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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.

81T05Axiomatic quantum field theory; operator algebras
81T08Constructive quantum field theory
Full Text: DOI arXiv
[1] Bakalov B. and Nikolov N.M. (2006). Jacobi identity for vertex algebras in higher dimensions. J. Math. Phys. 47: 053505 · Zbl 1111.17014 · doi:10.1063/1.2197687
[2] Baumann K. (1976). There are no scalar Lie fields in three or more dimensional space-time. Commun. Math. Phys. 47: 69--74 · Zbl 0318.53032 · doi:10.1007/BF01609354
[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); English translation, Funct. Anal. Appl. 5, 1--8 (1971)
[4] Boerner, H.: Representations of Groups, 2nd edition, Amsterdam: North-Holland Publishing Company 1970
[5] 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 · Zbl 0784.46060 · doi:10.1142/S0129055X92000157
[6] Carpi S. and Conti, R. (2005). Classification of subsystems for graded-local nets with trivial superselection structure. Commun. Math. Phys. 253: 423--449 · Zbl 1087.81039 · doi:10.1007/s00220-004-1135-2
[7] 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 · Zbl 0734.46042 · doi:10.1007/BF02097680
[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
[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
[10] Haag R. (1992). Local Quantum Physics. Springer, Berlin-New York · Zbl 0777.46037
[11] Jakobsen H.P. (1981). The last possible place of unitarity for certain highest weight modules. Math. Ann. 256: 439--447 · Zbl 0478.22007 · doi:10.1007/BF01450539
[12] Jordan P. (1935). Der Zusammenhang der symmetrischen und linearen Gruppen und das Mehrkörperproblem. Zeitschr. für Physik 94: 531 · Zbl 0011.18504 · doi:10.1007/BF01330618
[13] Kac V. and Radul A. (1996). Representation theory of the vertex algebra W 1+ Transform. Groups 1: 41--70 · Zbl 0862.17023 · doi:10.1007/BF02587735
[14] Kashiwara M. and Vergne M. (1978). On the Segal--Shale--Weil representations and harmonic polynomials. Invent. Math. 44: 1--47 · Zbl 0375.22009 · doi:10.1007/BF01389900
[15] Lowenstein J.H. (1967). The existence of scalar Lie fields. Commun. Math. Phys. 6: 49--60 · Zbl 0166.23504 · doi:10.1007/BF01646322
[16] Nikolov N.M. (2005). Vertex algebras in higher dimensions and globally conformal invariant quantum field theory. Commun. Math. Phys. 253: 283--322 · Zbl 1125.17010 · doi:10.1007/s00220-004-1133-4
[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. and Todorov I.T. (2005). Partial wave expansion and Wightman positivity in conformal field theory. Nucl. Phys. B 722: 266--296 · Zbl 1128.81320 · doi:10.1016/j.nuclphysb.2005.06.006
[19] 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 · Zbl 1041.81097 · doi:10.1088/0305-4470/35/12/319
[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. and Todorov I.T. (2001). Rationality of conformally invariant local correlation functions on compactified Minkowski space. Commun. Math. Phys. 218: 417--436 · Zbl 0985.81055 · doi:10.1007/s002200100414
[22] 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 · Zbl 1111.81112 · doi:10.1142/S0129055X0500239X
[23] Reeh H. and Schlieder S. (1961). Bemerkungen zur Unitäräquivalenz von Lorentz-invarianten Feldern. Nuovo Cim. 22: 1051--1068 · Zbl 0101.22402 · doi:10.1007/BF02787889
[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. (1990). Lowest weight representations of some infinite dimensional groups on Fock spaces. Acta Appl. Math. 18: 59--84 · Zbl 0729.22023 · doi:10.1007/BF00822205
[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); Errata, ibid. 628 · Zbl 0157.07604