Representation Space

  • Jȩdrzej Śniatycki
Part of the Applied Mathematical Sciences book series (AMS, volume 30)


In quantum mechanics one can represent the Hilbert space of states as the space of square integrable complex functions on the spectrum of any given complete set of commuting observables. In the process of quantization, however, one has only the classical phase space (X, ω) to work with, and one has to find a suitable classical counterpart of the notion of a complete set of commuting observables. A natural choice is a set of n = 1/2dim X functions f1,…, fn on X, independent at all points of X, satisfying
$$ [{{f}_{i}},{{f}_{j}}] = 0,\quad i,j \in \{ 1,2,...,n\} $$
such that their Hamiltonian vector fields \( {\xi_{{f_1}}},\,...,{\xi_{{f_n}}} \) are complete. However, for many phase spaces of interest there does not exist such a set. If one drops the assumption that the fi be real and globally defined, one is led to the notion of a “polarization” of (X, ω). Note that the Hamiltonian vector fields \( {\xi_{{f_i}}} \), i ∈ {1,…, n} span over C an involutive distribution F on X such that
$$ {\dim _{\text{C}}}{\text{F }} ={\text{}}\frac{1}{2}\dim {\mkern 1mu} {\text{X,}} $$
and also that ω restricted to F vanishes identically,
$$ \omega \left| {FxF = 0.} \right. $$


Representation Space Holonomy Group Integral Manifold Frame Field Hamiltonian Vector Field 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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Copyright information

© Springer-Verlag New York Inc. 1980

Authors and Affiliations

  • Jȩdrzej Śniatycki
    • 1
  1. 1.The University of CalgaryCalgaryCanada

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