A mathematical interpretation of Dirac's formalism for quantum mechanics

Manuel Gadella, F. Gómez-Cubillo

Jean-Pierre Antoine

Dirac's bra and ket formalism is investigated and incorporated into a complete mathematical theory. First the axiomatic foundations of quantum mechanics and von Neumann's spectral theory of observables are reviewed and several inadequacies are pointed out. These defects then are remedied by extending the usual Hilbert space to a rigged Hilbert space as introduced by Gel'fand, i.e., a triplet Φ⊂H⊂Φ′ where H is a Hilbert space, Φ a dense subspace of H provided with a new (finer) topology, Φ′ the dual of Φ. It is shown that this mathematical structure, together with the Schwartz nuclear theorem, allows us to reproduce Dirac's formalism in a completely rigorous way, without losing its transparency; this makes the theory easier to handle. The temporal evolution of the system and the wave equation are considered. Finally the probabilistic interpretation and the physical aspects of the theory are discussed; Φ is identified with the set of all physically accessible states of the system, Φ′ with the set of all possible experiments (apparatus) to which it can be subjected; this provides a direct connection with Feynman's formulation of quantum mechanics.

Ph. Droz-Vincent

François Gieres B

By a series of simple examples, we illustrate how the lack of mathematical concern can readily lead to surprising mathematical contradictions in wave mechanics. The basic mathematical notions allowing for a precise formulation of the theory are then summarized and it is shown how they lead to an elucidation and deeper understanding of the aforementioned problems. After stressing the equivalence between wave mechanics and the other formulations of quantum mechanics, i.e. matrix mechanics and Dirac's abstract Hilbert space formulation, we devote the second part of our paper to the latter approach: we discuss the problems and shortcomings of this formalism as well as those of the bra and ket notation introduced by Dirac in this context. In conclusion, we indicate how all of these problems can be solved or at least avoided. Dedicated to the memory of Tanguy Altherr (1963-94). I dedicate these notes to the memory of Tanguy Altherr who left us very unexpectedly in the mountains which he loved so much (and where I had some pleasant trips with him). As to the subject of these notes (which I had the pleasure to discuss with him), Tanguy greatly appreciated Dirac's formalism and certainly knew how to apply it to real physical problems (such as the ones he worked on with an impressive enthusiasm, energy and productivity). Even if he did not share my preoccupations in this field, he liked to discuss the problems related to the general formalism of quantum physics.

Roderick I. Sutherland

A weakness which has previously seemed unavoidable in particle interpretations of quantum mechanics (such as in the de Broglie-Bohm model) is addressed here and a resolution proposed. The weakness in question is the lack of action and reaction occurring between the model's field (or "pilot wave") and the particle. Although the field acts on the particle, the particle does not act back on the field. It is shown here that this rather artificial feature is, in fact, not necessary and can be fully eliminated while remaining consistent with the usual quantum predictions. Mathematically this amounts to demonstrating that there exists a suitable Lagrangian density function which generates equations coinciding with quantum mechanics yet incorporates the desired action and reaction. As a by-product, an appealing explanation emerges to another long-standing question, namely why the mathematical formalism of quantum mechanics seems only to be describing fields when measurements generally detect localised particles. A further bonus is that the hitherto unrelated concept of a gauge transformation is found to arise naturally as an essential part of the formalism. In particular, the phase S of the gauge transformation is seen to be the action function describing the hidden motion of the particle.

Manuel Gadella, F. Gómez-Cubillo