Abstract
Using the method of implementable one-particle Bogoliubov transformations, it is possible to explicitly define a local covariant net of quantum fields on the (universal covering of the) circle S 1 with braid group statistics. These anyon fields transform under a representation of \({\widetilde{U(1)}}\) for arbitrary real-valued spin and their commutation relations depend on the relative winding number of localization regions. By taking the tensor product with a local covariant field theory on \({{{\rm{I\!R}}}^2}\), one can obtain a (non-boost covariant) cone-localized field net for anyons in two dimensions.
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References
Adler C.: Braid group statistics in two-dimensional quantum field theory. Rev. Math. Phys. 8, 907–924 (1996)
Baker G.A., Canright G.S., Mulay S.B., Sundberg C.: On the spectral problem for anyons. Commun. Math. Phys. 153, 277–295 (1193)
Bros J., Mund J.: Braid group statistics implies scattering in three-dimensional local quantum physics. Commun. Math. Phys. 315, 465–488 (2012)
Buchholz D., Fredenhagen K.: Locality and the structure of particle states. Commun. Math. Phys. 84, 1–54 (1982)
Buchholz D., Mack G., Todorov I.: The current algebra on the circle as a germ of local field theories. Nucl. Phys. B 5, 20–56 (1988)
Carey A.L., Hurst C.A.: A note on the boson–fermion correspondence and infinite dimensional groups. Commun. Math. Phys. 98, 435–448 (1985)
Carey A.L., Langmann E.: Loop groups, anyons and the Calogero–Sutherland model. Commun. Math. Phys. 201, 1–34 (1999)
Carey, A.L., Langmann, E.: Loop groups and quantum fields. In: Bouwknegt, P., Wu, S. (eds.) Geometric Analysis and Applications to Quantum Field Theory. Progress in Mathematics, vol. 205, pp. 45–94. Birkhauser, Boston (2002)
Carey A.L., Ruijsenaars S.N.M.: On fermion gauge groups, current algebras and Kac–Moody algebras. Acta Appl. Math. 10, 1–86 (1987)
Dell’Antonio G., Figari R., Teta A.: Statistics in space dimension two. Lett. Math. Phys. 40, 235–256 (1997)
Fredenhagen K., Gaberdiel M., Rüger S.: Scattering states of plektons (particles with braid group statistics) in 2 + 1 dimensional quantum field theory. Commun. Math. Phys. 175, 319–335 (1996)
Fredenhagen, K., Rehren, K.-H., Schroer, B.: Superselection sectors with braid group statistics and exchange algebras II: geometric aspects and conformal covariance. Rev. Math. Phys. (special issue), 113–157 (1992)
Fröhlich J., Marchetti P.A.: Quantum field theories of vortices and anyons. Commun. Math. Phys. 121, 177–223 (1989)
Fröhlich J., Marchetti P.A.: Spin-statistics theorem and scattering in planar quantum field theories with braid statistics. Nucl. Phys. B 356, 533–573 (1991)
Graziano E., Rothe K.D.: Anyons, spin and statistics in (2 + 1)-dimensional U(1)-scalar Chern–Simons gauge field theory. Phys. Rev. D 49, 5512–5525 (1994)
Grosse H., Lechner G.: Noncommutative deformations of Wightman quantum field theories. JHEP 09, 131 (2008)
Langmann E.: Cocycles for boson and fermion Bogoliubov transformations. J. Math. Phys. 35, 96–112 (1994)
Langmann E.: Fermion current algebras and Schwinger terms in 3 + 1 dimensions. Commun. Math. Phys. 162, 1–32 (1994)
Lechner G.: Deformations of quantum field theories and integrable models. Commun. Math. Phys. 212, 265–302 (2012)
Liguori A., Mintchev M., Rossi M.: Anyon quantum fields without a Chern–Simons term. Phys. Lett. B 305, 52–58 (1993)
Lundholm D., Solovej J.P.: Hardy and Lieb–Thirring inequalities for anyons. Commun. Math. Phys. 322, 883–908 (2013)
Lundholm D., Solovej J.P.: Local exclusion and Lieb–Thirring inequalities for intermediate and fractional statistics. Ann. Henri Poincaré 15, 1061–1107 (2014)
Marino EC: Duality, quantum vortices and anyons in the Maxwell–Chrn–Simons–Higgs theories. Ann. Phys. 224, 225–274 (1993)
Mund J.: No-Go theorem for "free" relativistic anyons in d = 2 + 1. Lett. Math. Phys. 43, 319–328 (1998)
Mund J.: Modular localization of massive particles with any spin in d = 2 + 1. J. Math. Phys. 44, 2037–2057 (2003)
Mund J.: The spin-statistics theorem for anyons and plektons in d = 2 + 1. Commun. Math. Phys. 286, 1159–1180 (2009)
Plaschke M.: Wedge local deformations of charged fields leading to anyonic commutation relations. Lett. Math. Phys. 103, 507–532 (2013)
Pressley. A., Segal, G.: Loop Groups. Oxford Mathematical Monographs, Oxford (1986)
Ruijsenaars S.N.M.: On Bogoliubov transformations for systems of relativistic charged particles. J. Math. Phys. 18, 517 (1977)
Ruijsenaars S.N.M.: On Bogoliubov transformations. II. The general case. Ann. Phys. 116, 105–134 (1978)
Ruijsenaars S.N.M.: Integrable quantum field theories and Bogoliubov transformations. Ann. Phys. 132, 328–382 (1981)
Salvitti D: Generalized particle statistics in two-dimensions: examples from the theory of free massive Dirac field. Commun. Math. Phys. 269, 473–492 (2007)
Segal G.: Unitary representations of some infinite dimensional groups. Commun. Math. Phys. 80, 301–342 (1981)
Semenoff G.: Canonical quantum field theory with exotic statistics. Phys. Rev. Lett. 61, 517–520 (1988)
Wilczek F.: Quantum mechanics of fractional-spin particles. Phys. Rev. Lett. 49, 957 (1982)
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Plaschke, M. Local Anyonic Quantum Fields on the Circle Leading to Cone-Local Anyons in Two Dimensions. Lett Math Phys 105, 1033–1055 (2015). https://doi.org/10.1007/s11005-015-0767-9
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DOI: https://doi.org/10.1007/s11005-015-0767-9