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Spinors on Singular Spaces and the Topology of Causal Fermion Systems
About this Title
Felix Finster, Fakultät für Mathematik , Universität Regensburg , D-93040 Regensburg , Germany and Niky Kamran, Department of Mathematics and Statistics , McGill University , Montréal , Canada
Publication: Memoirs of the American Mathematical Society
Publication Year:
2019; Volume 259, Number 1251
ISBNs: 978-1-4704-3621-6 (print); 978-1-4704-5257-5 (online)
DOI: https://doi.org/10.1090/memo/1251
Published electronically: April 19, 2019
MSC: Primary 53-02; Secondary 53Z05, 53C80, 53C27, 57R22
Table of Contents
Chapters
- 1. Introduction
- 2. Basic Definitions and Simple Examples
- 3. Topological Structures
- 4. Topological Spinor Bundles
- 5. Further Examples
- 6. Tangent Cone Measures and the Tangential Clifford Section
- 7. The Topology of Discrete and Singular Fermion Systems
- 8. Basic Examples
- 9. Spinors on Singular Spaces
Abstract
Causal fermion systems and Riemannian fermion systems are proposed as a framework for describing non-smooth geometries. In particular, this framework provides a setting for spinors on singular spaces. The underlying topological structures are introduced and analyzed. The connection to the spin condition in differential topology is worked out. The constructions are illustrated by many simple examples like the Euclidean plane, the two-dimensional Minkowski space, a conical singularity, a lattice system as well as the curvature singularity of the Schwarzschild space-time. As further examples, it is shown how complex and Kähler structures can be encoded in Riemannian fermion systems.- C. Bär, P. Gauduchon, and A. Moroianu, Generalized cylinders in semi-Riemannian and spin geometry, arXiv:math/0303095 [math.DG], Math. Z. 249 (2005), no. 3, 545–580.
- Helga Baum, Spinor structures and Dirac operators on pseudo-Riemannian manifolds, Bull. Polish Acad. Sci. Math. 33 (1985), no. 3-4, 165–171 (English, with Russian summary). MR 805031
- L. Bäuml, F. Finster, H. von der Mosel, and D. Schiefeneder, Singular support of minimizers of the causal variational principle on the sphere, arXiv:1808.09754 [math.CA] (2018).
- Yann Bernard and Felix Finster, On the structure of minimizers of causal variational principles in the non-compact and equivariant settings, Adv. Calc. Var. 7 (2014), no. 1, 27–57. MR 3176583, DOI 10.1515/acv-2012-0109
- Glen E. Bredon, Sheaf theory, 2nd ed., Graduate Texts in Mathematics, vol. 170, Springer-Verlag, New York, 1997. MR 1481706, DOI 10.1007/978-1-4612-0647-7
- T. Bröcker, Analysis II, Bibliographisches Institut, Mannheim, 1992.
- Shiing Shen Chern, Complex manifolds without potential theory, 2nd ed., Universitext, Springer-Verlag, New York-Heidelberg, 1979. With an appendix on the geometry of characteristic classes. MR 533884
- A. Diethert, F. Finster, and D. Schiefeneder, Fermion systems in discrete space-time exemplifying the spontaneous generation of a causal structure, arXiv:0710.4420 [math-ph], Int. J. Mod. Phys. A 23 (2008), no. 27/28, 4579–4620.
- Felix Finster, Local $\rm U(2,2)$ symmetry in relativistic quantum mechanics, J. Math. Phys. 39 (1998), no. 12, 6276–6290. MR 1656964, DOI 10.1063/1.532638
- Felix Finster, The principle of the fermionic projector, AMS/IP Studies in Advanced Mathematics, vol. 35, American Mathematical Society, Providence, RI; International Press, Somerville, MA, 2006. MR 2200689, DOI 10.1090/amsip/035
- Felix Finster, Causal variational principles on measure spaces, J. Reine Angew. Math. 646 (2010), 141–194. MR 2719559, DOI 10.1515/CRELLE.2010.069
- Felix Finster, A formulation of quantum field theory realizing a sea of interacting Dirac particles, Lett. Math. Phys. 97 (2011), no. 2, 165–183. MR 2821236, DOI 10.1007/s11005-011-0473-1
- Felix Finster, The continuum limit of causal fermion systems, Fundamental Theories of Physics, vol. 186, Springer, [Cham], 2016. From Planck scale structures to macroscopic physics. MR 3525361, DOI 10.1007/978-3-319-42067-7
- —, Causal fermion systems: A primer for Lorentzian geometers, arXiv:1709.04781 [math-ph], J. Phys.: Conf. Ser. 968 (2018), 012004.
