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## Causal fermion systems and the ETH approach to quantum theory

 1 Fakultät für Mathematik, Universität Regensburg, D-93040 Regensburg, Germany 2 Institute of Theoretical Physics, ETH Zurich, Switzerland 3 Fakultät für Mathematik, Universität Regensburg, D-93040 Regensburg, Germany 4 Max Planck Institute for Gravitational Physics (Albert Einstein Institute), Am Mühlenberg 1, D-14476 Potsdam, Germany

* Corresponding author: Felix Finster

Received  April 2020 Revised  August 2020 Published  November 2020

After reviewing the theory of "causal fermion systems" (CFS theory) and the "Events, Trees, and Histories Approach" to quantum theory (ETH approach), we compare some of the mathematical structures underlying these two general frameworks and discuss similarities and differences. For causal fermion systems, we introduce future algebras based on causal relations inherent to a causal fermion system. These algebras are analogous to the algebras previously introduced in the ETH approach. We then show that the spacetime points of a causal fermion system have properties similar to those of "events", as defined in the ETH approach. Our discussion is underpinned by a survey of results on causal fermion systems describing Minkowski space that show that an operator representing a spacetime point commutes with the algebra in its causal future, up to tiny corrections that depend on a regularization length.

Citation: Felix Finster, Jürg Fröhlich, Marco Oppio, Claudio F. Paganini. Causal fermion systems and the ETH approach to quantum theory. Discrete & Continuous Dynamical Systems - S, doi: 10.3934/dcdss.2020451
##### References:
 [1] Link to web platform on causal fermion systems: http://www.causal-fermion-system.com. Google Scholar [2] L. Bäuml, F. Finster, D. Schiefeneder and H. von der Mosel, Singular support of minimizers of the causal variational principle on the sphere, Calc. Var. Partial Differential Equations, 58 (2019), 205, 27 pp. doi: 10.1007/s00526-019-1652-7.  Google Scholar [3] P. Blanchard, J. Fröhlich and B. Schubnel, A garden of forking paths – the quantum mechanics of histories of events, Nuclear Phys. B, 912 (2016), 463-484.  doi: 10.1016/j.nuclphysb.2016.04.010.  Google Scholar [4] D. Buchholz and J. E. Roberts, New light on infrared problems: Sectors, statistics, symmetries and spectrum, Commun. Math. Phys., 330 (2014), 935-972.  doi: 10.1007/s00220-014-2004-2.  Google Scholar [5] L. J. Bunce and J. D. Maitland Wright, The Mackey-Gleason problem, Bull. Amer. Math. Soc., 26 (1992), 288-293.  doi: 10.1090/S0273-0979-1992-00274-4.  Google Scholar [6] E. Curiel, F. Finster and J. M. Isidro, Two-dimensional area and matter flux in the theory of causal fermion systems, preprint, arXiv: 1910.06161, to appear in Internat. J. Modern Phys. D, (2020). Google Scholar [7] C. Dappiaggi and F. Finster, Linearized fields for causal variational principles: Existence theory and causal structure, Methods Appl. Anal., 27 (2020), 1-56.  doi: 10.4310/MAA.2020.v27.n1.a1.  Google Scholar [8] S. Doplicher, K. Fredenhagen and J. E. Roberts, The quantum structure of spacetime at the Planck scale and quantum fields, Commun. Math. Phys., 172 (1995), 187-220.  doi: 10.1007/BF02104515.  Google Scholar [9] A. Dvurečenskij, Gleason's Theorem and its Applications, Mathematics and its Applications (East European Series), vol. 60, Kluwer Academic Publishers Group, Dordrecht; Ister Science Press, Bratislava, 1993. doi: 10.1007/978-94-015-8222-3.  Google Scholar [10] F. Finster, 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. doi: 10.1090/amsip/035.  Google Scholar [11] F. Finster, On the regularized fermionic projector of the vacuum, J. Math. Phys., 49 (2008), 032304, 60 pp. doi: 10.1063/1.2888187.  Google Scholar [12] F. Finster, Causal variational principles on measure spaces, J. Reine Angew. Math., 646 (2010), 141-194.  doi: 10.1515/CRELLE.2010.069.  Google Scholar [13] F. Finster, Perturbative quantum field theory in the framework of the fermionic projector, J. Math. Phys., 55 (2014), 042301, 53 pp. doi: 10.1063/1.4871549.  Google Scholar [14] F. Finster, Causal fermion systems – an overview, in Quantum Mathematical Physics: A Bridge between Mathematics and Physics (F. Finster, J. Kleiner, C. R ken, and J. Tolksdorf, eds.), Birkhäuser Verlag, Basel, (2016), 313–380. doi: 10.1007/978-3-319-42067-7.  