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arXiv:quant-ph/9805079

On Quantum Mechanics

F. Ghaboussi

Published May 27, 1998• 0 citations
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Abstract

We discuss the axiomatic basis of quantum mechanics and show that it is neither general nor consistent, since its axioms are incompatible with each other and moreover it does not incorporate the magnetic quantization as in the cyclotron motion. A general and consistent system of axioms is conjectured which incorporates also the magnetic quantization.

DOI
10.48550/arxiv.quant-ph/9805079

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We discuss the axiomatic basis of quantum mechanics and show that it is neither general nor consistent, since its axioms are incompatible with each other and moreover it does not incorporate the magnetic quantization as in the cyclotron motion. A general and consistent system of axioms is conjectured which incorporates also the magnetic quantization.
We discuss the axiomatic basis of quantum mechanics and show that it is neither general nor consistent, since its axioms are incompatible with each other and moreover it does not incorporate the magnetic quantization as in the cyclotron motion. A general and consistent system of axioms is conjectured which incorporates also the magnetic quantization. 1 The neccessity of quantum mechanics (QM) was a result of experimental data for the energy spectrum of atoms which could not be explained in accord with the laws of classical mechanics. Nevertheless in the axiomatic basis of QM there are axioms which are not supported by the experimental results, since the whole system of axioms (1) is introduced only as a ”plausible generalization ” of the first axiom [1]:
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.
An axiomatic approach to quantum mechanics that has, as a theorem, the Schrödinger equation may be of enormous value to cope with interpretation issues, since all the interpretation constructs must be present in the axioms, or directly derived by them from their mathematical unfolding. Thus, it is critical to show that this axiomatic derivation is reliable beyond any possible doubt. To show this, it is possible to make generalizations and extensions of the axioms to derive the Schrödinger equation in the underlying generalized or extended formats. In previous papers, we have shown that the axiomatic approach we propose can be used to derive the Schrödinger equation as a direct axiom. Since then, we have also shown that it was possible to generalize that derivation to coordinate systems other than the Cartesian, as well as its relativistic extensions that lead to the relativistic wave equations. An extension to dissipative systems was also performed, allowing us to mathematically derive the Caldirola–Kanai equation from first principles. All these derivations were performed using pure states and in the absence of the electromagnetic field. This means that we can further generalize the approach to embrace these two possibilities. Being an axiomatic approach, we show that we need only to slightly modify the axioms to derive the Schrödinger equation for these two contexts. Despite being quite direct, the algebraic complexity of these derivations should give the reader the desired confidence in the proposed axioms.
This paper presents a comparison of the conceptual bases of the quantum theory and the theory of relativity, each considered as fundamental theories of elementary matter. From a study of the irreducible axiomatic bases of these respective approaches to elementary matter, it is concluded that they cannot peacefully co-exist because of logical incompatibility. Among the reasons discussed for this conclusion, one is the implication of the Hamiltonian form of quantum mechanics yielding an irreducible absolute frame of reference - that of the measuring apparatus - in the basic expression for the laws of elementary particles. This difficulty leads to logical inconsistency as well as a formulation of relativistic quantum field theory that is not demonstrably mathematically consistent. In spite of their logical incompatiability, it is argued that each of these theories logically requires the other, within its own framework. The conclusion is reached that, to make further progress in our understanding of elementary matter, it is necessary to abandon the axiomatic basis of one of these theories for the other, while keeping the formal structure of the abandoned theory as an accurate mathematical approximation (under special physical conditions) for a generalized form of the retained theory. An argument is presentedmore » for the author's retention of the theory of relativity as an authentic basis of elementary matter, and how its generalization is achieved by incorporating a fully (generally) covariant expression of the inertial manifestations of matter with its force manifestations.« less