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References

Published online by Cambridge University Press:  15 November 2025

Felix Finster ,
Sebastian Kindermann  and
Jan‐Hendrik Treude
Felix Finster
Affiliation:
Universität Regensburg, Germany
Sebastian Kindermann
Affiliation:
Comenius Gymnasium Deggendorf, Germany
Jan‐Hendrik Treude
Affiliation:
Universität Konstanz, Germany
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Causal Fermion Systems
An Introduction to Fundamental Structures, Methods and Applications
 , pp. 391 - 397
Publisher: Cambridge University Press
Print publication year: 2025
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Creative Common License - CCCreative Common License - BYCreative Common License - NC
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References

[1] Amann, H., and Escher, J. 2008. Analysis II. Basel: Birkhäuser Verlag.Google Scholar
[2] Arminjon, M., and Reifler, F. 2010. Basic quantum mechanics for three Dirac equations in a curved spacetime. Braz. J. Phys., 40(6), 242–255. arXiv:0807.0570 [gr-qc].CrossRefGoogle Scholar
[3] Arminjon, M., and Reifler, F. 2011. A non-uniqueness problem of the Dirac theory in a curved spacetime. Ann. Phys., 523(7), 1521–3889. arXiv:0905.3686 [gr-qc].CrossRefGoogle Scholar
[4] Bär, C., and Fredenhagen, K. (eds.) 2009. Quantum Field Theory on Curved Spacetimes. Lecture Notes in Physics, vol. 786. Berlin: Springer Verlag.CrossRefGoogle Scholar
[5] Bär, C., Gauduchon, P., and Moroianu, A. 2005. Generalized cylinders in semi-Riemannian and spin geometry. Math. Z., 249(3), 545–580. arXiv:math/0303095 [math.DG].CrossRefGoogle Scholar
[6] Bär, C., Ginoux, N., and Pfäffle, F. 2007. Wave Equations on Lorentzian Manifolds and Quantization. ESI Lectures in Mathematics and Physics. Zürich: European Mathematical Society (EMS). arXiv:0806.1036 [math.DG].CrossRefGoogle Scholar
[7] Barut, A. O. 1980. Electrodynamics and Classical Theory of Fields & Particles. Corrected reprint of the 1964 original. New York: Dover Publications, Inc.Google Scholar
[8] Bauer, H. 2001. Measure and Integration Theory. De Gruyter Studies in Mathematics, vol. 26. Translated from the German by Robert B. Burckel. Berlin: Walter de Gruyter & Co.CrossRefGoogle Scholar
[9] Baum, H. 1985. Spinor structures and Dirac operators on pseudo-Riemannian manifolds. Bull. Polish Acad. Sci. Math., 33(3–4), 165–171.Google Scholar
[10] Bäuml, L., Finster, F., von der Mosel, H., and Schiefeneder, D. 2019. Singular support of minimizers of the causal variational principle on the sphere. Calc. Var. Partial Differ. Equ., 58(6), 205. arXiv:1808.09754 [math.CA].CrossRefGoogle Scholar
[11] Beem, J. K., Ehrlich, P. E., and Easley, K. L. 1996. Global Lorentzian Geometry. 2nd ed. Monographs and Textbooks in Pure and Applied Mathematics, vol. 202. New York: Marcel Dekker, Inc.Google Scholar
[12] Bernal, A. N., and Sánchez, M. 2003. On smooth Cauchy hypersurfaces and Geroch’s splitting theorem. Commun. Math. Phys., 243(3), 461–470. arXiv:gr-qc/0306108.CrossRefGoogle Scholar
[13] Bernard, Y., and Finster, F. 2014. On the structure of minimizers of causal variational principles in the non-compact and equivariant settings. Adv. Calc. Var., 7(1), 27–57. arXiv:1205.0403 [math-ph].CrossRefGoogle Scholar
[14] Bjorken, J. D., and Drell, S. D. 1964. Relativistic Quantum Mechanics. New York: McGraw-Hill Book Co.Google Scholar
