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Birkhäuser Boston eBooks

Quantum Theory

José M. Gracia-Bondı́a, Joseph C. Várilly, Héctor Figueroa

Published Jan 1, 2001• 0 citations
Publisher

Abstract

In Part II, we expounded Shale-Stinespring theory and exhibited the abstract charged field as the simplest example of a summable Fredholm module, and thus a source of noncommutative geometry. Having come so far, it would be a pity not to develop fermion quantum dynamics in external fields, which comes straight from the spin representation. As on many occasions in this book, we set to work here on an enterprise of translation; in this case, to render in algebraic terms the quantization of wave equations of the Dirac type. Most of the footwork has already been laid in Section 6.4. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

DOI
10.1007/978-1-4612-0005-5_13

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In Part II, we expounded Shale-Stinespring theory and exhibited the abstract charged field as the simplest example of a summable Fredholm module, and thus a source of noncommutative geometry. Having come so far, it would be a pity not to develop fermion quantum dynamics in external fields, which comes straight from the spin representation. As on many occasions in this book, we set to work here on an enterprise of translation; in this case, to render in algebraic terms the quantization of wave equations of the Dirac type. Most of the footwork has already been laid in Section 6.4. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
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In this paper, which is closely connected with our previous study ”What is a particle from the standpoint of systems theory?” [3], we consider the case of pairs of commuting operators with one-dimensional imaginary parts and we develop a unified theory of waves and particles for the case of Dirac equations. Particles emerge in Space as localized fields with singularities at their centers and wave equations appear as compatibility conditions for an overdetermined pair of equations of corresponding open systems. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
Up to this point, our study of quantum mechanics has concerned itself with the behaviour of particles that inhabit a Galilean spacetime. For many purposes, in atomic, molecular and condensed matter physics, this theory is quite adequate. We saw in earlier chapters, however, that our actual spacetime has a structure which is much closer to that of the Minkowski spacetime of special relativity and that more general structures must be considered when gravitational phenomena are signicant. From a purely theoretical point of view, it is therefore important to formulate quantum theory in a way which is consistent with these more general spacetimes. The benets of constructing a relativistic quantum theory actually go far beyond the aesthetic satisfaction of making our geometrical and quantum-mechanical reasoning compatible. For one thing, we shall discover that the relativistic theory provides a deeper understanding of spin and the distinction between fermions and bosons, which in the non-relativistic theory appear simply as facts of life that we must strive to accommodate. Also, of course, there are many situations in which relativistic eects become observable, for which non-relativistic theory provides no explanation. The most obvious are high-energy scattering experiments, in which particles acquire kinetic energies comparable with or greater than their rest energies mc2, and the correct 4-momentum (3.33) must be used. There are, however, more subtle eects, such as the spin-orbit coupling that is essential for interpreting atomic spectra, which are also of relativistic origin.