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The Vertical Logic of Hamiltonian Methods

C. Baumgarten

Published Jan 10, 2020• 0 citations
OpenAlex

Abstract

We discuss the key role that Hamiltonian notions could play in physics. Five examples are given that illustrate the underestimated versatility and almost magical generality of Hamiltonian notions. The given examples concern the interconnection between quantum mechanics, special relativity and electromagnetism. We demonstrate that a derivation of these core concepts of modern physics requires little more than a proper formulation in terms of classical Hamiltonian theory.

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We discuss the key role that Hamiltonian notions could play in physics. Five examples are given that illustrate the underestimated versatility and almost magical generality of Hamiltonian notions. The given examples concern the interconnection between quantum mechanics, special relativity and electromagnetism. We demonstrate that a derivation of these core concepts of modern physics requires little more than a proper formulation in terms of classical Hamiltonian theory.
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 defined 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.
The current 50th issue of Philosophical Problems in Science (Zagadnienia Filozoficzne w Nauce) summarizes the efforts of both philosophers and scientists to understand how a broader philosophical context sets the stage for the development of scientific research, with physics playing a leading role. In particular, the paper reviews the content of the last twenty-five issues of the journal with an emphasis on the philosophical problems that arise in the practice of physics. The overview reveals that these problems reflect the main conceptual division in physics between the treatment of the micro-world described by quantum mechanics and the macro-world governed by the general theory of relativity. Both of these theories, taken separately, generate a host of philosophical concerns such as their proper interpretation (Bell’s Theorem and its consequences) or the meaning and the eventual validity of the notions of space and time. Other philosophical problems in physics, such as chaos and determinism, are also considered. The authors are well aware that the formulation of the future theory of quantum gravity will be a demanding task requiring profound philosophical reflection.