Foundations of Physics

, Volume 47, Issue 4, pp 532–552 | Cite as

Theory of Stochastic Schrödinger Equation in Complex Vector Space

  • Kundeti Muralidhar


A generalized Schrödinger equation containing correction terms to classical kinetic energy, has been derived in the complex vector space by considering an extended particle structure in stochastic electrodynamics with spin. The correction terms are obtained by considering the internal complex structure of the particle which is a consequence of stochastic average of particle oscillations in the zeropoint field. Hence, the generalised Schrödinger equation may be called stochastic Schrödinger equation. It is found that the second order correction terms are similar to corresponding relativistic corrections. When higher order correction terms are neglected, the stochastic Schrödinger equation reduces to normal Schrödinger equation. It is found that the Schrödinger equation contains an internal structure in disguise and that can be revealed in the form of internal kinetic energy. The internal kinetic energy is found to be equal to the quantum potential obtained in the Madelung fluid theory or Bohm statistical theory. In the rest frame of the particle, the stochastic Schrödinger equation reduces to a Dirac type equation and its Lorentz boost gives the Dirac equation. Finally, the relativistic Klein–Gordon equation is derived by squaring the stochastic Schrödinger equation. The theory elucidates a logical understanding of classical approach to quantum mechanical foundations.


Foundations of quantum mechanics Stochastic electrodynamics Zeropoint field Complex vector algebra 


