Abstract
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.
Similar content being viewed by others
References
Barut, A.O.: Electrodynamics and classical theory of fields & particles, Dover Publications, Inc., New York (1980, Corrected reprint of the 1964 original)
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)
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]
Davies, B.: Integral transforms and their applications, Texts in applied mathematics, vol. 41, 3rd edn. Springer-Verlag, New York (2002)
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)
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)
Finster, F.: On the regularized fermionic projector of the vacuum. J. Math. Phys. 49(3), 032304 (2008). arXiv:math-ph/0612003
Finster, F.: Causal variational principles on measure spaces. J. Reine Angew. Math. 646, 141–194 (2010). arXiv:0811.2666 [math-ph]
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]
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]
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]
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]
Finster, F., Grotz, A.: A Lorentzian quantum geometry. Adv. Theor. Math. Phys. 16(4), 1197–1290 (2012). arXiv:1107.2026 [math-ph]
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]
Finster, F., Kleiner, J.: The jet bundle dynamics of causal fermion systems (2016, in preparation)
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]
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]
Finster, F., Tolksdorf, J.: A microscopic derivation of quantum electrodynamics (2016, in preparation)
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]
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
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)
Goldstein, H.: Classical mechanics, 2nd edn. Addison-Wesley Publishing Co., Reading (1980)
Halmos, P.R.: Measure theory. Springer, New York (1974)
Hawking, S.W., Ellis, G.F.R.: The large scale structure of space-time. Cambridge University Press, London (1973)
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)
Noether, E.: Invariante Variationsprobleme, Nachr. d. König. Gesellsch. d. Wiss. Math-phys. Klasse, Berlin, pp. 235–257 (1918)
Pearle, P.: Combining stochastic dynamical state-vector reduction with spontaneous localization. Phys. Rev. A 39(5), 2277–2289 (1989)
Straumann, N.: General relativity, Texts and monographs in physics. Springer-Verlag, Berlin (2004)
Acknowledgments
J. K. gratefully acknowledges support by the “Studienstiftung des deutschen Volkes”.
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by J. Jost.
Rights and permissions
About this article
Cite this article
Finster, F., Kleiner, J. Noether-like theorems for causal variational principles. Calc. Var. 55, 35 (2016). https://doi.org/10.1007/s00526-016-0966-y
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s00526-016-0966-y