zbMATH — the first resource for mathematics

Local nets of von Neumann algebras in the sine-Gordon model. (English) Zbl 07333675
Summary: The Haag-Kastler net of local von Neumann algebras is constructed in the ultraviolet finite regime of the Sine-Gordon model, and its equivalence with the massive Thirring model is proved. In contrast to other authors, we do not add an auxiliary mass term, and we work completely in Lorentzian signature. The construction is based on the functional formalism for perturbative Algebraic Quantum Field Theory together with estimates originally derived within Constructive Quantum Field Theory and adapted to Lorentzian signature. The paper extends previous work by two of us.
81T05 Axiomatic quantum field theory; operator algebras
35Q55 NLS equations (nonlinear Schrödinger equations)
81T15 Perturbative methods of renormalization applied to problems in quantum field theory
81T08 Constructive quantum field theory
81U20 \(S\)-matrix theory, etc. in quantum theory
46L05 General theory of \(C^*\)-algebras
46L10 General theory of von Neumann algebras
46L60 Applications of selfadjoint operator algebras to physics
46L30 States of selfadjoint operator algebras
Full Text: DOI
[1] Araki, H.; Yamagami, S., On quasi-equivalence of quasifree states of the canonical commutation relations, Publ. Res. Inst. Math. Sci., 18, 703-758 (1982) · Zbl 0505.46052
[2] Bahns, D.; Rejzner, K., The Quantum Sine Gordon model in perturbative AQFT, Commun. Math. Phys. (2017) · Zbl 1386.81110
[3] Benfatto, G.; Falco, P.; Mastropietro, V., Massless Sine-Gordon and massive thirring models: proof of coleman’s equivalence, Commun. Math. Phys., 285, 713-762 (2009) · Zbl 1178.35326
[4] Benfatto, G.; Falco, P.; Mastropietro, V., Functional Integral Construction of the Thirring model: axioms verification and massless limit, Commun. Math. Phys., 273, 67 (2007) · Zbl 1124.81031
[5] Brunetti, R.; Fredenhagen, K., Remarks on time energy uncertainty relations, Rev. Math. Phys., 14, 897 (2002) · Zbl 1033.81010
[6] Brunetti, R.; Guido, D.; Longo, R., Modular localization and Wigner particles, Rev. Math. Phys., 14, 759-786 (2002) · Zbl 1033.81063
[7] Buchholz, D., Product states for local algebras, Commun. Math. Phys., 36, 287 (1974) · Zbl 0289.46050
[8] Buchholz, D.; Mack, G.; Todorov, I., The current algebra on the circle as a germ of local field theories, Nucl. Phys. Proc. Suppl., 5B, 20-56 (1988) · Zbl 0958.22500
[9] Cadamuro, D., Tanimoto, Y.: Wedge-local observables in the deformed Sine-Gordon model. arXiv:1612.02073 [math-ph] · Zbl 1393.81023
[10] Carey, AL; Ruijsenaars, SNM; Wright, JD, The massless thirring model: positivity of Klaiber’s n-point functions, Commun. Math. Phys., 99, 347-364 (1985)
[11] Coleman, S., Quantum Sine-Gordon equation as the massive Thirring model, Phys. Rev. D, 11, 1975, 2088-2097 (1992)
[12] Derezinski, J., Meissner, K.A.: Quantum massless field in 1+1 dimensions. Lect. Notes Phys. 690, 107 (2006). ([math-ph/0408057]) · Zbl 1168.81368
[13] Eckmann, JP; Fröhlich, J., Unitary equivalence of local algebras in the quasifree representation, Ann. Inst. H. Poincare Phys. Theor., 20, 201 (1974) · Zbl 0287.46077
[14] Faddeev, LD; Korepin, VE, Quantization of solitons, Theor. Math. Phys., 25, 103-1049 (1975)
[15] Fredenhagen, K., Rejzner, K.: Perturbative construction of models of quantum field theory. In: Advances in Algebraic Quantum Field Theory, R. Brunetti et al (eds), Mathematical Physics Studies, Springer (2015) · Zbl 1334.81059
[16] Fröhlich, J.; Seiler, E., The massive Thirring-Schwinger model (QED in two-dimensions): convergence of perturbation theory and particle structure, Helv. Phys. Acta, 49, 889 (1976)
[17] Fröhlich, J., Classical and quantum statistical mechanics in one and two dimensions: two-component Yukawa and Coulomb systems, Commun. Math. Phys., 47, 233-268 (1976) · Zbl 1092.82505
[18] Hadjiivanov, LK; Stoyanov, DT, Wightman functions in the thirring model, Theor. Math. Phys., 46, 236-242 (1981)
[19] Hollands, S.; Wald, R., Axiomatic quantum field theory in curved spacetime, Commun. Math. Phys., 293, 85 (2010) · Zbl 1193.81076
[20] Karowski, M.; Weisz, P., Exact form factors in (1 + 1)-dimensional field theoretic models with soliton behaviour, Nucl. Phys. B, 139, 455-476 (1978)
[21] Lechner, G., Construction of quantum field theories with factorizing S-matrices, Commun. Math. Phys., 277, 821-860 (2008) · Zbl 1163.81010
[22] Lowenstein, JH; Speer, ER, Existence of conserved currents in the perturbative sine-gordon and massive thirring models, Commun. Math. Phys., 63, 97-112 (1978)
[23] Mandelstam, S., Soliton operators for the quantized Sine-Gordon equation, Phys. Rev. D, 11, 3026-3030 (1975)
[24] Park, YM, Massless quantum sine-gordon equation in two space-time dimensions: correlation inequalities and infinite volume limit, J. Math. Phys., 18, 2423-2426 (1977)
[25] Requardt, M., Symmetry conservation and integrals over local charge densities in quantum field theory, Commun. Math. Phys., 50, 259 (1976) · Zbl 0336.46064
[26] Radzikowski, MJ, Micro-local approach to the Hadamard condition in quantum field theory on curved space-time, Commun. Math. Phys., 179, 529-553 (1996) · Zbl 0858.53055
[27] Rejzner, K., Perturbative Algebraic Quantum Field Theory?: An Introduction for Mathematicians (2016), New York: Springer, Mathematical Physics Studies, New York · Zbl 1347.81011
[28] Schroer, B., Modular wedge localization and the d = (1+1) form-factor program, Ann. Phys., 275, 190-223 (1999) · Zbl 0989.81057
[29] Schubert, S.: Über die Charakterisierung von Zuständen hinsichtlich der Erwartungswerte quadratischer Operatoren, Diplomarbeit Hamburg (2013). http://www.desy.de/uni-th/theses/Dipl_Schubert.pdf
[30] Smirnov, F.A.: Form Factors in Completely Integrable Models of Quantum Field Theory. World Scientific. Advanced Series in Mathematical Physics 14, (1992) · Zbl 0788.46077
[31] Wightman, A.S.: Introduction to some aspects of the relativistic dynamics of quantized fields. In: High energy electromagnetic interactions and field theory, pp. 171-289 Ltvy, M. ed. New York: Gordon and Breach, (1967)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.