[15] Bogachev, V. I. 2007. Measure Theory, vol. I. Berlin: Springer-Verlag.CrossRefGoogle Scholar
[16] Bognár, J. 1974. Indefinite Inner Product Spaces. Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 78. New York: Springer-Verlag.CrossRefGoogle Scholar
[17] Bombelli, L., Lee, J., Meyer, D., and Sorkin, R. D. 1987. Space-time as a causal set. Phys. Rev. Lett., 59(5), 521–524.CrossRefGoogle ScholarPubMed
[18] Bröcker, T. 1992. Analysis II. Mannheim: Bibliographisches Institut.Google Scholar
[19] Brunetti, R., Dappiaggi, C., Fredenhagen, K., and Yngvason, J. (eds.) 2015. Advances in Algebraic Quantum Field Theory. Mathematical Physics Studies. Berlin: Springer-Verlag.CrossRefGoogle Scholar
[20] Chandrasekhar, S. 1998. The Mathematical Theory of Black Holes. Oxford Classic Texts in the Physical Sciences. New York: Oxford University Press.CrossRefGoogle Scholar
[21] Curiel, E., Finster, F., and Isidro, J. M. 2020. Two-dimensional area and matter flux in the theory of causal fermion systems. Int. J. Mod. Phys. D, 29, 2050098. arXiv:1910.06161 [math-ph].CrossRefGoogle Scholar
[22] Dappiaggi, C., and Finster, F. 2020. Linearized fields for causal variational principles: Existence theory and causal structure. Methods Appl. Anal., 27(1), 1–56. arXiv:1811.10587 [math-ph].CrossRefGoogle Scholar
[23] Dappiaggi, C., Finster, F., and Oppio, M. 2022. Linear bosonic quantum field theories arising from causal variational principles. Lett. Math. Phys., 112, 38. arXiv:2112.10656 [math-ph].CrossRefGoogle Scholar
[24] Dappiaggi, C., Finster, F., Kamran, N., and Reintjes, M. 2025. Holographic mixing and Fock space dynamics of causal fermion systems. Ann. Henri Poincaré, arXiv:2410.18045 [math-ph].CrossRefGoogle Scholar
[25] Deckert, D.-A., and Merkl, F. 2014. Dirac equation with external potential and initial data on Cauchy Surfaces. J. Math. Phys., 55(12), 122305. arXiv:1404.1401 [math-ph].CrossRefGoogle Scholar
[26] Dieudonné, J. 1969. Foundations of Modern Analysis. Enlarged and corrected printing, Pure and Applied Mathematics, vol. 10-I. New York-London: Academic Press.Google Scholar
[27] Dimock, J. 1982. Dirac quantum fields on a manifold. Trans. Am. Math. Soc., 269(1), 133–147.CrossRefGoogle Scholar
[28] Dirac, P. A. M. 1930. A theory of electrons and protons. Proc. R. Soc. Lond. A, 126, 360–365.Google Scholar
[29] do Carmo, M. P. 1976. Differential Geometry of Curves and Surfaces. Englewood Cliffs, NJ: Prentice-Hall, Inc.Google Scholar
[30] Doković, D. Z., and Thang, N. Q. 1994. Conjugacy classes of maximal tori in simple real algebraic groups and applications. Canad. J. Math., 46(4), 699–717. Correction: Canad. J. Math., 46 (1994), no. 6, 1208–1210.Google Scholar
[31] Elstrodt, J. 2005. Maß- und Integrationstheorie. 4th ed. Grundwissen Mathematik. Berlin: Springer-Verlag.Google Scholar
[32] Evans, L. C. 2010. Partial Differential Equations. 2nd ed. Graduate Studies in Mathematics, vol. 19. Providence, RI: American Mathematical Society.Google Scholar
[33] Evans, L. C., and Gariepy, R. F. 1992. Measure Theory and Fine Properties of Functions. Studies in Advanced Mathematics. Boca Raton, FL: CRC Press.Google Scholar
[34] Fefferman, C. L. 2005. A sharp form of Whitney’s extension theorem. Ann. Math. (2), 161(1), 509–577.CrossRefGoogle Scholar
[35] Finster, F. 1996. Derivation of field equations from the principle of the fermionic projector. (unpublished preprint in German). arXiv:gr-qc/9606040.Google Scholar
