Fermionic Fock spaces and quantum states for causal fermion systems. (English) Zbl 1495.81108
Summary: 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.
MSC:
| 81V74 | Fermionic systems in quantum theory |
| 81V73 | Bosonic systems in quantum theory |
| 62D20 | Causal inference from observational studies |
| 51B20 | Minkowski geometries in nonlinear incidence geometry |
| 81P16 | Quantum state spaces, operational and probabilistic concepts |
| 30H20 | Bergman spaces and Fock spaces |
References:
| [1] | Arveson, W.: An Invitation to \(C^*\)-Algebras. Graduate Texts in Mathematics, vol. 39 Springer, New York (1976) · Zbl 0344.46123 |
| [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) · Zbl 1281.49039 · doi:10.1515/acv-2012-0109 |
| [3] | Bogachev, VI, Measure Theory (2007), Berlin: Springer, Berlin · Zbl 1120.28001 · doi:10.1007/978-3-540-34514-5 |
| [4] | Dappiaggi, C.; Finster, F., Linearized fields for causal variational principles: existence theory and causal structure, Methods Appl. Anal., 27, 1, 1-56 (2020) · Zbl 1451.35005 · doi:10.4310/MAA.2020.v27.n1.a1 |
| [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) · Zbl 1362.81064 · doi:10.1007/s00023-016-0521-6 |
| [6] | Finster, F., The Principle of the Fermionic Projector, hep-th/0001048, hep-th/0202059, hep-th/0210121, AMS/IP Studies in Advanced Mathematics (2006), Providence, RI: American Mathematical Society, Providence, RI · Zbl 1090.83002 |
| [7] | Finster, F., A variational principle in discrete space-time: existence of minimizers, Calc. Var. Partial Differ. Equ., 29, 4, 431-453 (2007) · Zbl 1131.49031 · doi:10.1007/s00526-006-0042-0 |
| [8] | Finster, F., Causal variational principles on measure spaces, J. Reine Angew. Math., 646, 141-194 (2010) · Zbl 1213.49006 |
| [9] | Finster, F., Perturbative quantum field theory in the framework of the fermionic projector, J. Math. Phys., 55, 4 (2014) · Zbl 1295.81111 · doi:10.1063/1.4871549 |
| [10] | Finster, F.: The Continuum Limit of Causal Fermion Systems. Fundamental Theories of Physics, vol. 186. Springer (2016). arXiv:1605.04742 [math-ph] · Zbl 1353.81001 |
| [11] | Finster, F., Causal fermion systems: a primer for Lorentzian geometers, J. Phys. Conf. Ser., 968 (2018) · doi:10.1088/1742-6596/968/1/012004 |
| [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] · Zbl 1458.83003 |
| [13] | Finster, F., Perturbation theory for critical points of causal variational principles, Adv. Theor. Math. Phys., 24, 3, 563-619 (2020) · Zbl 07435483 · doi:10.4310/ATMP.2020.v24.n3.a2 |
| [14] | Finster, F.: A notion of entropy for causal fermion systems. arXiv:2103.14980 [math-ph] (2021) · Zbl 1476.49057 |
| [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] | Finster, F., Kamran, N.: The quantum field theory limit of causal fermion systems. (in preparation) · Zbl 1445.53001 |
| [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) · Zbl 1479.58018 · doi:10.4310/PAMQ.2021.v17.n1.a3 |
| [18] | Finster, F.; Kamran, N.; Oppio, M., The linear dynamics of wave functions in causal fermion systems, J. Differ. Equ., 293, 115-187 (2021) · Zbl 1472.81334 · doi:10.1016/j.jde.2021.05.025 |
| [19] | Finster, F., Kamran, N., Reintjes, M.: Holographic mixing and bosonic loop diagrams for causal fermion systems (in preparation) |
| [20] | Finster, F.; Kindermann, S., A gauge fixing procedure for causal fermion systems, J. Math. Phys., 61, 8 (2020) · Zbl 1454.81143 · doi:10.1063/1.5125585 |
| [21] | Finster, F.; Kleiner, J., Causal fermion systems as a candidate for a unified physical theory, J. Phys. Conf. Ser., 626 (2015) · doi:10.1088/1742-6596/626/1/012020 |
| [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] · Zbl 1337.49076 |
| [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] · Zbl 1375.49060 |
| [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] | Finster, F.; Lottner, M., Banach manifold structure and infinite-dimensional analysis for causal fermion systems, Ann. Glob. Anal. Geom., 60, 2, 313-354 (2021) · Zbl 1471.58003 · doi:10.1007/s10455-021-09775-4 |
| [26] | Finster, F., Platzer, A.: A positive mass theorem for static causal fermion systems. arXiv:1912.12995 [math-ph] (2019) |
| [27] | Glimm, J.; Jaffe, A., Quantum Physics, a Functional Integral Point of View (1987), New York: Springer, New York · Zbl 0461.46051 |
| [28] | Greene, RE; Shiohama, K., Diffeomorphisms and volume-preserving embeddings of noncompact manifolds, Trans. Am. Math. Soc., 255, 403-414 (1979) · Zbl 0418.58002 · doi:10.1090/S0002-9947-1979-0542888-3 |
| [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 · Zbl 0965.43007 |
| [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) · Zbl 1334.81081 |
| [31] | Klaus, M.; Scharf, G., The regular external field problem in quantum electrodynamics, Helv. Phys. Acta, 50, 6, 779-802 (1977) |
| [32] | Link to web platform on causal fermion systems. http://www.causal-fermion-system.com |
| [33] | Nenciu, G.; Scharf, G., On regular external fields in quantum electrodynamics, Helv. Phys. Acta, 51, 3, 412-424 (1978) |
| [34] | Pokorski, S.: Gauge Field Theories. Cambridge Monographs on Mathematical Physics, 2nd edn. Cambridge University Press, Cambridge (2000) · Zbl 0998.81504 |
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