Hostname: page-component-586b7cd67f-t7fkt Total loading time: 0 Render date: 2024-11-29T20:35:56.777Z Has data issue: false hasContentIssue false
  • English
  • Français

Massless Wigner particles in conformal field theory are free

Part of: Scattering theory Quantum field theory; related classical field theories

Published online by Cambridge University Press:  22 August 2014

YOH TANIMOTO
YOH TANIMOTO*
Affiliation:
Graduate School of Mathematical Sciences, The University of Tokyo, Institut für Theoretische Physik, Göttingen University, 3-8-1 Komaba Meguro-ku, Tokyo 153-8914, Japan; hoyt@ms.u-tokyo.ac.jp

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We show that the massless particle spectrum in a four-dimensional conformal Haag–Kastler net is generated by a free field subnet. If the massless particle spectrum is scalar, then the free field subnet decouples as a tensor product component.

MSC classification

Secondary: 81T05: Axiomatic quantum field theory; operator algebras 81T40: Two-dimensional field theories, conformal field theories, etc. 81U99: None of the above, but in this section
Type
Research Article
Information
Forum of Mathematics, Sigma , Volume 2 , 01 July 2014 , e21
Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/3.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
© The Author 2014

References

Araki, H., ‘von Neumann algebras of local observables for free scalar field’, J. Math. Phys. 5 (1964), 1–13.Google Scholar
Bakalov, B., Nikolov, N. M., Rehren, K.-H. and Todorov, I., ‘Unitary positive-energy representations of scalar bilocal quantum fields’, Commun. Math. Phys. 271 (1) (2007), 223–246.Google Scholar
Baumann, K., ‘All massless, scalar fields with trivial S-matrix are Wick-polynomials’, Commun. Math. Phys. 86 (2) (1982), 247–256.Google Scholar
Baumgärtel, H., Operator algebraic methods in quantum field theory, (Akademie Verlag, Berlin, 1995).Google Scholar
Bischoff, M. and Tanimoto, Y., ‘Construction of Wedge-Local Nets of Observables through Longo–Witten Endomorphisms. II’, Commun. Math. Phys. 317 (3) (2013), 667–695.Google Scholar
Borchers, H.-J., Buchholz, D. and Schroer, B., ‘Polarization-free generators and the S-matrix’, Commun. Math. Phys. 219 (1) (2001), 125–140.CrossRefGoogle Scholar
Brunetti, R., Guido, D. and Longo, R., ‘Modular structure and duality in conformal quantum field theory’, Commun. Math. Phys. 156 (1) (1993), 201–219.Google Scholar
Buchholz, D. and Jacobi, P., ‘On the nuclearity condition for massless fields’, Lett. Math. Phys. 13 (4) (1987), 313–323.Google Scholar
Buchholz, D., ‘Collision theory for waves in two dimensions and a characterization of models with trivial S-matrix’, Commun. Math. Phys. 45 (1) (1975), 1–8.Google Scholar
Buchholz, D., ‘Collision theory for massless bosons’, Commun. Math. Phys 52 (2) (1977), 147–173.Google Scholar
Buchholz, D., ‘The physical state space of quantum electrodynamics’, Commun. Math. Phys. 85 (1) (1982), 49–71.Google Scholar
Buchholz, D. and Fredenhagen, K., ‘Dilations and interaction’, J. Math. Phys. 18 (5) (1977), 1107–1111.Google Scholar
Buchholz, D. and Wichmann, E. H., ‘Causal independence and the energy-level density of states in local quantum field theory’, Commun. Math. Phys. 106 (2) (1986), 321–344.CrossRefGoogle Scholar
Carpi, S. and Conti, R., ‘Classification of subsystems for local nets with trivial superselection structure’, Commun. Math. Phys. 217 (1) (2001), 89–106.Google Scholar
Driessler, W., ‘Duality and absence of locally generated superselection sectors for CCR-type algebras’, Commun. Math. Phys. 70 (3) (1979), 213–220.Google Scholar
Dymarsky, A., Komargodski, Z., Schwimmer, A. and Theisen, S., On scale and conformal invariance in four dimensions. 2013. arXiv:1309.2921.Google Scholar