  1. 1.
    Boyer, T.H.: Random electrodynamics: the theory of classical electrodynamics with classical electromagnetic zeropoint radiation. Phys. Rev. D 11, 790–808 (1975)CrossRefADSGoogle Scholar
  2. 2.
    Boyer, T.H.: A brief survey of stochastic electrodynamics. In: Barut, O.A. (ed.) Foundations of Radiation Theory and Quantum Electrodynamics, pp. 49–63. Springer, New York (1980)CrossRefGoogle Scholar
  3. 3.
    Cole, D.C.: Reviewing and extending some recent work on stochastic electrodynamics. In: Lakhtakia, A. (ed.) Essays on the Formal Aspects of Electromagnetic Theory, pp. 501–532. World Scientific Publ. Co., Singapore (1993)CrossRefGoogle Scholar
  4. 4.
    de la Peña, L., Cetto, A.M.: Quantum Dice: An Introduction to Stochastic Electrodynamics. Kluwer, Dordrecht (1996)CrossRefGoogle Scholar
  5. 5.
    Schrödinger, E.: Quantisation as a Problem of Proper Values (Part I). Collected Papers of Wave Mechanics, pp. 1–12. Blackie and Son, London (1928)Google Scholar
  6. 6.
    Fenyes, I.: Eine wahrscheinlichkeitstheoretische begründung und interpretation der quantenmechanik. Zeit. Phys. 132, 81 (1952)Google Scholar
  7. 7.
    Nelson, E.: Quantum Fluctuations. Princeton University Press, Princeton (1985)MATHGoogle Scholar
  8. 8.
    Nelson, E.: Derivation of Schrödinger equation from Newtonian mechanics. Phys. Rev. 150, 1079–1085 (1966)CrossRefADSGoogle Scholar
  9. 9.
    Della Riccia, G., Wiener, N.: Wave mechanics in classical phase space, Brownian motion, and quantum theory. J. Math. Phys. 7, 1372 (1966)MathSciNetCrossRefADSGoogle Scholar
  10. 10.
    Favella, I.F.: Brownian motions in quantum mechanics. Ann. Inst. Henri Poincáre 7, 77 (1967)MATHGoogle Scholar
  11. 11.
    de la Peña, L., Cetto, A.M.: Stochastic theory for classical and quantum mechanical systems. Found. Phys. 5, 355 (1975)MathSciNetCrossRefADSGoogle Scholar
  12. 12.
    de la Peña-Auerback, L.: New formulation of stochastic theory and quantum mechanics. J. Math. Phys. 10, 1620–1630 (1969)CrossRefMATHADSGoogle Scholar
  13. 13.
    de la Peña-Auerback, L.: Stochastic theory of quantum mechanics for particles with spin. J. Math. Phys. 12, 453–461 (1971)CrossRefADSGoogle Scholar
  14. 14.
    de la Peña, L., Cetto, A.M., Hernández, A.V.: The Emerging Quantum: The Physics Behind Quantum Mechanics. Springer, Cham (2015)MATHGoogle Scholar
  15. 15.
    Cetto, A.M., de la Peña, L.: Specificity of Schrödinger equation. Quantum Stud. Math. Found. 2, 275–287 (2015)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Cavalleri, G.: Schrödinger’s equation as a consequence of of zitterbewegung. Lett. Nuovo Cimento. 43, 285 (1985)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Cavalleri, G., Mauri, G.: Integral expansion often reducing the density gradient expansion, extended to non-Markov stochastic process: Consequent non-Markovian stochastic equation whose leading terms coincide with Schrödinger’s. Phys. Rev. B 41, 6751–6758 (1990)MathSciNetCrossRefADSGoogle Scholar
  18. 18.
    Cavalleri, G., Zecca, A.: Interpretation of Schrödinger like equation derived from non-Markovian process. Phys. Rev. B 43, 3223–3227 (1991)CrossRefADSGoogle Scholar
  19. 19.
    Wigner, E.: On the quantum correction for thermodynamic equilibrium. Phys. Rev. 40, 749 (1932)CrossRefMATHADSGoogle Scholar
  20. 20.
    Dechoum, K., França, H.M., Malta, C.P.: Towards a classical reinterpretation of the Schrödinger equation according to stochastic electrodynamics. In: Amoroso, R.L., et al. (eds.) Gravitation and Cosmology: From the Hubble Radius to the Planck Scale, pp. 393–400. Kluwer, Dordrecht (2002)CrossRefGoogle Scholar
  21. 21.
    Faria, A.J., França, H.M., Gomes, G.G., Sponchiado, R.C.: The vacuum electromagnetic fields and the Schrödinger equation. Found. Phys. 37, 1296–1305 (2007)CrossRefMATHADSGoogle Scholar
  22. 22.
    Dechoum, K., França, H.M., Malta, C.P.: Classical aspects of Pauli-Schrödinger equation. Phys. Lett. A 248, 93–102 (1998)MathSciNetCrossRefMATHADSGoogle Scholar
  23. 23.
    Olavo, L.S.F.: Foundations of quantum mechanics: non-relativistic theory. Physica A 262, 197–214 (1999)MathSciNetCrossRefADSGoogle Scholar
  24. 24.
    Olavo, L.S.F.: Foundations of quantum mechanics: Connection with stochastic processes. Phys. Rev. A 61, 052109-1–052109-14 (2000)CrossRefADSGoogle Scholar
  25. 25.
    Hall, M.J.W., Reginatto, M.: Schrödinger equation from exact uncertainty principle. J. Phys. A 35, 3289–3303 (2002)MathSciNetCrossRefMATHADSGoogle Scholar
  26. 26.
    Schleich, W.P., Greenberger, D.M., Kobe, D.H., Scully, M.O.: Schrödinger equation revisited. Proc. Natl. Acad. Sci. USA 110, 5374–5379 (2013)MathSciNetCrossRefMATHADSGoogle Scholar
  27. 27.
    Schleich, W.P., Greenberger, D.M., Kobe, D.H., Scully, M.O.: A wave equation interpolating between classical and quantum mechanics. Phys. Scr. 90, 108009 (2015)CrossRefADSGoogle Scholar
