Abstract
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.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
H. Araki, Mathematical Theory of Quantum Fields. Oxford University Press (1999)
H. Araki, E.J. Woods, Representations of the canonical commutation relations describing a nonrelativistic free Bose gas. J. Math. Phys. 1, 637 (1963)
G. Barton, Elements of Green’s Functions and Propagation (Clarendon Press, Oxford, 1989)
N. Bohr, L. Rosenfeld, On the question of measurability of electromagnetic field strengths. Kgl. Danske Vidensk. Selsk. Mat.-Fys. Med. 12, 8 (1933)
O. Bratelli, D.W Robinson, Operator Algebras and Quantum Statistical Mechanics II. Springer, 2nd edition (1997)
D. Buchholz, J. Yngvason, There are no causality problems for Fermi’s two atom system. Phys. Rev. Lett. 73, 613 (1994)
D. Deutsch, P. Candelas, Phys. Rev. D. 20, 3063 (1979)
J. Dimock, The non relativistic limit of P(ϕ)2 quantum field theory: two-particle phenomena. Comm. Math. Phys. 57, 51–66 (1977)
T.C. Dorlas, Orthogonality and completeness of the Bethe Ansatz eigenstates of the nonlinear Schrödinger model. Comm. Math. Phys. 154, 347–376 (1993)
E. Fermi, Rev. Mod. Phys. 4, 87 (1932)
G.Bimonte, Commutation relations for the electromagnetic fields in the presence of dielectrics. J. Phys. A. 43, 155402 (2010)
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)
J. Glimm, A. Jaffe, The Lambda Phi 4 quantum field theory without cutoffs III—the physical vacuum. Acta Math. 125, 203–267 (1970)
J. Glimm, A.M. Jaffe, Quantum Physics - a Functional Integral Point of View. Springer Verlag (1981)
P. Garbaczewski, W. Karkowski, Impenetrable barriers and cononical quantization. Am. J. Phys. 72, 924 (2004)
R. Haag, Local Quantum Physics - Fields, Particles, Algebras. Springer Verlag (1996)
G.C. Hegerfeldt, Instantaneous spreading and Einstein causality in quantum theory. Ann. Phys. Leipzig. 7, 716–725 (1998)
K. Hepp, The classical limit of quantum mechanical correlation functions. Comm. Math. Phys. 35, 265–277 (1974)
E. Heifets, E.P. Osipov, The energy momentum spectrum in the P(Phi)2 quantum field theory. Comm. Math. Phys. 56, 161–172 (1977)
N.M. Hugenholtz, in Mathematics of Contemporary Physics, ed. R.F. Streater (Academic Press, 1972)
E. Inönü, E.P. Wigner, On the contraction of groups and their representations. Proc. nat. Acad. Sci. USA. 35, 510–524 (1953)
C. Jäkel, W.F. Wreszinski, Stability and related properties of vacua and ground states. Ann. Phys. 323, 251–266 (2008)
N. Kawakami, M.C. Nemes, W.F. Wreszinski. J. Math. Phys. 48, 102302 (2007)
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)
E.H. Lieb, D.W. Robinson, The finite group velocity of quantum spin systems. Comm. Math. Phys. 28, 251 (1972)
P.M. Morse, H. Feshbach, Methods of Theoretical Physics - Part I. McGraw Hill Book Co.Inc and Kogakusha Co. Ltd. (1953)
P. Milonni, Phys. Rev. A. 25, 1315 (1982)
Ph. A. Martin, F Rothen, Many Body Problems and Quantum Field Theory - an Introduction. Springer (2004)
D.H.U. Marchetti, W.F. Wreszinski, Asymptotic Time Decay in Quantum Physics. World Scientific (2013)
B. Nachtergaele, R. Sims, Lieb-Robinson bounds and the exponential clustering theorem. Comm. Math. Phys. 265, 119–130 (2006)
H.M. Nussenzveig, Dispersion Relations and Causality. Academic Press, (1972)
J.F. Perez, I.F. Wilde, Localization and causality in relativistic quantum mechanics. Phys. Rev. D. 16, 315–317 (1977)
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)
D.W. Robinson, The Thermodynamic Pressure in Quantum Statistical Mechanics (Springer, Berlin-Heidelberg-New York, 1971)
M. Reed, B. Simon, Methods of Modern Mathematical Physics - v.2, Fourier Analysis, Self-adjointness. Academic Press (1975)
D. Ruelle, Statistical Mechanics - Rigorous Results. W. A. Benjamin Inc. (1969)
J.J. Sakurai, Advanced Quantum Mechanics. Addison Wesley Publishing Co (1967)
R.F. Streater, A.S. Wightman, PCT, Spin and Statistics and all that. W. A. Benjamin, Inc. (1964)
G. Guralnik, C.R. Hagen (eds.), Proceedings of the International Conference on Particles and Fields, Rochester 1967 (Wiley, N.Y., 1967)
J. Yngvason, Localization and entanglement in relativistic quantum physics. arXiv:1401.2652v1 12 january (2014)
Acknowledgments
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].
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
B. Coutinho, F.A., Wreszinski, W.F. Instantaneous Spreading Versus Space Localization for Nonrelativistic Quantum Systems. Braz J Phys 46, 462–470 (2016). https://doi.org/10.1007/s13538-016-0426-3
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s13538-016-0426-3