This monograph contains a functional analytic introduction to Dirac's formalism. The first part presents some new mathematical notions in the setting of triples of Hilbert spaces, mentioning the concept of Dirac basis. The second part introduces a conceptually new theory of generalized functions, integrating the notions of the first part. The last part of the book is devoted to a mathematical interpretation of the main features of Dirac's formalism. It involves a pairing between distributional bras and kets, continuum expansions and continuum matrices.

Table of Contents

Some Basic Concepts and Developments in Sobolev Triples. Carleman Operators. A Measure Theoretical Sobolev Lemma. Dirac Bases. The Generalized Eigenvalue Problem for Self-Adjoint Operators. Direct Resolutions in Sobolev Triples. A Theory of Generalized Functions. Analyticity Spaces, Trajectory Spaces and Their Duality. Linear Mappings, Tensor Products and Kernel Theorems. Illustrations of Analyticity Spaces and Trajectory Spaces. The Concept of Dirac Bases Lifted to Trajectory Spaces. A Mathematical Interpretation of Dirac's Formalism. Dirac's Formalism According to Dirac and its Relations with Linear Algebra. A Mathematical Interpretation of Dirac's Bracket Formalism. The Free Field Operator Formalism.


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© 1986
North Holland
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