[36] Finster, F. 1997. Light-cone expansion of the Dirac sea with light cone integrals. unpublished preprint. arXiv:funct-an/9707003.Google Scholar
[37] Finster, F. 1998. Definition of the Dirac sea in the presence of external fields. Adv. Theor. Math. Phys., 2(5), 963–985. arXiv:hep-th/9705006.CrossRefGoogle Scholar
[38] Finster, F. 1998. Local U(2, 2) symmetry in relativistic quantum mechanics. J. Math. Phys., 39(12), 6276–6290. arXiv:hep-th/9703083.CrossRefGoogle Scholar
[39] Finster, F. 1999. Light-cone expansion of the Dirac sea to first order in the external potential. Michigan Math. J., 46(2), 377–408. arXiv:hep-th/9707128.CrossRefGoogle Scholar
[40] Finster, F. 2000. Light-cone expansion of the Dirac sea in the presence of chiral and scalar potentials. J. Math. Phys., 41(10), 6689–6746. arXiv:hep-th/9809019.CrossRefGoogle Scholar
[41] Finster, F. 2006. The Principle of the Fermionic Projector. AMS/IP Studies in Advanced Mathematics, vol. 35. Providence, RI: American Mathematical Society. hep-th/0001048, hep-th/0202059, hep-th/0210121.Google Scholar
[42] Finster, F. 2007. A variational principle in discrete space-time: Existence of minimizers. Calc. Var. Partial. Differ. Equ., 29(4), 431–453. arXiv:math-ph/0503069.CrossRefGoogle Scholar
[43] Finster, F. 2010. Causal variational principles on measure spaces. J. Reine Angew. Math., 646, 141–194. arXiv:0811.2666 [math-ph].Google Scholar
[44] Finster, F. 2014. Perturbative quantum field theory in the framework of the fermionic projector. J. Math. Phys., 55(4), 042301. arXiv:1310.4121 [math-ph].CrossRefGoogle Scholar
[45] Finster, F. 2016. The Continuum Limit of Causal Fermion Systems. Fundamental Theories of Physics, vol. 186. Cham: Springer. arXiv:1605.04742 [math-ph].Google Scholar
[46] Finster, F. 2017. The chiral index of the fermionic signature operator. Math. Res. Lett., 24(1), 37–66. arXiv:1404.6625 [math-ph].CrossRefGoogle Scholar
[47] Finster, F. 2018. Causal fermion systems: A primer for Lorentzian geometers. J. Phys.: Conf. Ser., 968, 012004. arXiv:1709.04781 [math-ph].Google Scholar
[48] Finster, F. 2019. Causal fermion systems: Discrete space-times, causation and finite propagation speed. J. Phys.: Conf. Ser., 1275, 012009. arXiv:1812.00238 [math-ph].Google Scholar
[49] Finster, F. 2019. Positive functionals induced by minimizers of causal variational principles. Vietnam J. Math., 47, 23–37. arXiv:1708.07817 [math-ph].CrossRefGoogle Scholar
[50] Finster, F. 2020. The causal action in Minkowski space and surface layer integrals. SIGMA Symmetry Integrability Geom. Methods Appl., 16(091), 83. arXiv:1711.07058 [math-ph].Google Scholar
[51] Finster, F. 2020. Perturbation theory for critical points of causal variational principles. Adv. Theor. Math. Phys., 24(3), 563–619. arXiv:1703.05059 [math-ph].CrossRefGoogle Scholar
[52] Finster, F. 2023. Solving the linearized field equations of the causal action principle in Minkowski space. Adv. Theor. Math. Phys., 27(7), 2087–2217. arXiv:2304.00965 [math-ph].CrossRefGoogle Scholar
[53] Finster, F., and Fischer, P. 2025. A canonical construction of the extended Hilbert space for causal fermion systems. arXiv:2504.18276 [math-ph (2025)].Google Scholar
[54] Finster, F., and Grotz, A. 2010. The causal perturbation expansion revisited: Rescaling the interacting Dirac sea. J. Math. Phys., 51(7), 072301. arXiv:0901.0334 [math-ph].CrossRefGoogle Scholar
