Advertisement

Evolution of quantum observables: from non-commutativity to commutativity

  • S. Fortin
  • M. Gadella
  • F. Holik
  • M. LosadaEmail author
Focus
  • 32 Downloads

Abstract

A fundamental aspect of the quantum-to-classical limit is the transition from a non-commutative algebra of observables to commutative one. However, this transition is not possible if we only consider unitary evolutions. One way to describe this transition is to consider the Gamow vectors, which introduce exponential decays in the evolution. In this paper, we give two mathematical models in which this transition happens in the infinite time limit. In the first one, we consider operators acting on the space of the Gamow vectors, which represent quantum resonances. In the second one, we use an algebraic formalism from scattering theory. We construct a non-commuting algebra which commutes in the infinite time limit.

Keywords

Foundations of quantum mechanics Gamow vectors Rigged Hilbert space Classical limit Quantum resonances 

Notes

Acknowledgements

M. Gadella acknowledges partial financial support to the Spanish Government Grant MTM2014-57129-C2-1-P, the Junta de Castilla y León Grants BU229P18, VA137G18. S. Fortin, F. Holik and M. Losada wish to acknowledge the financial support of the Universidad de Buenos Aires, the Grant PICT-2014-2812 from the Consejo Nacional de Investigaciones Científicas y Técnicas of Argentina.

Compliance with ethical standards

Conflict of interest

The authors declare that they have no conflict of interest.

Ethical approval

This article does not contain any studies with human participants or animals performed by any of the authors.