- Felix Finster and Andreas Grotz, A Lorentzian quantum geometry, Adv. Theor. Math. Phys. 16 (2012), no. 4, 1197–1290. MR 3053970, DOI 10.4310/ATMP.2012.v16.n4.a3
- Felix Finster and Andreas Grotz, On the initial value problem for causal variational principles, J. Reine Angew. Math. 725 (2017), 115–141. MR 3630119, DOI 10.1515/crelle-2014-0080
- Felix Finster, Andreas Grotz, and Daniela Schiefeneder, Causal fermion systems: a quantum space-time emerging from an action principle, Quantum field theory and gravity, Birkhäuser/Springer Basel AG, Basel, 2012, pp. 157–182. MR 3074851, DOI 10.1007/978-3-0348-0043-3_{9}
- Felix Finster and Christian Hainzl, Quantum oscillations can prevent the big bang singularity in an Einstein-Dirac cosmology, Found. Phys. 40 (2010), no. 1, 116–124. MR 2575733, DOI 10.1007/s10701-009-9380-z
- Felix Finster, Niky Kamran, Joel Smoller, and Shing-Tung Yau, Nonexistence of time-periodic solutions of the Dirac equation in an axisymmetric black hole geometry, Comm. Pure Appl. Math. 53 (2000), no. 7, 902–929. MR 1752438, DOI 10.1002/(SICI)1097-0312(200007)53:7<902::AID-CPA4>3.0.CO;2-4
- Felix Finster and Johannes Kleiner, A Hamiltonian formulation of causal variational principles, Calc. Var. Partial Differential Equations 56 (2017), no. 3, Paper No. 73, 33. MR 3641920, DOI 10.1007/s00526-017-1153-5
- F. Finster and J. Kleiner, Causal fermion systems as a candidate for a unified physical theory, arXiv:1502.03587 [math-ph], J. Phys.: Conf. Ser. 626 (2015), 012020.
- Felix Finster and Johannes Kleiner, Noether-like theorems for causal variational principles, Calc. Var. Partial Differential Equations 55 (2016), no. 2, Art. 35, 41. MR 3475674, DOI 10.1007/s00526-016-0966-y
- Felix Finster and Moritz Reintjes, The Dirac equation and the normalization of its solutions in a closed Friedmann-Robertson-Walker universe, Classical Quantum Gravity 26 (2009), no. 10, 105021, 20. MR 2532058, DOI 10.1088/0264-9381/26/10/105021
- Felix Finster and Moritz Reintjes, A non-perturbative construction of the fermionic projector on globally hyperbolic manifolds I—space-times of finite lifetime, Adv. Theor. Math. Phys. 19 (2015), no. 4, 761–803. MR 3454231, DOI 10.4310/ATMP.2015.v19.n4.a3
- Felix Finster and Daniela Schiefeneder, On the support of minimizers of causal variational principles, Arch. Ration. Mech. Anal. 210 (2013), no. 2, 321–364. MR 3101787, DOI 10.1007/s00205-013-0649-1
- M. Fischer, Tangentialer Clifford-Schnitt und tangentiales Kegelmaß am Beispiel eines zweidimensionalen kausalen Fermionsystems, Masterarbeit Mathematik, Universität Regensburg (2015).
