Brazilian Journal of Physics

, Volume 46, Issue 4, pp 462–470 | Cite as

Instantaneous Spreading Versus Space Localization for Nonrelativistic Quantum Systems

  • F. A. B. Coutinho
  • W. F. WreszinskiEmail author
Condensed Matter


A theorem of Hegerfeldt (Kielanowski et al. 1998) establishes, for a class of quantum systems, a dichotomy between those which are permanently localized in a bounded region of space, and those exhibiting instantaneous spreading. We analyze in some detail the physical inconsistencies which follow from both of these options, and formulate which, in our view, are the basic open problems.


Instantaneous spreading Space localization Einstein causality Nonrelativistic limit 



The idea of a part of this review arose at the meeting of operator algebras and quantum physics, satellite conference to the XVIII international congress of mathematical physics. We thank the organizers for making the participation of one of us (W. F. W.) possible, and Prof. J. Dimock for discussions there on matters related to Section 3. We also thank Christian Jäkel for critical remarks concerning possible changes of viewpoint, and for recalling some relevant references. W.F.W. also thanks J. Froehlich for calling his attention to the reference [40].


  1. 1.
    H. Araki, Mathematical Theory of Quantum Fields. Oxford University Press (1999)Google Scholar
  2. 2.
    H. Araki, E.J. Woods, Representations of the canonical commutation relations describing a nonrelativistic free Bose gas. J. Math. Phys. 1, 637 (1963)ADSMathSciNetCrossRefGoogle Scholar
  3. 3.
    G. Barton, Elements of Green’s Functions and Propagation (Clarendon Press, Oxford, 1989)zbMATHGoogle Scholar
  4. 4.
    N. Bohr, L. Rosenfeld, On the question of measurability of electromagnetic field strengths. Kgl. Danske Vidensk. Selsk. Mat.-Fys. Med. 12, 8 (1933)Google Scholar
  5. 5.
    O. Bratelli, D.W Robinson, Operator Algebras and Quantum Statistical Mechanics II. Springer, 2nd edition (1997)Google Scholar
  6. 6.
    D. Buchholz, J. Yngvason, There are no causality problems for Fermi’s two atom system. Phys. Rev. Lett. 73, 613 (1994)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    D. Deutsch, P. Candelas, Phys. Rev. D. 20, 3063 (1979)ADSMathSciNetCrossRefGoogle Scholar
  8. 8.
    J. Dimock, The non relativistic limit of P(ϕ)2 quantum field theory: two-particle phenomena. Comm. Math. Phys. 57, 51–66 (1977)ADSMathSciNetCrossRefGoogle Scholar
  9. 9.
    T.C. Dorlas, Orthogonality and completeness of the Bethe Ansatz eigenstates of the nonlinear Schrödinger model. Comm. Math. Phys. 154, 347–376 (1993)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    E. Fermi, Rev. Mod. Phys. 4, 87 (1932)ADSCrossRefGoogle Scholar
  11. 11.
    G.Bimonte, Commutation relations for the electromagnetic fields in the presence of dielectrics. J. Phys. A. 43, 155402 (2010)ADSMathSciNetCrossRefGoogle Scholar
  12. 12.
    P. Kielanowski, A. Bohm, H.D. Doebner (eds.), Irreversibility and causality in quantum theory - semigroups and rigged Hilbert spaces. Lect. Notes in Phys., Vol. 504 (Springer Verlag , 1998)Google Scholar
  13. 13.
    J. Glimm, A. Jaffe, The Lambda Phi 4 quantum field theory without cutoffs III—the physical vacuum. Acta Math. 125, 203–267 (1970)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    J. Glimm, A.M. Jaffe, Quantum Physics - a Functional Integral Point of View. Springer Verlag (1981)Google Scholar
  15. 15.
    P. Garbaczewski, W. Karkowski, Impenetrable barriers and cononical quantization. Am. J. Phys. 72, 924 (2004)ADSCrossRefGoogle Scholar
  16. 16.
    R. Haag, Local Quantum Physics - Fields, Particles, Algebras. Springer Verlag (1996)Google Scholar
  17. 17.
