The Geometry of Ensembles on Configuration Space

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


A description of ensembles on configuration space incorporates at least two geometrical structures which arise in a natural way: a metric structure, which derives from the natural geometry associated with a space of probabilities, and a symplectic structure, which derives from the symplectic geometry associated with a Hamiltonian description of motion. We show that these two geometrical structures give rise to a Kähler geometry. We first consider probabilities P and introduce the information metric. This leads to information geometry, a Riemannian geometry defined on the space of probabilities. We then bring in dynamics via a Hamiltonian formalism defined on a phase space with canonically conjugate coordinates P and S. This leads to more geometrical structure, a symplectic geometry defined on this phase space. The next step is to extend the information metric, which is defined over the space of probabilities only, to a metric over the full phase space. This requires satisfying certain conditions which ensure the compatibility of the metric and symplectic structures. These conditions are equivalent to requiring that the space have a Kähler structure. In this way, we are led to a Kähler geometry. This rich geometrical structure allows for a reconstruction of the geometric formulation of quantum theory. One may associate a Hilbert space with the Kähler space and this leads to the standard version of quantum theory. Thus the theory of ensembles on configuration space permits a geometric derivation of quantum theory.


Theinformation Rich Geometric Structure Symplectic Structure Meta Information Ensemble Hamiltonian 
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© Springer International Publishing Switzerland 2016

Authors and Affiliations

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

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