[55] Finster, F., and Grotz, A. 2012. A Lorentzian quantum geometry. Adv. Theor. Math. Phys., 16(4), 1197–1290. arXiv:1107.2026 [math-ph].CrossRefGoogle Scholar
[56] Finster, F., and Kamran, N. 2019. Spinors on singular spaces and the topology of causal fermion systems. Mem. Amer. Math. Soc., 259(1251), v+83. arXiv:1403.7885 [math-ph].Google Scholar
[57] Finster, F., and Kamran, N. 2021. Complex structures on jet spaces and bosonic Fock space dynamics for causal variational principles. Pure Appl. Math. Q., 17(1), 55–140. arXiv:1808.03177 [math-ph].CrossRefGoogle Scholar
[58] Finster, F., and Kamran, N. 2022. Fermionic Fock spaces and quantum states for causal fermion systems. Ann. Henri Poincaré, 23(4), 1359–1398. arXiv:2101.10793 [math-ph].CrossRefGoogle Scholar
[59] Finster, F., and Kamran, N. 2023. A positive quasilocal mass for causal variational principles. Calc. Var., 64(3), 91. arXiv:2310.07544 [math-ph].Google Scholar
[60] Finster, F., and Kindermann, S. 2020. A gauge fixing procedure for causal fermion systems. J. Math. Phys., 61(8), 082301. arXiv:1908.08445 [math-ph].CrossRefGoogle Scholar
[61] Finster, F., and Kleiner, J. 2016. Noether-like theorems for causal variational principles. Calc. Var. Partial Differ. Equ., 55:35(2), 41. arXiv:1506.09076 [math-ph].Google Scholar
[62] Finster, F., and Kleiner, J. 2017. A Hamiltonian formulation of causal variational principles. Calc. Var. Partial Differ. Equ., 56:73(3), 33. arXiv:1612.07192 [math-ph].Google Scholar
[63] Finster, F., and Kleiner, J. 2019. A class of conserved surface layer integrals for causal variational principles. Calc. Var. Partial Differ. Equ., 58:38(1), 34. arXiv:1801.08715 [math-ph].Google Scholar
[64] Finster, F., and Kraus, M. 2020. The regularized Hadamard expansion. J. Math. Anal. Appl., 491(2), 124340. arXiv:1708.04447 [math-ph].CrossRefGoogle Scholar
[65] Finster, F., and Kraus, M. 2023. Construction of global solutions to the linearized field equations for causal variational principles. Methods Appl. Anal., 30(2), 77–94. arXiv:2210.16665 [math-ph].CrossRefGoogle Scholar
[66] Finster, F., and Langer, C. 2022. Causal variational principles in the σ-locally compact setting: Existence of minimizers. Adv. Calc. Var., 15(3), 551–575. arXiv:2002.04412 [math-ph].CrossRefGoogle Scholar
[67] Finster, F., and Lottner, M. 2021. Banach manifold structure and infinite-dimensional analysis for causal fermion systems. Ann. Global Anal. Geom., 60(2), 313–354. arXiv:2101.11908 [math-ph].CrossRefGoogle Scholar
[68] Finster, F., and Lottner, M. 2022. Elliptic methods for solving the linearized field equations of causal variational principles. Calc. Var. Partial Differ. Equ., 61(4), 133. arXiv:2111.08261 [math-ph].CrossRefGoogle Scholar
[69] Finster, F., and Müller, O. 2016. Lorentzian spectral geometry for globally hyperbolic surfaces. Adv. Theor. Math. Phys., 20(4), 751–820. arXiv:1411.3578 [math-ph].CrossRefGoogle Scholar
[70] Finster, F., and Reintjes, M. 2015. A non-perturbative construction of the fermionic projector on globally hyperbolic manifolds I – Space-times of finite lifetime. Adv. Theor. Math. Phys., 19(4), 761–803. arXiv:1301.5420 [math-ph].CrossRefGoogle Scholar
[71] Finster, F., and Reintjes, M. 2016. A non-perturbative construction of the fermionic projector on globally hyperbolic manifolds II – Space-times of infinite lifetime. Adv. Theor. Math. Phys., 20(5), 1007–1048. arXiv:1312.7209 [math-ph].CrossRefGoogle Scholar