Google Scholar [15] F. Finster, The Continuum Limit of Causal Fermion Systems, Fundamental Theories of Physics, vol. 186, Springer, 2016. doi: 10.1007/978-3-319-42067-7.  Google Scholar [16] F. Finster, Causal fermion systems: Discrete space-times, causation and finite propagation speed, J. Phys.: Conf. Ser., 1275 (2019), 012009. Google Scholar [17] F. Finster, Perturbation theory for critical points of causal variational principles, Adv. Theor. Math. Phys., 24 (2020), 563-619.  doi: 10.4310/ATMP.2020.v24.n3.a2.  Google Scholar [18] F. Finster and A. Grotz, A Lorentzian quantum geometry, Adv. Theor. Math. Phys., 16 (2012), 1197-1290.  doi: 10.4310/ATMP.2012.v16.n4.a3.  Google Scholar [19] F. Finster and A. Grotz, On the initial value problem for causal variational principles, J. Reine Angew. Math., 725 (2017), 115-141.  doi: 10.1515/crelle-2014-0080.  Google Scholar [20] F. Finster, A. Grotz and D. Schiefeneder, Causal fermion systems: A quantum space-time emerging from an action principle, in Quantum Field Theory and Gravity (F. Finster, O. Müller, M. Nardmann, J. Tolksdorf, and E. Zeidler, eds.), Birkhäuser Verlag, Basel, (2012), 157–182. doi: 10.1007/978-3-0348-0043-3_9.  Google Scholar [21] F. Finster and M. Jokel, Causal fermion systems: An elementary introduction to physical ideas and mathematical concepts, in Progress and Visions in Quantum Theory in View of Gravity (F. Finster, D. Giulini, J. Kleiner, and J. Tolksdorf, eds.), Birkhäuser Verlag, Basel, (2020), 63–92. doi: 10.1007/978-3-030-38941-3_2.  Google Scholar [22] F. Finster and N. Kamran, Complex structures on jet spaces and bosonic Fock space dynamics for causal variational principles, preprint, arXiv: 1808.03177, to appear in Pure Appl. Math. Q., (2020). Google Scholar [23] F. Finster and J. Kleiner, Causal fermion systems as a candidate for a unified physical theory, J. Phys.: Conf. Ser., 626 (2015), 012020. Google Scholar [24] F. Finster and J. Kleiner, A Hamiltonian formulation of causal variational principles, Calc. Var. Partial Differential Equations, 56 (2017), no. 73, 33pp. doi: 10.1007/s00526-017-1153-5.  Google Scholar [25] F. Finster and C. Langer, Causal variational principles in the $\sigma$-locally compact setting: Existence of minimizers, preprint, arXiv: 2002.04412, to appear in Adv. Calc. Var., (2020). Google Scholar [26] F. Finster and M. Oppio, Local algebras for causal fermion systems in Minkowski space, preprint, arXiv: 2004.00419, (2020). Google Scholar [27] F. Finster and D. Schiefeneder, On the support of minimizers of causal variational principles, Arch. Ration. Mech. Anal., 210 (2013), 321-364.  doi: 10.1007/s00205-013-0649-1.  Google Scholar [28] J. Fröhlich, The quest for laws and structure, Mathematics and Society, (2016), 101–129. Google Scholar [29] J. Fröhlich, A brief review of the "ETH-approach to quantum mechanics", preprint, arXiv: 1905.06603, (2019). Google Scholar [30] J. Fröhlich, Relativistic quantum theory, preprint, arXiv: 1912.00726, (2019). Google Scholar [31] J. Fröhlich, "Diminishing potentialities", entanglement, "purification" and the emergence of events in quantum mechanics – a simple model, Sect. 5.6 of Notes for a course on Quantum Theory at LMU-Munich (Nov./Dec. 2019). Google Scholar [32] J. Fröhlich and B. Schubnel, Quantum probability theory and the foundations of quantum mechanics, in The Message of Quantum Science, Springer, 899 (2015), 131–193. doi: 10.1007/978-3-662-46422-9_7.  Google Scholar [33] A. M. Gleason, Measures on the closed subspaces of a Hilbert space, J. Math. Mech., 6 (1957), 885-893.  doi: 10.1512/iumj.1957.6.56050.  Google Scholar [34] G. C. Hegerfeldt, Remark on causality and particle localization, Physical Review D, 10 (1974), no. 10, 3320. doi: 10.1103/PhysRevD.10.3320.  Google Scholar [35] J. Kleiner, Dynamics of Causal Fermion Systems – Field Equations and Correction Terms for a New Unified Physical Theory, Dissertation, Universität Regensburg, 2017. Google Scholar [36] H. Lin, Almost commuting selfadjoint matrices and applications, in Operator Algebras and their Applications (Waterloo, ON, 1994/1995), Fields Inst. Commun., vol. 13, Amer. Math. Soc., Providence, RI, (1997), 193–233.  Google Scholar [37] B. Thaller, The Dirac Equation, Texts and Monographs in Physics, Springer-Verlag, Berlin, 1992. doi: 10.1007/978-3-662-02753-0.  Google Scholar

