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Foundations of Physics

, Volume 43, Issue 4, pp 424–439 | Cite as

Clifford Algebras in Symplectic Geometry and Quantum Mechanics

  • Ernst Binz
  • Maurice A. de Gosson
  • Basil J. Hiley
Article

Abstract

The necessary appearance of Clifford algebras in the quantum description of fermions has prompted us to re-examine the fundamental role played by the quaternion Clifford algebra, C 0,2. This algebra is essentially the geometric algebra describing the rotational properties of space. Hidden within this algebra are symplectic structures with Heisenberg algebras at their core. This algebra also enables us to define a Poisson algebra of all homogeneous quadratic polynomials on a two-dimensional sub-space, \(\mathbb{F}^{a}\) of the Euclidean three-space. This enables us to construct a Poisson Clifford algebra, ℍ F , of a finite dimensional phase space which will carry the dynamics. The quantum dynamics appears as a realisation of ℍ F in terms of a Clifford algebra consisting of Hermitian operators.

Keywords

Quantum mechanics Clifford algebra Poisson algebra Symplectic geometry 

Notes

Acknowledgement

We are deeply indebted to the generosity of Georg Wikman in sponsoring the Askloster Seminars where different aspects of this work were presented and discussed. Without those meetings the work presented in this paper would not have been possible.

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Copyright information

© Springer Science+Business Media, LLC 2012

Authors and Affiliations

  • Ernst Binz
    • 1
  • Maurice A. de Gosson
    • 2
  • Basil J. Hiley
    • 3
  1. 1.Fakultät f. Mathematik und Informatik A5/6MannheimGermany
  2. 2.NuHAG, Fakultät für MatematikUniversität WienViennaAustria
  3. 3.TPRU, BirkbeckUniversity of LondonLondonUK

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