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RETRACTED: Non-Hermiticity of tunneling time in a spacetime-symmetric extension of nonrelativistic quantum mechanics

Arlans J. S. de Lara* and Marcus W. Beims†

  • Departamento de Física, Universidade Federal do Paraná, 81531-990 Curitiba, Paraná, Brazil

  • *arlansslara@gmail.com
  • †mbeims@fisica.ufpr.br
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Phys. Rev. A 107, 052220 – Published 30 May, 2023Retraction Phys. Rev. A 109, 049903 (2024)

DOI: https://doi.org/10.1103/PhysRevA.107.052220

Abstract

This article has been retracted: see Phys. Rev. A 109, 049903 (2024)

Time continues to be an intriguing physical property in the modern era. On the one hand, we have the classical and relativistic notions of time, where space and time have the same hierarchy, which is essential in describing events in spacetime. On the other hand, in quantum mechanics time appears as a classical parameter, meaning that it does not have an uncertainty relation with its canonical conjugate. In this work, we use a recent spacetime-symmetric proposal [Phys. Rev. A 95, 032133 (2017)] that tries to solve the unbalance in nonrelativistic quantum mechanics by extending the usual Hilbert space, having the time parameter t and the position operator X̂ in one subspace and the position parameter x and time operator T in the other subspace. Time as an operator is better suitable for describing tunneling processes. We then solve the 1/2-fractional integrodifferential equation for a particle subjected to strong and weak potential limits and obtain an analytical expression for the tunneling time through a rectangular barrier. Using a Gaussian energy distribution, we demonstrate that, for wavepackets well resolved in time, the expectation value of the operator T is the energy average of the classical time Tclass=∂S/∂E, where S is the classical action, which can be real or imaginary. For wavepackets not well resolved in time, the contribution of Tclass consistently vanishes, and solely properties of the energy distribution contribute to T. We show that the time of travel for nontunneling particles is purely real. When tunneling is involved, complex arrival times emerge, becoming a signature of tunneling. Furthermore, we apply our results to a constant energy distribution, obtaining pure imaginary times for energies below the barrier while obtaining complex times for particles with a wavepacket spreading energies below and above the barrier, and show a comparison to previous works.

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Physics Subject Headings (PhySH)

Erratum

Retraction: Non-Hermiticity of tunneling time in a spacetime-symmetric extension of nonrelativistic quantum mechanics [Phys. Rev. A 107, 052220 (2023)]

Arlans J. S. de Lara and Marcus W. Beims
Phys. Rev. A 109, 049903 (2024)

See Also

Traveling time in a spacetime-symmetric extension of nonrelativistic quantum mechanics

Arlans J. S. de Lara and Marcus W. Beims
Phys. Rev. A 110, 012216 (2024)

Article Text

References (86)

