Local Representations of Rotations on Discrete Configuration Spaces

  • Michael J. W. HallEmail author
  • Marcel Reginatto
Part of the Fundamental Theories of Physics book series (FTPH, volume 184)


A spin-half system may be characterised as having a set of two-valued observables which generate infinitesimal rotations in three dimensions. This abstract formulation can be given a concrete realization using ensembles on configuration space. We derive very general probabilistic models for ensembles that consist of one and two spin-half systems. In the case of a single spin-half system, there are two main requirements that need to be satisfied: the configuration space must be a discrete set, labelling the outcomes of two-valued spin observables, and these observables must provide a representation of so(3). These two requirements are sufficient to lead to a model which is equivalent to the quantum theory of a single qubit. The case of a pair of spin-half systems is more complicated, in that additional physical requirements concerning locality and subsystem independence must also be taken into account, and now the observables must provide a representation of \(so(3) \oplus so(3)\). We show in this case that, in addition to a model equivalent to the quantum theory of a pair of qubits, it may also be possible to have non-quantum local models.


Poisson Bracket Configuration Space Canonical Transformation Bell Inequality Spin Measurement 
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Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  1. 1.Centre for Quantum DynamicsGriffith UniversityBrisbaneAustralia
  2. 2.Physikalisch-Technische BundesanstaltBraunschweigGermany

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