Abstract
The superselection sectors of two classes of scalar bilocal quantum fields in D ≥ 4 dimensions are explicitly determined by working out the constraints imposed by unitarity. The resulting classification in terms of the dual of the respective gauge groups U(N) and O(N) confirms the expectations based on general results obtained in the framework of local nets in algebraic quantum field theory, but the approach using standard Lie algebra methods rather than abstract duality theory is complementary. The result indicates that one does not lose interesting models if one postulates the absence of scalar fields of dimension D−2 in models with global conformal invariance. Another remarkable outcome is the observation that, with an appropriate choice of the Hamiltonian, a Lie algebra embedded into the associative algebra of observables completely fixes the representation theory.
Similar content being viewed by others
References
Bakalov B. and Nikolov N.M. (2006). Jacobi identity for vertex algebras in higher dimensions. J. Math. Phys. 47: 053505
Baumann K. (1976). There are no scalar Lie fields in three or more dimensional space-time. Commun. Math. Phys. 47: 69–74
Bernstein, I.N., Gelfand, I.M., Gelfand, S.I.: Structure of representations that are generated by vectors of highest weight (Russian). Funk. Anal. i Prilozen. 5, 1–9 (1971); English translation, Funct. Anal. Appl. 5, 1–8 (1971)
Boerner, H.: Representations of Groups, 2nd edition, Amsterdam: North-Holland Publishing Company 1970
Buchholz D., Doplicher S., Longo R. and Roberts J.E. (1992). A new look at Goldstone’s theorem. Rev. Math. Phys. SI 1: 49–84
Carpi S. and Conti, R. (2005). Classification of subsystems for graded-local nets with trivial superselection structure. Commun. Math. Phys. 253: 423–449
Doplicher S. and Roberts J.E. (1990). Why there is a field algebra with a compact gauge group describing the superselection structure in particle physics. Commun. Math. Phys. 131: 51–107
Enright, T.J., Parthasarathy, R.: A proof of a conjecture of Kashiwara and Vergne. In: Noncommutative harmonic analysis and Lie groups (Marseille, 1980), Lecture Notes in Math. 880, Berlin-New York: Springer, 1981, pp. 74–90
Enright, T.J., Howe, R., Wallach, N.: A classification of unitary highest weight modules. In: Representation theory of reductive groups (Park City, Utah, 1982), Progr. Math. 40, Boston, MA: Birkhäuser, 1983, pp. 97–143
Haag R. (1992). Local Quantum Physics. Springer, Berlin-New York
Jakobsen H.P. (1981). The last possible place of unitarity for certain highest weight modules. Math. Ann. 256: 439–447
Jordan P. (1935). Der Zusammenhang der symmetrischen und linearen Gruppen und das Mehrkörperproblem. Zeitschr. für Physik 94: 531
Kac V. and Radul A. (1996). Representation theory of the vertex algebra W 1+∞. Transform. Groups 1: 41–70
Kashiwara M. and Vergne M. (1978). On the Segal–Shale–Weil representations and harmonic polynomials. Invent. Math. 44: 1–47
Lowenstein J.H. (1967). The existence of scalar Lie fields. Commun. Math. Phys. 6: 49–60
Nikolov N.M. (2005). Vertex algebras in higher dimensions and globally conformal invariant quantum field theory. Commun. Math. Phys. 253: 283–322
Nikolov, N.M., Rehren, K.-H., Todorov, I.T.: Harmonic bilocal fields generated by globally conformal invariant scalar fields. (In preparation)
Nikolov N.M., Rehren K.-H. and Todorov I.T. (2005). Partial wave expansion and Wightman positivity in conformal field theory. Nucl. Phys. B 722: 266–296
Nikolov N.M., Stanev Ya.S. and Todorov I.T. (2002). Four dimensional CFT models with rational correlation functions. J. Phys. A: Math. Gen. 35: 2985–3007
Nikolov, N.M., Stanev, Ya.S., Todorov, I.T.: Globally conformal invariant gauge field theory with rational correlation functions. Nucl. Phys. B670[FS], 373–400 (2003)
Nikolov N.M. and Todorov I.T. (2001). Rationality of conformally invariant local correlation functions on compactified Minkowski space. Commun. Math. Phys. 218: 417–436
Nikolov N.M. and Todorov I.T. (2005). Elliptic thermal correlation functions and modular forms in a globally conformal invariant QFT. Rev. Math. Phys. 17: 613–667
Reeh H. and Schlieder S. (1961). Bemerkungen zur Unitäräquivalenz von Lorentz-invarianten Feldern. Nuovo Cim. 22: 1051–1068
Roberts, J.E.: Lectures on algebraic quantum field theory. In: The Algebraic Theory of Superselection Sectors, D. Kastler (ed.), Singapore: World Scientific, 1990, pp. 1–112
Schmidt M.U. (1990). Lowest weight representations of some infinite dimensional groups on Fock spaces. Acta Appl. Math. 18: 59–84
Schwinger, J.: On angular momentum. In: Quantum Theory of Angular Momentum, L.C. Biedenharn, H. Van Dam (eds.), New York: Academic Press, 1965, pp. 229–279
Todorov, I.T.: Infinite-dimensional Lie algebras in conformal QFT models. In: A.O. Barut, H.-D. (eds.), Conformal Groups and Related Symmetries. Physical Results and Mathematical Background, Lecture Notes in Physics 261, Berlin: Springer, 1986, pp. 387–443
Verma, D.-N.: Structure of certain induced representations of complex semisimple Lie algebras. Bull. Amer. Math. Soc. 74, 160–166 (1968); Errata, ibid. 628
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Y. Kawahigashi
Rights and permissions
About this article
Cite this article
Bakalov, B., Nikolov, N.M., Rehren, KH. et al. Unitary Positive-Energy Representations of Scalar Bilocal Quantum Fields. Commun. Math. Phys. 271, 223–246 (2007). https://doi.org/10.1007/s00220-006-0182-2
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00220-006-0182-2