    G.C. Hegerfeldt, Instantaneous spreading and Einstein causality in quantum theory. Ann. Phys. Leipzig. 7, 716–725 (1998)ADSCrossRefzbMATHGoogle Scholar
  18. 18.
    K. Hepp, The classical limit of quantum mechanical correlation functions. Comm. Math. Phys. 35, 265–277 (1974)ADSMathSciNetCrossRefGoogle Scholar
  19. 19.
    E. Heifets, E.P. Osipov, The energy momentum spectrum in the P(Phi)2 quantum field theory. Comm. Math. Phys. 56, 161–172 (1977)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    N.M. Hugenholtz, in Mathematics of Contemporary Physics, ed. R.F. Streater (Academic Press, 1972)Google Scholar
  21. 21.
    E. Inönü, E.P. Wigner, On the contraction of groups and their representations. Proc. nat. Acad. Sci. USA. 35, 510–524 (1953)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    C. Jäkel, W.F. Wreszinski, Stability and related properties of vacua and ground states. Ann. Phys. 323, 251–266 (2008)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    N. Kawakami, M.C. Nemes, W.F. Wreszinski. J. Math. Phys. 48, 102302 (2007)ADSMathSciNetCrossRefGoogle Scholar
  24. 24.
    E.H. Lieb, W. Liniger, Exact analysis of an interacting Bose gas I—the general solution and the ground state. Phys. Rev. 130, 1605–1616 (1963)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    E.H. Lieb, D.W. Robinson, The finite group velocity of quantum spin systems. Comm. Math. Phys. 28, 251 (1972)ADSMathSciNetCrossRefGoogle Scholar
  26. 26.
    P.M. Morse, H. Feshbach, Methods of Theoretical Physics - Part I. McGraw Hill Book Co.Inc and Kogakusha Co. Ltd. (1953)Google Scholar
  27. 27.
    P. Milonni, Phys. Rev. A. 25, 1315 (1982)ADSCrossRefGoogle Scholar
  28. 28.
    Ph. A. Martin, F Rothen, Many Body Problems and Quantum Field Theory - an Introduction. Springer (2004)Google Scholar
  29. 29.
    D.H.U. Marchetti, W.F. Wreszinski, Asymptotic Time Decay in Quantum Physics. World Scientific (2013)Google Scholar
  30. 30.
    B. Nachtergaele, R. Sims, Lieb-Robinson bounds and the exponential clustering theorem. Comm. Math. Phys. 265, 119–130 (2006)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    H.M. Nussenzveig, Dispersion Relations and Causality. Academic Press, (1972)Google Scholar
  32. 32.
    J.F. Perez, I.F. Wilde, Localization and causality in relativistic quantum mechanics. Phys. Rev. D. 16, 315–317 (1977)ADSCrossRefGoogle Scholar
  33. 33.
    M. Requardt, Reeh-Schlieder type density results in one and n body Schrödinger theory and the ”unique continuation problem”. J. Math. Phys. 27, 1571–1577 (1986)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    D.W. Robinson, The Thermodynamic Pressure in Quantum Statistical Mechanics (Springer, Berlin-Heidelberg-New York, 1971)CrossRefGoogle Scholar
  35. 35.
    M. Reed, B. Simon, Methods of Modern Mathematical Physics - v.2, Fourier Analysis, Self-adjointness. Academic Press (1975)Google Scholar
  36. 36.
    D. Ruelle, Statistical Mechanics - Rigorous Results. W. A. Benjamin Inc. (1969)Google Scholar
  37. 37.
    J.J. Sakurai, Advanced Quantum Mechanics. Addison Wesley Publishing Co (1967)Google Scholar
  38. 38.
    R.F. Streater, A.S. Wightman, PCT, Spin and Statistics and all that. W. A. Benjamin, Inc. (1964)Google Scholar
  39. 39.
    G. Guralnik, C.R. Hagen (eds.), Proceedings of the International Conference on Particles and Fields, Rochester 1967 (Wiley, N.Y., 1967)Google Scholar
  40. 40.
    J. Yngvason, Localization and entanglement in relativistic quantum physics. arXiv:1401.2652v1 12 january (2014)

Copyright information

© Sociedade Brasileira de Física 2016

Authors and Affiliations

  1. 1.School of MedicineUniversity of São PauloSão PauloBrazil
  2. 2.LIM 01 - HCFMUSPSão PauloBrazil
  3. 3.Instituto de Fisica USPSão PauloBrazil

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