An investigation of the mathematical formulation of quantum theory and its physical interpretation, 1900-1927

Armin Hermann, Arthur E. Ruark

David Böhm

The usual interpretation of the quantum theory is self-consistent, but it involves an assumption that cannot be tested experimentally, viz., that the most complete possible specification of an individual system is in terms of a wave function that determines only probable results of actual measurement processes. The only way of investigating the truth of this assumption is by trying to find some other interpretation of the quantum theory in terms of at present "hidden" variables, which in principle determine the precise behavior of an individual system, but which are in practice averaged over in measurements of the types that can now be carried out. In this paper and in a subsequent paper, an interpretation of the quantum theory in terms of just such "hidden" variables is suggested. It is shown that as long as the mathematical theory retains its present general form, this suggested interpretation leads to precisely the same results for all physical processes as does the usual interpretation. Nevertheless, the suggested interpretation provides a broader conceptual framework than the usual interpretation, because it makes possible a precise and continuous description of all processes, even at the quantum level. This broader conceptual framework allows more general mathematical formulations of the theory than those allowed by the usual interpretation. Now, the usual mathematical formulation seems to lead to insoluble difficulties when it is extrapolated into the domain of distances of the order of ${10}^{\ensuremath{-}13}$ cm or less. It is therefore entirely possible that the interpretation suggested here may be needed for the resolution of these difficulties. In any case, the mere possibility of such an interpretation proves that it is not necessary for us to give up a precise, rational, and objective description of individual systems at a quantum level of accuracy.

P. V. Kurakin

Previously suggested hidden time interpretation of quantum mechanics allows to reproduce the same predictions as standard quantum mechanics provides, since it is based on Feynman many - paths formulation of QM. While new experimental consequences of this interpretation are under investigation, some advantages can be enumerated. (1) The interpretation is much field theoretic - like in classical sense, so it is local in mathematical sense, though quantum (physical) non-locality is preserved. (2) The interpretation is based on one type of mathematical objects, rather than two different (Hilbert space vectors and operators). (3) The interpretation, as it was argued, overcomes the problem of hidden variables in a radically new way, with no conflict to Bell's theorem. Recently an important argument against hidden variables - like formulations of quantum theory was risen - "no protocol" argument. It is argued in the paper, that hidden time interpretation successfully overcomes this argument.

Christian Joas

Borros Arneth

Modern theoretical physics is formulated through several highly successful mathematical frameworks, including Green-function theory, quantum mechanics, quantum field theory, gauge theory and statistical mechanics. Although these theories employ different mathematical objects and were developed independently, they exhibit remarkable mathematical compatibility. Green functions naturally appear in quantum field theory, Hilbert-space methods underlie quantum mechanics, gauge theories are formulated geometrically, and statistical mechanics connects microscopic dynamics with macroscopic thermodynamics. The mathematical origin of this compatibility remains an open conceptual question. This paper investigates whether these apparently different frameworks can be understood as representations of a common underlying mathematical structure. We introduce the Architecture Principle, according to which the primitive object is not a Green function, wave function or Hilbert space, but a Green-Function Architecture describing propagation, interaction, closed transport and structural consistency. Green functions, quantum-mechanical states, gauge transport and statistical ensembles are interpreted as different mathematical representations of the same underlying architecture. The proposed framework does not modify the established equations of physics. Instead, it provides a common structural language that relates several major branches of theoretical physics while preserving their existing mathematical formalisms. The paper develops the basic concepts of the Architecture Principle, introduces an architectural invariant, formulates the Representation Principle and discusses representative applications. Finally, the principal mathematical problems required for a complete representation theory are identified.

Clayton A. Gearhart