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The effectiveness of mathematics in physics of the unknown

  • Alexei Grinbaum
Article

Abstract

If physics is a science that unveils the fundamental laws of nature, then the appearance of mathematical concepts in its language can be surprising or even mysterious. This was Eugene Wigner’s argument in 1960. I show that another approach to physical theory accommodates mathematics in a perfectly reasonable way. To explore unknown processes or phenomena, one builds a theory from fundamental principles, employing them as constraints within a general mathematical framework. The rise of such theories of the unknown, which I call blackbox models, drives home the unsurprising effectiveness of mathematics. I illustrate it on the examples of Einstein’s principle theories, the S-matrix approach in quantum field theory, effective field theories, and device-independent approaches in quantum information.

Keywords

Mathematics Physics Wigner Principle theory S-matrix Effective field theory Device-independence Unknown 

Notes

Acknowledgements

Many thanks to Bryan Roberts for helpful remarks.

References

  1. Aharon, N., Massar, S., Pironio, S., & Silman, J. (2016). Device-independent bit commitment based on the CHSH inequality. New Journal of Physics, 18, 025014. arXiv:1511.06283.
  2. Appelquist, T., & Bjorken, J. D. (1971). Weak interactions at high energies. Physical Review D, 4(12), 3726–3737.CrossRefGoogle Scholar
  3. Bancal, J.-D. (2013). On the device-independent approach to quantum physics: Advances in quantum nonlocality and multipartite entanglement detection. Geneva: Springer.Google Scholar
  4. Bancal, J.-D., Gisin, N., Liang, Y.-C., & Pironio, S. (2011). Device-independent witnesses of genuine multipartite entanglement. Physical Review Letters, 106, 250404.CrossRefGoogle Scholar
  5. Bardyn, C.-E., Liew, T. C. H., Massar, S., McKague, M., & Scarani, V. (2009). Device-independent state estimation based on Bell’s inequalities. Physical Review A, 80, 062327.CrossRefGoogle Scholar
  6. Barrett, J., Hardy, L., & Kent, A. (2005). No signaling and quantum key distribution. Physical Review Letters, 95(1), 010503.CrossRefGoogle Scholar
  7. Baumeler, Ä., & Wolf, S. (2014). Perfect signaling among three parties violating predefined causal order. In Proceedings of 2014 IEEE international symposium on information theory (ISIT) (pp. 526–530). Red Hook, NY: Institute of Electrical and Electronics Engineers (IEEE).Google Scholar
  8. Bell, J. (1964). On the Einstein–Podolsky–Rosen paradox. Physica, 1, 195–200.Google Scholar
  9. Bilaniuk, O. M. P, & Sudarshan, E. C. G. (1969). Particles beyond the light barrier. Physics Today, 22, 43–51. This is the first known reference in press. Attribution to Gell-Mann is however indisputable.Google Scholar
  10. Brown, H. (2005). Physical relativity: Space-time structure from a dynamical perspective. Oxford: Oxford University Press.CrossRefGoogle Scholar
  11. Brown, H., & Timpson, C. (2006). Why special relativity should not be a template for a fundamental reformulation of quantum mechanics. In W. Demopoulous, & I. Pitowsky (Eds.), Physical theory and its interpretation (pp. 29–42). Amsterdam: Springer. arXiv:quant-ph/0601182.
  12. Cirel’son, B. S. (1980). Quantum generalizations of Bell’s inequality. Letters in Mathematical Physics, 4(2), 93–100.CrossRefGoogle Scholar
  13. Clauser, J., Holt, R., Horne, M., & Shimony, A. (1969). Proposed experiment to test local hidden-variable theories. Physical Review Letters, 23, 880–884.CrossRefGoogle Scholar