- Thomas Friedrich, Dirac operators in Riemannian geometry, Graduate Studies in Mathematics, vol. 25, American Mathematical Society, Providence, RI, 2000. Translated from the 1997 German original by Andreas Nestke. MR 1777332, DOI 10.1090/gsm/025
- Nicolas Ginoux, The Dirac spectrum, Lecture Notes in Mathematics, vol. 1976, Springer-Verlag, Berlin, 2009. MR 2509837, DOI 10.1007/978-3-642-01570-0
- J. N. Goldberg, A. J. Macfarlane, E. T. Newman, F. Rohrlich, and E. C. G. Sudarshan, Spin-$s$ spherical harmonics and $\dbar$, J. Mathematical Phys. 8 (1967), 2155–2161. MR 241084, DOI 10.1063/1.1705135
- Paul R. Halmos, Measure Theory, D. Van Nostrand Co., Inc., New York, 1950. MR 33869
- Allen Hatcher, Algebraic topology, Cambridge University Press, Cambridge, 2002. MR 1867354
- S. W. Hawking and G. F. R. Ellis, The large scale structure of space-time, Cambridge Monographs on Mathematical Physics, No. 1, Cambridge University Press, London-New York, 1973. MR 424186
- Oussama Hijazi, A conformal lower bound for the smallest eigenvalue of the Dirac operator and Killing spinors, Comm. Math. Phys. 104 (1986), no. 1, 151–162. MR 834486
- Friedrich Hirzebruch, Topological methods in algebraic geometry, Classics in Mathematics, Springer-Verlag, Berlin, 1995. Translated from the German and Appendix One by R. L. E. Schwarzenberger; With a preface to the third English edition by the author and Schwarzenberger; Appendix Two by A. Borel; Reprint of the 1978 edition. MR 1335917
- Nigel Hitchin, Harmonic spinors, Advances in Math. 14 (1974), 1–55. MR 358873, DOI 10.1016/0001-8708(74)90021-8
- D. Husemoller, Fibre bundles, third ed., Graduate Texts in Mathematics, vol. 20, Springer-Verlag, New York, 1994.
- Robion C. Kirby and Laurence C. Siebenmann, Foundational essays on topological manifolds, smoothings, and triangulations, Annals of Mathematics Studies, No. 88, Princeton University Press, Princeton, NJ; University of Tokyo Press, Tokyo, 1977. With notes by John Milnor and Michael Atiyah. MR 645390
- Steven G. Krantz, Function theory of several complex variables, AMS Chelsea Publishing, Providence, RI, 2001. Reprint of the 1992 edition. MR 1846625, DOI 10.1090/chel/340
- H. Blaine Lawson Jr. and Marie-Louise Michelsohn, Spin geometry, Princeton Mathematical Series, vol. 38, Princeton University Press, Princeton, NJ, 1989. MR 1031992
- M. Lüscher, Topology of lattice gauge fields, Comm. Math. Phys. 85 (1982), no. 1, 39–48. MR 667766
- John McCleary, User’s guide to spectral sequences, Mathematics Lecture Series, vol. 12, Publish or Perish, Inc., Wilmington, DE, 1985. MR 820463
- John W. Milnor and James D. Stasheff, Characteristic classes, Annals of Mathematics Studies, No. 76, Princeton University Press, Princeton, NJ; University of Tokyo Press, Tokyo, 1974. MR 440554
- F. W. J. Olver, D. W. Lozier, R. F. Boisvert, and C. W. Clark (eds.), Digital Library of Mathematical Functions, National Institute of Standards and Technology from http://dlmf.nist.gov/ (release date 2011-07-01), Washington, DC, 2010.
- A. Ranicki, Algebraic and Geometric Surgery, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, Oxford, 2002.
- Edwin H. Spanier, Algebraic topology, Springer-Verlag, New York-Berlin, 1981. Corrected reprint. MR 666554
- Norman Steenrod, The topology of fibre bundles, Princeton Landmarks in Mathematics, Princeton University Press, Princeton, NJ, 1999. Reprint of the 1957 edition; Princeton Paperbacks. MR 1688579
- Robert M. Wald, General relativity, University of Chicago Press, Chicago, IL, 1984. MR 757180, DOI 10.7208/chicago/9780226870373.001.0001
- Peter Woit, Topology and lattice gauge fields, Nuclear Phys. B 262 (1985), no. 2, 284–298. MR 819657, DOI 10.1016/0550-3213(85)90287-1