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Luis Santaló and classical field theory

  • Mariano Galvagno
  • Gaston GiribetEmail author
Article

Abstract

Considered one of the founding fathers of integral geometry, Luis Santaló has contributed to various areas of mathematics. His work has applications in number theory, in the theory of differential equations, in stochastic geometry, in functional analysis, and also in theoretical physics. Between the 1950’s and the 1970’s, he wrote a series of papers on general relativity and on the attempts at generalizing Einstein’s theory to formulate a unified field theory. His main contribution in this subject was to provide a classification theorem for the plethora of tensors that were populating Einstein’s generalized theory. This paper revisits his work on theoretical physics.

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References

  1. 1.
    L. Santaló, Un invariante afin para los cuerpos convexos del espacio des n dimensiones, Portugaliae Math. 8:155 (1949). MathSciNetGoogle Scholar
  2. 2.
    S. Chern, Review: Luis A. Santaló, Integral geometry and geometric probability, Bull. Am. Math. Soc. 83:1289 (1977). CrossRefGoogle Scholar
  3. 3.
    R. Abt, J. Erdmenger, M. Gerbershagen, C. Melby-Thompson, C. Northe, Holographic Subregion Complexity from Kinematic Space, JHEP 1901:12 (2019). ADSMathSciNetCrossRefGoogle Scholar
  4. 4.
    V. Balasubramanian, C. Rabideau, The dual of non-extremal area: differential entropy in higher dimensions [http://arXiv:1812.06985].
  5. 5.
    B. Czech, L. Lamprou, S. McCandlish, J. Sully, Integral Geometry and Holography, JHEP 1510:175 (2015). ADSMathSciNetCrossRefGoogle Scholar
  6. 6.
    R. Gardner, Geometric tomography, Cambridge University Press (1995). Google Scholar
  7. 7.
    J. Girbau, Discurs llegit en la cerimònia del doctorat honoris causa de Luis Antoni Santaló, Universitat Autònoma de Barcelona (1986). Google Scholar
  8. 8.
    A. Naveira and A. Reventós (Eds.), Luís Antonio Santaló: Selected works, SpringerCollected Works in Mathematics (2009). Google Scholar
  9. 9.
    L. Santaló, La última teoría del campo unificado de Einstein, Ciencia e Investigación 9:300 (1953). Google Scholar
  10. 10.
    L. Santaló, El problema de la unificación de los campos, Mundo Atómico 4:11 (1953). Google Scholar
  11. 11.
    L. Santaló, Sobre algunos tensores análogos al de curvatura en espacios con una conexión afín no-simétrica, Revista de la Universidad Nacional de Tucumán 10:19 (1954). Google Scholar
  12. 12.
    L. Santaló, El legado einsteiniano, Ciencia e Investigación 7:289 (1955). Google Scholar
  13. 13.
    L. Santaló, Influencia de Einstein en el campo matemático, Ciencia e Investigación 11:304 (1955). Google Scholar
  14. 14.
    L. Santaló, Sobre las ecuaciones del campo unificado de Einstein, Revista de la Universidad Nacional de Tucumán 12:31 (1959). Google Scholar
  15. 15.
    L. Santaló, Sobre las ecuaciones del campo unificado de Einstein, Revista de la Unión Matemática Argentina 19:196 (1960). MathSciNetGoogle Scholar
  16. 16.
    L. Santaló, On Einstein’s unified field theory, in Prospects in Geometry and Relativity (1966), p. 343. Google Scholar
  17. 17.
    L. Santaló, Sobre algunas teorías asimétricas del campo unificado. Revista de la Real Academia de Ciencias Exactas de Madrid 66:395 (1972). Google Scholar
  18. 18.
    L. Santaló, Unified field theory of Einstein’s type deduced from variational principle, Tensor 25:383 (1972). MathSciNetzbMATHGoogle Scholar
  19. 19.
    A. Einstein, The meaning of relativity, 4th edn., Princeton University Press (1953), p. 321. Google Scholar
  20. 20.
    H. Goenner, On the History of Unified Field Theories. Part II, Living Rev. Rel. 17:5 (2014). CrossRefGoogle Scholar
  21. 21.
    A. Einstein, A generalized theory of gravitation, Rev. Mod. Phys. 20:35 (1948). ADSMathSciNetCrossRefGoogle Scholar