[72] Finster, F., and Röken, C. 2016. Self-adjointness of the Dirac Hamiltonian for a class of non-uniformly elliptic boundary value problems. Ann. Math. Sci. Appl., 1(2), 301–320. arXiv:1512.00761 [math-ph].CrossRefGoogle Scholar
[73] Finster, F., and Röken, C. 2019. The fermionic signature operator in the exterior Schwarzschild geometry. Ann. Henri Poincaré, 20(10), 3389–3418. arXiv:1812.02010 [math-ph].CrossRefGoogle Scholar
[74] Finster, F., and Schiefeneder, D. 2013. On the support of minimizers of causal variational principles. Arch. Ration. Mech. Anal., 210(2), 321–364. arXiv:1012.1589 [math-ph].CrossRefGoogle Scholar
[75] Finster, F., and Tolksdorf, J. 2014. Perturbative description of the fermionic projector: Normalization, causality and Furry’s theorem. J. Math. Phys., 55(5), 052301. arXiv:1401.4353 [math-ph].CrossRefGoogle Scholar
[76] Finster, F., Smoller, J., and Yau, S.-T. 2000. Non-existence of time-periodic solutions of the Dirac equation in a Reissner-Nordström black hole background. J. Math. Phys., 41(4), 2173–2194. arXiv:gr-qc/9805050.CrossRefGoogle Scholar
[77] Finster, F., Kamran, N., Smoller, J., and Yau, S.-T. 2000. Nonexistence of time-periodic solutions of the Dirac equation in an axisymmetric black hole geometry. Comm. Pure Appl. Math., 53(7), 902–929. arXiv:gr-qc/9905047.3.0.CO;2-4>CrossRefGoogle Scholar
[78] Finster, F., Kamran, N., Smoller, J., and Yau, S.-T. 2002. Decay rates and probability estimates for massive Dirac particles in the Kerr-Newman black hole geometry. Commun. Math. Phys., 230(2), 201–244. arXiv:gr-qc/0107094.CrossRefGoogle Scholar
[79] Finster, F., Kamran, N., Smoller, J., and Yau, S.-T. 2003. The long-time dynamics of Dirac particles in the Kerr-Newman black hole geometry. Adv. Theor. Math. Phys., 7(1), 25–52. arXiv:gr-qc/0005088.CrossRefGoogle Scholar
[80] Finster, F., Grotz, A., and Schiefeneder, D. 2012. Causal fermion systems: A quantum space-time emerging from an action principle. Pages 157–182 of: Finster, F., Müller, O., Nardmann, M., Tolksdorf, J., and Zeidler, E. (eds), Quantum Field Theory and Gravity. Basel: Birkhäuser Verlag. arXiv:1102.2585 [math-ph].CrossRefGoogle Scholar
[81] Finster, F., Murro, S., and Röken, C. 2016. The fermionic projector in a time-dependent external potential: Mass oscillation property and Hadamard states. J. Math. Phys., 57(7), 072303. arXiv:1501.05522 [math-ph].CrossRefGoogle Scholar
[82] Finster, F., Kamran, N., and Oppio, M. 2021. The linear dynamics of wave functions in causal fermion systems. J Differ. Equ., 293, 115–187. arXiv:2101.08673 [math-ph].CrossRefGoogle Scholar
[83] Finster, F., Jonsson, R. H., and Kilbertus, N. 2025. Numerical analysis of the causal action principle in low dimensions. Comm. Math. Sci. arXiv:2201.06382 [math-ph].CrossRefGoogle Scholar
[84] Finster, F., Kamran, N., and Reintjes, M. 2023. Entangled quantum states of causal fermion systems and unitary group integrals. Adv. Theor. Math. Phys., 27(5), 1463–1589. arXiv:2207.13157 [math-ph].CrossRefGoogle Scholar
[85] Finster, F., Frankl, M., and Langer, C. 2024. The homogeneous causal action principle on a compact domain in momentum space. Adv. Calc. Var., 17(3), 559–585. arXiv:2205.04085 [math-ph].CrossRefGoogle Scholar
[86] Finster, F., Guendelman, E., and Paganini, C. 2024. Modified measures as an effective theory for causal fermion systems. Class. Quant. Gravity, 41(3), 035007, 25. arXiv:2303.16566 [gr-qc].CrossRefGoogle Scholar
