Localization and Entanglement in Relativistic Quantum Physics

Part of the Lecture Notes in Physics book series (LNP, volume 899)


These notes are a slightly expanded version of a lecture presented in February 2012 at the workshop “The Message of Quantum Science—Attempts Towards a Synthesis” held at the ZIF in Bielefeld. The participants were physicists with a wide range of different expertise and interests. The lecture was intended as a survey of a small selection of the insights into the structure of relativistic quantum physics that have accumulated through the efforts of many people over more than 50 years. (Including, among many others, R. Haag, H. Araki, D. Kastler, H.-J. Borchers, A. Wightman, R. Streater, B. Schroer, H. Reeh, S. Schlieder, S. Doplicher, J. Roberts, R. Jost, K. Hepp, J. Fröhlich, J. Glimm, A. Jaffe, J. Bisognano, E. Wichmann, D. Buchholz, K. Fredenhagen, R. Longo, D. Guido, R. Brunetti, J. Mund, S. Summers, R. Werner, H. Narnhofer, R. Verch, G. Lechner, ….) This contribution discusses some facts about relativistic quantum physics, most of which are quite familiar to practitioners of Algebraic Quantum Field Theory (AQFT) [Also known as Local Quantum Physics (Haag, Local quantum physics. Springer, Berlin, 1992).] but less well known outside this community. No claim of originality is made; the goal of this contribution is merely to present these facts in a simple and concise manner, focusing on the following issues:
  • Explaining how quantum mechanics (QM) combined with (special) relativity, in particular an upper bound on the propagation velocity of effects, leads naturally to systems with an infinite number of degrees of freedom (relativistic quantum fields).

  • A brief summary of the differences in mathematical structure compared to the QM of finitely many particles that emerge form the synthesis with relativity, in particular different localization concepts, type III von Neumann algebras rather than type I, and “deeply entrenched” (Clifton and Halvorson, Stud Hist Philos Mod Phys 32:1–31, 2001) entanglement,

  • Comments on the question whether these mathematical differences have significant consequences for the physical interpretation of basic concepts of QM.



I thank the organizers of the Bielefeld workshop, Jürg Fröhlich and Philippe Blanchard, for the invitation that lead to these notes, Detlev Buchholz for critical comments on the text, Wolfgang L. Reiter for drawing my attention to [76], and the Austrian Science Fund (FWF) for support under Project P 22929-N16.


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Authors and Affiliations

  1. 1.Faculty of PhysicsUniversity of ViennaViennaAustria

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