Abstract
The modular structure of the von Neumann algebra of local observables associated with a double cone in the vacuum representation of the free massless scalar field theory of any number of dimensions is described. The modular automorphism group is induced by the unitary implementation of a family of generalized fractional linear transformations on Minkowski space and is a subgroup of the conformal group. The modular conjugation operator is the anti-unitary implementation of a product of time reversal and relativistic ray inversion. The group generated by the modular conjugation operators for the local algebras associated with the family of double cone regions is the group of proper conformal transformations. A theorem is presented asserting the unitary equivalence of local algebras associated with lightcones, double cones, and wedge regions. For the double cone algebras, this provides an explicit realization of spacelike duality and establishes the known typeIII 1 factor property. It is shown that the timelike duality property of the lightcone algebras does not hold for the double cone algebras. A different definition of the von Neumann algebras associated with a region is introduced which agrees with the standard one for a lightcone or a double cone region but which allows the timelike duality property for the double cone algebras. In the case of one spatial dimension, the standard local algebras associated with the double cone regions satisfy both spacelike and timelike duality.
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References
Bisognano, J. J., Wichmann, E. H.: On the duality condition for a Hermitian scalar field. J. Math. Phys.16, 985–1007 (1975)
Buchholz, D.: On the structure of local quantum fields with non-trivial interaction. In: Proc. of the Int. Conf. on Operator Algebras, Ideals and their Application in Theoretical Physics, Baumgärtel, Lassner, Pietsch, Uhlmann, (eds.) pp. 146–153. Leipzig: Teubner Verlagsgesell schaft 1978
Bisognano, J. J., Wichmann, E. H.: On the duality condition for quantum fields. J. Math. Phys.17, 303–321 (1976)
Swieca, J. A., Völkel, A. H.: Remarks on conformal invariance. Commun. Math. Phys.24, 319–342 (1973)
Courant, R., Hilbert, D.: Methods of mathematical physics, Vol. II. New York: Interscience Publishers, pp. 668–691 1962
Driessler, W.: On the type of local algebras in quantum field theory. Commun. Math. Phys.53, 295–297 (1977)
Longo, R.: Notes on algebraic invariants for non-commutative dynamical systems. Commun. Math. Phys.69, 195–207 (1979)
Araki, H.: A lattice of von Neumann algebras associated with the quantum theory of a free bose field. J. Math. Phys.4, 1343–1362 (1963)
Benfatto, G., Nicolò, F.: The local von Neumann algebras for the massless scalar free field and the free electromagnetic field. J. Math. Phys.19, 653–660 (1978)
Araki, H., Woods, E. J.: A classification of factors. Publ. R.I.M.S., Kyoto Univ.4A, 51–130 (1968)
Eckmann, J. P., Fröhlich, J.: Unitary equivalence of local algebras in the quasifree representation. Ann. Inst. Henri Poincaré20A, 201–209 (1974)
Glimm, J., Jaffe, A.: The λ(φ4)2 quantum field theory without cutoffs: III. The physical vacuum. Acta Math. (Uppsala)125, 203–261 (1970)
Araki, H.: Von Neumann algebras of local observables for free scalar field. J. Math. Phys.5, 1–13 (1964)
Vladimirov, V. S.: Les fonctions de plusieurs variables complexes et leur application à la théorie quantique des champs, Lagowski, N. (tr.) Paris: Dunod 1967
Hortaçsu, M., Seiler, R., Schroer, B.: Conformal symmetry and reverberations. Phys. Rev.5D, 2519–2534 (1972)
D. Buchholz: private communication
Gross, L.: Norm invariance of mass-zero equations under the conformal group. J. Math. Phys.5, 687–695 (1964)
Borchers, H. J.: Field operators asC ∞ functions in spacelike directions. Nuovo Cimento33, 1600–1613 (1964)
Doplicher, S., Haag, R., Roberts, J. E.: Fields, observables and gauge transformations I. Commun. Math. Phys.13, 1–23 (1969)
Rieffel, M. A.: A commutation theorem and duality for free bose fields. Commun. Math. Phys.39, 153–164 (1974)
Araki, H.: “A generalization of Borchers theorem. Helv. Phys. Acta36, 132–139 (1963)
Leyland, P., Roberts, J. E., Testard, D.: Duality for quantum free fields. Marseille Preprint 1978
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Communicated by R. Haag
Supported by the National Science Foundation under Grant No. PHY-79-23251
Supported in part by C. N. R.
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Hislop, P.D., Longo, R. Modular structure of the local algebras associated with the free massless scalar field theory. Commun.Math. Phys. 84, 71–85 (1982). https://doi.org/10.1007/BF01208372
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DOI: https://doi.org/10.1007/BF01208372