References

  1. Antoine JP (1969) Dirac formalism and symmetry problems in quantum mechanics. I. General Dirac formalism. J Math Phys 10:53–69MathSciNetzbMATHCrossRefGoogle Scholar
  2. Antoniou I, Laura R, Suchanecki Z, Tasaki S (1997) Intrinsic irreversibility of quantum systems with diagonal singularity. Phys A 241:737–772CrossRefGoogle Scholar
  3. Antoniou IE, Gadella M, Pronko GP (1998) Gamow vectors for degenerate scattering resonances. J Math Phys 39:2459–2475MathSciNetzbMATHCrossRefGoogle Scholar
  4. Antoniou I, Gadella M, Suchanecki Z (1998) Some general properties of Liouville spaces. In: Bohm A, Doebner HD, Kielanowski P (eds) Irreversibility and causality. Lecture notes in physics, vol 504. Springer, Berlin, pp 38–56Google Scholar
  5. Birkhoff G, von Neumann J (1936) The logic of quantum mechanics. Ann Math 37:823–843MathSciNetzbMATHCrossRefGoogle Scholar
  6. Bleistein N, Handelsman R (1986) Asymptotic expansion of integrals. Dover Inc., New YorkzbMATHGoogle Scholar
  7. Bohm A (1978) The rigged Hilbert space and quantum mechanics. Springer lecture notes in physics, vol 78. Springer, New YorkCrossRefGoogle Scholar
  8. Bohm A (1981) Resonance poles and Gamow vectors in the rigged Hilbert space formulation of quantum mechanics. J Math Phys 22:2813–2823MathSciNetCrossRefGoogle Scholar
  9. Bohm A (1993) Quantum mechanics: foundations and applications. Springer, BerlinzbMATHCrossRefGoogle Scholar
  10. Bohm A, Gadella M (1989) Dirac kets, Gamow vectors and Gelfand triplets. Springer lecture notes in physics, vol 348. Springer, New YorkCrossRefGoogle Scholar
  11. Bohm A, Erman F, Uncu H (2011) Resonance phenomena and time asymmetric quantum mechanics. Turk J Phys 35:209–240Google Scholar
  12. Castagnino M, Fortin S (2013) Formal features of a general theoretical framework for decoherence in open and closed systems. Int J Theor Phys 52:1379–1398MathSciNetzbMATHCrossRefGoogle Scholar
  13. Castagnino M, Gadella M (2006) The problem of the classical limit of quantum mechanics and the role of self-induced decoherence. Found Phys 36:920–952MathSciNetzbMATHCrossRefGoogle Scholar
  14. Castagnino M, Lombardi O (2005) Self-induced decoherence and the classical limit of quantum mechanics. Philos Sci 72:764–776MathSciNetCrossRefGoogle Scholar
  15. Castagnino M, Gadella M, Gaioli F, Laura R (1999) Gamow vectors and time asymmetry. Int J Theor Phys 38:2823–2865MathSciNetzbMATHCrossRefGoogle Scholar
  16. Castagnino M, Gadella M, Betán RI, Laura R (2001) Gamow functionals on operator algebras. J Phys A Math Gen 34:10067–10083MathSciNetzbMATHCrossRefGoogle Scholar
  17. Castagnino M, Fortin S, Lombardi O (2010) The effect of random coupling coefficients on decoherence. Mod Phys Lett A 25:611–617zbMATHCrossRefGoogle Scholar
  18. Celeghini E, Gadella M, del Olmo MA (2016) Applications of rigged Hilbert spaces in quantum mechanics and signal processing. J Math Phys 57:072105MathSciNetzbMATHCrossRefGoogle Scholar
  19. Celeghini E, Gadella M, del Olmo MA (2017) Lie algebra representations and rigged Hilbert spaces: the SO(2) case. Acta Polytech (Prag) 57:379–384CrossRefGoogle Scholar
  20. Celeghini E, Gadella M, del Olmo MA (2018) Spherical harmonics and rigged Hilbert spaces. J Math Phys 59:053502MathSciNetzbMATHCrossRefGoogle Scholar
  21. Civitarese O, Gadella M (2004) Physical and mathematical aspects of Gamow states. Phys Rep 396:41–113MathSciNetCrossRefGoogle Scholar
  22. Dalla Chiara ML, Giuntini R, Greechie R (2004) Reasoning in quantum theory. Kluwer Academic, DordrechtzbMATHCrossRefGoogle Scholar
  23. Exner P (1984) Open quantum systems and Feynman integrals. Reidel, DordrechtzbMATHGoogle Scholar
  24. Fischer MC, Gutiérrez-Medina B, Reizen MG (2001) Observation of the quantum Zeno and anti-Zeno effects in an unstable system. Phys Rev Lett 87:40402CrossRefGoogle Scholar
  25. Fonda L, Ghirardi GC, Rimini A (1978) Decay theory of unstable quantum systems. Rep Prog Phys 41:587–631CrossRefGoogle Scholar
  26. Fortin S, Vanni L (2014) Quantum decoherence: a logical perspective. Found Phys 44:1258–1268MathSciNetzbMATHCrossRefGoogle Scholar
  27. Fortin S, Holik F, Vanni L (2016) Non-unitary evolution of quantum logics. In: Bagarello F, Passante R, Trapani C (eds) Non-hermitian Hamiltonians in quantum physics. Springer proceedings in physics, vol 184. Springer, ChamzbMATHGoogle Scholar
  28. Friedrichs KO (1948) On the perturbation of continuous spectra. Commun Appl Math 1:361–406MathSciNetzbMATHCrossRefGoogle Scholar
  29. Gadella M (2014) Quantum resonances: theory and models. In: Kielanowski P, Bieliavsky P, Odesskii A, Odzijewicz A, Schlichenmaier M, Voronov T (eds) Geometric methods in physics, XXXII workshop Bialowieza. Springer Basel AG, Poland, pp 99–118Google Scholar
  30. Gadella M (2015) A discussion on the properties of Gamow states. Found Phys 45:177–197MathSciNetzbMATHCrossRefGoogle Scholar
