Foundations for Quantum Mechanics

Abstract

A discussion is given of the structure of a physical theory and an "ideal form" for such a theory is proposed. The essential feature is that all concepts should be defined in operational terms. Quantum (and classical) mechanics is then formulated in this way (the formulation being, however, restricted to the kinematical theory). This requires the introduction of the concept of a mixed test, related to a pure test (or "question") just as a mixed state is related to a pure state. In the new formulation, the primitive concepts are not states and observables but certain operationally accessible mixed states and tests called physical. The notion of a C·-system is introduced; each such system is characterized by a certain C*-algebra. The structure of a general C*-system is then studied, all concepts being dei1.ned in terms of physical states and tests. It is shown first how pure states and tests can be so defined. The quantum analog 'of the phase space of classical mechanics is then constructed and on it is built a mathematical structure, called a q-topology, which is a quantum analog of the topology of classical phase space. Mathematically, a q-topology is related to a noncommutative C·-algebra as an ordinary topology is related to a commutative C*-algebra. Some properties of the q-topology of a C*-system are given. An appendix contains some physically motivated examples illustrating the theory.

A general scheme for microscopic theories

“…the concept of a microscopic object or of a space-time region) should be defined in terms of primitive concepts. These ideas are clearly explained in Giles (1970).…”

“…One of the aims of these investigations was to formulate the basic assumptions separating, as clearly as possible, the assumptions which have a direct and natural physical justification from the assumptions which are peculiar of quantum mechanics and are justified only indirectly by its success. Considering only the first class of assumptions, we get a class of formalism which we call 'general quantum theories' (Giles, 1970;Birkhoff & yon Neumann, 1936;Segal, 1947;Mackey, 1963;Haag & Kastler, 1964;Jauch, 1968;Varadarajan, 1968;Ludwig, 1970). The formalism described in the present paper is of this kind.…”

Bridges Between Contextual Linguistic Models of Vagueness and T-Norm Based Fuzzy Logic

“…On the linguistic side context and precisification based approaches suggested, e.g., by Kennedy [16], Kyburg and Moreau [18], and already earlier by Pinkal [25] and Bosch [3] are certainly worth investigating from this perspective. On the fuzzy logic side we just mention similarity semantics [27,17,30], Robin Giles's dialogue and betting game based characterisation of Lukasiewicz logic [9,8] (extended to other logics in [4,6]), acceptability semantics [22], rerandomising semantics [13,11], and approximation semantics [2,23] as alternative candidates for corresponding bridge heads. We plan to explore at least some of these options in future work.…”

Dynamic Resonant Tunneling

“…This phenomenon was investigated in electronics [13], nanotechnology [14][15][16], the foundations of quantum mechanics [17][18][19][20][21][22][23][24][25][26][27][28][29][30][31][32][33], and even biology and biochemistry [34][35][36][37][38][39].…”