Ge, L. and Kadison, R., ‘On tensor products for von Neumann algebras’, Invent. Math. 123 (3) (1996), 453–466.Google Scholar
Haag, R., Local quantum physics, second edition, Texts and Monographs in Physics (Springer-Verlag, Berlin, 1996).Google Scholar
Hislop, P. D., ‘Conformal covariance, modular structure, and duality for local algebras in free massless quantum field theories’, Ann. Phys. 185 (2) (1988), 193–230.Google Scholar
Hislop, P. D. and Longo, R., ‘Modular structure of the local algebras associated with the free massless scalar field theory’, Commun. Math. Phys. 84 (1) (1982), 71–85.Google Scholar
Kawahigashi, Y., Longo, R. and Müger, M., ‘Multi-interval subfactors and modularity of representations in conformal field theory’, Commun. Math. Phys. 219 (3) (2001), 631–669.Google Scholar
Landau, L. J., ‘Asymptotic locality and the structure of local internal symmetries’, Commun. Math. Phys. 17 (1970), 156–176.Google Scholar
Lechner, G., ‘Construction of quantum field theories with factorizing S-matrices’, Commun. Math. Phys. 277 (3) (2008), 821–860.Google Scholar
Longo, R., Real Hilbert subspaces, modular theory, SL(2, R) and CFT, Von Neumann algebras in Sibiu: Conference Proceedings (Theta, Bucharest, 2008), 33–91.Google Scholar
Mack, G., ‘All unitary ray representations of the conformal group SU(2, 2) with positive energy’, Commun. Math. Phys. 55 (1) (1977), 1–28.Google Scholar
Mandelstam, S., ‘Light-cone superspace and the ultraviolet finiteness of the n = 4 model’, Nucl. Phys. B 213 (1) (1983), 149–168.Google Scholar
Mund, J., ‘An algebraic Jost-Schroer theorem for massive theories’, Commun. Math. Phys. 315 (2) (2012), 445–464.Google Scholar
Nakayama, Y., A lecture note on scale invariance vs conformal invariance. 2013 arXiv:1302.0884.Google Scholar
Nikolov, N. M. and Todorov, I. T., ‘Conformal invariance and rationality in an even dimensional quantum field theory’, Intl. J. Mod. Phys. A 19 (22) (2004), 3605–3636.Google Scholar
Nikolov, N. M. and Todorov, I. T., ‘Rationality of conformally invariant local correlation functions on compactified Minkowski space’, Commun. Math. Phys. 218 (2) (2001), 417–436.Google Scholar
Reed, M. and Simon, B., Methods of modern mathematical physics. II. Fourier analysis, self-adjointness, (Academic Press, New York, 1975).Google Scholar
Reed, M. and Simon, B., Methods of modern mathematical physics. I. Functional analysis, second edition, (Academic Press, New York, 1980).Google Scholar
Stanev, Y., ‘Constraining conformal field theory with higher spin symmetry in four dimensions’, Nucl. Phys. B 876 (2013), 651–666.Google Scholar
Takesaki, M., Theory of Operator Algebras. II, Encyclopaedia of Mathematical Sciences, 125 (Springer-Verlag, Berlin, 2003), 6 Operator algebras and non-commutative geometry.Google Scholar
Tanimoto, Y., ‘Construction of Wedge-Local Nets of Observables Through Longo–Witten Endomorphisms’, Commun. Math. Phys. 314 (2) (2012), 443–469.Google Scholar
Tanimoto, Y., ‘Noninteraction of waves in two-dimensional conformal field theory’, Commun. Math. Phys. 314 (2) (2012), 419–441.Google Scholar
Tanimoto, Y., ‘Construction of two-dimensional quantum field models through Longo–Witten endomorphisms’, Forum of Mathematics, Sigma 2 e7. doi:10.1017/fms.2014.3.Google Scholar
Todorov, I., ‘Vertex algebras and conformal field theory models in four dimensions’, Fortschr. Phys. 54 (5–6) (2006), 496–504.Google Scholar
Weinberg, S., ‘Minimal fields of canonical dimensionality are free’, Phys. Rev. D 86 (2012), 105015.Google Scholar
Weiner, Mihály, ‘An algebraic version of Haag’s theorem’, Commun. Math. Phys. 305 (2) (2011), 469–485.Google Scholar