show all references

##### References:
 [1] Link to web platform on causal fermion systems: http://www.causal-fermion-system.com. Google Scholar [2] L. Bäuml, F. Finster, D. Schiefeneder and H. von der Mosel, Singular support of minimizers of the causal variational principle on the sphere, Calc. Var. Partial Differential Equations, 58 (2019), 205, 27 pp. doi: 10.1007/s00526-019-1652-7.  Google Scholar [3] P. Blanchard, J. Fröhlich and B. Schubnel, A garden of forking paths – the quantum mechanics of histories of events, Nuclear Phys. B, 912 (2016), 463-484.  doi: 10.1016/j.nuclphysb.2016.04.010.  Google Scholar [4] D. Buchholz and J. E. Roberts, New light on infrared problems: Sectors, statistics, symmetries and spectrum, Commun. Math. Phys., 330 (2014), 935-972.  doi: 10.1007/s00220-014-2004-2.  Google Scholar [5] L. J. Bunce and J. D. Maitland Wright, The Mackey-Gleason problem, Bull. Amer. Math. Soc., 26 (1992), 288-293.  doi: 10.1090/S0273-0979-1992-00274-4.  Google Scholar [6] E. Curiel, F. Finster and J. M. Isidro, Two-dimensional area and matter flux in the theory of causal fermion systems, preprint, arXiv: 1910.06161, to appear in Internat. J. Modern Phys. D, (2020). Google Scholar [7] C. Dappiaggi and F. Finster, Linearized fields for causal variational principles: Existence theory and causal structure, Methods Appl. Anal., 27 (2020), 1-56.  doi: 10.4310/MAA.2020.v27.n1.a1.  Google Scholar [8] S. Doplicher, K. Fredenhagen and J. E. Roberts, The quantum structure of spacetime at the Planck scale and quantum fields, Commun. Math. Phys., 172 (1995), 187-220.  doi: 10.1007/BF02104515.  Google Scholar [9] A. Dvurečenskij, Gleason's Theorem and its Applications, Mathematics and its Applications (East European Series), vol. 60, Kluwer Academic Publishers Group, Dordrecht; Ister Science Press, Bratislava, 1993. doi: 10.1007/978-94-015-8222-3.  Google Scholar [10] F. Finster, 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. doi: 10.1090/amsip/035.  Google Scholar [11] F. Finster, On the regularized fermionic projector of the vacuum, J. Math. Phys., 49 (2008), 032304, 60 pp. doi: 10.1063/1.2888187.  Google Scholar [12] F. Finster, Causal variational principles on measure spaces, J. Reine Angew. Math., 646 (2010), 141-194.  doi: 10.1515/CRELLE.2010.069.  Google Scholar [13] F. Finster, Perturbative quantum field theory in the framework of the fermionic projector, J. Math. Phys., 55 (2014), 042301, 53 pp. doi: 10.1063/1.4871549.  Google Scholar [14] F. Finster, Causal fermion systems – an overview, in Quantum Mathematical Physics: A Bridge between Mathematics and Physics (F. Finster, J. Kleiner, C. R ken, and J. Tolksdorf, eds.), Birkhäuser Verlag, Basel, (2016), 313–380. doi: 10.1007/978-3-319-42067-7.  Google Scholar [15] F. Finster, The Continuum Limit of Causal Fermion Systems, Fundamental Theories of Physics, vol. 186, Springer, 2016. doi: 10.1007/978-3-319-42067-7.  Google Scholar [16] F. Finster, Causal fermion systems: Discrete space-times, causation and finite propagation speed, J. Phys.: Conf. Ser., 1275 (2019), 012009. Google Scholar [17] F. Finster, Perturbation theory for critical points of causal variational principles, Adv. Theor. Math. Phys., 24 (2020), 563-619.  doi: 10.4310/ATMP.2020.v24.n3.a2.  Google Scholar [18] F. Finster and A. Grotz, A Lorentzian quantum geometry, Adv. Theor. Math. Phys., 16 (2012), 1197-1290.  doi: 10.4310/ATMP.2012.v16.n4.a3.  Google Scholar [19] F. Finster and A. Grotz, On the initial value problem for causal variational principles, J. Reine Angew. Math., 725 (2017), 115-141.  doi: 10.1515/crelle-2014-0080.  