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International Journal of Modern Physics A

FROM RELATIVISTIC VECTOR MESONS IN CONSTANT MAGNETIC FIELDS TO NONRELATIVISTIC (PSEUDO)SUPERSYMMETRIES

J. Beckers, N. Debergh

Published Jul 30, 1995• 13 citations
Publisher

Abstract

Results coming from the study of relativistic vector mesons interacting with a constant magnetic field are examined through Johnson-Lippmann implications on one-dimensional oscillatorlike systems. We obtain specific nonrelativistic Hamiltonians showing new properties in quantum mechanics and leading to superpositions of bosons and pseudofermions. Moreover, two “potentials” are introduced and discussed in comparison with recent developments usually obtained in p=2 parasupersymmetric quantum mechanics. Pseudofermions are also examined, particularly with respect to orthofermions.

DOI
10.1142/s0217751x95001315

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Results coming from the study of relativistic vector mesons interacting with a constant magnetic field are examined through Johnson-Lippmann implications on one-dimensional oscillatorlike systems. We obtain specific nonrelativistic Hamiltonians showing new properties in quantum mechanics and leading to superpositions of bosons and pseudofermions. Moreover, two “potentials” are introduced and discussed in comparison with recent developments usually obtained in p=2 parasupersymmetric quantum mechanics. Pseudofermions are also examined, particularly with respect to orthofermions.
Pseudosupersymmetric quantum mechanics (PsSSQM), based upon the use of pseudofermions, was introduced in the context of a new Kemmer equation describing charged vector mesons interacting with an external constant magnetic field. Here we construct the complete explicit solution for its realization in terms of two superpotentials, both equal or unequal. We prove that any orthosupersymmetric quantum mechanical system has a pseudosupersymmetry and give conditions under which a pseudosupersymmetric one may be described by orthosupersymmetries of order two. We propose two new matrix realizations of PsSSQM in terms of the generators of a generalized deformed oscillator algebra (GDOA) and relate them to the cases of equal or unequal superpotentials, respectively. We demonstrate that these matrix realizations are fully reducible and that their irreducible components provide two distinct sets of bosonized operators realizing PsSSQM and corresponding to nonlinear spectra. We relate such results to some previous ones obtained for a GDOA connected with a C3-extended oscillator algebra (where C3 = ℤ3) in the case of linear spectra.
The (group and spin space) matrix Hamiltonian describing the dynamics of a non-relativistic spin-½ particle moving in a static, but spatially dependent, non-Abelian magnetic field in two spatial dimensions is shown to take the form of an anticommutator of a nilpotent operator and its Hermitian conjugate. Consequently, the (group space) matrix Hamiltonians for the two different spin projections form partners of a supersymmetric quantum mechanical system. The resulting supersymmetry algebra is exploited to explicitly construct the exact zero energy ground state wave function(s) for the system. The remaining eigenstates and eigenvalues of the two partner Hamiltonians form positive energy degenerate pairs.
By a canonical transformation on the Dirac Hamiltonian for a free particle, a representation of the Dirac theory is obtained in which positive and negative energy states are separately represented by two-component wave functions. Playing an important role in the new representation are new operators for position and spin of the particle which are physically distinct from these operators in the conventional representation. The components of the time derivative of the new position operator all commute and have for eigenvalues all values between $\ensuremath{-}c$ and ccc. The new spin operator is a constant of the motion unlike the spin operator in the conventional representation. By a comparison of the new Hamiltonian with the non-relativistic Pauli-Hamiltonian for particles of spin \textonehalf{}, one finds that it is these new operators rather than the conventional ones which pass over into the position and spin operators in the Pauli theory in the non-relativistic limit. The transformation of the new representation is also made in the case of interaction of the particle with an external electromagnetic field. In this way the proper non-relativistic Hamiltonian (essentially the Pauli-Hamiltonian) is obtained in the non-relativistic limit. The same methods may be applied to a Dirac particle interacting with any type of external field (various meson fields, for example) and this allows one to find the proper non-relativistic Hamiltonian in each such case. Some light is cast on the question of why a Dirac electron shows some properties characteristic of a particle of finite extension by an examination of the relationship between the new and the conventional position operators.