Why there is a field algebra with a compact gauge group describing the superselection structure in particle physics. (English) Zbl 0734.46042

Utilizing earlier obtained results [Invent. Math. 98, No.1, 157-218 (1989; Zbl 0691.22002); Ann. Math., II. Ser. 130, No.1, 75-119 (1989; Zbl 0702.46044)] the authors solve a long standing problem by showing that the superselection structure of local quantum theory is determined in terms of a compact group of gauge automorphisms of a field net with normal Bose and Fermi commutation rules. We quote most enlightening authors’ abstract:
“Given the local observables in the vacuum sector fulfilling a few basic principles of local quantum theory, we show that the superselection structure intrinsically determined a priori, can always be described by a unique compact global gauge group acting on a field algebra generated by field operators which commute or anticommute at spacelike separations. The field algebra and the gauge group are constructed simultaneously from the local observables. There will be sectors obeying parastatistics, an intrinsic notion derived from the observables, if and only if the gauge group is non-Abelian. Topological charges would manifest themselves in field operators associated with spacelike cones but not localizable in bounded regions of Minkowski space. No assumptions on the particle spectrum or even on the covariance of the theory is made.However the notion of superselection sector is tailored to theories without massless particles. When translation or Poincaré covariance of the vacuum sector is assumed, our construction leads to a covariant field algebra describing all covariant sectors.”


46L60 Applications of selfadjoint operator algebras to physics
81T05 Axiomatic quantum field theory; operator algebras
Full Text: DOI


