Skip to main content

Fermionic Fock Spaces and Quantum States for Causal Fermion Systems


It is shown for causal fermion systems describing Minkowski-type spacetimes that an interacting causal fermion system at time t gives rise to a distinguished state on the algebra generated by fermionic and bosonic field operators. The proof of positivity of the state is given, and representations are constructed.

This is a preview of subscription content, access via your institution.

Fig. 1
Fig. 2


  1. 1.

    Arveson, W.: An Invitation to \(C^*\)-Algebras. Graduate Texts in Mathematics, vol. 39 Springer, New York (1976)

  2. 2.

    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]

    MathSciNet  Article  Google Scholar 

  3. 3.

    Bogachev, V.I.: Measure Theory, vol. I. Springer, Berlin (2007)

    Book  Google Scholar 

  4. 4.

    Dappiaggi, C., Finster, F.: Linearized fields for causal variational principles: existence theory and causal structure. Methods Appl. Anal. 27(1), 1–56 (2020). arXiv:1811.10587 [math-ph]

    MathSciNet  MATH  Google Scholar 

  5. 5.

    Drago, N., Hack, T.-P., Pinamonti, N.: The generalised principle of perturbative agreement and the thermal mass. Ann. Henri Poincaré 18(3), 807–868 (2017). arXiv:1502.02705 [math-ph]

    ADS  MathSciNet  Article  Google Scholar 

  6. 6.

    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)

    Google Scholar 

  7. 7.

    Finster, F.: A variational principle in discrete space-time: existence of minimizers. Calc. Var. Partial Differ. Equ. 29(4), 431–453 (2007). arXiv:math-ph/0503069

    ADS  MathSciNet  Article  Google Scholar 

  8. 8.

    Finster, F.: Causal variational principles on measure spaces. J. Reine Angew. Math. 646, 141–194 (2010). arXiv:0811.2666 [math-ph]

    MathSciNet  MATH  Google Scholar 

  9. 9.

    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]

    ADS  MathSciNet  Article  Google Scholar 

  10. 10.

    Finster, F.: The Continuum Limit of Causal Fermion Systems. Fundamental Theories of Physics, vol. 186. Springer (2016). arXiv:1605.04742 [math-ph]

  11. 11.

    Finster, F.: Causal fermion systems: a primer for Lorentzian geometers. J. Phys. Conf. Ser. 968, 012004 (2018). arXiv:1709.04781 [math-ph]

    MathSciNet  Article  Google Scholar 

  12. 12.

    Finster, F.: The causal action in Minkowski space and surface layer integrals. SIGMA Symmetry Integrability Geom. Methods Appl. 16(091) (2020). arXiv:1711.07058 [math-ph]

  13. 13.

    Finster, F.: Perturbation theory for critical points of causal variational principles. Adv. Theor. Math. Phys. 24(3), 563–619 (2020). arXiv:1703.05059 [math-ph]

    MathSciNet  Article  Google Scholar 

  14. 14.

    Finster, F.: A notion of entropy for causal fermion systems. arXiv:2103.14980 [math-ph] (2021)

  15. 15.

    Finster, F., Jokel, M.: Progress and visions in quantum theory in view of gravity. In: Finster, F., Giulini, D., Kleiner, J., Tolksdorf, J. (eds.) Causal Fermion Systems: An Elementary Introduction to Physical Ideas and Mathematical Concepts, pp. 63–92. Birkhäuser Verlag, Basel (2020) . arXiv:1908.08451 [math-ph]

  16. 16.

    Finster, F., Kamran, N.: The quantum field theory limit of causal fermion systems. (in preparation)

  17. 17.

    Finster, F.: Complex structures on jet spaces and bosonic Fock space dynamics for causal variational principles. Pure Appl. Math. Q. 17(1), 55–140 (2021). arXiv:1808.03177 [math-ph]

    MathSciNet  Article  Google Scholar 

  18. 18.

    Finster, F., Kamran, N., Oppio, M.: The linear dynamics of wave functions in causal fermion systems. J. Differ. Equ. 293, 115–187 (2021). arXiv:2101.08673 [math-ph]

    ADS  MathSciNet  Article  Google Scholar 

  19. 19.

