nLab Dirac equation

Idea

The differential equation encoded by a Dirac operator.

The equations of motion of the Dirac field.

Related concepts

• Dirac spinor, Dirac conjugate

• Dirac field

• heat equation

• wave equation, Klein-Gordon equation

• Schrödinger equation

References

Monographs on the relativistic Dirac equation in 3+1d Minkowski spacetime:

• James D. Bjorken, Sidney D. Drell: Relativistic Quantum Mechanics, McGrawHill (1964) [ark:/13960/t5fc2v05h, pdf, pdf]

• Bernd Thaller, The Dirac Equation, Texts and Monographs in Physics, Springer (1992) (doi:10.1007/978-3-662-02753-0)

See also:

• Wikipedia, Dirac equation

Discussion for curved spacetime:

• M. D. Pollock, On the Dirac equation in curved space-time, Acta Physica Polonica B 41 (2010) [InSpire:874211, pdf]

• Peter Collas, David Klein: The Dirac equation in general relativity, a guide for calculations, Springer Briefs in Physics, Springer (2019) [arXiv:1809.02764, doi:10.1007/978-3-030-14825-6]

• Miguel Alcubierre: The Dirac equation in General Relativity and the 3+1 formalism [https://arxiv.org/abs/2503.03918]

Specifically in Schwarzschild spacetime background:

• Paul M. Alsing, Bound states of the Dirac equation in Schwarzschild spacetime: an exploration of intuition for the curious student [arXiv:2207.00905]

The path integral approach to the Dirac equation:

• Takashi Ichinose, Hiroshi Tamura, Path Integral Approach to Relativistic Quantum Mechanics: Two-Dimensional Dirac Equation, Progress of Theoretical Physics Supplement, Volume 92, April 1987, Pages 144–175, doi.

• Pierre Gosselin, Janos Polonyi, Path Integral for Relativistic Equations of Motion, arXiv:hep-th/9708121.

• Janos Polonyi, Path Integral for the Dirac Equation, arXiv:hep-th/9809115.

• Wataru Ichinose, On the Feynman Path Integral for the Dirac Equation in the General Dimensional Spacetime, Communications in Mathematical Physics 329, 83–508 (2014), doi.

• Wataru Ichinose, Notes on the Feynman path integral for the Dirac equation, Journal of Pseudo-Differential Operators and Applications 9, 789–809 (2018), doi.

On quantum simulation of the Dirac equation:

• L. Lamata, J. León, T. Schätz, Enrique Solano: Dirac Equation and Quantum Relativistic Effects in a Single Trapped Ion, Phys. Rev. Lett. 98 253005 (2007) [doi:10.1103/PhysRevLett.98.253005]

• R. Gerritsma, G. Kirchmair, F. Zähringer, Enrique Solano, R. Blatt & C. F. Roos: Quantum simulation of the Dirac equation, Nature 463 68–71 (2010) [doi:10.1038/nature08688]