ◽

10.2298/fupct1501029j ◽

2015 ◽

Vol 13 (1) ◽

pp. 29-38

Author(s):

Jasmina Jeknic-Dugic

Keyword(s):

Quantum Theory ◽

Structural Realism ◽

Mechanical Analysis ◽

Quantum Systems ◽

Quantum Mechanical ◽

Ontic Structural Realism

A quantum mechanical analysis of the decomposability of quantum systems into subsystems provides support for the so-called "attenuated Eliminative Ontic Structural Realism" within Categorical Structuralism studies in physics. Quantum subsystems are recognized as non-individual, relationally defined objects that deflate or relax some standard objections against Eliminative Ontic Structural Realism. Our considerations assume the universally valid quantum theory without tackling interpretational issues.

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Irreversibility in quantum mechanics

Discrete Dynamics in Nature and Society ◽

10.1155/s1026022604401046 ◽

2004 ◽

Vol 2004 (1) ◽

pp. 75-83 ◽

Cited By ~ 1

Author(s):

R. C. Bishop ◽

A. Bohm ◽

M. Gadella

Keyword(s):

Quantum Mechanics ◽

Quantum Theory ◽

Boundary Conditions ◽

Liouville Equation ◽

Quantum Systems ◽

Arrow Of Time ◽

Time Asymmetry ◽

Classical Physics ◽

Von Neumann ◽

Signal Features

Time asymmetry and irreversibility are signal features of our world. They are the reason of our aging and the basis for our belief that effects are preceded by causes. These features have many manifestations called arrows of time. In classical physics, some of these arrows are described by the increase of entropy or probability, and others by time-asymmetric boundary conditions of time-symmetric equations (e.g., Maxwell or Einstein). However, there is some controversy over whether probability or boundary conditions are more fundamental. For quantum systems, entropy increase is usually associated with the effects of an environment or measurement apparatus on a quantum system and is described by the von Neumann-Liouville equation. But since the traditional (von Neumann) axioms of quantum mechanics do not allow time-asymmetric boundary conditions for the dynamical differential equations (Schrödinger or Heisenberg), there is no quantum analogue of the radiation arrow of time. In this paper, we review consequences of a modification of a fundamental axiom of quantum mechanics. The new quantum theory is time asymmetric and accommodates an irreversible time evolution of isolated quantum systems.

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An Entropy Based Treatment of Gaussian Communication Process for General Quantum Systems

Open Systems & Information Dynamics ◽

10.1142/s123016121340009x ◽

2013 ◽

Vol 20 (03) ◽

pp. 1340009

Author(s):

Noboru Watanabe

Keyword(s):

Von Neumann Algebras ◽

Quantum Systems ◽

Communication Process ◽

Quantum Mechanical ◽

Input State ◽

Von Neumann ◽

Output State ◽

Amount Of Information ◽

General Quantum ◽

Mutual Entropy

The quantum entropy introduced by von Neumann around 1932 describes the amount of information of the quantum state itself. It was extended by Ohya for C*-systems before Conne-Narnhoffer-Thirring (CNT) entropy. The quantum relative entropy was first defined by Umegaki for σ-finite von Neumann algebras and it was subsequently extended by Araki and Uhlmann for general von Neumann algebras and *-algebras, respectively. By introducing a new notion, the so-called compound state, in 1983 Ohya succeeded to construct the mutual entropy in a complete quantum mechanical system (i.e., input state, output state and channel are all quantum mechanical) describing the amount of information correctly transmitted through the quantum channel. In this paper, we briefly review Ohya's S-mixing entropy and the quantum mutual entropy for general quantum systems. Based on a concept of structure equivalent, we apply the general framework of quantum communication to the Gaussian communication processes.

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Reconstructing quantum theory from diagrammatic postulates

Quantum ◽

10.22331/q-2021-04-28-445 ◽

2021 ◽

Vol 5 ◽

pp. 445

Author(s):

John H. Selby ◽

Carlo Maria Scandolo ◽

Bob Coecke

Keyword(s):

Quantum Theory ◽

Information Gain ◽

Quantum Systems ◽

Von Neumann ◽

Finite Dimensional ◽

Process Theories ◽

Classical Quantum ◽

The Right ◽

Theoretic Description ◽

Probabilistic Theories

A reconstruction of quantum theory refers to both a mathematical and a conceptual paradigm that allows one to derive the usual formulation of quantum theory from a set of primitive assumptions. The motivation for doing so is a discomfort with the usual formulation of quantum theory, a discomfort that started with its originator John von Neumann. We present a reconstruction of finite-dimensional quantum theory where all of the postulates are stated in diagrammatic terms, making them intuitive. Equivalently, they are stated in category-theoretic terms, making them mathematically appealing. Again equivalently, they are stated in process-theoretic terms, establishing that the conceptual backbone of quantum theory concerns the manner in which systems and processes compose. Aside from the diagrammatic form, the key novel aspect of this reconstruction is the introduction of a new postulate, symmetric purification. Unlike the ordinary purification postulate, symmetric purification applies equally well to classical theory as well as quantum theory. Therefore we first reconstruct the full process theoretic description of quantum theory, consisting of composite classical-quantum systems and their interactions, before restricting ourselves to just the ‘fully quantum’ systems as the final step. We propose two novel alternative manners of doing so, ‘no-leaking’ (roughly that information gain causes disturbance) and ‘purity of cups’ (roughly the existence of entangled states). Interestingly, these turn out to be equivalent in any process theory with cups & caps. Additionally, we show how the standard purification postulate can be seen as an immediate consequence of the symmetric purification postulate and purity of cups. Other tangential results concern the specific frameworks of generalised probabilistic theories (GPTs) and process theories (a.k.a. CQM). Firstly, we provide a diagrammatic presentation of GPTs, which, henceforth, can be subsumed under process theories. Secondly, we argue that the ‘sharp dagger’ is indeed the right choice of a dagger structure as this sharpness is vital to the reconstruction.