  22. 22.
    A. Einstein, The Bianchi identities in the generalized theory of gravitation, Can. J. Math. 2:120 (1950). MathSciNetCrossRefGoogle Scholar
  23. 23.
    A. Einstein, A generalization of the relativistic theory of gravitation, Ann. Math. 46:578 (1945). MathSciNetCrossRefGoogle Scholar
  24. 24.
    A. Einstein, E. Straus, A generalization of the relativistic theory of gravitation II, Ann. Math. 47:731 (1946). MathSciNetCrossRefGoogle Scholar
  25. 25.
    A. Einstein, B. Kaufman, Sur l’état actuel de la théorie générale de la gravitation in Louis de Broglie, Physicien et Penseur, Volume in honor to Louis de Broglie, Paris (1952), p. 321. Google Scholar
  26. 26.
    A. Einstein, B. Kaufman, Algebraic properties of the field theory of the asymmetric field, Ann. Math. 59:230 (1954). MathSciNetCrossRefGoogle Scholar
  27. 27.
    A. Einstein, B. Kaufman, A new form of the general relativistic field equations, Ann. Math. 62:128 (1955). MathSciNetCrossRefGoogle Scholar
  28. 28.
    L. Santaló, Integral Geometry on surfaces, Duke Math. J. 16:361 (1949). MathSciNetCrossRefGoogle Scholar
  29. 29.
    L. Santaló, On parallel hypersurfaces in the elliptic and hyperbolic n-dimensional space, Proc. Amer. Math. Soc. 1:325 (1950). MathSciNetCrossRefGoogle Scholar
  30. 30.
    L. Santaló, Integral Geometry in general spaces, Proceedings of the International Congress of Mathematicians, Cambridge Mass. Amer. Math. Soc. R. I. 1:482 (1950). Google Scholar
  31. 31.
    L. Santaló, Integral Geometry in projective and affine spaces, Ann. Math. 51:739 (1950). MathSciNetCrossRefGoogle Scholar
  32. 32.
    E. Cartan, Sur les équations de la gravitation d’Einstein, J. Math. Pures Appl. 1:141 (1922). ADSzbMATHGoogle Scholar
  33. 33.
    E. Schrödinger, Generalizations of Einstein theory, in Space-Time Structure, Cambridge University Press (1950), p. 106. Google Scholar
  34. 34.
    L. Eisenhart, The Einstein generalized Riemannian geometry, Proc. Natl. Acad. Sci. USA 50:190 (1963). ADSMathSciNetCrossRefGoogle Scholar
  35. 35.
    V. Hlavatý, Geometry of Einstein’s Unified Field Theory, Noordhoff, Groningen (1957). Google Scholar
  36. 36.
    B. Kaufan, Mathematical struture of the non-symmetric field theory, Proceeding of the 50 anniversary conference on relativity, Bern 1950, Helvetica Physica Acta IV (1956) 227. Google Scholar
  37. 37.
    A. Lichnerowicz, Théories relativistes de la gravitation et l’electromagnétisme, Paris, Masson (1955). Google Scholar
  38. 38.
    M. Tonnelat, La théorie du champ unifié d’Einstein et quelques-uns de ses développements, Paris, Gauthier-Villars (1955). Google Scholar
  39. 39.
    J. Winogradzki, Le group relativiste de la théorie unitaire d’Einstein-Schrödinger, J. Phys. Radium 16:438 (1955). MathSciNetCrossRefGoogle Scholar
  40. 40.
    L. Eisenhart, Non-Riemannian Geometry, Am. Math. Soc. Coll. Pubns VIII (1927). Google Scholar
  41. 41.
    M. Gualtieri, Generalized complex geometry. Oxford University D. Phil thesis. [http://arXiv:math/0401221 [math.DG]].
  42. 42.
    W. Siegel, Superspace duality in low-energy superstrings, Phys. Rev. D 48:2826 (1993). ADSMathSciNetCrossRefGoogle Scholar
  43. 43.
    W. Siegel, Two vierbein formalism for string inspired axionic gravity, Phys. Rev. D 47:5453 (1993). ADSMathSciNetCrossRefGoogle Scholar
  44. 44.
    C. Hull, B. Zwiebach, Double Field Theory, JHEP 0909:099 (2009). ADSMathSciNetCrossRefGoogle Scholar

Copyright information

© EDP Sciences, Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Faculty of Civil and Environmental Engineering, Technion, Israel Institute of TechnologyTechnion City, HaifaIsrael
  2. 2.Physics Department, University of Buenos Aires and IFIBA-CONICET, Ciudad UniversitariaBuenos AiresArgentina
  3. 3.Abdus Salam International Centre for Theoretical Physics, ICTPTriesteItaly

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