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Compact quantum systems: Internal geometry of relativistic systems. (English) Zbl 0606.22018
A generalization is presented of the kinematical algebra \({\mathfrak so}(5)\), shown previously to be relevant for the description of the internal dynamics (Zitterbewegung) of Dirac’s electron. The algebra \({\mathfrak so}(n+2)\) is proposed for the case of a compact quantum system with n degrees of freedom. Associated wave equations follow from boosting these compact quantum systems. There exists a contraction to the kinematical algebra of a system with n degrees of freedom of the usual type, by which the commutation relations between n coordinate operators \(Q_ i\) and corresponding momentum operators \(p_ i\), occurring within the \({\mathfrak so}(n+2)\) algebra, go over into the usual canonical commutation relations.
The \({\mathfrak so}(n+2)\) algebra is contrasted with the \({\mathfrak sl}(1,n)\) superalgebra introduced recently by Palev in a similar context: because \({\mathfrak so}(n+2)\) has spinor representations, its use allows the possibility of interpreting the half-integral spin in terms of the angular momentum of internal finite quantum systems. Connection is made with the ideas of Weyl on the possible use in quantum mechanics of ray representation of finite abelian groups, and so also with other recent works on finite quantum systems. Possible directions of future research are indicated.

MSC:
22E70 Applications of Lie groups to the sciences; explicit representations
22E60 Lie algebras of Lie groups
17A70 Superalgebras
81T60 Supersymmetric field theories in quantum mechanics
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