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Canadian Journal of Physics

Etude de la transcription de la quasi-particule dans l'espace idéal

M. Banville, P. A. Simard

Published Apr 1, 1970• 4 citations
Publisher

Abstract

The transcription of the quasi-particle in the ideal space has been studied in such a way that all terms in the Hamiltonian in the QTD approximation for the system of four quasi-particles can be found. It is found that an infinity of solutions exists which verify the commutator {α, α+} while yielding correct Hamiltonian matrix elements. Finally, particular solutions which do not verify this commutator are found. Their particular invariance properties under canonical transformations make them relatively easy to obtain. Nonphysical states are eliminated by these transcriptions and all the physical states appear properly antisymmetrized.

DOI
10.1139/p70-104

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The transcription of the quasi-particle in the ideal space has been studied in such a way that all terms in the Hamiltonian in the QTD approximation for the system of four quasi-particles can be found. It is found that an infinity of solutions exists which verify the commutator {α, α+} while yielding correct Hamiltonian matrix elements. Finally, particular solutions which do not verify this commutator are found. Their particular invariance properties under canonical transformations make them relatively easy to obtain. Nonphysical states are eliminated by these transcriptions and all the physical states appear properly antisymmetrized.
An antisymmetric basis has been set up in the ideal space as a combination of one boson and one ideal quasi-particle state; in this way a three quasi-particle system may be studied. The knowledge of solutions of linearized equations for two quasi-particles is a requisite for the setting up of this basis. The exclusion principle is thus introduced correctly. The spurious states coming from the non-conservation of the number of particles are separated out and can be rejected easily. The method can be used only in QTD approximation. The one-shell case is discussed as a verification of the method.
The quasi-chemical equilibrium, or pair correlation, approximation to statistical mechanics is written in terms of second-quantization formalism. This involves an entension of the original theory to include an Ansatz for off-diagonal elements of the density matrix. The formal expressions involve “labelling operators” acting in a purely formal Hilbert space. These labelling operators obey Bose commutation rules for correlations between even numbers particles (e.g. pair correlations), Fermi commutation rules for correlations between odd numbers of particles. The labelling operators provide an algebraic way of formulating the restriction to certain types of “graphs”. We also show how to include higher-order correlations into the Ansatz. The main results of the original theory can be derived more rapidly with the new formalism. However, certain correction terms obtained earlier are shown to be in error, necessitating a reinvestigation of the nature of the condensation phenomenon. This re-investigation is carried out in two following papers, and proves to lead to no essential modifications of the previous results. The correlation matrices which enter into the Ansatz need to be related to the Hamiltonian of the system, and to the thermodynamic variables which define its thermodynamic state. A variational formulation is developed for this purpose, but no explicit calculations are carried out in this paper.
Cet article présente une étude algébrique du Hamiltonien non hermitien de Swanson fondée sur le formalisme de la quantification canonique. En suivant la formulation algébrique proposée par da Providência et al. , une paire généralisée d’opérateurs d’annihilation et de création satisfaisant la relation de commutation bosonique est construite afin de diagonaliser le Hamiltonien et de générer la représentation correspondante dans l’espace de Fock. Le spectre exact d’énergie et les fonctions d’onde associées sont obtenus analytiquement. En outre, les générateurs de l’algèbre de Lie sont construits à partir des opérateurs d’échelle généralisés et utilisés pour dériver les états cohérents du système. Enfin, la valeur moyenne du Hamiltonien est évaluée. La formulation proposée fournit un cadre algébrique cohérent pour l’étude de cette classe de systèmes quantiques non hermitiens.
There are two stochastic descriptions of relativistic quantum spinless particles. In the first one, suggested by Feynman's path integral approach to Klein-Gordon propagator, the main ingredients are diffusions in Minkowski space parametrized by some kind of proper time. The second description, on the contrary, is based upon jumb Markov processes giving the space position of the particle in the sense of Newton and Wigner with the time as parameter. In this paper we bridge the gap between these different probabilistic scenarios by constructing the space jump processes by means of space-time diffusions. We give some aplications to the nonrelativistic limit of the hamiltonian semigroup t→exp−(t/¬h)H where H=√−c 2 ¬h 2 Δ+(M 2 c 4 /¬h 2 )−Mc 2 +V(.) Il y a deux descriptions stochastiques de la mecanique quantique relativiste pour une particule sans spin. Dans la premiere, suggere par la theorie de Feynman, les ingredients principaux sont des diffusions dans l'espace de Minkowski parametrise par une sorte de temps propre. La deuxieme description, par contre, emploie des processus de Markov a sauts donnant la position de la particule dans l'espace au sens de Newton et Wigner avec le temps comme parametre. Notre papier va relier les deux prescriptions probabilistes car nous construisons les processus a sauts dans l'espace par les diffusions dans l'espace-temps. Nous donnons aussi des applications a l'etude de la limite non relativiste du semi-groupe hamiltonien t→exp−(t/¬h)H ou H=√−c 2 ¬h 2 Δ+(M 2 c 4 /¬h 2 )−Mc 2 +V(.)