A philosophically consistent axiomatic approach to classical and quantum mechanics is given. The approach realizes a strong formal implementation of Bohr's correspondence principle. In all instances, classical and quantum concepts are fully parallel: the same general theory has a classical realization and a quantum realization. Extending the `probability via expectation' approach of Whittle to noncommuting quantities, this paper defines quantities, ensembles, and experiments as mathematical concepts and shows how to model complementarity, uncertainty, probability, nonlocality and dynamics in these terms. The approach carries no connotation of unlimited repeatability; hence it can be applied to unique systems such as the universe. Consistent experiments provide an elegant solution to the reality problem, confirming the insistence of the orthodox Copenhagen interpretation on that there is nothing but ensembles, while avoiding its elusive reality picture. The weak law of large numbers explains the emergence of classical properties for macroscopic systems.A philosophically consistent axiomatic approach to classical and quantum mechanics is given. The approach realizes a strong formal implementation of Bohr's correspondence principle. In all instances, classical and quantum concepts are fully parallel: the same general theory has a classical realization and a quantum realization. Extending the `probability via expectation' approach of Whittle to noncommuting quantities, this paper defines quantities, ensembles, and experiments as mathematical concepts and shows how to model complementarity, uncertainty, probability, nonlocality and dynamics in these terms. The approach carries no connotation of unlimited repeatability; hence it can be applied to unique systems such as the universe. Consistent experiments provide an elegant solution to the reality problem, confirming the insistence of the orthodox Copenhagen interpretation on that there is nothing but ensembles, while avoiding its elusive reality picture. The weak law of large numbers explains the emergence of classical properties for macroscopic systems.

Jan 1, 20039 citations

Noncommutative analysis and quantum physics I. Quantities, ensembles and states

Arnold Neumaier

A unified conceptual foundation of classical and quantum physics is given, free of undefined terms. Ensembles are defined by extending the `probability via expectation' approach of Whittle to noncommuting quantities. This approach carries no connotations of unlimited repeatability; hence it can be applied to unique systems such as the universe. Precise concepts and traditional results about complementarity, uncertainty and nonlocality follow with a minimum of technicalities. Probabilities are introduced in a generality supporting so-called effects (i.e., fuzzy events). States are defined as partial mappings that provide reference values for certain quantities. An analysis of sharpness properties yields well-known no-go theorems for hidden variables. By dropping the sharpness requirement, hidden variable theories such as Bohmian mechanics can be accommodated, but so-called ensemble states turn out to be a more natural realization of a realistic state concept. The weak law of large numbers explains the emergence of classical properties for macroscopic systems.A unified conceptual foundation of classical and quantum physics is given, free of undefined terms. Ensembles are defined by extending the `probability via expectation' approach of Whittle to noncommuting quantities. This approach carries no connotations of unlimited repeatability; hence it can be applied to unique systems such as the universe. Precise concepts and traditional results about complementarity, uncertainty and nonlocality follow with a minimum of technicalities. Probabilities are introduced in a generality supporting so-called effects (i.e., fuzzy events). States are defined as partial mappings that provide reference values for certain quantities. An analysis of sharpness properties yields well-known no-go theorems for hidden variables. By dropping the sharpness requirement, hidden variable theories such as Bohmian mechanics can be accommodated, but so-called ensemble states turn out to be a more natural realization of a realistic state concept. The weak law of large numbers explains the emergence of classical properties for macroscopic systems.

arXiv:quant-ph/0001096Jan 26, 20000 citations

Noncommutative analysis and quantum physics I. States and ensembles

Arnold Neumaier

In this sequence of papers, noncommutative analysis is used to give a consistent axiomatic approach to a unified conceptual foundation of classical and quantum physics. The present Part I defines the concepts of observables, states and ensembles, clarifies the logical relations and operations for them, and shows how they give rise to dynamics and probabilities. States are identified with maximal consistent sets of weak equalities in the algebra of observables (instead of, as usual, with the rays in a Hilbert space). This leads to a concise foundation of quantum mechanics, free of undefined terms, separating in a clear way the deterministic and the stochastic features of quantum physics. The traditional postulates of quantum mechanics are derived from well-motivated axiomatic assumptions. No special quantum logic is needed to handle the peculiarities of quantum mechanics. Foundational problems associated with the measurement process, such as the reduction of the state vector, disappear. The new interpretation of quantum mechanics contains `elements of physical reality' without the need to introduce a classical framework with hidden variables. In particular, one may talk about the state of the universe without the need of an external observer and without the need to assume the existence of multiple universes.In this sequence of papers, noncommutative analysis is used to give a consistent axiomatic approach to a unified conceptual foundation of classical and quantum physics. The present Part I defines the concepts of observables, states and ensembles, clarifies the logical relations and operations for them, and shows how they give rise to dynamics and probabilities. States are identified with maximal consistent sets of weak equalities in the algebra of observables (instead of, as usual, with the rays in a Hilbert space). This leads to a concise foundation of quantum mechanics, free of undefined terms, separating in a clear way the deterministic and the stochastic features of quantum physics. The traditional postulates of quantum mechanics are derived from well-motivated axiomatic assumptions. No special quantum logic is needed to handle the peculiarities of quantum mechanics. Foundational problems associated with the measurement process, such as the reduction of the state vector, disappear. The new interpretation of quantum mechanics contains `elements of physical reality' without the need to introduce a classical framework with hidden variables. In particular, one may talk about the state of the universe without the need of an external observer and without the need to assume the existence of multiple universes.