  28. 28.
    Grössing, G.: Sub-quantum thermodynamics as a basis of emergent quantum mechanics. Entropy 12, 1975–2044 (2010)MathSciNetCrossRefMATHADSGoogle Scholar
  29. 29.
    Grössing, G.: The vacuum fluctuation theorem: exact Schrödinger equation via nonequilibrium thermodynamics. Phys. Let. A 372, 4556–4563 (2008)MathSciNetCrossRefMATHADSGoogle Scholar
  30. 30.
    Grössing, G.: On the thermodynamic origin of quantum potential. Physica A 388, 811–823 (2009)MathSciNetCrossRefADSGoogle Scholar
  31. 31.
    Sakurai, J.J.: Advanced Quantum Mechanics. Pearson Education, New Delhi (2007)Google Scholar
  32. 32.
    Barut, A.O., Bracken, A.J.: Zitterbewegung and the internal geometry of electron. Phys. Rev. D 23, 2454 (1981)MathSciNetCrossRefADSGoogle Scholar
  33. 33.
    Bhabha, H.J., Corben, H.C.: General classical theory of spinning particles in a Maxwell’s field. Proc. R. Soc. Lond. A 178(974), 273–314 (1941)MathSciNetCrossRefMATHADSGoogle Scholar
  34. 34.
    Corben, H.C.: Spin in classical and quantum theories. Phys. Rev. 121, 1833–1839 (1961)CrossRefMATHADSGoogle Scholar
  35. 35.
    Corben, H.C.: Classical and Quantum Theories of Spinning Particles. Holden and Day, New York (1968)Google Scholar
  36. 36.
    Mathisson, M.: Neue mekhanik materietter system. Acta Phys. Pol. 6, 163–200 (1937)Google Scholar
  37. 37.
    Barut, O.A., Zanghi, A.J.: Classical model of the dirac electron. Phys. Rev. Lett. 52, 2009–2012 (1984)MathSciNetCrossRefADSGoogle Scholar
  38. 38.
    Salesi, G.: The spin and Madelung fluid. Mod. Phys. Lett. A 11, 1815–1853 (1996)MathSciNetCrossRefMATHADSGoogle Scholar
  39. 39.
    Recami, E., Salesi, G.: Kinematics and hydrodynamics of spinning particles. Phys. Rev. A 57, 98–105 (1998)CrossRefMATHADSGoogle Scholar
  40. 40.
    Salesi, G., Recami, E.: A veleocity field and operator for spinning particles in (nonrelativistic) quantum mechanics. Found. Phys. 28, 763–773 (1998)MathSciNetCrossRefADSGoogle Scholar
  41. 41.
    Salesi, G., Recami, E.: Hydrodynamical reformulation and quantum limit of the Barut-Zanghi theory. Found. Phys. Lett. 10, 533–546 (1997)MathSciNetCrossRefMATHGoogle Scholar
  42. 42.
    Pavšič, M., Recami, E., Rodrigues, W.A., Maccarrone, G.D., Raciti, F., Saleci, G.: Spin and electron structure. Phys. Lett. B 318, 481 (1993)MathSciNetCrossRefADSGoogle Scholar
  43. 43.
    Muralidhar, K.: Complex vector formalism of harmonic oscillator in geometric algebra: particle mass, spin and dynamics in complex vector space. Found. Phys. 44, 265–295 (2014)MathSciNetCrossRefMATHADSGoogle Scholar
  44. 44.
    Muralidhar, K.: Algebra of complex vectors and applications in electromagnetic theory and quantum mechanics. Mathematics 3, 781–842 (2015)CrossRefMATHGoogle Scholar
  45. 45.
    Muralidhar, K.: Classical origin of quantum spin. Apeiron 6, 146–160 (2011)Google Scholar
  46. 46.
    Muralidhar, K.: The spin bivector and zeropoint energy in geometric algebra. Adv. Stud. Theor. Phys. 6, 675–686 (2012)Google Scholar
  47. 47.
    Hestenes, D.: Zitterbewegung in quantum mechanics. Found. Phys. 40, 1–54 (2010)MathSciNetCrossRefMATHADSGoogle Scholar
  48. 48.
    Bohm, D.: A suggested interpretation of the quantum theory in terms of hidden variables I. Phys. Rev. 85, 166–179 (1952)MathSciNetCrossRefMATHADSGoogle Scholar
  49. 49.
    Bohm, D.: A suggested interpretation of the quantum theory in terms of hidden variables II. Phys. Rev. 85, 180–193 (1952)MathSciNetCrossRefMATHADSGoogle Scholar
  50. 50.
    Muralidhar, K.: Classical approach to quantum condition and biaxial spin connection to the Schrödinger equation. Quantum Stud. Math. Found. 3, 31–39 (2016)MathSciNetCrossRefMATHGoogle Scholar
  51. 51.
    Hestenes, D.: Spin and uncertainty in the interpretation of quantum mechanics. Am. J. Phys. 47, 399–415 (1979)MathSciNetCrossRefADSGoogle Scholar
  52. 52.
    Snyder, H.S.: Quantized space-time. Phys. Rev. 71, 38–41 (1947)MathSciNetCrossRefMATHADSGoogle Scholar
  53. 53.
    Kempf, A., Mangano, G., Mann, R.B.: Hilbert space representation of the minimal length uncertainty relation. Phys. Rev. D 52, 1108 (1995)MathSciNetCrossRefADSGoogle Scholar
  54. 54.
    Hestenes, D.: Oersted medal lecture 2002: Reforming the mathematical language of physics. Am. J. Phys. 71, 104 (2003)CrossRefADSGoogle Scholar
  55. 55.
    Hestenes, D.: Space-Time Algebra. Gordon and Breach Science Publishers, New York (1966)MATHGoogle Scholar
  56. 56.
    Doran, C., Lasenby, A.: Geometric Algebra for Physicists. Cambridge University Press, Cambridge (2003)CrossRefMATHGoogle Scholar

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© Springer Science+Business Media New York 2017

Authors and Affiliations

  1. 1.Physics DepartmentNational Defence Academy KhadakwaslaPuneIndia

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