  1. Y. Aharanov and D. Bohm, Phys. Rev. 122, 1649 (1961).
  2. M. Razavy, Am. J. Phys. 35, 955 (1967).
  3. M. Razavy, Il N. Cim. B 63, 271 (1969).
  4. G. R. Allcock, Ann. Phys. (NY) 53, 253 (1969).
  5. G. R. Allcock, Ann. Phys. (NY) 53, 286 (1969).
  6. G. R. Allcock, Ann. Phys. (NY) 53, 311 (1969).
  7. J. Kijowski, Rep. Math. Phys. 6, 361 (1974).
  8. A. L. Xavier and M. A. M. de Aguiar, Phys. Rev. Lett. 79, 3323 (1997).
  9. J. S. Briggs and J. M. Rost, Eur. Phys. J. D 10, 311 (2000).
  10. J. S. Briggs and J. M. Rost, Found. of Phys. 31, 693 (2001).
  11. J. S. Briggs, J. Phys.: Conf. Ser. 99, 012002 (2008).
  12. R. Brunetti, K. Fredenhagen, and M. Hoge, Found. Phys. 40, 1368 (2010).
  13. Time in Quantum Mechanics, 2nd ed., edited by J. G. Muga, R. S. Mayato, and I. L. Gusquiza (Springer, Berlin, 2008), Vol. 1.
  14. N. Grot, C. Rovelli, and R. S. Tate, Phys. Rev. A 54, 4676 (1996).
  15. V. Delgado and J. G. Muga, Phys. Rev. A 56, 3425 (1997).
  16. J. G. Muga, C. R. Leavens, and J. P. Palao, Phys. Rev. A 58, 4336 (1998).
  17. I. L. Egusquiza and J. G. Muga, Phys. Rev. A 61, 012104 (1999).
  18. J. G. Muga and C. R. Leavens, Phys. Rep. 338, 353 (2000).
  19. Classical and Quantum Gravity: Theory, Analysis and Applications, 1st ed., edited by E. Anderson (Nova, New York, 2012).
  20. E. A. Galapon, R. F. Caballar, and R. T. Bahague, Phys. Rev. Lett. 93, 180406 (2004).
  21. E. A. Galapon, R. F. Caballar, and R. T. Bahague, Phys. Rev. A 72, 062107 (2005).
  22. E. A. Galapon, F. Delgado, J. G. Muga, and I. Egusquiza, Phys. Rev. A 72, 042107 (2005).
  23. E. A. Galapon, Int. J. Mod. Phys. A 21, 6351 (2006).
  24. E. A. Galapon and A. Villaneuva, J. Phys. A: Math. Theor. 41, 455302 (2008).
  25. E. A. Galapon, Proc. Roy. Soc. A 465, 71 (2009).
  26. E. A. Galapon and J. J. Magadan, Ann. Phys. (NY) 397, 278 (2018).
  27. P. C. Flores and E. A. Galapon, Phys. Rev. A 105, 062208 (2022).
  28. C. Rovelli, Class. Quantum Grav. 8, 317 (1991).
  29. B. Dittrich, Class. Quantum Grav. 23, 6155 (2006).
  30. B. Dittrich, Gen. Relativ. Gravit. 39, 1891 (2007).
  31. M. Bojowald, P. A. Höhn, and A. Tsobanjan, Class. Quantum Grav. 28, 035006 (2011).
  32. P. Małkiewicz, Class. Quantum Grav. 34, 145012 (2017).
  33. F. Giacomini, E. Castro-Ruiz, and C. Brukner, Nat. Commun. 10, 494 (2019).
  34. R. Gambini and J. Pullin, Universe 6, 236 (2020).
  35. M. Bojowald and A. Tsobanjan, Commun. Math. Phys. 382, 547 (2021).
  36. A. Singh, Quantum Stud.: Math. Found. 9, 35 (2022).
  37. G. Czelusta and J. Mielczarek, Phys. Rev. D 105, 126005 (2022).
  38. A. P. French, Special Relativity, 1st ed. (CRC, London, 1968).
  39. C. Cohen-Tannoudji, B. Diu, and F. Laloë, Quantum Mechanics, 1st ed. (Wiley-VCH, New York, 1991), Vol. 1.
  40. J. J. Sakurai and J. Napolitano, Mecânica Quântica Moderna, Portuguese ed. (Bookman, Porto Alegre, Brazil, 2013).
  41. E. Merzbacher, Quantum Mechanics, 3rd ed. (John Wiley & Sons, New York, 1999).
  42. W. Pauli, Handbuch der Physik, 1st ed. (Springer, New York, 1933), Vol. 24.
  43. D. N. Page and W. K. Wootters, Phys. Rev. D 27, 2885 (1983).
  44. V. Giovannetti, S. Lloyd, and L. Maccone, Phys. Rev. D 92, 045033 (2015).
  45. L. Maccone and K. Sacha, Phys. Rev. Lett. 124, 110402 (2020).
  46. L. R. S. Mendes and D. O. Soares-Pinto, Proc. R. Soc. A. 475, 20190470 (2019).
  47. B. S. DeWitt, Phys. Rev. 160, 1113 (1967).
  48. L. Mandelstam and I. G. Tamm, J. Phys. USSR 9, 249 (1945).
  49. N. Margolus and L. B. Levitin, Physica D 120, 188 (1998).
  50. E. O. Dias, Phys. Rev. A 103, 012219 (2021).
