Dispersion relations alone cannot guarantee causality
Abstract: We show that linear superpositions of plane waves involving a single-valued, covariantly stable dispersion relation always propagate outside the lightcone, unless . This implies that there is no notion of causality for individual dispersion relations, since no mathematical condition on the function (such as the front velocity or the asymptotic group velocity conditions) can serve as a sufficient condition for subluminal propagation in dispersive media. Instead, causality can only emerge from a careful cancellation that occurs when one superimposes all the excitation branches of a physical model. This is shown to happen automatically in local theories of matter that are covariantly stable. Hence, we find that the need for nonhydrodynamic modes in relativistic fluid mechanics is analogous to the need for antiparticles in relativistic quantum mechanics.
- A. Einstein, Jahrbuch der Radioaktivität und Elektronik 4, 411 (1908).
- R. Tolman, The Theory of the Relativity of Motion (University of Califormia Press, Berkeley, 1917).
- L. Landau and E. Lifshitz, Course of Theoretical Physics v.2: Classical Theory of Fields (Butterworth Heinemann, Amsterdam, 1994).
- S. W. Hawking and G. F. R. Ellis, The Large Scale Structure of Space-Time, Cambridge Monographs on Mathematical Physics (Cambridge University Press, 2011).
- R. M. Wald, General relativity (Chicago Univ. Press, Chicago, IL, 1984).
- J. Rauch, Partial Differential Equations, Graduate Texts in Mathematics (Springer, New York, NY, 1991).
- M. E. Peskin and D. V. Schroeder, An introduction to quantum field theory (Addison-Wesley, Reading, USA, 1995).
- P. H. Eberhard and R. R. Ross, Found. Phys. 2, 127 (1989).
- B. D. Keister and W. N. Polyzou, Phys. Rev. C 54, 2023 (1996), arXiv:hep-th/9605100 [hep-th] .
- L. Brillouin, Wave propagation and group velocity (Academic Press, New York, NY, 1960).
- S. Koide and R. Morino, Phys. Rev. D 84, 083009 (2011), arXiv:1109.1856 [astro-ph.HE] .
- E. Krotscheck and W. Kundt, Communications in Mathematical Physics 60, 171 (1978).
- L. Gavassino, Physics Letters B 840, 137854 (2023).
- L. Gavassino, Physical Review X 12, 041001 (2022), arXiv:2111.05254 [gr-qc] .
- W. Hiscock and L. Lindblom, Physical review D: Particles and fields 31, 725 (1985).
- P. Kostädt and M. Liu, Phys. Rev. D 62, 023003 (2000), arXiv:cond-mat/0010276 [cond-mat.stat-mech] .
- P. K. Kovtun and A. O. Starinets, Phys. Rev. D 72, 086009 (2005).
- P. Kovtun, Journal of High Energy Physics 2019, 34 (2019), arXiv:1907.08191 [hep-th] .
- G. Perna and E. Calzetta, Phys. Rev. D 104, 096005 (2021), arXiv:2108.01114 [hep-ph] .
- L. Hormander, The Analysis of Linear Partial Differential Operators I, Second Edition, Comprehensive Studies in Mathematics (Springer-Verlag, Berlin, 1990).
- G. C. Hegerfeldt, Phys. Rev. D 10, 3320 (1974).
- B. Thaller, The Dirac equation (Springer-Werlag, 1992).
- P. M. Morse and H. Feshbach, Methods of theoretical physics, Internation Series in Pure and Applied Mathematics (McGraw-Hill Book Company, New York, 1953).
- T. Kato, Perturbation Theory for Linear Operators, Classics in Mathematics (Springer-Verlag, Berlin, 1995).
- N. Barat and J. Kimball, Physics Letters A 308, 110 (2003).
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