Notes on the Feynman path integral for the Dirac equation

  • Wataru IchinoseEmail author


This paper is a continuation of the author’s preceding one. In the preceding paper the author has rigorously constructed the Feynman path integral for the Dirac equation in the form of the sum-over-histories, satisfying the superposition principle, over all paths of one electron in space-time that goes in any direction at any speed, forward and backward in time with a finite number of turns. In the present paper, first we will generalize the results in the preceding paper and secondly prove in a direct way that our Feynman path integral satisfies the unitarity principle and the causality one.


The Feynman path integral Dirac equation Unitarity Causality 

Mathematics Subject Classification

81Q30 35Q40 



This work is partially supported by JSPS KAKENHI Grant Number 2640016.


  1. 1.
    Dirac, P.A.M.: The Principles of Quantum Mechanics, 4th edn. Oxford University Press, Oxford (1958)zbMATHGoogle Scholar
  2. 2.
    Dyson, F.: Comment on the topic “Beyond the black hole”. In: Some Strangeness in the Proportion: A Centennial Symposium to Celebrate the Achievements of Albert Einstein, pp. 376–380. Addison-Wesely, Reading (1980)Google Scholar
  3. 3.
    Feynman, R.P.: Theory of positrons. Phys. Rev. 76, 749–759 (1949)CrossRefGoogle Scholar
  4. 4.
    Feynman, R.P., Hibbs, A.R.: Quantum Mechanics and Path Integrals. McGraw-Hill, New York (1965)zbMATHGoogle Scholar
  5. 5.
    Fujiwara, D., Kumano-go, N.: Phase space Feynman path integrals via piecewise bicharacteristic paths and their semiclassical approximations. Bull. Sci. Math. 132, 313–357 (2008)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Fujiwara, D.: An integration by parts formula for Feynman path integrals. J. Math. Soc. Jpn. 65, 1273–1318 (2013)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Ichinose, W.: A note on the existence and \(\hbar \)-dependency of the solution of equations in quantum mechanics. Osaka J. Math. 32, 327–345 (1995)MathSciNetzbMATHGoogle Scholar
  8. 8.
    Ichinose, W.: On the formulation of the Feynman path integral through broken line paths. Commun. Math. Phys. 189, 17–33 (1997)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Ichinose, W.: On convergence of the Feynman path integral formulated through broken line paths. Rev. Math. Phys. 11, 1001–1025 (1999)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Ichinose, W.: On the Feynman path integral for the Dirac equation in the general dimensional spacetime. Commun. Math. Phys. 329, 483–508 (2014)MathSciNetCrossRefGoogle Scholar
  11. 11.
    John, F.: Partial Differential Equations. 4th edn. Springer, New York (1982)Google Scholar
  12. 12.
    Kumano-go, H.: Pseudo-Differential Operators. MIT Press, Cambridge (1981)zbMATHGoogle Scholar
  13. 13.
    Mizohata, S.: The Theory of Partial Differential Equations. Cambridge University Press, New York (1973)zbMATHGoogle Scholar
  14. 14.
    Nicola, F.: Convergence in \(L^p\) for Feynman path integrals. Adv. Math. 294, 384–409 (2016)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Reed, M., Simon, B.: Methods of Modern Mathematical Physics I: Functional Analysis. Academic Press, San Diego (1980)zbMATHGoogle Scholar
  16. 16.
    Schweber, S.S.: QED and the Men Who Made It: Dyson, Feynman, Schwinger and Tomonaga. Princeton University Press, Princeton (1994)zbMATHGoogle Scholar
  17. 17.
    Taylor, M.E.: Pseudodifferential Operators. Princeton University Press, Princeton (1981)zbMATHGoogle Scholar
  18. 18.
    Yajima, K.: Schrödinger evolution equations with magnetic fields. J. Anal. Math. 56, 29–76 (1991)CrossRefGoogle Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.Department of MathematicsShinshu UniversityMatsumotoJapan

Personalised recommendations