[87] Folland, G. B. 1999. Real Analysis: Modern Techniques and Their Applications. 2nd edn. Pure and Applied Mathematics (New York). New York: John Wiley & Sons, Inc.Google Scholar
[88] Friedlander, F. G. 1975. The Wave Equation on a Curved Space-Time. Cambridge: Cambridge University Press.Google Scholar
[89] Friedlander, F. G. 1998. Introduction to the Theory of Distributions. 2nd edn. Cambridge: Cambridge University Press, With additional material by M. Joshi.Google Scholar
[90] Friedrich, T. 2000. Dirac Operators in Riemannian Geometry. Graduate Studies in Mathematics, vol. 25. Providence, RI: American Mathematical Society.Google Scholar
[91] Friedrichs, K. O. 1954. Symmetric hyperbolic linear differential equations. Comm. Pure Appl. Math., 7, 345–392.CrossRefGoogle Scholar
[92] Friedrichs, K. O. 1958. Symmetric positive linear differential equations. Comm. Pure Appl. Math., 11, 333–418.CrossRefGoogle Scholar
[93] Glimm, J., and Jaffe, A. 1987. Quantum Physics: A Functional Integral Point of View. 2nd edn. New York: Springer-Verlag.CrossRefGoogle Scholar
[94] Gohberg, I., Lancaster, P., and Rodman, L. 2005. Indefinite Linear Algebra and Applications. Basel: Birkhäuser Verlag.Google Scholar
[95] Goldstein, H. 1980. Classical Mechanics. 2nd edn. Reading, MA: Addison-Wesley Publishing Co.Google Scholar
[96] Gradshteyn, I. S., and Ryzhik, I. M. 1965. Table of Integrals, Series, and Products. Fourth edition prepared by Ju. V. Geronimus and M. Ju. Tseytlin. New York: Academic Press.Google Scholar
[97] Greene, R. E., and Shiohama, K. 1979. Diffeomorphisms and volume-preserving embeddings of noncompact manifolds. Trans. Amer. Math. Soc., 255, 403–414.CrossRefGoogle Scholar
[98] Hack, T.-P. 2010. On the Backreaction of Scalar and Spinor Quantum Fields in Curved Spacetimes – From the Basic Foundations to Cosmological Applications. Dissertation Universität Hamburg, viii+254. arXiv:1008.1776 [gr-qc].Google Scholar
[99] Hadamard, J. 1953. Lectures on Cauchy’s Problem in Linear Partial Differential Equations. New York: Dover Publications.CrossRefGoogle Scholar
[100] Halmos, P. R. 1958. Finite-Dimensional Vector Spaces. The University Series in Undergraduate Mathematics. 2nd ed. Princeton-Toronto-New York-London: D. Van Nostrand Co., Inc.Google Scholar
[101] Halmos, P. R. 1974. Measure Theory. New York: Springer.Google Scholar
[102] Hawking, S. W., and Ellis, G. F. R. 1973. The Large Scale Structure of Space-Time. London: Cambridge University Press.CrossRefGoogle Scholar
[103] Hawking, S. W., King, A. R., and McCarthy, P. J. 1976. A new topology for curved space-time which incorporates the causal, differential, and conformal structures. J. Math. Phys., 17(2), 174–181.CrossRefGoogle Scholar
[104] Helgason, S. 2000. Groups and Geometric Analysis. Mathematical Surveys and Monographs, vol. 83. Providence, RI: American Mathematical Society. Integral geometry, invariant differential operators, and spherical functions, Corrected reprint of the 1984 original.Google Scholar
[105] Hörmander, L. 1990. The Analysis of Linear Partial Differential Operators. I. 2nd edn. Grundlehren der Mathematischen Wissenschaften, vol. 256. Berlin: Springer-Verlag.Google Scholar
[106] Jänich, K. 1984. Topology. Undergraduate Texts in Mathematics. New York: Springer-Verlag, With a chapter by Theodor Bröcker, Translated from the German by Levy, Silvio.CrossRefGoogle Scholar