arXiv:quant-ph/9909080Sep 26, 19990 citations

Consistent Interpretation of Quantum and Classical Mechanics

Andrew Das Arulsamy

Interpreting quantum mechanics is a hard problem basically because it means explaining why and how the mathematics exploited to formulate wave-particle duality are related to observations or reality in classical physics (due to Newtonian mechanics, special relativity, gravity and entropy). Here, we first prove the limitation of the Heisenberg uncertainty principle by invoking the wave-particle duality and apply it to revise and/or extend the Copenhagen postulates. In particular, even though the said uncertainty exists due to non-commuting operators, but these operators do not commute simply because one of the operators is well-defined in classical physics (particle-like), while the other is wave-like (well-defined in quantum mechanics), or the eigenvalue is not at all defined in classical physics. We also construct a new quantum mechanical postulate (Postulate 9) to deduce why the forbidden gap between discrete energy levels should exist in atoms, molecules and condensed matter phases. In fact, we tackle all the problems arising from the Copenhagen interpretation without violating established experiments and without proposing ideas that violate physical reality.Interpreting quantum mechanics is a hard problem basically because it means explaining why and how the mathematics exploited to formulate wave-particle duality are related to observations or reality in classical physics (due to Newtonian mechanics, special relativity, gravity and entropy). Here, we first prove the limitation of the Heisenberg uncertainty principle by invoking the wave-particle duality and apply it to revise and/or extend the Copenhagen postulates. In particular, even though the said uncertainty exists due to non-commuting operators, but these operators do not commute simply because one of the operators is well-defined in classical physics (particle-like), while the other is wave-like (well-defined in quantum mechanics), or the eigenvalue is not at all defined in classical physics. We also construct a new quantum mechanical postulate (Postulate 9) to deduce why the forbidden gap between discrete energy levels should exist in atoms, molecules and condensed matter phases. In fact, we tackle all the problems arising from the Copenhagen interpretation without violating established experiments and without proposing ideas that violate physical reality.

May 15, 20230 citations

3. Uncertainty, statistics, probability

Arnold Neumaier

In this chapter, we consider the way quantum theory represents deterministic, statistical, and probabilistic aspects of Nature. After a discussion in Section 3.1 of uncertainty in general, we discuss in Sections 3.2-3.4 several formal notions of classical probability, their relation to the probability concept used in applied statistics, and their dependence on the description used. Section 3.5 briefly discusses various examples, where probabilistic features emerge from fully deterministic situations-a theme to be taken up in much more detail in Chapter 11. Section 3.6 then shows how the statistical aspects of the quantum formalism naturally follow from the weak law of large numbers. The notions of c-probability in the classical case and of q-probability in the quantum case are formally defined in Section 3.7. Historically, the concept of classical probability (including its use in stochastic processes) was given an undisputed formal mathematical foundation in 1933 in terms of the measure-theoretic setting of Kolmogorov [164]. Apart from this traditional axiomatic foundation of classical probability theory, there is a less well-known equivalent axiomatic treatment by Whittle [302] in terms of expectations. Here probabilities appear as the expectations of statements, {0, 1}-valued random variables. von Plato [240] discusses the history of the concept of probability. Krüger et al. [169] discuss the history of probability in the various fields of application. Sklar [275] discusses the philosophical problems of the probability concept, with an emphasis on statistical mechanics.In this chapter, we consider the way quantum theory represents deterministic, statistical, and probabilistic aspects of Nature. After a discussion in Section 3.1 of uncertainty in general, we discuss in Sections 3.2-3.4 several formal notions of classical probability, their relation to the probability concept used in applied statistics, and their dependence on the description used. Section 3.5 briefly discusses various examples, where probabilistic features emerge from fully deterministic situations-a theme to be taken up in much more detail in Chapter 11. Section 3.6 then shows how the statistical aspects of the quantum formalism naturally follow from the weak law of large numbers. The notions of c-probability in the classical case and of q-probability in the quantum case are formally defined in Section 3.7. Historically, the concept of classical probability (including its use in stochastic processes) was given an undisputed formal mathematical foundation in 1933 in terms of the measure-theoretic setting of Kolmogorov [164]. Apart from this traditional axiomatic foundation of classical probability theory, there is a less well-known equivalent axiomatic treatment by Whittle [302] in terms of expectations. Here probabilities appear as the expectations of statements, {0, 1}-valued random variables. von Plato [240] discusses the history of the concept of probability. Krüger et al. [169] discuss the history of probability in the various fields of application. Sklar [275] discusses the philosophical problems of the probability concept, with an emphasis on statistical mechanics.