  31. Gadella M, Gómez F (2002) A unified mathematical formalism for the Dirac formulation of quantum mechanics. Found Phys 32:815–869MathSciNetCrossRefGoogle Scholar
  32. Gadella M, Gómez F (2003) On the mathematical basis of the Dirac formulation of quantum mechanics. Int J Theor Phys 42:2225–2254MathSciNetzbMATHCrossRefGoogle Scholar
  33. Gadella M, Laura R (2001) Gamow dyads and expectation values. Int J Quantum Chem 81:307–320CrossRefGoogle Scholar
  34. Gadella M, de la Madrid R (1999) Resonances and time reversal operator in rigged Hilbert spaces. Int J Theor Phys 38:93–113MathSciNetzbMATHCrossRefGoogle Scholar
  35. Gadella M, Pronko GP (2011) The Friedrichs model and its use in resonance phenomena. Fortschr Phys 59:795–859MathSciNetzbMATHCrossRefGoogle Scholar
  36. Gadella M, Kuru Ş, Negro J (2017) The hyperbolic step potential: antibound states, SUSY partners and Wigner time delays. Ann Phys 379:86–101zbMATHCrossRefGoogle Scholar
  37. Gelfand IM, Vilenkin NY (1964) Generalized functions: applications to harmonic analysis. Academic, New YorkGoogle Scholar
  38. Gell-Mann M, Hartle JB (1990) Quantum mechanics in the light of quantum cosmology. In: Zurek WH (ed) Complexity, entropy and the physics of information. Addison-Wesley, ReadingGoogle Scholar
  39. Gell-Mann M, Hartle JB (1993) Classical equations for quantum systems. Phys Rev D 47:3345–3382MathSciNetCrossRefGoogle Scholar
  40. Griffiths RB (2002) Consistent quantum theory. Cambridge University Press, CambridgezbMATHGoogle Scholar
  41. Horvath J (1966) Topological vector spaces and distributions. Addison-Wesley, ReadingzbMATHGoogle Scholar
  42. Khalfin LA (1972) CPT invariance of CP-noninvariant theory of K0 and Kbar0 Mesons and permissible mass distributions of the KS and KL Mesons. JETP Lett 15:388–392Google Scholar
  43. Kiefer C, Polarski D (2009) Why do cosmological perturbations look classical to us? Adv Sci Lett 2:164–173CrossRefGoogle Scholar
  44. Losada M, Fortin S, Holik F (2018) Classical limit and quantum logic. Int J Theor Phys 57:465–475MathSciNetzbMATHCrossRefGoogle Scholar
  45. Losada M, Fortin S, Gadella M, Holik F (2018) Dynamics of algebras in quantum unstable systems. Int J Mod Phys A 33:1850109MathSciNetzbMATHCrossRefGoogle Scholar
  46. Melsheimer O (1974) Rigged Hilbert space formalism as an extended mathematical formalism for quantum systems. I. General theory. J Math Phys 15:902–916MathSciNetCrossRefGoogle Scholar
  47. Misra B, Sudarshan ECG (1977) The Zeno’s paradox in quantum theory. J Math Phys 18:756–763MathSciNetCrossRefGoogle Scholar
  48. Mondragón A, Hernández E (1993) Degeneracy and crossing of resonance energy surfaces. J Phys A Math Gen 26:5595–5611MathSciNetCrossRefGoogle Scholar
  49. Nakanishi N (1958) A theory of clothed unstable particles. Progr Theor Phys 19:607–621MathSciNetzbMATHCrossRefGoogle Scholar
  50. Nussenzveig HM (1972) Causality and dispersion relations. Academic Press, New YorkGoogle Scholar
  51. Omnès R (1999) Understanding quantum mechanics. Princeton University Press, PrincetonzbMATHGoogle Scholar
  52. Ramírez R, Reboiro M (2019) Dynamics of finite dimensional non-hermitian systems with indefinite metric. J Math Phys 60:012106MathSciNetzbMATHCrossRefGoogle Scholar
  53. Ramírez R, Reboiro M (2019) Optimal spin squeezed steady state induced by the dynamics of non-hermitian Hamiltonians. Phys Scr 94:085220CrossRefGoogle Scholar
  54. Reed M, Simon B (1978) Analysis of operators. Academic Press, New YorkzbMATHGoogle Scholar
  55. Reed M, Simon B (1981) Functional analysis. Academic Press, New YorkGoogle Scholar
  56. Roberts JE (1966) Rigged Hilbert spaces in quantum mechanics. Commun Math Phys 3:98–119MathSciNetzbMATHCrossRefGoogle Scholar
  57. Rothe C, Hintschich SI, Monkman AP (2006) Violation of the exponential-decay law at long times. Phys Rev Lett 96:163601CrossRefGoogle Scholar
  58. Schlosshauer M (2007) Decoherence and the quantum-to-classical transition. Springer, BerlinGoogle Scholar
  59. Urbanowski K (2009) General properties of the evolution of unstable states at long times. Eur Phys J D 54:25–29CrossRefGoogle Scholar
  60. Wigner EP (1967) Symmetries and reflections. Indiana University Press, Bloomington, pp 38–39Google Scholar
  61. Wigner EP (1994) Group theoretical concepts and methods in elementary particle physics. Gordon and Breach, New York, pp 37–38Google Scholar
  62. Zurek WH (2009) Quantum darwinism. Nat Phys 5:181–188CrossRefGoogle Scholar

Copyright information

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

Authors and Affiliations

  1. 1.CONICETUniversidad de Buenos AiresBuenos AiresArgentina
  2. 2.Departamento de Física Teórica, Atómica y Óptica and IMUVAUniversidad de ValladolidValladolidSpain
  3. 3.Instituto de Física La PlataConsejo Nacional de Investigaciones Científicas y TécnicasLa PlataArgentina

Personalised recommendations