[107] John, F. 1991. Partial Differential Equations. 4th edn. Applied Mathematical Sciences, vol. 1. New York: Springer-Verlag.Google Scholar
[108] Kato, T. 1995. Perturbation Theory for Linear Operators. Classics in Mathematics. Berlin: Springer-Verlag.Google Scholar
[109] Kilbertus, N. 2015. Numerical analysis of causal fermion systems on ℝ × S3. Masterarbeit Mathematik, Universität Regensburg.Google Scholar
[110] Kleinert, H. 2006. Path Integrals in Quantum Mechanics, Statistics, Polymer Physics, and Financial Markets. 4th edn. Hackensack, NJ: World Scientific Publishing Co. Pte. Ltd.CrossRefGoogle Scholar
[111] Landau, L. D., and Lifshitz, E. M. 1958. Quantum Mechanics: Non-Relativistic Theory. Course of Theoretical Physics, vol. 3. Translated from the Russian by J. B. Sykes and J. S. Bell, Addison-Wesley Series in Advanced Physics. London-Paris: Pergamon Press Ltd.Google Scholar
[112] Landau, L. D., and Lifshitz, E. M. 1962. The Classical Theory of Fields. Revised 2nd edition. Course of Theoretical Physics, vol. 2. Translated from the Russian by M. Hamermesh. Oxford: Pergamon Press.Google Scholar
[113] Lang, S. 1999. Fundamentals of Differential Geometry. Graduate Texts in Mathematics, vol. 191. New York: Springer-Verlag.Google Scholar
[114] Finster, F. and Gmeineder, F. 2025. Action-driven flows for causal variational principles. arXiv:2503.00526 [math-ph].Google Scholar
[115] Lawson, Jr., H. B., and Michelsohn, M.-L. 1989. Spin Geometry. Princeton Mathematical Series, vol. 38. Princeton, NJ: Princeton University Press.Google Scholar
[116] Lax, P. D. 2002. Functional Analysis. Pure and Applied Mathematics (New York). New York: Wiley-Interscience [John Wiley & Sons].Google Scholar
[117] Lee, J. M. 2009. Manifolds and Differential Geometry. Graduate Studies in Mathematics, vol. 107. Providence, RI: American Mathematical Society.Google Scholar
[118] Lee, J. M. 2013. Introduction to Smooth Manifolds. 2nd edn. Graduate Texts in Mathematics, vol. 218. New York: Springer.Google Scholar
[119] Malament, D. B. 1977. The class of continuous timelike curves determines the topology of spacetime. J. Math. Phys., 18(7), 1399–1404.CrossRefGoogle Scholar
[120] Milnor, J. W., and Stasheff, J. D. 1974. Characteristic Classes. Princeton, NJ: Princeton University Press. Annals of Mathematics Studies, No. 76.CrossRefGoogle Scholar
[121] Naber, G. L. 2012. The Geometry of Minkowski Spacetime. Applied Mathematical Sciences, vol. 92. New York: Springer, An introduction to the mathematics of the special theory of relativity, 2nd edition [of MR1174969].Google Scholar
[122] Noether, E. 1918. Invariante Variationsprobleme. Nachr. d. König. Gesellsch. d. Wiss. Math-phys. Klasse, Berlin, 1918, 235–257.Google Scholar
[123] Nomizu, K., and Ozeki, H. 1961. The existence of complete Riemannian metrics. Proc. Amer. Math. Soc., 12, 889–891.CrossRefGoogle Scholar
[124] Olver, F. W. J., Lozier, D. W., Boisvert, R. F., and Clark, C. W. (eds). 2010. Digital Library of Mathematical Functions. Washington, DC: National Institute of Standards and Technology from http://dlmf.nist.gov/ (release date December 15, 2019).Google Scholar
[125] O’Neill, B. 1983. Semi-Riemannian Geometry. Pure and Applied Mathematics, vol. 103. New York: Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers].Google Scholar
[126] Palais, R. S. 1979. The principle of symmetric criticality. Commun. Math. Phys., 69(1), 19–30.CrossRefGoogle Scholar
