Abstract
We present a simple isomorphism between the algebra of one real chiral Fermi field and the algebra of n real chiral Fermi fields. The isomorphism preserves the vacuum state. This is possible by a “change of localization”, and gives rise to new multilocal symmetries generated by the corresponding multilocal current and stress–energy tensor. The result gives a common underlying explanation of several remarkable recent results on the representation of the free Bose field in terms of free Fermi fields (Anguelova, arXiv:1112.3913, 2011; Anguelova, arXiv:1206.4026, 2012), and on the modular theory of the free Fermi algebra in disjoint intervals (Casini and Huerta, Class Quant Grav 26:185005, 2009; Longo et al., Rev Math Phys 22:331–354, 2010)
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Anguelova, I.: Boson–fermion correspondence of type B and twisted vertex algebras (2011). arXiv:1112.3913
Anguelova, I.: Twisted vertex algebras, bicharacter construction and boson-fermion correspondences (2012). arXiv:1206.4026
Bisognano J.J., Wichmann E.H.: On the duality condition for quantum fields. J. Math. Phys. 17, 303–321 (1976)
Borchers H.-J.: The CPT theorem in two-dimensional theories of local observables. Commun. Math. Phys. 143, 315–322 (1992)
Brunetti R., Guido D., Longo R.: Modular structure and duality in conformal quantum field theory. Commun. Math. Phys. 156, 201–219 (1993)
Carey A.L., Ruijsenaars S.N.M., Wright J.D.: The massless Thirring model: positivity of Klaiber’s N-point functions. Commun. Math. Phys. 99, 347–364 (1985)
Casini H., Huerta M.: Reduced density matrix and internal dynamics for multicomponent regions. Class. Quant. Gravity 26, 185005 (2009)
Date E., Jimbo M., Kashiwara M., Miwa T.: Transformation groups for soliton equations. 4. A new hierarchy of soliton equations of KP type. Physica D4, 343–365 (1982)
Fredenhagen, K.: Superselection sectors with infinite statistical dimension (1994, preprint DESY-94-071)
Fuchs J.: Affine Lie Algebras and Quantum Groups. Cambridge University Press, London (1992)
Fröhlich J., Gabbiani F.: Operator algebras and conformal field theory. Commun. Math. Phys. 155, 569–640 (1993)
Guido D., Longo R., Wiesbrock H.-W.: Extensions of conformal nets and superselection structures. Commun. Math. Phys. 192, 217–244 (1998)
Haag R., Hugenholtz N., Winnink M.: On the equilibrium states in quantum statistical mechanics. Commun. Math. Phys. 5, 215–236 (1967)
Kähler R., Wiesbrock H.-W.: Modular theory and the reconstruction of four-dimensional quantum field theories. J. Math. Phys. 42, 74–86 (2001)
Longo R., Martinetti P., Rehren K.-H.: Geometric modular action for disjoint intervals and boundary conformal field theory. Rev. Math. Phys. 22, 331–354 (2010)
Mandelstam S.: Soliton operators for the quantized sine-Gordon equation. Phys. Rev. D11, 3026 (1975)
Stone, M. (ed): Bosonization. World Scientific, Singapore (1994)
Acknowledgments
This work was supported in part by the German Research Foundation [Deutsche Forschungsgemeinschaft (DFG)] through the Institutional Strategy of the University of Göttingen, and by the DFG Research Training School 1493 “Mathematical Structures in Modern Quantum Physics”. KHR is grateful to I. Anguelova for interesting discussions, which triggered this work, and to Y. Tanimoto and M. Bischoff, who raised interesting questions.
Open Access
This article is distributed under the terms of the Creative Commons Attribution License which permits any use, distribution, and reproduction in any medium, provided the original author(s) and the source are credited.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License (https://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
About this article
Cite this article
Rehren, KH., Tedesco, G. Multilocal Fermionization. Lett Math Phys 103, 19–36 (2013). https://doi.org/10.1007/s11005-012-0582-5
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11005-012-0582-5