Bohm A. The Rigged Hilbert Space and Quantum Mechanics

Berlin, Heidelberg, New York: Springer-Verlag, 1978. — VIII, 69 p. — (Lecture Notes in Physics Series, Volume 78).The purpose of this paper is not to present new results, but to make a little-known area in the mathematical foundations of quantum mechanics accessable to a wider audience. Therefore the discussion of the physics and mathematics will be limited to the bare essentials. The whole subject is developed in terms of the simple, well-known example of the harmonic oscillator, and the mathematical notions have been introduced in the least general form which is compatible with the purpose of this paper. After one has once met these notions for this well known particular case, one will more easily absorb them in their general form. Most of the results stated here are known to be true in much more general situations.One can argue that the subject described here is very useful for physics, because it makes the Dirac formalism rigorous and therewith gives a mathematical justification for all the mathematically undefined operations which physicists have been using for generations. It also gives a slightly different framework for quantum mechanics than the von Neumann axioms, in which a physical state is described by a collection of square integrable functions whose elements differ on a set of measure zero. In a physically more practical description one would associate with the physical state one test function (an element of the Schwark space) which is given by the resolution function of the experimental apparatus that is used in the preparation of the state. The subject presented here provides just such a description. Thus the use of the mathematical structure set forth here leads to a physically more practical and mathematically simpler description, even though it requires one to learn initially a little bit more mathematics than that for the Hilbert space formulation of quantum mechanics.IntroductionThe Algebraic Structure of the Space of States
The Topological Structure of the Space of States
Conjugate Space of Φ
Generalized Eigenvectors and Nuclear Spectral TheoremAppendiciesReferences and Footnotes