Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 16 (2020), 091, 83 pages      arXiv:1711.07058      https://doi.org/10.3842/SIGMA.2020.091

The Causal Action in Minkowski Space and Surface Layer Integrals

Felix Finster
Fakultät für Mathematik, Universität Regensburg, D-93040 Regensburg, Germany

Received September 19, 2019, in final form September 11, 2020; Published online September 27, 2020

Abstract
The Lagrangian of the causal action principle is computed in Minkowski space for Dirac wave functions interacting with classical electromagnetism and linearized gravity in the limiting case when the ultraviolet cutoff is removed. Various surface layer integrals are computed in this limiting case.

Key words: causal action; surface layer integral; special relativity; Dirac field; Maxwell field.

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References

  1. Bäuml L., Finster F., Schiefeneder D., von der Mosel H., Singular support of minimizers of the causal variational principle on the sphere, Calc. Var. Partial Differential Equations 58 (2019), 205, 27 pages, arXiv:1808.09754.
  2. Bernard Y., Finster F., On the structure of minimizers of causal variational principles in the non-compact and equivariant settings, Adv. Calc. Var. 7 (2014), 27-57, arXiv:1205.0403.
  3. Bogachev V.I., Measure theory, Vol. I, Springer-Verlag, Berlin, 2007.
  4. Curiel E., Finster F., Isidro J.M., Two-dimensional area and matter flux in the theory of causal fermion systems, Internat. J. Modern Phys. D 29 (2020), 2050098, 23 pages, arXiv:1910.06161.
  5. Deligne P., Freed D.S., Classical field theory, in Quantum Fields and Strings: a Course for Mathematicians, Vols. 1, 2 (Princeton, NJ, 1996/1997), Amer. Math. Soc., Providence, RI, 1999, 137-225.
  6. Finster F., Light-cone expansion of the Dirac sea with light cone integrals, arXiv:funct-an/9707003.
  7. Finster F., Light-cone expansion of the Dirac sea in the presence of chiral and scalar potentials, J. Math. Phys. 41 (2000), 6689-6746, arXiv:hep-th/9809019.
  8. Finster F., The principle of the fermionic projector, AMS/IP Studies in Advanced Mathematics, Vol. 35, Amer. Math. Soc., Providence, RI, 2006, arXiv:hep-th/0001048.
  9. Finster F., A variational principle in discrete space-time: existence of minimizers, Calc. Var. Partial Differential Equations 29 (2007), 431-453, arXiv:math-ph/0503069.
  10. Finster F., On the regularized fermionic projector of the vacuum, J. Math. Phys. 49 (2008), 032304, 60 pages, arXiv:math-ph/0612003.
  11. Finster F., Causal variational principles on measure spaces, J. Reine Angew. Math. 646 (2010), 141-194, arXiv:0811.2666.
  12. Finster F., Perturbative quantum field theory in the framework of the fermionic projector, J. Math. Phys. 55 (2014), 042301, 53 pages, arXiv:1310.4121.
  13. Finster F., The continuum limit of causal fermion systems. From Planck scale structures to macroscopic physics, Fundamental Theories of Physics, Vol. 186, Springer, Cham, 2016, arXiv:1605.04742.
  14. Finster F., Causal Fermion systems: a primer for Lorentzian geometers, J. Phys. Conf. Ser. 968 (2018), 012004, 14 pages, arXiv:1709.04781.
  15. Finster F., Positive functionals induced by minimizers of causal variational principles, Vietnam J. Math. 47 (2019), 23-37, arXiv:1708.07817.
  16. Finster F., Perturbation theory for critical points of causal variational principles, Adv. Theor. Math. Phys. 24 (2020), 563-619, arXiv:1703.05059.
  17. Finster F., Hoch S., An action principle for the masses of Dirac particles, Adv. Theor. Math. Phys. 13 (2009), 1653-1711, arXiv:0712.0678.
  18. Finster F., Jokel M., Causal fermion systems: an elementary introduction to physical ideas and mathematical concepts, in Progress and Visions in Quantum Theory in View of Gravity, Editors F. Finster, D. Giulini, J. Kleiner, J. Tolksdorf, Birkhäuser, Cham, 2020, 63-92, arXiv:1908.08451.
  19. Finster F., Kamran N., Complex structures on jet spaces and bosonic Fock space dynamics for causal variational principles, Pure Appl. Math. Q., to appear, arXiv:1808.03177.
  20. Finster F., Kamran N., Fermionic Fock space dynamics and quantum states for causal fermion systems, in preparation.
  21. Finster F., Kleiner J., Causal fermion systems as a candidate for a unified physical theory, J. Phys. Conf. Ser. 626 (2015), 012020, 17 pages, arXiv:1502.03587.
  22. Finster F., Kleiner J., Noether-like theorems for causal variational principles, Calc. Var. Partial Differential Equations 55 (2016), 35, 41 pages, arXiv:1506.09076.
  23. Finster F., Kleiner J., A Hamiltonian formulation of causal variational principles, Calc. Var. Partial Differential Equations 56 (2017), 73, 33 pages, arXiv:1612.07192.
  24. Finster F., Kleiner J., A class of conserved surface layer integrals for causal variational principles, Calc. Var. Partial Differential Equations 58 (2019), 38, 34 pages, arXiv:1801.08715.
  25. Finster F., Kleiner J., Treude J.H., An introduction to the fermionic projector and causal fermion systems, in preparation, available at https://causal-fermion-system.com/intro-public.pdf.
  26. Finster F., Schiefeneder D., On the support of minimizers of causal variational principles, Arch. Ration. Mech. Anal. 210 (2013), 321-364, arXiv:1012.1589.
  27. Finster F., Tolksdorf J., Perturbative description of the fermionic projector: normalization, causality, and Furry's theorem, J. Math. Phys. 55 (2014), 052301, 31 pages, arXiv:1012.1589.
  28. John F., Partial differential equations, 4th ed., Applied Mathematical Sciences, Vol. 1, Springer-Verlag, New York, 1991.
  29. The theory of causal fermion systems, see https://www.causal-fermion-system.com.
  30. Weinberg S., The quantum theory of fields, Vol. I. Foundations, Cambridge University Press, Cambridge, 1996.

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