De Gruyter eBooksOct 24, 20190 citations

Colloquium: An introduction to consistent quantum theory

Pierre C. Hohenberg

This paper presents an elementary introduction to consistent quantum theory, as developed by Griffiths and others over the past $25\phantom{\rule{0.3em}{0ex}}\text{years}$ . The theory is a version of orthodox (Copenhagen) quantum mechanics based on the notion that the unique and mysterious feature of quantum, as opposed to classical systems, is the simultaneous existence of multiple incompatible representations of reality, referred to as ``frameworks.'' A framework is a maximal set of properties of a system for which probabilities can be consistently defined. This notion is expressed by stating that a framework provides an exhaustive set of exclusive alternatives, but no single framework suffices to fully characterize a quantum system. Any prediction of the theory must be confined to a single framework and combining elements from different frameworks leads to quantum mechanically meaningless statements. This ``single-framework rule'' is the precise mathematical statement of Bohr's complementarity. It is shown that if the microscopic description is assumed to incorporate these elements in a local setting, then distant entanglements, macroscopic measurements, wave-function collapse, and other features of quantum behavior follow in a logical manner. The essential elements of the theory are first explained using the simplest quantum system, a single spin- $\frac{1}{2}$ degree of freedom at one time.This paper presents an elementary introduction to consistent quantum theory, as developed by Griffiths and others over the past $25\phantom{\rule{0.3em}{0ex}}\text{years}$. The theory is a version of orthodox (Copenhagen) quantum mechanics based on the notion that the unique and mysterious feature of quantum, as opposed to classical systems, is the simultaneous existence of multiple incompatible representations of reality, referred to as ``frameworks.'' A framework is a maximal set of properties of a system for which probabilities can be consistently defined. This notion is expressed by stating that a framework provides an exhaustive set of exclusive alternatives, but no single framework suffices to fully characterize a quantum system. Any prediction of the theory must be confined to a single framework and combining elements from different frameworks leads to quantum mechanically meaningless statements. This ``single-framework rule'' is the precise mathematical statement of Bohr's complementarity. It is shown that if the microscopic description is assumed to incorporate these elements in a local setting, then distant entanglements, macroscopic measurements, wave-function collapse, and other features of quantum behavior follow in a logical manner. The essential elements of the theory are first explained using the simplest quantum system, a single spin-$\frac{1}{2}$ degree of freedom at one time.

arXiv:0909.2359Reviews of Modern PhysicsOct 5, 201095 citations

This paper presents an elementary introduction to consistent quantum theory, as developed by Griffiths and others over the past

25\phantom{\rule{0.3em}{0ex}}\text{years}

. The theory is a version of orthodox (Copenhagen) quantum mechanics based on the notion that the unique and mysterious feature of quantum, as opposed to classical systems, is the simultaneous existence of multiple incompatible representations of reality, referred to as ``frameworks.'' A framework is a maximal set of properties of a system for which probabilities can be consistently defined. This notion is expressed by stating that a framework provides an exhaustive set of exclusive alternatives, but no single framework suffices to fully characterize a quantum system. Any prediction of the theory must be confined to a single framework and combining elements from different frameworks leads to quantum mechanically meaningless statements. This ``single-framework rule'' is the precise mathematical statement of Bohr's complementarity. It is shown that if the microscopic description is assumed to incorporate these elements in a local setting, then distant entanglements, macroscopic measurements, wave-function collapse, and other features of quantum behavior follow in a logical manner. The essential elements of the theory are first explained using the simplest quantum system, a single spin-

\frac{1}{2}

degree of freedom at one time.