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Journal of Philosophical Logic

Constructive mathematics and quantum mechanics: Unbounded operators and the spectral theorem

Geoffrey Paul Hellman

Published Jun 1, 1993• 42 citations
Publisher
DOI
10.1007/bf01049303

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A major outstanding issue in the foundations and philosophy of mathematics concerns the indispensability of classical infinitistic mathematics for the empirical sciences. Claims of such indispensability form a modern cornerstone of mathematical platonism and alternative classical realist conceptions as well (e.g. modal structuralism) (cf. e.g., Quine [1953], Putnam [1967, 1971], and Hellman [1989a]). This has posed a corresponding challenge to constructivist views (intuitionistic, Bishop-constructivist [Bishop 1967], and related approaches). How much of the mathematics actually employed in the empirical sciences, especially physics, can be carried out constructively (in the various relevant senses)? It is probably no exaggeration to say that the viability of a constructivist philosophy of mathematics is here at stake (cf. Burgess [1984]).

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A major outstanding issue in the foundations and philosophy of mathematics concerns the indispensability of classical infinitistic mathematics for the empirical sciences. Claims of such indispensability form a modern cornerstone of mathematical platonism and alternative classical realist conceptions as well (e.g. modal structuralism) (cf. e.g., Quine [1953], Putnam [1967, 1971], and Hellman [1989a]). This has posed a corresponding challenge to constructivist views (intuitionistic, Bishop-constructivist [Bishop 1967], and related approaches). How much of the mathematics actually employed in the empirical sciences, especially physics, can be carried out constructively (in the various relevant senses)? It is probably no exaggeration to say that the viability of a constructivist philosophy of mathematics is here at stake (cf. Burgess [1984]).
Quantum mechanics and the theory of operators on Hilbert space have been deeply linked since their beginnings in the early twentieth century. States of a quantum system correspond to certain elements of the configuration space and observables correspond to certain operators on the space. This book is a brief, but self-contained, introduction to the mathematical methods of quantum mechanics, with a view towards applications to Schrodinger operators. Part 1 of the book is a concise introduction to the spectral theory of unbounded operators. Only those topics that will be needed for later applications are covered. The spectral theorem is a central topic in this approach and is introduced at an early stage. Part 2 starts with the free Schrodinger equation and computes the free resolvent and time evolution. Position, momentum, and angular momentum are discussed via algebraic methods. Various mathematical methods are developed, which are then used to compute the spectrum of the hydrogen atom. Further topics include the nondegeneracy of the ground state, spectra of atoms, and scattering theory. This book serves as a self-contained introduction to spectral theory of unbounded operators in Hilbert space with full proofs and minimal prerequisites: Only a solid knowledge of advanced calculus and a one-semester introduction to complex analysis are required. In particular, no functional analysis and no Lebesgue integration theory are assumed. It develops the mathematical tools necessary to prove some key results in nonrelativistic quantum mechanics. Mathematical Methods in Quantum Mechanics is intended for beginning graduate students in both mathematics and physics and provides a solid foundation for reading more advanced books and current research literature. This new edition has additions and improvements throughout the book to make the presentation more student friendly.
Mathematical Applications of Science Fiction We present a comprehensive operator-theoretic framework for modeling stochastic cognitive evolutionwithin the context of non-separable Hilbert spaces, denoted HΨ. By constructing a Gelfand-triplestructure Φ ⊂ HΨ ⊂ Φ∗, we define the unbounded generator G of a C0-semigroup T (t) governing thetime-evolution of cognitive states. The central contribution of this monograph is the rigorous derivationof a Spectral Mapping Theorem for G , overcoming the traditional failure of such theorems for non-analytic semigroups in infinite dimensions. We introduce the concept of Phenomenological Decoherenceas a projection-valued measure associated with the essential spectrum of G . Through a series of tenfundamental lemmas and five major theorems, we prove that the asymptotic stability of the cognitivesemigroup is determined by the spectral bound s(G ), and that the non-separability of the underlyingspace is a necessary condition for the emergence of robust qualia. We conclude with applications totrans-universal constant variation and speculative engineering of synthetic consciousness.