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Revista Brasileira de Ensino de Física

Schwinger's oscillator method, supersymmetric quantum mechanics and massless particles

F. M. Mejía, V. Pleitez

Published Mar 1, 2002• 2 citations
Publisher

Abstract

We consider Schwinger's method of angular momentum addition using the SU(2) algebra with both a fermionic and a bosonic oscillator. We show that the total spin states obtained are: one boson singlet state and an arbitrary number of spin-1/2 states, the later ones are energy degenerate. It means that we have in this case supersymmetric quantum mechanics and also the addition of angular momentum for massless particles. We review too the cases of two bosonic and two fermionic oscillators.

DOI
10.1590/s1806-11172002000100007

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We consider Schwinger's method of angular momentum addition using the SU(2) algebra with both a fermionic and a bosonic oscillator. We show that the total spin states obtained are: one boson singlet state and an arbitrary number of spin-1/2 states, the later ones are energy degenerate. It means that we have in this case supersymmetric quantum mechanics and also the addition of angular momentum for massless particles. We review too the cases of two bosonic and two fermionic oscillators.
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Two complementary approaches of N = 2 fractional supersymmetric quantum mechanics of order k are studied in this article. The first one, based on a generalized Weyl-Heisenberg algebra W(k) (that comprizes the affine quantum algebra Uq(sl(2)) with q to k = 1 as a special case), apparently contains solely one bosonic degree of freedom. The second one uses generalized bosonic and k-fermionic degrees of freedom. As an illustration, a particular emphasis is put on the fractional supersymmetric oscillator of order k.
The Schwinger's representation of angular momentum(AM) relates two important fundamental models, that of AM and that of harmonic oscillator(HO). However, the representation offers only the relations of operators but not states. Here, by developing a graphic method to calculate the states, we show that the Schwinger's model is valid only with certain spin. With the relation of states, we also demonstrate how the superposition states in AM map to entangled states in HO. This study has promising applications in quantum computation, and may cast light on the relation between superposition and entanglement.
Various definitions of the coherent states of the angular momentum are shown to be special cases of the oscillator-like coherent states of the groups SU(2) and SO(3) obtained by Mikhailov on the basis of a generalized Schwinger boson realization of the angular momentum algebra. This realization is then generalized to that of the angular momentum algebra of an asymmetric top by means of a transformation from the Euler angles to the Cayley–Klein parameters. The oscillator-like coherent states of an asymmetric top, analogous to those of Mikhailov, are then constructed. It is, then, shown that Janssen's and Mostowski's definitions of the coherent states of a top are special cases of these.
In this paper we study the number of states with given spin for identical particles, denoted as DIn(l){D}_{I}^{n}(l)DIn​(l) for bosons and as DIn(j){\mathcal{D}}_{I}^{n}(j)DIn​(j) for fermions, where III is the total angular momentum of the system, nnn is the particle number, lll is the spin of a single boson, and jjj is the spin of a single fermion. We derive relations of DIn(l){D}_{I}^{n}(l)DIn​(l) and DIn(j){\mathcal{D}}_{I}^{n}(j)DIn​(j); in particular, we obtain recursive and explicit formulas of DIn(l){D}_{I}^{n}(l)DIn​(l) and DIn(j){\mathcal{D}}_{I}^{n}(j)DIn​(j) in terms of those for $n\ensuremath{-}1$ particles and apply them to systems of two, three, and five identical particles.
This paper deals with the features of quantum computing for a system containing a relatively large number of qubits n, which is characterized by a half-integer effective spin S in the Schwinger representation of paired bosons. It was shown that for n = 70, the number of boson states corresponding to lowest 2S + 1 excited energy levels of each of the two harmonic oscillators implementing the two-boson representation of the effective spin S is N = 1070 = 1.18 x 1021. Under such conditions, it can be assumed with great precision that for n ≥ 70 all boson states of both harmonic oscillators participate in the implementation of the two-boson representation of the effective spin S. This allows one to apply the quantum field theory methods when performing quantum computations for multi-qubit systems. Another advantage of the Schwinger representation of paired bosons compared to the spinor representation when performing quantum computations for multi-qubit systems is that in the first case the form of the spin projection operators of the effective spin S does not depend on the number of qubits n.