Foundations of Physics

, Volume 23, Issue 9, pp 1239–1264 | Cite as

States and operators in the spacetime algebra

  • Chris Doran
  • Anthony Lasenby
  • Stephen Gull
Part I. Invited Papers Dedicated To David Hestenes


The spacetime algebra (STA) is the natural, representation-free language for Dirac's theory of the electron. Conventional Pauli, Dirac, Weyl, and Majorana spinors are replaced by spacetime multivectors, and the quantum σ- and γ-matrices are replaced by two-sided multivector operations. The STA is defined over the reals, and the role of the scalar unit imaginary of quantum mechanics is played by a fixed spacetime bivector. The extension to multiparticle systems involves a separate copy of the STA for each particle, and it is shown that the standard unit imaginary induces correlations between these particle spaces. In the STA, spinors and operators can be manipulated without introducing any matrix representation or coordinate system. Furthermore, the formalism provides simple expressions for the spinor bilinear covariants which dispense with the need for the Fierz identities. A reduction to2+1 dimensions is given, and applications beyond the Dirac theory are discussed.


Coordinate System Quantum Mechanic Matrix Representation Simple Expression Standard Unit 
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Copyright information

© Plenum Publishing Corporation 1993

Authors and Affiliations

  • Chris Doran
    • 1
  • Anthony Lasenby
    • 2
  • Stephen Gull
    • 2
  1. 1.DAMTPCambridgeUnited Kingdom
  2. 2.MRAO, Cavendish LaboratoryCambridgeUnited Kingdom

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