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arXiv:hep-th/9807047 · Modern Physics Letters A

DIRAC'S CONTOUR REPRESENTATION FOR PARAPARTICLES

SICONG JING, CHARLES A. NELSON

Published Nov 10, 1998• 2 citations
arXivHTMLPDF

Abstract

Dirac's contour representation is extended to para-Bose and para-Fermi systems by use of deformed algebra techniques. In this analytic representation, the action of the para-particle annihilation operator is equivalent to a deformed differentiation which encodes the statistics of the paraparticle. In the para-Fermi case, the derivative's ket-domain is degree p polynomial.

DOI
10.1142/s0217732398002953

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Dirac's contour representation is extended to para-Bose and para-Fermi systems by use of deformed algebra techniques. In this analytic representation, the action of the para-particle annihilation operator is equivalent to a deformed differentiation which encodes the statistics of the paraparticle. In the para-Fermi case, the derivative's ket-domain is degree p polynomial.
This research explores analytic and Dirac contour representation systems with finite d-dimensional Hilbert space, where the dual variables, position x, and momentum p, take values in Z(d)(the integer modulo d). It introduces the SU(2) formalism, defining angular momentum states and operators, and investigates the SU(2) formalism in the harmonic oscillator context and SU(2) coherent states and their resolution of identity. Analytical representations based on SU(2) coherent states are developed and numerically implemented in the extended complex plane. The investigation explores time evolution in periodic and non-periodic systems, examining special cases like joining two paths and the behavior of zeros of the analytic representation function approaching zero and ±∞. Furthermore, the research explores Dirac contour representation in a finite dimensional Hilbert space in the extended complex plane. It defines the scalar product of states and proves the relationship between Dirac ket and Dirac bra representations. Time evolution in periodic and non-periodic systems is investigated with numerical implementations and visualizations using MATLAB. Additionally, this study applies Dirac contour representation to finite-dimensional Hilbert spaces at positive and negative temperatures, denoted as H(2j+1)+ and H(2j+1)−, respectively. It defines operators within these spaces and introduces the scalar product of two states in each system. The investigation explores L operators capable of mapping states from H(2j+1)+ to H(2j+1)− and vice versa, examining their Dirac contour representation and obtaining intriguing numerical results visualized using MATLAB.
Eigenstates of the parabose and parafermi creation operators are constructed. In the Dirac contour representation, the parabose eigenstates correspond to the dual vectors of the parabose coherent states. In order p=2p=2p=2, conserved-charge parabose creation operator eigenstates are also constructed. The contour forms of the associated resolutions of unity are obtained.
A new formulation of the many-electron correlation problem is presented based on the use of parastatistics. It is shown that second-order para-Fermi creation and annihilation operators, which correspond to the creation and annihilation of spin-averaged paraparticles, occur naturally for the spin-independent many-electron problem. Moreover, the spin-independent many-electron Hamiltonian is directly expressible in terms of the parafield operators. The structure of the general para-Fermi algebra is also investigated from the viewpoint of the pseudo-orthogonal group O(2n+1, 1). Finally, an explicit matrix representation for the para-Fermi algebra of order 2, which enables one to handle even particle-number-nonconserving operators, is obtained in a canonical U(n)-adapted basis.