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Birkhäuser Basel eBooks

Commuting Nonselfadjoint Operators and a Unified Theory of Waves and Corpuscles

M. S. Livšic

Published Jan 1, 1997• 3 citations
Publisher

Abstract

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.

DOI
10.1007/978-3-0348-8910-0_12

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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.
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.
We consider a system of three one-dimensional particles with one of pair potentials V α ( x α ) decaying at infinity as ∣xα∣−ρ,0<ρ<1/2 |{{x}^{\alpha }}{{|}^{{ - \rho }}},0 < \rho < 1/2 ∣xα∣−ρ,0<ρ<1/2 . It is shown that such a system can possess channels of scattering not included in the usual list of channels called the asymptotic completeness. Similar channels arise in a two-body system with a long-range potential if a traditional condition on its derivatives is relaxed. 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.
It is well known that the analysis of a relativistic n -body problem invariant under the transformations of the Poincaré group and involving one time has only been done for n =1. For n >1 one uses the second quantization formalism of field theory. In this paper we state it in the ordinary space time coordinates associated with the n-bodies as Dirac did it in the one body case. We apply the formalism first to the two body problem if the interaction is of an harmonic oscillator form and then extend it to n particles. 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.