    Finster, F., Kamran, N., Reintjes, M.: Holographic mixing and bosonic loop diagrams for causal fermion systems (in preparation)

  20. 20.

    Finster, F., Kindermann, S.: A gauge fixing procedure for causal fermion systems. J. Math. Phys. 61(8), 082301 (2020). arXiv:1908.08445 [math-ph]

    ADS  MathSciNet  Article  Google Scholar 

  21. 21.

    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]

    Article  Google Scholar 

  22. 22.

    Finster, F.: Noether-like theorems for causal variational principles. Calc. Var. Partial Differ. Equ. 55:35(2), 41 (2016). arXiv:1506.09076 [math-ph]

  23. 23.

    Finster, F.: A Hamiltonian formulation of causal variational principles. Calc. Var. Partial Differ. Equ. 56:73(3), 33 (2017). arXiv:1612.07192 [math-ph]

  24. 24.

    Finster, F., Langer, C.: Causal variational principles in the \(\sigma \)-locally compact setting: existence of minimizers. (to appear in Adv. Calc. Var.) (2021). arXiv:2002.04412 [math-ph]

  25. 25.

    Finster, F., Lottner, M.: Banach manifold structure and infinite-dimensional analysis for causal fermion systems. Ann. Glob. Anal. Geom. 60(2), 313–354 (2021). arXiv:2101.11908 [math-ph]

    MathSciNet  Article  Google Scholar 

  26. 26.

    Finster, F., Platzer, A.: A positive mass theorem for static causal fermion systems. arXiv:1912.12995 [math-ph] (2019)

  27. 27.

    Glimm, J., Jaffe, A.: Quantum Physics, a Functional Integral Point of View, 2nd edn. Springer, New York (1987)

    MATH  Google Scholar 

  28. 28.

    Greene, R.E., Shiohama, K.: Diffeomorphisms and volume-preserving embeddings of noncompact manifolds. Trans. Am. Math. Soc. 255, 403–414 (1979)

    MathSciNet  Article  Google Scholar 

  29. 29.

    Helgason, S.: Groups and Geometric Analysis. Mathematical Surveys and Monographs, vol. 83, American Mathematical Society, Providence, RI (2000). Integral geometry, invariant differential operators, and spherical functions, Corrected reprint of the 1984 original

  30. 30.

    Khavkine, I., Moretti, V.: Algebraic QFT in curved spacetime and quasifree Hadamard states: an introduction. Adv. Algebraic Quantum Field Theo. Math. Phys. Stud. Springer, Cham, pp. 191–251. arXiv:1412.5945 [math-ph] (2015)

  31. 31.

    Klaus, M., Scharf, G.: The regular external field problem in quantum electrodynamics. Helv. Phys. Acta 50(6), 779–802 (1977)

    MathSciNet  Google Scholar 

  32. 32.

    Link to web platform on causal fermion systems.

  33. 33.

    Nenciu, G., Scharf, G.: On regular external fields in quantum electrodynamics. Helv. Phys. Acta 51(3), 412–424 (1978)

    MathSciNet  Google Scholar 

  34. 34.

    Pokorski, S.: Gauge Field Theories. Cambridge Monographs on Mathematical Physics, 2nd edn. Cambridge University Press, Cambridge (2000)

Download references


We would like to thank Claudio Dappiaggi, Jürg Fröhlich, Marco Oppio, Claudio Paganini, Moritz Reintjes and Jürgen Tolksdorf for helpful discussions. We are grateful to the referees for valuable feedback and suggestions. We would like to thank the “Universitätsstiftung Hans Vielberth” for support. N.K.’s research was also supported by the NSERC Grant RGPIN 105490-2018.

Author information



Corresponding author

Correspondence to Felix Finster.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Communicated by Karl-Henning Rehren.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Finster, F., Kamran, N. Fermionic Fock Spaces and Quantum States for Causal Fermion Systems. Ann. Henri Poincaré (2021).

Download citation