Noether-like theorems for causal variational principles

  • Felix FinsterEmail author
  • Johannes Kleiner


The connection between symmetries and conservation laws as made by Noether’s theorem is extended to the context of causal variational principles and causal fermion systems. Different notions of continuous symmetries are introduced. It is proven that these symmetries give rise to corresponding conserved quantities, expressed in terms of so-called surface layer integrals. In a suitable limiting case, the Noether-like theorems for causal fermion systems reproduce charge conservation and the conservation of energy and momentum in Minkowski space. Thus the conservation of charge and energy-momentum are found to be special cases of general conservation laws which are intrinsic to causal fermion systems.

Mathematics Subject Classification

49Q20 49S05 58C35 58Z05 83C40 49K21 49K27 



J. K. gratefully acknowledges support by the “Studienstiftung des deutschen Volkes”.


  1. 1.
    Barut, A.O.: Electrodynamics and classical theory of fields & particles, Dover Publications, Inc., New York (1980, Corrected reprint of the 1964 original)Google Scholar
  2. 2.
    Bassi, A., Lochan, K., Satin, S., Singh, T.P., Ulbricht, H.: Models of wave-function collapse, underlying theories, and experimental tests. Rev. Mod. Phys. 85, 471–527 (2013)CrossRefGoogle Scholar
  3. 3.
    Bernard, Y., Finster, F.: On the structure of minimizers of causal variational principles in the non-compact and equivariant settings. Adv. Calc. Var. 7(1), 27–57 (2014). arXiv:1205.0403 [math-ph]
  4. 4.
    Davies, B.: Integral transforms and their applications, Texts in applied mathematics, vol. 41, 3rd edn. Springer-Verlag, New York (2002)Google Scholar
  5. 5.
    Finster, F.: The continuum limit of causal fermion systems. Book based on the preprints. arXiv:0908.1542 [math-ph], arXiv:1211.3351 [math-ph] and arXiv:1409.2568 [math-ph] (2016, in preparation)
  6. 6.
    Finster, F.: The Principle of the Fermionic Projector, hep-th/0001048, hep-th/0202059, hep-th/0210121, AMS/IP Studies in Advanced Mathematics, vol. 35. American Mathematical Society, Providence, RI (2006)
  7. 7.
    Finster, F.: On the regularized fermionic projector of the vacuum. J. Math. Phys. 49(3), 032304 (2008). arXiv:math-ph/0612003
  8. 8.
    Finster, F.: Causal variational principles on measure spaces. J. Reine Angew. Math. 646, 141–194 (2010). arXiv:0811.2666 [math-ph]
  9. 9.
    Finster, F.: Entanglement and second quantization in the framework of the fermionic projector. J. Phys. A Math. Theor. 43, 395302 (2010). arXiv:0911.0076 [math-ph]
  10. 10.
    Finster, F.: The fermionic projector, entanglement, and the collapse of the wave function. J. Phys. Conf. Ser. 306, 012024 (2011). arXiv:1011.2162 [quant-ph]
  11. 11.
    Finster, F.: Perturbative quantum field theory in the framework of the fermionic projector. J. Math. Phys. 55(4), 042301 (2014). arXiv:1310.4121 [math-ph]
  12. 12.
    Finster, F.: Causal fermion systems—an overview. In:  Finster, F., Kleiner, J., Röken, C., Tolksdorf J. (eds.) Quantum mathematical physics: a bridge between mathematics and physics. Birkhäuser Verlag, Basel (2016). arXiv:1505.05075 [math-ph]
  13. 13.
    Finster, F., Grotz, A.: A Lorentzian quantum geometry. Adv. Theor. Math. Phys. 16(4), 1197–1290 (2012). arXiv:1107.2026 [math-ph]
  14. 14.
    Finster, F., Hoch, S.: An action principle for the masses of Dirac particles. Adv. Theor. Math. Phys. 13(6), 1653–1711 (2009). arXiv:0712.0678 [math-ph]
  15. 15.
    Finster, F., Kleiner, J.: The jet bundle dynamics of causal fermion systems (2016, in preparation)Google Scholar
  16. 16.
    Finster, F., Kleiner, J.: Causal fermion systems as a candidate for a unified physical theory. J. Phys.: Conf. Ser. 626, 012020 (2015). arXiv:1502.03587 [math-ph]
  17. 17.
    Finster, F., Schiefeneder, D.: On the support of minimizers of causal variational principles. Arch. Ration. Mech. Anal. 210(2), 321–364 (2013). arXiv:1012.1589 [math-ph]
  18. 18.
    Finster, F., Tolksdorf, J.: A microscopic derivation of quantum electrodynamics (2016, in preparation)Google Scholar
  19. 19.
    Finster, F., Tolksdorf, J.: Perturbative description of the fermionic projector: normalization, causality and Furry’s theorem. J. Math. Phys. 55(5), 052301 (2014). arXiv:1401.4353 [math-ph]
  20. 20.
    Forger, M., Römer, H.: Currents and the energy-momentum tensor in classical field theory: a fresh look at an old problem. Ann. Phys. 309(2), 306–389 (2004). arXiv:hep-th/0307199
  21. 21.
    Ghirardi, G.C., Pearle, P., Rimini, A.: Stochastic processes in Hilbert space: a consistent formulation of quantum mechanics. In: Foundations of quantum mechanics in the light of new technology (Tokyo, 1989) Phys. Soc. Japan, Tokyo, pp. 181–189 (1990)Google Scholar
  22. 22.
    Goldstein, H.: Classical mechanics, 2nd edn. Addison-Wesley Publishing Co., Reading (1980)Google Scholar
  23. 23.
    Halmos, P.R.: Measure theory. Springer, New York (1974)zbMATHGoogle Scholar
  24. 24.
    Hawking, S.W., Ellis, G.F.R.: The large scale structure of space-time. Cambridge University Press, London (1973)CrossRefzbMATHGoogle Scholar
  25. 25.
    Landau, L.D., Lifshitz, E.M.: The classical theory of fields, Revised second edition. Course of theoretical physics, vol. 2. Translated from the Russian by Morton Hamermesh, Pergamon Press, Oxford (1962)Google Scholar
  26. 26.
    Noether, E.: Invariante Variationsprobleme, Nachr. d. König. Gesellsch. d. Wiss. Math-phys. Klasse, Berlin, pp. 235–257 (1918)Google Scholar
  27. 27.
    Pearle, P.: Combining stochastic dynamical state-vector reduction with spontaneous localization. Phys. Rev. A 39(5), 2277–2289 (1989)CrossRefGoogle Scholar
  28. 28.
    Straumann, N.: General relativity, Texts and monographs in physics. Springer-Verlag, Berlin (2004)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  1. 1.Fakultät für MathematikUniversität RegensburgRegensburgGermany

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