Google Scholar [20] F. Finster, A. Grotz and D. Schiefeneder, Causal fermion systems: A quantum space-time emerging from an action principle, in Quantum Field Theory and Gravity (F. Finster, O. Müller, M. Nardmann, J. Tolksdorf, and E. Zeidler, eds.), Birkhäuser Verlag, Basel, (2012), 157–182. doi: 10.1007/978-3-0348-0043-3_9.  Google Scholar [21] F. Finster and M. Jokel, Causal fermion systems: An elementary introduction to physical ideas and mathematical concepts, in Progress and Visions in Quantum Theory in View of Gravity (F. Finster, D. Giulini, J. Kleiner, and J. Tolksdorf, eds.), Birkhäuser Verlag, Basel, (2020), 63–92. doi: 10.1007/978-3-030-38941-3_2.  Google Scholar [22] F. Finster and N. Kamran, Complex structures on jet spaces and bosonic Fock space dynamics for causal variational principles, preprint, arXiv: 1808.03177, to appear in Pure Appl. Math. Q., (2020). Google Scholar [23] F. Finster and J. Kleiner, Causal fermion systems as a candidate for a unified physical theory, J. Phys.: Conf. Ser., 626 (2015), 012020. Google Scholar [24] F. Finster and J. Kleiner, A Hamiltonian formulation of causal variational principles, Calc. Var. Partial Differential Equations, 56 (2017), no. 73, 33pp. doi: 10.1007/s00526-017-1153-5.  Google Scholar [25] F. Finster and C. Langer, Causal variational principles in the $\sigma$-locally compact setting: Existence of minimizers, preprint, arXiv: 2002.04412, to appear in Adv. Calc. Var., (2020). Google Scholar [26] F. Finster and M. Oppio, Local algebras for causal fermion systems in Minkowski space, preprint, arXiv: 2004.00419, (2020). Google Scholar [27] F. Finster and D. Schiefeneder, On the support of minimizers of causal variational principles, Arch. Ration. Mech. Anal., 210 (2013), 321-364.  doi: 10.1007/s00205-013-0649-1.  Google Scholar [28] J. Fröhlich, The quest for laws and structure, Mathematics and Society, (2016), 101–129. Google Scholar [29] J. Fröhlich, A brief review of the "ETH-approach to quantum mechanics", preprint, arXiv: 1905.06603, (2019). Google Scholar [30] J. Fröhlich, Relativistic quantum theory, preprint, arXiv: 1912.00726, (2019). Google Scholar [31] J. Fröhlich, "Diminishing potentialities", entanglement, "purification" and the emergence of events in quantum mechanics – a simple model, Sect. 5.6 of Notes for a course on Quantum Theory at LMU-Munich (Nov./Dec. 2019). Google Scholar [32] J. Fröhlich and B. Schubnel, Quantum probability theory and the foundations of quantum mechanics, in The Message of Quantum Science, Springer, 899 (2015), 131–193. doi: 10.1007/978-3-662-46422-9_7.  Google Scholar [33] A. M. Gleason, Measures on the closed subspaces of a Hilbert space, J. Math. Mech., 6 (1957), 885-893.  doi: 10.1512/iumj.1957.6.56050.  Google Scholar [34] G. C. Hegerfeldt, Remark on causality and particle localization, Physical Review D, 10 (1974), no. 10, 3320. doi: 10.1103/PhysRevD.10.3320.  Google Scholar [35] J. Kleiner, Dynamics of Causal Fermion Systems – Field Equations and Correction Terms for a New Unified Physical Theory, Dissertation, Universität Regensburg, 2017. Google Scholar [36] H. Lin, Almost commuting selfadjoint matrices and applications, in Operator Algebras and their Applications (Waterloo, ON, 1994/1995), Fields Inst. Commun., vol. 13, Amer. Math. Soc., Providence, RI, (1997), 193–233.  Google Scholar [37] B. Thaller, The Dirac Equation, Texts and Monographs in Physics, Springer-Verlag, Berlin, 1992. doi: 10.1007/978-3-662-02753-0.  Google Scholar
The spacetime restricted causal future $I^\vee_\rho(x)$ of the causal fermion system and the future light cone $\mathcal{I}^\vee(x)$ of Minkowski space
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