A solution to the problem of unifying the General Relativistic and Quantum Theoretical formalisms is given which introduces a new non-axiomatic symbolic method and an algebraic generalization of the Calculus to non-finite symbolisms without reference to the concept of a limit. An essential feature of the non-axiomatic method is the inadequacy of any (finite) statements: Identifying this aspect of the theory with the "existence of an external physical reality" both allows for the consistency of the method with the results of experiments and avoids the so-called "measurement problem" of quantum theory.
A solution to the problem of unifying the General Relativistic and Quantum Theoretical formalisms is given which introduces a new non-axiomatic symbolic method and an algebraic generalization of the Calculus to non-finite symbolisms without reference to the concept of a limit. An essential feature of the non-axiomatic method is the inadequacy of any (finite) statements: Identifying this aspect of the theory with the "existence of an external physical reality" both allows for the consistency of the method with the results of experiments and avoids the so-called "measurement problem" of quantum theory.
Motivated by Hilbert's sixth problem on axiomatization of physics, the author is proposing a rather provocative abstract axiomatic framework called S - formalization, where S is an arbitrary physical system .The proposal is an attempt to provide a general axiomatic framework , from which mathematical frameworks of new physical theories can be formulated on firm axiomatic basis and the mathematical frameworks of subjects such as standard (nonrelativistic) quantum mechanics are naturally derived as special cases of this general axiomatic framework. Introduction
The classical concept of 'physical entity', be it particle, wave, fleld or system, has become a problematic concept since the advent of relativity theory and quantum mechanics. The recent developments in modern quantum mechanics, with the performance of delicate and precise experiments involving single quantum entities, manifesting explicit non-local behavior for these entities, brings essential new information about the nature of the concept of entity. Such fundamental categories as space and time are put into question, and only a recourse to more axiomatic descriptions seems possible. In this contribution we want to put forward a 'picture' of what an 'entity' might be, taking into account these recent experimental and theoretical results, and using fundamental results of the axiomatic physical theories (describing classical as well as quantum entities) such as they have been developed during the last decade. We call our approach the 'creation-discovery view' because it considers measurements as physical interactions that in general entail two aspects: (1) a discovery of an already existing reality and (2) a creation of new aspects of reality during the act of measurement. We analyze the paradoxes of orthodox quantum mechanics in this creation-discovery view and point out the pre-scientiflc preconceptions that are contained in the well-known orthodox interpretations of quantum mechanics. Finally we identify orthodox quantum mechanics as a flrst order non classical theory, and explain in this way why it is so successful in its numerical predictions
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.
Authoritative appraisals qualified this book as an axiomatic theory. However, being its essential content no more than an analogy, its theoretical organization cannot be an axiomatic one. In fact, in the first edition Dirac declares to avoid an axiomatic presentation. Moreover, I show that the text is aimed at solving a basic problem: How quantum mechanics is similar to classical mechanics. A previous paper analyzed all past theories of physics, chemistry and mathematics, presented by the respective authors in a non-axiomatic way. Four characteristic features of a new model of organizing a theory have been recognized. An accurate inspection of Dirac text proves that it actually applied this kind of organization of a theory. This fact gives a formal reason of what Kronz and Lupher suggested through intuitive categories, pragmatism and rigor, i.e. Dirac formulation of Quantum mechanics represents a distinct theoretical attitude from von Neumann axiomatic attitude. However, since the second edition Dirac changed his attitude; although relying again on an analogy, his theory was referred to the axiomatic method. Some considerations on the odd paths which led to the present formulation of QM are added. They suggest to look for a new foundation of this theory.
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.
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.