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On the connection between quantum fields and von Neumann algebras of local operators. (English) Zbl 0595.46062

Each of the two main approaches to the theory of particles, i.e. local fields and nets of local von Neumann algebras, has its own strengths and problems. In order to pool the first and to minimize the second we should know more about the relation between these two approaches. The present paper first considers a local field and a so-called AB-system (somewhat more general than a local net) action on the same Hilbert space and having the same localization - meaning that field operators and operators in the AB-system ”belonging” to causally disjoint regions of space-time commute weakly. It shows - very roughly speaking - that in this situation there is an almost canonical relation between the two structures.
The paper then defines ”intrinsic locality” for field operators and shows that any single such operator (if one exists) can be used to construct just such an AB-system as investigated in part one. Using an additional mild regularity condition - the generalized H-bound shown to be significant in other contexts, too - the AB-system so constructed turns out to be unique. This considerably reduces the problem of relating local fields to local nets.
In addition to the above the paper discusses various questions related to the main results.

MSC:

46L60 Applications of selfadjoint operator algebras to physics
46N99 Miscellaneous applications of functional analysis
81T05 Axiomatic quantum field theory; operator algebras
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[1] Araki, H.: On the algebra of all local observables. Prog. Theor. Phys.32, 844-854 (1964) · Zbl 0125.21904
[2] Bisognano, J.J., Wichmann, E.H.: On the duality condition for a Hermitian scalar field. J. Math. Phys.16, 985-1007 (1975) · Zbl 0316.46062
[3] Bisognano, J.J., Wichmann, E.H.: On the duality condition for quantum fields. J. Math. Phys.17, 303-321 (1976)
[4] Borchers, H.J.: Über die Mannigfaltigkeit der interpolierenden Felder zu einer kausalenS-Matrix. Il Nuovo Cimento15, 784-794 (1960) · Zbl 0093.44002
[5] Borchers, H.J., Zimmermann, W.: On the self-adjointness of field operators. Il Nuovo Cimento31, 1047-1059 (1963) · Zbl 0151.44303
[6] Borchers, H.J.: A remark on a theorem of B. Misra. Commun. Math. Phys.4, 315-323 (1967) · Zbl 0155.32401
[7] Doplicher, S., Haag, R., Roberts, J.E.: Fields, observables, and gauge transformations. I. Commun. Math. Phys.13, 1-23 (1969); II. Commun. Math. Phys.15, 173-200 (1969) · Zbl 0175.24704
[8] Driessler, W.: Comments on lightlike translations and applications in relativistic quantum field theory. Commun. Math. Phys.44, 133-141 (1975) · Zbl 0306.46076
[9] Driessler, W., Fröhlich, J.: The reconstruction of local observable algebras from the Euclidean Green’s functions of relativistic quantum field theory. Ann. Inst. Henri Poincaré27, 221-236 (1977) · Zbl 0364.46051
[10] Driessler, W., Summers, S.J.: On the decomposition of relativistic quantum field theories into pure phases (to appear in Helv. Phys. Acta)
[11] Driessler, W., Summers, S.J.: Central decomposition of Poincaré-invariant nets of local field algebras and absence of spontaneous breaking of the Lorentz group. Ann. Inst. Henri Poincaré43 A, 147-166 (1985) · Zbl 0609.47059
[12] Epstein, H.: On the Borchers class of a free field. Il Nuovo Cimento27, 886-893 (1963) · Zbl 0135.44401
[13] Fredenhagen, K., Hertel, J.: Local algebras of observables and pointlike localized fields. Commun. Math. Phys.80, 555-561 (1981) · Zbl 0472.46051
[14] Fredenhagen, K.: On the modular structure of local algebras of observables. Commun. Math. Phys.97, 79-89 (1985) · Zbl 0582.46067
[15] Glimm, J., Jaffe, A.: Quantum physics. Berlin, Heidelberg, New York: Springer 1981 · Zbl 0461.46051
[16] Haag, R.: In: Colloque international sur les problèmes mathématiques sur la théorie quantique des champs, Lille, 1957. Centre National de la Recherche Scientifique, Paris, 1959
[17] Haag, R., Kastler, D.: An algebraic approach to quantum field theory. J. Math. Phys.5, 848-861 (1964) · Zbl 0139.46003
[18] Hertel, J.: Lokale Quantentheorie und Felder am Punkt, DESY T-81/01, 1981 (preprint)
[19] Jaffe, A.M.: High-energy behavior in quantum field theory. I. Strictly localizable fields. Phys. Rev.158, 1454-1461 (1967)
[20] Jørgensen, P.E.T.: Selfadjoint extension operators commuting with an algebra. Math. Z.169, 41-62 (1979) · Zbl 0406.47015
[21] Jost, R.: The general theory of quantized fields. Providence, R.I.: Am. Math. Soc. 1965 · Zbl 0127.19105
[22] Landau, L.J.: On local functions of fields. Commun. Math. Phys.39, 49-62 (1974) · Zbl 0309.46055
[23] Langerholc, J., Schroer, B.: On the structure of the von Neumann algebras generated by local functions of the free Bose fields. Commun. Math. Phys.1, 215-239 (1965) · Zbl 0138.45103
[24] Longo, R.: Notes on algebraic invariants for non-commutative dynamical systems. Commun. Math. Phys.69, 195-207 (1979) · Zbl 0421.46053
[25] Murray, F.J., von Neumann, J.: On rings of operators. Ann. Math.37, 116-229 (1936) · Zbl 0014.16101
[26] Powers, R.T.: Self-adjoint algebras of unbounded operators. I. Commun. Math. Phys.21, 85-124 (1971); II. Trans. Am. Math. Soc.187, 261-293 (1974) · Zbl 0214.14102
[27] Rehberg, J., Wollenberg, M.: Quantum fields as pointlike localized objects (to appear in Math. Nachr.) · Zbl 0617.46080
[28] Streater, R.F., Wightman, A.S.:PCT, spin and statistics, and all that. New York: Benjamin 1964 · Zbl 0135.44305
[29] Summers, S.J.: From algebras of local observables to quantum fields: generalizedH-bounds (preprint, 1986)
[30] Wichmann, E.H.: On systems of local operators and the duality condition. J. Math. Phys.24, 1633-1644 (1983) · Zbl 0542.46037
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