The paper proposes a new approach in the foundations of Quantum Mechanics.It does not make any assumption about the physical world, but looks at the consequences of the formalism used in models.Whenever a system is represented by variables which meet precise, but common, mathematical properties, one can prove theorems which are very close to the axioms of Quantum Mechanics (Hilbert spaces, observables, eigen values,...).It is then possible to explore the conditions of the validity of these axioms and to give a firm ground to the usual computations.Moreover this approach sheds a new ligth on the issues of determism, and interacting systems.In the third edition of this paper developments have been added about the statistical procedures used to detect anomalies in Physical Laws.
The paper proposes a new approach in the foundations of Quantum Mechanics.It does not make any assumption about the physical world, but looks at the consequences of the formalism used in models.Whenever a system is represented by variables which meet precise, but common, mathematical properties, one can prove theorems which are very close to the axioms of Quantum Mechanics (Hilbert spaces, observables, eigen values,...).It is then possible to explore the conditions of the validity of these axioms and to give a firm ground to the usual computations.Moreover this approach sheds a new ligth on the issues of determism, and interacting systems.In the third edition of this paper developments have been added about the statistical procedures used to detect anomalies in Physical Laws.
The paper proposes a new approach in the foundations of Quantum Mechanics.It does not make any assumption about the physical world, but looks at the consequences of the formalism used in models.Whenever a system is represented by variables which meet precise, but common, mathematical properties, one can prove theorems which are very close to the axioms of Quantum Mechanics (Hilbert spaces, observables, eigen values,...).It is then possible to explore the conditions of the validity of these axioms and to give a firm ground to the usual computations.Moreover this approach sheds a new ligth on the issues of determism, and interacting systems.In the third edition of this paper developments have been added about the statistical procedures used to detect anomalies in Physical Laws.
The purpose of the paper is to study the foundations of the main axioms of Quantum Mechanics. From a general study of the mathematical properties of the models used in Physics to represent systems, we prove that the states of a system can be represented in a Hilbert space, that a self-adjoint operator is associated to any observable, that the result of a measure must be the eigen value of the operator and appear with the usual probability. Furthermore an equivalent of the Wigner's theorem holds, which leads to the Schr{ö}dinger equation. These results are based on well known mathematics, and do not involve any specific hypothesis in Physics. They validate and explain the methods currently used, which are made simpler and safer, and open new developments. In the third edition of this paper developments have been added about the estimation of physical anomalies.
"Apart from providing a lucid introduction to the mathematical formalism and conceptual foundations of quantum mechanics, we explain why the laws of physics have the form that they do. In addition, we present a new and unique look at the quantum world, steering clear of two common errors: the error of the ψ-ontologists, who reify a calculational tool; and the error of the anti-realists, for whom physical theories are simply devices for expressing regularities among observations. The new edition of this acclaimed text adds around 200 pages on a variety of topics, such as how the founders sought to make sense of quantum mechanics, Kant's theory of science, QBism, Everettian quantum mechanics, de Broglie-Bohm theory, environmental decoherence, contextuality, nonlocality, and the paradox of subjectivity - the curious fact that the world seems to exist twice, once for us, in our minds, and once by itself, independently of us."--
The final version of a new approach to quantum theory is formulated in this paper. The basis is taken to be theoretical variables, variables that may be accessible or inaccessible, i.e., it may be possible or impossible for an observer to assign arbitrarily sharp numerical values to them. In an epistemic process, the accessible variables are just ideal observations connected to an observer or to some communicating observers. Group actions are defined on these variables, and group representation theory is the basis for developing the Hilbert space formalism here. Operators corresponding to accessible theoretical variables are derived, and in the finite-dimensional case, it is proved that the possible physical values are the eigenvalues of these operators. The focus of the paper is some mathematical theorems paving the ground for the proposed foundation of quantum theory. It is indicated here that the groups and transformations needed in this approach can be constructed explicitly in the case where the accessible variables are finite-dimensional. In case, this simplifies the theory considerably: To reproduce the Hilbert space formulation, it is enough to assume the existence of two complementary variables. What is new in this version of the article, is that the assumptions given behind the main theorems are simplified considerably, and that all proofs are given explicitly or refered to in other recent papers.
The list of basic axioms of quantum mechanics as it was formulated by von Neumann includes only the mathematical formalism of the Hilbert space and its statistical interpretation. We point out that such an approach is too general to be considered as the foundation of quantum mechanics. In particular in this approach any finite-dimensional Hilbert space describes a quantum system. Though such a treatment might be a convenient approximation it can not be considered as a fundamental description of a quantum system and moreover it leads to some paradoxes like Bell's theorem. I present a list from seven basic postulates of axiomatic quantum mechanics. In particular the list includes the axiom describing spatial properties of quantum system. These axioms do not admit a nontrivial realization in the finite-dimensional Hilbert space. One suggests that the axiomatic quantum mechanics is consistent with local realism.
Abstract The paper, summarizing the results of recent papers of the author, is devoted to a detailed analysis of postulates of nonrelativistic quantum theory appearing within two classical axiomatic frameworks the “quantum logic” and the “algebraic approach” respectively, which are presently the two main alternatives for quantum axiomatics. The first part of the paper concerns the structure of two important sets, the set of questions (propositional logic) and the set of pure states. It is shown how the quantum logic axiomatic scheme can be modified and improved in order to overcome the well known troubles connected with the lack of satisfactory physical justification for some postulates of this approach. In particular, an axiom system is developed, related closely to the quantum logic axiomatic scheme, in which the questions of the complete lattice structure, the atomisticity, and the validity of the covering law in the propositional logic do not appear so problematic. The particular attention is directed to the covering law, which is here obtained as a consequence of physically clear properties of the experimental procedures (“filters”) associated with the quantum‐mechanical propositions. In Section 1.5 we formulate an axiom system for quantum mechanics based exclusively on the concepts of pure state, transition probability, and pure filter. In the second part of the paper the axiomatic scheme presented in Part 1 is developed with the aim to analyse the algebraic structure of some subspaces of the set of simple observables. The quantum logic axiomatic framework is here shown closely connected with the Jordan‐Banach algebraic scheme. In the third part of the paper, similarly as it was done in Parts 1 and 2, the correspondence between the propositions and the experimental (measuring) procedures verifying these propositions is put as the basic assumption of the axiomatization of quantum mechanics. As a consequence of this assumption, there is established the structure of a real Jordan‐Banach algebra in the set Ob of the bounded observables associated with a physical system, so that we can apply the GNS representation theorem proved recently for such algebras by Alfsen et al. (1978) to obtain the Hilbert space (or, to be more precise, the C*‐algebraic) representation for Ob.