  14. Colbeck, R. (2006). Quantum and relativistic protocols for secure multi-party computation. PhD thesis, University of Cambridge.Google Scholar
  15. Darrigol, O. (2014). Physics and necessity. Oxford: Oxford University Press.CrossRefGoogle Scholar
  16. Dyson, F. (1949a). The radiation theories of Tomonaga, Schwinger, and Feynman. Physical Review, 75, 486–502.CrossRefGoogle Scholar
  17. Dyson, F. (1949b). The \(S\)-matrix in quantum electrodynamics. Physical Review, 75, 1736–1755.CrossRefGoogle Scholar
  18. Einstein, A. (1905). Über einen die Erzeugung und Verwandlung des Lichtes betreffenden heuristischen Gesichtspunkt. Annalen der Physik, 17, 132–148.CrossRefGoogle Scholar
  19. Einstein, A. (1949). Autobiographical notes. In P. Schlipp (Ed.), Albert Einstein: Philosopher–scientist (Vol. 7, pp. 1–94)., The library of living philosophers IL: Open Court.Google Scholar
  20. Einstein, A. (1982). What is the theory of relativity? London Times, 1919. Reprinted in: A. Einstein, Ideas and opinions, Crown Publishers, New York.Google Scholar
  21. Einstein, A. (1987). Letter to Maurice Solovine, May 7, 1952. In Letters to Solovine (pp. 121–125). New York: Philosophical Library.Google Scholar
  22. Einstein, A. (2004). Address to a scientific meeting in Zurich, 1911. Cited in: P. Galison, Einstein’s Clocks, Poincaré’s Maps. Empires of Time (p. 268). London: Hodder and Stoughton.Google Scholar
  23. Euler, H. (1936). Über die Streuung von Licht an Licht nach der Diracschen Theorie. Ann. der Physik, 418, 398–448.CrossRefGoogle Scholar
  24. Euler, H., & Kockel, B. (1935). Über die Streuung von Licht an Licht nach der Diracschen Theorie. Naturwissenschaften, 23, 246.CrossRefGoogle Scholar
  25. Fock, V. (1959). The theory of space, time and gravitation. Gostekhizdat, Moscow, 1955. English edition: Pergamon Press.Google Scholar
  26. Fock, V. (1971). The principle of relativity with respect to observation in modern physics. Vestnik AN SSSR, 4, 8–12.Google Scholar
  27. Friedman, M. (2001). Dynamics of reason. Stanford: CSLI Publications.Google Scholar
  28. Frisch, M. (2005). Mechanisms, principles, and Lorentz’s cautious realism. Studies in the History and Philosophy of Modern Physics, 36, 659–679.CrossRefGoogle Scholar
  29. Giddings, S. B. (2013). The gravitational \(S\)-matrix. In 48th course of the Erice International School of Subnuclear Physics, volume 48 of Subnuclear Ser. (pp. 93–147). arXiv:1105.2036.
  30. Grinbaum, A. (2017). How device-independent approaches change the meaning of physical theory. Studies in the History and Philosophy of Modern Physics. doi: 10.1016/j.shpsb.2017.03.003. arXiv:1512.01035.
  31. Grythe, I. (1982). Some remarks on the early \(S\)-matrix. Centaurus, 26, 198–203.CrossRefGoogle Scholar
  32. Heisenberg, W. (1925). Über quantentheoretische Umdeutung kinematischer und mechanischer Beziehungen. Zeit. f. Phys., 33, 879–893.CrossRefGoogle Scholar
  33. Heisenberg, W. (1938). Über die in der Theorie der Elementarteilchen auftretende universelle Länge. Annals of Physics, 424, 20–33.CrossRefGoogle Scholar
  34. Heisenberg, W. (1942). Die “beobachbaren Grössen” in der Theorie der Elementarteilchen. Zeit. f. Phys., 120, 513–539. English translation in [38, pp. 1030–1031].Google Scholar
  35. Heisenberg, W., & Euler, H. (1936). Folgerungen aus der Diracschen Theorie des Positrons. Z. Phys., 98, 714.CrossRefGoogle Scholar
  36. Lange, M. (2014). Did Einstein really believe that principle theories are explanatorily powerless? Perspectives on Science, 22(4), 449–463.CrossRefGoogle Scholar