[127] Penrose, R., and Rindler, W. 1987. Spinors and Space-Time, vol. 1. Cambridge Monographs on Mathematical Physics. Cambridge: Cambridge University Press.Google Scholar
[128] Peskin, M. E., and Schroeder, D. V. 1995. An Introduction to Quantum Field Theory. Reading, MA: Addison-Wesley Publishing Company Advanced Book Program.Google Scholar
[129] Radzikowski, M. J. 1996. Micro-local approach to the Hadamard condition in quantum field theory on curved space-time. Commun. Math. Phys., 179(3), 529–553.CrossRefGoogle Scholar
[130] Rauch, J. 1991. Partial Differential Equations. Graduate Texts in Mathematics, vol. 128. New York: Springer-Verlag.CrossRefGoogle Scholar
[131] Reed, M., and Simon, B. 1980. Methods of Modern Mathematical Physics. I, Functional Analysis. 2nd edn. New York: Academic Press Inc.Google Scholar
[132] Rejzner, K. 2016. Perturbative Algebraic Quantum Field Theory. Mathematical Physics Studies Springer.CrossRefGoogle Scholar
[133] Rendall, A. D. 2008. Partial Differential Equations in General Relativity. Oxford Graduate Texts in Mathematics, vol. 16. Oxford: Oxford University Press.Google Scholar
[134] Rindler, W. 1991. Introduction to Special Relativity. 2nd edn. Oxford Science Publications. New York: The Clarendon Press, Oxford University Press.Google Scholar
[135] Ringström, H. 2009. The Cauchy Problem in General Relativity. ESI Lectures in Mathematics and Physics. Zürich: European Mathematical Society (EMS).Google Scholar
[136] Rudin, W. 1987. Real and Complex Analysis. 3rd edn. New York: McGraw-Hill Book Co.Google Scholar
[137] Rudin, W. 1991. Functional Analysis. 2nd edn. International Series in Pure and Applied Mathematics. New York: McGraw-Hill Inc.Google Scholar
[138] Sahlmann, H., and Verch, R. 2001. Microlocal spectrum condition and Hadamard form for vector-valued quantum fields in curved spacetime. Rev. Math. Phys., 13(10), 1203–1246. arXiv:math-ph/0008029.CrossRefGoogle Scholar
[139] Sakurai, J. J., and Napolitano, J. 1994. Advanced Quantum Mechanics. 2nd edn. San Francisco: Addison-Wesley Publishing Company.Google Scholar
[140] Schubert, H. 1968. Topology. Boston, Mass: Allyn and Bacon, Inc., Translated from the German by Siegfried Moran.Google Scholar
[141] Schwabl, F. 2002. Quantum Mechanics. 3rd edn. Berlin: Springer-Verlag.Google Scholar
[142] Strang, G. 1980. Linear Algebra and Its Applications. 2nd edn. New York-London: Academic Press.Google Scholar
[143] Taylor, M. E. 1996. Partial Differential Equations. I. Applied Mathematical Sciences, vol. 115. New York: Springer-Verlag.Google Scholar
[144] Taylor, M. E. 1997. Partial Differential Equations. III. Applied Mathematical Sciences, vol. 117. New York: Springer-Verlag.Google Scholar
[145] Taylor, M. E. 2011. Partial Differential Equations. II. 2nd edn. Applied Mathematical Sciences, vol. 116. New York: Springer.CrossRefGoogle Scholar
[146] Thaller, B. 1992. The Dirac Equation. Texts and Monographs in Physics. Berlin: Springer-Verlag.CrossRefGoogle Scholar
[147] Weinberg, S. 1996. The Quantum Theory of Fields, vol. I. Cambridge: Cambridge University Press. Foundations, Corrected reprint of the 1995 original.CrossRefGoogle Scholar
[148] Finster, F., Murro, S., and Schmid, G. 2025. The Cauchy problem for symmetric hyperbolic systems with nonlocal potentials. arXiv:2507.05004 [math.AP].Google Scholar

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