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A Novel Interpretation of the Klein-Gordon Equation

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Abstract

The covariant Klein-Gordon equation requires twice the boundary conditions of the Schrödinger equation and does not have an accepted single-particle interpretation. Instead of interpreting its solution as a probability wave determined by an initial boundary condition, this paper considers the possibility that the solutions are determined by both an initial and a final boundary condition. By constructing an invariant joint probability distribution from the size of the solution space, it is shown that the usual measurement probabilities can nearly be recovered in the non-relativistic limit, provided that neither boundary constrains the energy to a precision near ℏ/t 0 (where t 0 is the time duration between the boundary conditions). Otherwise, deviations from standard quantum mechanics are predicted.

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References

  1. DeWitt, B.S.: Quantum theory of gravity I: the canonical theory. Phys. Rev. 160, 1113 (1967)

    Article  MATH  ADS  Google Scholar 

  2. Horton, G., Dewdney, C., Nesteruk, A.: Time-like flows of energy momentum and particle trajectories for the Klein-Gordon equation. J. Phys. A 33, 7337 (2000)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  3. Mostafazadeh, A., Zamani, F.: Quantum mechanics of Klein-Gordon fields, I: Hilbert space, localized states, and chiral symmetry. Ann. Phys. 321, 2183 (2006)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  4. Kleefeld, F.: On some meaningful inner product for real Klein-Gordon fields with positive semi-definite norm. arXiv:quant-ph/0606070 (2006)

  5. Nikolić, H.: Probability in relativistic quantum mechanics and foliation of space-time. Int. J. Mod. Phys. A 22, 6243 (2007)

    Article  MATH  ADS  Google Scholar 

  6. Spekkens, R.W.: Evidence for the epistemic view of quantum states: a toy theory. Phys. Rev. A 75, 032110 (2007)

    Article  ADS  Google Scholar 

  7. Aharonov, Y., Vaidman, L.: Complete description of a quantum system at a given time. J. Phys. A 24, 2315 (1991)

    Article  MathSciNet  ADS  Google Scholar 

  8. Aharonov, Y., Vaidman, L.: The two-state vector formalism: an updated review. In: Muga, J.G., et al. (ed.): Time in Quantum Mechanics. Springer, Berlin (2002)

    Google Scholar 

  9. Sutherland, R.: Causally symmetric Bohm model. arXiv:quant-ph/0601095

  10. Wharton, K.B.: Time-symmetric quantum mechanics. Found. Phys. 37, 159 (2007)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  11. Oeckl, R.: States on timelike hypersurfaces in quantum field theory. Phys. Lett. B 622, 172 (2005). arXiv:hep-th/0505267

    Article  MathSciNet  ADS  Google Scholar 

  12. Oeckl, R.: Probabilities in the general boundary formulation. J. Phys., Conf. Ser. 67, 12049 (2007). arXiv:hep-th/0612076

    Article  ADS  Google Scholar 

  13. Gell-Mann, M., Hartle, J.B.: Time symmetry and asymmetry in quantum mechanics and quantum cosmology. In: Halliwell, J.J., Perez-Mercader, J., Zurek, W. (eds.) Proceedings of the NATO Workshop on the Physical Origins of Time Asymmetry. Cambridge University Press, Cambridge (1994)

    Google Scholar 

  14. Schulman, L.S.: Time’s Arrows and Quantum Measurement. Cambridge University Press, Cambridge (1997)

    Google Scholar 

  15. Cramer, J.G.: Generalized absorber theory and the Einstein-Podolsky-Rosen paradox. Phys. Rev. D 22, 362 (1980)

    Article  MathSciNet  ADS  Google Scholar 

  16. Cramer, J.G.: The transactional interpretation of quantum mechanics. Rev. Mod. Phys. 58, 647 (1986)

    Article  MathSciNet  ADS  Google Scholar 

  17. Rovelli, C.: Quantum Gravity. Cambridge University Press, Cambridge (2004)

    MATH  Google Scholar 

  18. Sakurai, J.J.: Modern Quantum Mechanics. Addison-Wesley, Redwood City (1985)

    Google Scholar 

  19. Hardy, L.: Towards quantum gravity: a framework for probabilistic theories with non-fixed causal structure. J. Phys. A 40, 3081 (2007)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  20. Wharton, K.B.: Extending Hamilton’s principle to quantize classical fields. arXiv:0906.5409 (2009)

  21. Nikolić, H.: Quantum mechanics: myths and facts. Found. Phys. 37, 1563 (2007)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  22. Aharonov, Y., Bohm, D.: Time in the quantum theory and the uncertainty relation for time and energy. Phys. Rev. 122, 1649 (1961)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  23. Montina, A.: Exponential complexity and ontological theories of quantum mechanics. Phys. Rev. A 77, 22104 (2008)

    Article  ADS  Google Scholar 

  24. Bell, J.S.: On the problem of hidden variables in quantum mechanics. Rev. Mod. Phys. 38, 447 (1966)

    Article  MATH  ADS  Google Scholar 

  25. Dirac, P.A.M.: The electron wave equation in de-Sitter space. Ann. Math. 36, 657 (1935)

    Article  MathSciNet  Google Scholar 

  26. Nakanishi, N.: Covariant formulation of the complex-ghost relativistic field theory and the Lorentz noninvariance of the S matrix. Phys. Rev. D 5, 1968 (1972)

    Article  ADS  Google Scholar 

  27. Kleefeld, F.: On symmetries in (anti)causal (non)Abelian quantum theories. Proc. Inst. Math. NAS Ukr. 50, 1367 (2004)

    Google Scholar 

  28. Evans, J., Alsing, P.M., Giorgetti, S., Nandi, K.K.: Matter waves in a gravitational field: an index of refraction for massive particles in general relativity. Am. J. Phys. 69, 1103 (2001)

    Article  ADS  Google Scholar 

  29. Miller, D.J.: Quantum mechanics as a consistency condition on initial and final boundary conditions. arXiv:quant-ph/0607169 (2006)

  30. Cody, W.J.: Chebyshev approximations for the Fresnel integrals. Math. Comput. 22, 450 (1968)

    Article  MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Physics and Astronomy, San José State University, San José, CA, 95192-0106, USA

    K. B. Wharton

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Wharton, K.B. A Novel Interpretation of the Klein-Gordon Equation. Found Phys 40, 313–332 (2010). https://doi.org/10.1007/s10701-009-9398-2

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