Foundations of Physics

, Volume 42, Issue 1, pp 192–208

Clifford Algebras and the Dirac-Bohm Quantum Hamilton-Jacobi Equation


DOI: 10.1007/s10701-011-9558-z

Cite this article as:
Hiley, B.J. & Callaghan, R.E. Found Phys (2012) 42: 192. doi:10.1007/s10701-011-9558-z


In this paper we show how the dynamics of the Schrödinger, Pauli and Dirac particles can be described in a hierarchy of Clifford algebras, \({\mathcal{C}}_{1,3}, {\mathcal{C}}_{3,0}\), and \({\mathcal{C}}_{0,1}\). Information normally carried by the wave function is encoded in elements of a minimal left ideal, so that all the physical information appears within the algebra itself. The state of the quantum process can be completely characterised by algebraic invariants of the first and second kind. The latter enables us to show that the Bohm energy and momentum emerge from the energy-momentum tensor of standard quantum field theory. Our approach provides a new mathematical setting for quantum mechanics that enables us to obtain a complete relativistic version of the Bohm model for the Dirac particle, deriving expressions for the Bohm energy-momentum, the quantum potential and the relativistic time evolution of its spin for the first time.


Clifford algebras Schrödinger, Pauli and relativistic Dirac-Bohm model Relativistic quantum potential Spin evolution 


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Copyright information

© Springer Science+Business Media, LLC 2011

Authors and Affiliations

  1. 1.TPRU, BirkbeckUniversity of LondonLondonUK

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