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Localization and Entanglement in Relativistic Quantum Physics

  • Chapter
The Message of Quantum Science

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

Abstract

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.

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Notes

  1. 1.

    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, ….

  2. 2.

    Also known as Local Quantum Physics [55].

  3. 3.

    More precisely, also representations “up to a phase” are allowed, which amounts to replacing \(\mathcal{P}_{+}^{\uparrow }\) by its universal covering group \(\mathit{ISL}(2, \mathbb{C})\).

  4. 4.

    For simplicity of the exposition we refrain from discussing the possibility that the Lorentz transformations act only as automorphisms on the algebra of observables but are not unitarily implemented on the Hilbert space of states under consideration, as can be expected in charged superselection sectors of theories with massless particles [23, 50].

  5. 5.

    Here, and in the following, units are chosen so that Planck’s constant, ℏ, and the velocity of light, c, are equal to 1. The metric on Minkowski space is g μ ν  = diag (1, −1, −1, −1).

  6. 6.

    In this form the result was first published by J.F. Perez and I.F. Wilde in [75]. See also [65] for the same conclusion under slightly weaker premises.

  7. 7.

    This follows from the “edge of the wedge” theorem, that is a generalization of the Schwarz reflection principle to several complex variables, see, e.g., [83].

  8. 8.

    This objection does not exclude approximate localization in the sense of Newton and Wigner [73].

  9. 9.

    Field operators at a point can, however, be defined as quadratic forms on vectors with sufficiently nice high energy behavior.

  10. 10.

    More generally, a representation of the covering group \(\mathit{ISL}(2, \mathbb{C})\).

  11. 11.

    For mathematical convenience we assume that the operators are bounded and that the algebras are closed in the weak operator topology, i.e., that they are von Neumann algebras. The generation of such algebras from unbounded quantum field operators \(\Phi _{\alpha }(f)\) is in general a nontrivial issue that is dealt with, e.g., in [18, 45]. In cases when the real and imaginary parts of the field operators are essentially self-adjoint, one may think of the \(\mathcal{F}(\mathcal{O})\) as generated by bounded functions (e.g., spectral projectors, resolvents, or exponentials) of these operators smeared with test functions having support in \(\mathcal{O}\). More generally, the polar decomposition of the unbounded operators can be taken as a starting point for generating the local net of von Neumann algebras.

  12. 12.

    In the theory of superselection sectors, initiated by Borchers in [14] and further developed in particular by Doplicher et al. in [39–42], the starting point is the net of observables while the field net and the gauge group are derived objects. For a very recent development, applicable to theories with long range forces, see [26].

  13. 13.

    Here and in the sequel, a state means a positive, normalized linear functional on the algebra in question, i.e., a linear functional such that ω(A ∗ A) ≥ 0 for all A and ω(1) = 1. We shall also restrict the attention to normal states, i.e., ω(A) = trace (ρ A) with a nonnegative trace class operator ρ on \(\mathcal{H}\) with trace 1.

  14. 14.

    For simplicity we have assumed local commutativity. In the case of Fermi fields the same conclusion is drawn by splitting the operators into their bosonic and fermionic parts.

  15. 15.

    Equation (15.7) is, strictly speaking, only claimed for A,B in a the dense subalgebra of “smooth” elements of \(\mathcal{A}\) obtained by integrating \(\Delta ^{\mathrm{i}t}A\Delta ^{-\mathrm{i}t}\) with a test functions in t.

  16. 16.

    Due to a sign convention in modular theory the temperature is formally − 1, but by a scaling of the parameter t, including an inversion of the sign, can produce any value of the temperature.

  17. 17.

    Fermi fields can be included by means of a “twist” that turns anticommutators into commutators as in [11].

  18. 18.

    Such real subspaces of a complex Hilbert space are called standard in the spatial version of Tomita–Takesaki theory [81].

  19. 19.

    The hyperfiniteness, i.e., the approximability by finite dimensional matrix algebras, follows from the split property considered in Sect. 15.5.1.

  20. 20.

    Recall that “state” means here always normal state, i.e. a positive linear functional given by a density matrix in the Hilbert space where \(\mathcal{A}\) operates. As a C ∗ algebra \(\mathcal{A}\) has pure states, but these correspond to disjoint representations on different (non separable) Hilbert spaces.

  21. 21.

    If the algebra is represented in a “standard form” in the sense of modular theory the vector can be uniquely fixed by taking it from the corresponding “positive cone” [19].

  22. 22.

    This holds because t ↦ P 1∕2 e −iHt ψ is analytic in the complex lower half plane for all vectors \(\psi \in \mathcal{H}\) if H ≥ 0.

  23. 23.

    Already the Corollary to the Reeh–Schlieder Theorem in Sec. 15.3.3. implies that excitation cannot be measured by a local positive operator since the expectation value of such an operator cannot be zero in a state with bounded energy spectrum. The nonexistence of any positive operator satisfying (15.36) is a stronger statement.

  24. 24.

    See [12] for important steps in this direction and [49] for a thorough analysis of foundational issues of QM.

  25. 25.

    Already Max Planck in his Leiden lecture of 1908 speaks of the “Emanzipierung von den antrophomorphen Elementen” as a goal, see [76], p. 49.

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Acknowledgements

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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    Jakob Yngvason

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    Jürg Fröhlich

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Yngvason, J. (2015). Localization and Entanglement in Relativistic Quantum Physics. In: Blanchard, P., Fröhlich, J. (eds) The Message of Quantum Science. Lecture Notes in Physics, vol 899. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-46422-9_15

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