[1] Doplicher, S., Haag, R., Roberts, J. E.: Local observables and particle statistics I. Commun. Math. Phys.23, 199–230 (1971) · doi:10.1007/BF01877742
[2] Doplicher, S., Roberts, J. E.: A new duality theory for compact groups. Inventiones Math.98, 157–218 (1989) · Zbl 0691.22002 · doi:10.1007/BF01388849
[3] Wick, G. C., Wightman, A. S., Wigner, E. P.: The intrinsic parity of elementary particles. Phys. Rev.88, 101–105 (1952) · Zbl 0046.43906 · doi:10.1103/PhysRev.88.101
[4] Haag, R., Kastler, D.: An algebraic approach to quantum field theory. J. Math. Phys.5, 848–861 (1964) · Zbl 0139.46003 · doi:10.1063/1.1704187
[5] Borchers, H. J.: Local rings and the connection of spin with statistics. Commun. Math. Phys.1, 281–307 (1965) · Zbl 0138.45202 · doi:10.1007/BF01645905
[6] Buchholz, D., Fredenhagen, K.: Locality and the structure of particle states. Commun. Math. Phys.84, 1–54 (1982) · Zbl 0498.46061 · doi:10.1007/BF01208370
[7] Fredenhagen, K., Marcu, M.: Charged states inZ 2 gauge theories. Commun. Math. Phys.92, 81–119 (1983) · Zbl 0535.46052 · doi:10.1007/BF01206315
[8] Buchholz, D.: The physical state space of quantum electrodynamics. Commun. Math. Phys.85, 49–71 (1982) · Zbl 0506.46052 · doi:10.1007/BF02029133
[9] Doplicher, S., Haag, R., Roberts, J. E.: Local observables and particle statistics II. Commun. Math. Phys.35, 49–85 (1974) · doi:10.1007/BF01646454
[10] Doplicher, S., Haag, R., Roberts, J. E.: Fields, observables and gauge transformations I. Commun. Math. Phys.13, 1–23 (1969) · Zbl 0175.24704 · doi:10.1007/BF01645267
[11] Doplicher, S., Roberts, J. E.: Fields, statistics and non-Abelian gauge groups. Commun. Math. Phys.28, 331–348 (1972) · doi:10.1007/BF01645634
[12] Doplicher, S., Haag, R., Roberts, J. E.: Field, observables and gauge transformations II. Commun. Math. Phys.15, 173–200 (1969) · Zbl 0186.58205 · doi:10.1007/BF01645674
[13] Fröhlich, J.: New super-selection sectors (”soliton-states”) in two dimensional Bose quantum field models. Commun. Math. Phys.47, 269–310 (1976) · doi:10.1007/BF01609844
[14] Doplicher, S., Roberts, J. E.: Endomorphisms ofC *-algebras, cross products and duality for compact groups. Ann. Math.130, 75–119 (1989) · Zbl 0702.46044 · doi:10.2307/1971477
[15] Cuntz, J.: SimpleC *-algebras generated by isometries. Commun. Math. Phys.57, 173–185 (1977) · Zbl 0399.46045 · doi:10.1007/BF01625776
[16] Bisognano, J. J., Wichmann, E. H.: On the duality condition for quantum fields. J. Math. Phys.17, 303–321 (1976) · doi:10.1063/1.522898
[17] Doplicher, S., Roberts, J. E.: Duals of compact Lie groups realized in the Cuntz algebras and their actions onC *-algebras. J. Funct. Anal.74, 96–120 (1987) · Zbl 0619.46053 · doi:10.1016/0022-1236(87)90040-1
[18] Borchers, H. J.: A remark on a theorem of B. Misra. Commun. Math. Phys.4, 315–323 (1967) · Zbl 0155.32401 · doi:10.1007/BF01653645
[19] Fredenhagen, K.: On the existence of antiparticles. Commun. Math. Phys.79, 141–151 (1981) · doi:10.1007/BF01208291
[20] Drühl, K., Haag, R., Roberts, J. E.: On Parastatistics. Commun. Math. Phys.18, 204–226 (1970) · Zbl 0195.55904 · doi:10.1007/BF01649433
[21] Roberts, J. E.: Statistics and the intertwiner calculus. In:C *-Algebras and their applications to statistical mechanics and quantum field theory. Kastler, D. (ed.) pp. 203–225. Amsterdam, New York, Oxford: North Holland 1976
[22] Roberts, J. E.: Cross products of von Neumann algebras by group duals. Symposia Mathematica20, 335–363 (1976) · Zbl 0441.46053
[23] Doplicher, S., Roberts, J. E.: Compact group actions onC *-algebras. J. Operator Theory19, 283–305 (1988) · Zbl 0689.46020
[24] Tannaka, T.: Über den Dualitätssatz der nichtkommutativen topologischen Gruppen. Tôhoku Math. J.45, 1–12 (1939) · Zbl 0020.00904
[25] Roberts, J. E.: Localization in algebraic field theory. Commun. Math. Phys.85, 87–98 (1982) · Zbl 0509.47036 · doi:10.1007/BF02029135
[26] Doplicher, S.: Local aspects of superselection rules. Commun. Math. Phys.85, 73–86 (1982); Doplicher, S., Longo, R.: Local aspects of superselection rules II. Commun. Math. Phys.88, 399–409 (1983); Buchholz, D., Doplicher, S., Longo, R.: On Noether’s theorem in quantum field theory. Ann. Phys.170, 1–17 (1986) · Zbl 0515.46065 · doi:10.1007/BF02029134
[27] Pedersen, G. K.:C *-algebras and their automorphism groups. London, New York, San Francisco: Academic Press 1979 · Zbl 0416.46043
[28] Borchers, H. J., Buchholz, D.: The energy-momentum spectrum in local field theories with broken Lorentz-symmetry. Commun. Math. Phys.97, 169–185 (1985) · Zbl 0578.46061 · doi:10.1007/BF01206185
[29] Buchholz, D., Epstein, H.: Spin and statistics of quantum topological charges. Fizika17, 329–343 (1985)
[30] Roberts, J. E.: Spontaneously broken gauge symmetries and superselection rules. Proceedings of the International School of Mathematical Physics, Camerino 1974. Gavallotti, G. (ed.). Università di Camerino 1976
[31] Roberts, J. E.: Net cohomology and its applications to field theory. In: Quantum Fields–Algebras, Processes. Streit, L. (ed.). Wien, New York: Springer 1980 · Zbl 0484.57024
[32] Buchholz, D., Doplicher, S., Longo, R., Roberts, J. E.: Broken symmetries and degeneracy of the vacuum in quantum field theory, (in preparation)
[33] Fredenhagen, K., Rehren, K.-H., Schroer, B.: Superselection sectors with braid group statistics and exchange algebras I. Commun. Math. Phys.125, 201–226 (1989) · Zbl 0682.46051 · doi:10.1007/BF01217906
[34] Fröhlich, J.: Statistics of fields, the Yang-Baxter equation, and the theory of knots and links. In: Non-Perturbative Quantum Field Theory. ’t Hooft, G. et al. (eds.). New York: Plenum Press 1988; Fröhlich, J., Marchetti, P. A.: Quantum field theory of vortices and anyons. Commun. Math. Phys.121, 177–224 (1989) · Zbl 0819.58045
[35] Joyal, A., Street, R.: Braided monoidal categories. Macquarie Mathematics Studies, 1986 · Zbl 0817.18007
[36] Deligne, P.: Categories Tannakiennes. Grothendieck Festschrift (to appear) · Zbl 0727.14010
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