  51. M. Fadel and L. Maccone, Phys. Rev. A 104, L050204 (2021).
  52. C. Foti, A. Coppo, G. Barni, A. Cuccoli, and P. Verrucchi, Nat. Commun. 12, 1787 (2021).
  53. P. A. Höhn, A. R. H. Smith, and M. P. E. Lock, Phys. Rev. D 104, 066001 (2021).
  54. P. A. Höhn, A. R. H. Smith, and M. P. E. Lock, Front. Phys. 9, 587083 (2021).
  55. E. O. Dias and F. Parisio, Phys. Rev. A 95, 032133 (2017).
  56. R. Ximenes, F. Parisio, and E. O. Dias, Phys. Rev. A 98, 032105 (2018).
  57. M. Büttiker and R. Landauer, Phys. Rev. Lett. 49, 1739 (1982).
  58. M. Büttiker, Phys. Rev. B 27, 6178 (1983).
  59. E. P. Wigner, Phys. Rev. 98, 145 (1955).
  60. L. A. MacColl, Phys. Rev. 40, 621 (1932).
  61. E. H. Hauge and J. A. Støvneng, Rev. Mod. Phys. 61, 917 (1989).
  62. R. Landauer and T. Martin, Rev. Mod. Phys. 66, 217 (1994).
  63. T. Rivlin, E. Pollak, and R. S. Dumont, New J. Phys. 23, 063044 (2021).
  64. The motivation for the minus signal in this definition is similar to the position-momentum relation in the usual quantum mechanics: The application of the momentum operator on a plane-wave exp[−i(Et−px)/ℏ] gives +p. In contrast, a similar energy operator should give the result E. Since p is written as −i∂x/ℏ in the position representation, the negative signal of the commutator allows us to write the energy operator as +i∂t/ℏ in the new Hilbert space.
  65. R. Carnap, Logical Foundations of Probability, 2nd ed. (The University of Chicago Press, Chicago, 1962).
  66. M. F. Savi and R. M. Angelo, Braz. J. Phys. 47, 333 (2017).
  67. In our opinion, the Caputo fractional derivative has an adequate physical interpretation since its derivative of a constant vanishes. Besides that, the Laplace transform of the Caputo derivative depends only on initial conditions of integer order.
  68. K. B. Oldham and J. Spanier, The Fractional Calculus: Theory and Applications of Differentiation and Integration to Arbitrary Order, 1st ed. (Dover, New York, 2006).
  69. Fractional Quantum Mechanics, 1st ed., edited by N. Laskin (World Scientific, Singapore, 2018).
  70. A. M. Mathai, R. K. Saxena, and H. J. Haubold, The H-Function: Theory and Applications, 1st ed. (Springer, New York, 2010).
  71. R. Herrmann, Fractional Calculus: An Introduction for Physicists, 2nd ed. (World Scientific, Singapore, 2014).
  72. S. G. Samko, A. A. Kilbas, and O. I. Marichev, Fractional Integrals and Derivatives: Theory and Applications, 1st ed. (Gordon and Breach, Singapore, 1993).
  73. M. C. Gutzwiller, Chaos in Classical and Quantum Mechanics, 1st ed. (Springer, Berlin, 1990).
  74. R. Ramos, D. Spierings, I. Racicot, and A. M. Steinberg, Nature (London) 583, 529 (2020).
  75. G. Arfken, Mathematical Methods for Physicists, 3rd ed. (Academic, New York, 1985).
  76. A. M. Steinberg, Phys. Rev. Lett. 74, 2405 (1995).
  77. H. G. Winful, Phys. Rep. 436, 1 (2006).
  78. M. Naber, J. Math. Phys. 45, 3339 (2004).
  79. A. A. Stanislavsky, Eur. Phys. J. B 49, 93 (2006).
  80. N. Laskin, Phys. Rev. E 62, 3135 (2000).
  81. N. Laskin, Phys. Lett. A 268, 298 (2000).
  82. N. Laskin, Phys. Rev. E 66, 056108 (2002).
  83. V. E. Tarasov, Int. J. Mod. Phys. B 27, 1330005 (2013).
  84. R. Metzler and J. Klafter, Phys. Rep. 339, 1 (2000).
  85. S. Holm and S. P. Näsholm, Journ. Acoust. Soc. America 130, 2195 (2011).
  86. S. Yusofsani and M. Kolesik, Phys. Rev. A 101, 052121 (2020).

Outline

Information

Phys. Rev. A 107, 052220– Published 30 May, 2023Retraction: Phys. Rev. A 109, 049903 (2024)
  • Received 27 December 2022
  • Revised 20 April 2023
  • Accepted 18 May 2023
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