May  2021, 14(5): 1717-1746. doi: 10.3934/dcdss.2020451

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  May 2021 Early access  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 and Continuous Dynamical Systems - S, 2021, 14 (5) : 1717-1746. doi: 10.3934/dcdss.2020451
References:
[1]

Link to web platform on causal fermion systems: http://www.causal-fermion-system.com.

[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.

[3]

P. BlanchardJ. 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.

[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.

[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.

[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).

[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.

[8]

S. DoplicherK. 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.

[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.

[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.

[11]

F. Finster, On the regularized fermionic projector of the vacuum, J. Math. Phys., 49 (2008), 032304, 60 pp. doi: 10.1063/1.2888187.

[12]

F. Finster, Causal variational principles on measure spaces, J. Reine Angew. Math., 646 (2010), 141-194.  doi: 10.1515/CRELLE.2010.069.

[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.

[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.

[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.

[16]

F. Finster, Causal fermion systems: Discrete space-times, causation and finite propagation speed, J. Phys.: Conf. Ser., 1275 (2019), 012009.

[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.

[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.

[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.

[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.

[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.

[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).

[23]

F. Finster and J. Kleiner, Causal fermion systems as a candidate for a unified physical theory, J. Phys.: Conf. Ser., 626 (2015), 012020.

[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.

[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).

[26]

F. Finster and M. Oppio, Local algebras for causal fermion systems in Minkowski space, preprint, arXiv: 2004.00419, (2020).

[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.

[28]

J. Fröhlich, The quest for laws and structure, Mathematics and Society, (2016), 101–129.

[29]

J. Fröhlich, A brief review of the "ETH-approach to quantum mechanics", preprint, arXiv: 1905.06603, (2019).

[30]

J. Fröhlich, Relativistic quantum theory, preprint, arXiv: 1912.00726, (2019).

[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).

[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.

[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.

[34]

G. C. Hegerfeldt, Remark on causality and particle localization, Physical Review D, 10 (1974), no. 10, 3320. doi: 10.1103/PhysRevD.10.3320.

[35]

J. Kleiner, Dynamics of Causal Fermion Systems – Field Equations and Correction Terms for a New Unified Physical Theory, Dissertation, Universität Regensburg, 2017.

[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.

[37]

B. Thaller, The Dirac Equation, Texts and Monographs in Physics, Springer-Verlag, Berlin, 1992. doi: 10.1007/978-3-662-02753-0.

show all references

References:
[1]

Link to web platform on causal fermion systems: http://www.causal-fermion-system.com.

[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.

[3]

P. BlanchardJ. 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.

[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.

[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.

[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).

[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.

[8]

S. DoplicherK. 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.

[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.

[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.

[11]

F. Finster, On the regularized fermionic projector of the vacuum, J. Math. Phys., 49 (2008), 032304, 60 pp. doi: 10.1063/1.2888187.

[12]

F. Finster, Causal variational principles on measure spaces, J. Reine Angew. Math., 646 (2010), 141-194.  doi: 10.1515/CRELLE.2010.069.

[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.

[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.

[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.

[16]

F. Finster, Causal fermion systems: Discrete space-times, causation and finite propagation speed, J. Phys.: Conf. Ser., 1275 (2019), 012009.

[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.

[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.

[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.

[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.

[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.

[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).

[23]

F. Finster and J. Kleiner, Causal fermion systems as a candidate for a unified physical theory, J. Phys.: Conf. Ser., 626 (2015), 012020.

[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.

[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).

[26]

F. Finster and M. Oppio, Local algebras for causal fermion systems in Minkowski space, preprint, arXiv: 2004.00419, (2020).

[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.

[28]

J. Fröhlich, The quest for laws and structure, Mathematics and Society, (2016), 101–129.

[29]

J. Fröhlich, A brief review of the "ETH-approach to quantum mechanics", preprint, arXiv: 1905.06603, (2019).

[30]

J. Fröhlich, Relativistic quantum theory, preprint, arXiv: 1912.00726, (2019).

[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).

[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.

[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.

[34]

G. C. Hegerfeldt, Remark on causality and particle localization, Physical Review D, 10 (1974), no. 10, 3320. doi: 10.1103/PhysRevD.10.3320.

[35]

J. Kleiner, Dynamics of Causal Fermion Systems – Field Equations and Correction Terms for a New Unified Physical Theory, Dissertation, Universität Regensburg, 2017.

[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.

[37]

B. Thaller, The Dirac Equation, Texts and Monographs in Physics, Springer-Verlag, Berlin, 1992. doi: 10.1007/978-3-662-02753-0.

Figure 1.  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
Figure 2.  The approximate center and the loss of access to information
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