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Asymmetric Time Evolution and Reduced Dynamics

Open Systems & Information Dynamics ◽

10.1142/s1230161211000108 ◽

2011 ◽

Vol 18 (02) ◽

pp. 157-163

Author(s):

Peter W. Bryant

Keyword(s):

Quantum Theory ◽

Time Evolution ◽

Open Quantum Systems ◽

Quantum Systems ◽

Quantum Mechanical ◽

Time Coordinate ◽

Mechanical Environment ◽

Measurable Difference ◽

Asymmetric Quantum ◽

Symmetric Theory

When using a time asymmetric quantum theory, one must identify the time evolution parameter with a duration in time rather than with a time coordinate value. This identification restricts the options for the quantum mechanical environment of open quantum systems. The restriction may be important for interpretational questions concerning irreversibility or entanglement, but there is no measurable difference between a reduced dynamics within a time symmetric theory or within a time asymmetric theory.

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Operators and meaning of wave function

JOURNAL OF ADVANCES IN PHYSICS ◽

10.24297/jap.v11i8.216 ◽

2016 ◽

pp. 4039-4042

Author(s):

Viliam Malcher

Keyword(s):

Wave Function ◽

Quantum Theory ◽

State Vector ◽

The State ◽

Physical Significance ◽

Central Problem ◽

Quantum Mechanical ◽

Mechanical Description ◽

Quantum Mechanical Description ◽

Interpretation Of Quantum Theory

The interpretation problems of quantum theory are considered. In the formalism of quantum theory the possible states of a system are described by a state vector. The state vector, which will be represented as |Ïˆ> in Dirac notation, is the most general form of the quantum mechanical description. The central problem of the interpretation of quantum theory is to explain the physical significance of the |Ïˆ>. In this paper we have shown that one of the best way to make of interpretation of wave function is to take the wave function as an operator.

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Entanglement

10.1093/oso/9780198714057.003.0003 ◽

2017 ◽

Author(s):

Richard Healey

Keyword(s):

Quantum Theory ◽

Quantum State ◽

Physical Condition ◽

Physical System ◽

Polarization State ◽

Quantum Systems ◽

Ghz State ◽

Mathematical Representation ◽

Entangled State ◽

Physical Relation

Often a pair of quantum systems may be represented mathematically (by a vector) in a way each system alone cannot: the mathematical representation of the pair is said to be non-separable: Schrödinger called this feature of quantum theory entanglement. It would reflect a physical relation between a pair of systems only if a system’s mathematical representation were to describe its physical condition. Einstein and colleagues used an entangled state to argue that its quantum state does not completely describe the physical condition of a system to which it is assigned. A single physical system may be assigned a non-separable quantum state, as may a large number of systems, including electrons, photons, and ions. The GHZ state is an example of an entangled polarization state that may be assigned to three photons.

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John von Neumann and the Evolutionary Growth of Complexity: Looking Backward, Looking Forward

Artificial Life ◽

10.1162/106454600300103674 ◽

2000 ◽

Vol 6 (4) ◽

pp. 347-361 ◽

Cited By ~ 53

Author(s):

Barry McMullin

Keyword(s):

Subsequent Development ◽

Early Death ◽

Problem Situation ◽

John Von Neumann ◽

Von Neumann ◽

Comprehensive Theory ◽

History Of ◽

The Subject ◽

To Come ◽

Evolutionary Growth

In the late 1940s John von Neumann began to work on what he intended as a comprehensive “theory of [complex] automata.” He started to develop a book length manuscript on the subject in 1952. However, he put it aside in 1953, apparently due to pressure of other work. Due to his tragically early death in 1957, he was never to return to it. The draft manuscript was eventually edited, and combined for publication with some related lecture transcripts, by Burks in 1966. It is clear from the time and effort that von Neumann invested in it that he considered this to be a very significant and substantial piece of work. However, subsequent commentators (beginning even with Burks) have found it surprisingly difficult to articulate this substance. Indeed, it has since been suggested that von Neumann's results in this area either are trivial, or, at the very least, could have been achieved by much simpler means. It is an enigma. In this paper I review the history of this debate (briefly) and then present my own attempt at resolving the issue by focusing on an analysis of von Neumann's problem situation. I claim that this reveals the true depth of von Neumann's achievement and influence on the subsequent development of this field, and further that it generates a whole family of new consequent problems, which can still serve to inform—if not actually define—the field of artificial life for many years to come.

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