×

Causal fermion systems and the ETH approach to quantum theory. (English) Zbl 1475.83003

Summary: 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.

MSC:

83A05 Special relativity
81T05 Axiomatic quantum field theory; operator algebras
81P15 Quantum measurement theory, state operations, state preparations
47N50 Applications of operator theory in the physical sciences
81R15 Operator algebra methods applied to problems in quantum theory
49S05 Variational principles of physics
PDFBibTeX XMLCite
Full Text: DOI arXiv

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. · Zbl 1429.49047
[3] P. Blanchard; J. Fröhlich; B. Schubnel, A garden of forking paths – the quantum mechanics of histories of events, Nuclear Phys. B, 912, 463-484 (2016) · Zbl 1349.81024
[4] D. Buchholz; J. E. Roberts, New light on infrared problems: Sectors, statistics, symmetries and spectrum, Commun. Math. Phys., 330, 935-972 (2014) · Zbl 1296.81044
[5] L. J. Bunce; J. D. Maitland Wright, The Mackey-Gleason problem, Bull. Amer. Math. Soc., 26, 288-293 (1992) · Zbl 0759.46054
[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; F. Finster, Linearized fields for causal variational principles: Existence theory and causal structure, Methods Appl. Anal., 27, 1-56 (2020) · Zbl 1451.35005
[8] S. Doplicher; K. Fredenhagen; J. E. Roberts, The quantum structure of spacetime at the Planck scale and quantum fields, Commun. Math. Phys., 172, 187-220 (1995) · Zbl 0847.53051
[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. · Zbl 0795.46045
[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. · Zbl 1090.83002
[11] F. Finster, On the regularized fermionic projector of the vacuum, J. Math. Phys., 49 (2008), 032304, 60 pp. · Zbl 1153.81358
[12] F. Finster, Causal variational principles on measure spaces, J. Reine Angew. Math., 646, 141-194 (2010) · Zbl 1213.49006
[13] F. Finster, Perturbative quantum field theory in the framework of the fermionic projector, J. Math. Phys., 55 (2014), 042301, 53 pp. · Zbl 1295.81111
[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. · Zbl 1336.81002
[15] F. Finster, The Continuum Limit of Causal Fermion Systems, Fundamental Theories of Physics, vol. 186, Springer, 2016. · Zbl 1353.81001
[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, 563-619 (2020) · Zbl 07435483
[18] F. Finster; A. Grotz, A Lorentzian quantum geometry, Adv. Theor. Math. Phys., 16, 1197-1290 (2012) · Zbl 1346.82002
[19] F. Finster; A. Grotz, On the initial value problem for causal variational principles, J. Reine Angew. Math., 725, 115-141 (2017) · Zbl 1365.49044
[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. · Zbl 1246.81174
[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.
[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. · Zbl 1375.49060
[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). · Zbl 1458.81029
[27] F. Finster; D. Schiefeneder, On the support of minimizers of causal variational principles, Arch. Ration. Mech. Anal., 210, 321-364 (2013) · Zbl 1306.49060
[28] J. Fröhlich, The quest for laws and structure, Mathematics and Society, (2016), 101-129. · Zbl 1354.00067
[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.
[33] A. M. Gleason, Measures on the closed subspaces of a Hilbert space, J. Math. Mech., 6, 885-893 (1957) · Zbl 0078.28803
[34] G. C. Hegerfeldt, Remark on causality and particle localization, Physical Review D, 10 (1974), no. 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. · Zbl 0881.46042
[37] B. Thaller, The Dirac Equation, Texts and Monographs in Physics, Springer-Verlag, Berlin, 1992.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.