Abstract
We introduce a class of causal manifolds which contains the globally hyperbolic spacetimes and prove global propagation theorems for sheaves on such manifolds. As an application, we solve globally the Cauchy problem for hyperfunction solutions of hyperbolic systems.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Bär, C., Ginoux, N., Pfäffle, F.: Wave equations on Lorentzian manifolds and quantization, ESI Lectures in Mathematics and Physics. European Mathematical Society (EMS), Zürich (2007)
Beem, J.K., Ehrlich, P.E., Easley, K.L.: Global Lorentzian geometry, 2nd edn. Monographs and Textbooks in Pure and Applied Mathematics, vol. 202. Marcel Dekker, Inc., New York (1996)
Bernal A.N., Sánchez M.: Smoothness of time functions and the metric splitting of globally hyperbolic spacetimes. Commun. Math. Phys. 257(1), 43–50 (2005)
Bony, J.; Schapira, P.: Prolongement et existence des solutions des systémes hyperboliques non stricts á coefficients analytiques, Partial differential equations. Proc. Sympos. Pure Math., Vol. XXIII, Univ. California, Berkeley, Calif., vol. 1971. Amer. Math. Soc., Providence, pp.85–95 (1973) (French)
Brunetti, B., Fredenhagen, K.: Microlocal analysis and interacting quantum field theories: renormalization on physical backgrounds. Commun. Math. Phys., 623–661 (2000)
Brunetti, B., Fredenhagen, K.: Algebraic approach to Quantum Field Theory. ar**v:math-ph/0411072 (2011)
D’Agnolo, A., Schapira, P.: Global propagation on causal manifolds. Asian J. Math. 2(4), 641–653. Mikio Sato: a great Japanese mathematician of the twentieth century (1998)
Fathi, A., Siconolfi, A.: On smooth time functions. In: Mathematical Proceedings, CUP (2011). https://hal.inria.fr/hal-00660452
Geroch R.: Domain of dependence. J. Math. Phys. 11, 37–449 (1970)
Guillermou, S., Schapira, P.: Microlocal theory of sheaves and Tamarkin’s non displaceability theorem, LN of the UMI, pp. 43–85 (2014). ar**v:1106.1576
Hawking, S.W., Ellis, G.F.R.: The large scale structure of space-time, Cambridge University Press, London-New York, Cambridge Monographs on Mathematical Physics, No. 1 (1973)
Kashiwara, M.: Algebraic study of systems of partial differential equations, Mem. Soc. Math. France, vol. 63. Thesis, Tokyo 1970 (1970, in Japanese)
Kashiwara, M.: D-modules and microlocal calculus, Translations of Mathematical Monographs, vol. 217 (2003)
Kashiwara, M., Schapira, P.: Sheaves on manifolds, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 292. Springer, Berlin (1990)
Leray, J.: Hyperbolic Differential Equations. The Institute for Advanced Study, Princeton (1953)
Minguzzi, E., Sánchez, M.: The causal hierarchy of spacetimes, Recent developments in pseudo-Riemannian geometry, ESI Lect. Math. Phys. Eur. Math. Soc., Zürich, pp. 299–358 (2008)
Sato, M., Kawai, T., Kashiwara, M.: Microfunctions and pseudo-differential equations, Hyperfunctions and pseudo-differential equations. In: Proc. Conf., Katata (1971); dedicated to the memory of André Martineau, Lecture Notes in Math., vol. 287. Springer, Berlin, pp. 265–529 (1973)
Schapira, P.: Hyperbolic systems and propagation on causal manifolds. Lett. Math. Phys. 103(10), 1149–1164 (2013). ar**v:1305.3535
Sorkin, R.D., Woolgar, E.: A causal order for spacetimes with \({C^0}\) Lorentzian metrics: proof of compactness of the space of causal curves. Class. Quant. Gravity 13(7), 1971–1993 (1996)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Jubin, B., Schapira, P. Sheaves and D-Modules on Lorentzian Manifolds. Lett Math Phys 106, 607–648 (2016). https://doi.org/10.1007/s11005-016-0832-z
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11005-016-0832-z