Unitary positive-energy representations of scalar bilocal quantum fields. (English) Zbl 1156.81422

Summary: The superselection sectors of two classes of scalar bilocal quantum fields in \(D \geq 4\) 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.


81T05 Axiomatic quantum field theory; operator algebras
81T08 Constructive quantum field theory
Full Text: DOI arXiv


[1] Bakalov, B.; Nikolov, N. M., Jacobi identity for vertex algebras in higher dimensions, J. Math. Phys., 47, 053505 (2006) · Zbl 1111.17014
[2] Baumann, K., There are no scalar Lie fields in three or more dimensional space-time, Commun. Math. Phys., 47, 69-74 (1976) · Zbl 0318.53032
[3] 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)
[4] Boerner, H.: Representations of Groups, 2^nd edition, Amsterdam: North-Holland Publishing Company 1970 · Zbl 0112.26301
[5] Buchholz, D.; Doplicher, S.; Longo, R.; Roberts, J. E., A new look at Goldstone’s theorem, Rev. Math. Phys. SI, 1, 49-84 (1992) · Zbl 0784.46060
[6] Carpi, S.; Conti, R., Classification of subsystems for graded-local nets with trivial superselection structure, Commun. Math. Phys., 253, 423-449 (2005) · Zbl 1087.81039
[7] Doplicher, S.; Roberts, J. E., Why there is a field algebra with a compact gauge group describing the superselection structure in particle physics, Commun. Math. Phys., 131, 51-107 (1990) · Zbl 0734.46042
[8] 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 · Zbl 0492.22012
[9] 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 · Zbl 0535.22012
[10] Haag, R., Local Quantum Physics (1992), Berlin-New York: Springer, Berlin-New York · Zbl 0777.46037
[11] Jakobsen, H. P., The last possible place of unitarity for certain highest weight modules, Math. Ann., 256, 439-447 (1981) · Zbl 0478.22007
[12] Jordan, P., Der Zusammenhang der symmetrischen und linearen Gruppen und das Mehrkörperproblem, Zeitschr. für Physik, 94, 531 (1935) · Zbl 0011.18504
[13] Kac, V.; Radul, A., Representation theory of the vertex algebra W_1+∞, Transform. Groups, 1, 41-70 (1996) · Zbl 0862.17023
[14] Kashiwara, M.; Vergne, M., On the Segal-Shale-Weil representations and harmonic polynomials, Invent. Math., 44, 1-47 (1978) · Zbl 0375.22009
[15] Lowenstein, J. H., The existence of scalar Lie fields, Commun. Math. Phys., 6, 49-60 (1967) · Zbl 0166.23504
[16] Nikolov, N. M., Vertex algebras in higher dimensions and globally conformal invariant quantum field theory, Commun. Math. Phys., 253, 283-322 (2005) · Zbl 1125.17010
[17] Nikolov, N.M., Rehren, K.-H., Todorov, I.T.: Harmonic bilocal fields generated by globally conformal invariant scalar fields. (In preparation) · Zbl 1144.81028
[18] Nikolov, N. M.; Rehren, K.-H.; Todorov, I. T., Partial wave expansion and Wightman positivity in conformal field theory, Nucl. Phys. B, 722, 266-296 (2005) · Zbl 1128.81320
[19] Nikolov, N. M.; Stanev, Ya. S.; Todorov, I. T., Four dimensional CFT models with rational correlation functions, J. Phys. A: Math. Gen., 35, 2985-3007 (2002) · Zbl 1041.81097
[20] 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) · Zbl 1058.81054
[21] Nikolov, N. M.; Todorov, I. T., Rationality of conformally invariant local correlation functions on compactified Minkowski space. Commun, Math. Phys., 218, 417-436 (2001) · Zbl 0985.81055
[22] Nikolov, N. M.; Todorov, I. T., Elliptic thermal correlation functions and modular forms in a globally conformal invariant QFT, Rev. Math. Phys., 17, 613-667 (2005) · Zbl 1111.81112
[23] Reeh, H.; Schlieder, S., Bemerkungen zur Unitäräquivalenz von Lorentz-invarianten Feldern, Nuovo Cim., 22, 1051-1068 (1961) · Zbl 0101.22402
[24] 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
[25] Schmidt, M. U., Lowest weight representations of some infinite dimensional groups on Fock spaces, Acta Appl. Math., 18, 59-84 (1990) · Zbl 0729.22023
[26] 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
[27] 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
[28] Verma, D.-N., Structure of certain induced representations of complex semisimple Lie algebras, Bull. Amer. Math. Soc., 74, 160-166 (1968) · Zbl 0157.07604
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.