  37. Mayers, D. & Yao, A. (1998).Quantum cryptography with imperfect apparatus. In FOCS 1998: Proceedings of the 39th annual symposium on foundations of computer science (pp. 503–509). Los Alamitos, CA, USA: IEEE Computer Society.Google Scholar
  38. Mehra, J., & Rechenberg, H. (2001). The conceptual completion and the extensions of quantum mechanics 1932–1941. Epilogue: Aspects of the Further Development of Quantum Theory 1942–1999, volume 6 of The Historical Development of Quantum Theory. New York: Springer.Google Scholar
  39. Noyes, H. P. (ed.) (1954). Proceedings of the fourth annual Rochester conference on high energy nuclear physics. The University of Rochester and The National Science Foundation.Google Scholar
  40. Pauli, W. (1946). Letter to W. Heisenberg, 9 September 1946. Cited in Rechenberg (1989, p. 566).Google Scholar
  41. Pauli, W. (1948). Letter to W. Heisenberg, 20 October 1948. Cited in Rechenberg (1989, p. 568).Google Scholar
  42. Pich, A. (1999). Effective field theory. In R. Gupta, et al. (Eds.), Proceedings of Les Houches summer school of theoretical physics ‘Probing the Standard Model of Particle Interactions’ (vol. II, p. 949). Amsterdam: Elsevier. arXiv:hep-ph/9806303.
  43. Pironio, S., et al. (2010). Random numbers certified by Bell’s theorem. Nature, 464, 1021–1024.CrossRefGoogle Scholar
  44. Popescu, S. (2014). Nonlocality beyond quantum mechanics. Nature Physics, 10, 264–270.CrossRefGoogle Scholar
  45. Popescu, S., & Rohrlich, D. (1994). Nonlocality as an axiom for quantum theory. Foundations of Physics, 24, 379. arXiv:quant-ph/9508009.
  46. Rechenberg, H. (1989). The early \(S\)-matrix theory and its propagation (1942–1952). In L. M. Brown, M. Dresden, L. Hoddeson, & M. West (Eds.), Pions to quarks: Particle physics in the 1950s (pp. 551–578). Cambridge: Cambridge University Press.CrossRefGoogle Scholar
  47. Shankar, R. (1999). Effective field theory in condensed matter physics. In T. Y. Cao (Ed.), Conceptual foundations of quantum field theory (pp. 47–55). Cambridge: Cambridge University Press.Google Scholar
  48. Veneziano, G. (1968). Construction of a crossing-symmetric, Regge behaved amplitude for linearly rising trajectories. Nuovo Cimento A, 57, 190–197.CrossRefGoogle Scholar
  49. Weinberg, S. (1967). Dynamical approach to current algebra. Physical Review Letters, 18, 188–191.CrossRefGoogle Scholar
  50. Weinberg, S. (1979). Phenomenological lagrangians. Physica, 96A, 327–340.CrossRefGoogle Scholar
  51. Weinberg, S. (1989). Precision tests of quantum mechanics. Physical Review Letters, 62, 485–488.CrossRefGoogle Scholar
  52. Weinberg, S. (1996). Sokal’s hoax. The New York Review of Books, XLII, I(13), 11–15.Google Scholar
  53. Wentzel, G. (1947). Recent research in meson theory. Reviews of Modern Physics, 19, 1–18.CrossRefGoogle Scholar
  54. Wheeler, J. (1937). On the mathematical description of light nuclei by the method of resonating group structure. Physical Review, 52, 1107–1122.CrossRefGoogle Scholar
  55. Wigner, E. (1960). The unreasonable effectiveness of mathematics in the natural sciences. Communications in Pure and Applied Mathematics, 13, 1–14.CrossRefGoogle Scholar
  56. Wilson, K. G. (1971). The renormalization group and strong interactions. Physical Review D, 3, 1818.CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media B.V. 2017

Authors and Affiliations

  1. 1.CEA-Saclay/IRFU/LARSIMGif-sur-YvetteFrance

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