Abstract
We study solutions of the equations (△ -λ )φ = 0 and (△ -λ )2φ = 0 in global coordinates on the covering space CAdS d of the d-dimensional Anti de-Sitter space subject to various boundary conditions and their connection to the unitary irreducible representations of \(\overline {{\text{SO}}} _0 \)(d-1,2). The ‘vanishing flux’ boundary conditions at spatial infinity lead to the standard quantization scheme for CAdS d in which solutions of the second- and the fourth-order equations are equivalent. To include fields realizing the singleton unitary representation in the bulk of CAdS d one has to relax the boundary conditions thus allowing for the nontrivial space of solutions of the dipole equation known as the Gupta–Bleuler triplet. We obtain explicit expressions for the modes of the Gupta–Bleuler triplet and the corresponding two-point function. To avoid negative-energy states one must also introduce an additional constraint in the space of solutions of the dipole equation.
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Aharony, O., Gubser, S. S., Maldacena, J., Ooguri, H. and Oz, Y.: Large N field theories, string theory and gravity, hep-th/9905111.
Balasubramanian, V., Kraus, P. and Lawrence, A.: Bulk vs. boundary dynamics in anti-de Sitter space-time, Phys. Rev. D 59 (1999), 046003, hep-th/9805171.
Klebanov, I. R. and Witten, E.: AdS/CFT correspondence and symmetry breaking, hep-th/9905104.
Flato, M., Frønsdal, C. and Sternheimer, D.: Singleton physics, hep-th/9901043; Singletons, physics in AdS universe and oscillations of composite neutrinos, Lett. Math. Phys. 48 (1999), 109.
Kogan, I. I.: Singletons and logarithmic CFT in AdS/CFT correspondence, Phys. Lett. B 458 (1999), 66, hep-th/9903162.
Frønsdal, C.: Elementary particles in curved space. 4. Massless particles, Phys.Rev. D 12 (1975), 3819.
Avis, S. J., Isham, C. J. and Storey, D.: Quantum field theory in anti-de Sitter spacetime, Phys. Rev. D 18 (1978), 3565.
Breitenlohner, P. and Freedman, D. Z.: Stability in gauged extended supergravity, Ann. Phys. 144 (1982), 197.
Angelopoulos, E., Flato, M., Frønsdal, C. and Sternheimer, D.: Massless particles, conformal group and De Sitter universe, Phys. Rev. D 23 (1981), 1278.
Castell, L. and Heidenreich, W.: SO(3,2) invariant scattering and dirac singletons, Phys. Rev. D 24 (1981), 371.
Vilenkin, N. and Klimuk, A.: Representations of Lie Groups and Special Functions, Kluwer Acad. Publ., Dordrecht, 1993; Barut, A. and Rãczka, R.: Theory of Group Representations and Applications, Polish Scientific Publishers, Warsaw, 1977.
Minwalla, S.: Restrictions imposed by superconformal invariance on quantum field theories, Adv. Theoret. Math. Phys. 2 (1998), 781, hep-th/9712074.
Evans, N.T.: Discrete series for the universal covering group of the 3 + 2 de Sitter group, J. Math. Phys. 8 (1967), 170. Angelopoulos, E.: \(\overline {SO} \);0 (3, 2): linear and unitary irreducible representations, In: Quantum Theory, Groups, Fields and Particles, Math. Phys. Stud. 4, D. Reidel, Dordrecht, (1983), pp. 101-148.
Angelopoulos, E.: Sur les représentations unitaires irréductibles de \(\overline {SO} \) 0(p, 2), C.R. Acad. Sci. Paris Sér. I Math. 292 (1981), 469-471; Angelopoulos, E. and Laoues, M.: Masslessness in n-dimensions, Rev. Math. Phys. 10 (1998), 271.
Dirac, P. A.M.: A remarkable representation of the 3+2 de Sitter group, J. Math. Phys. 4 (1963), 901.
Günaydin, M. and Warner, N. P.: Unitary supermultiplets of Osp(8/4,R) and the spectrum of the S(7) compactification of eleven-dimensional supergravity, Nuclear Phys. B 272 (1986), 99; Bergshoeff, E., Duff, M. J., Pope, C. N. and Sezgin, E.: Supersymmetric supermembrane vacua and singletons, Phys. Lett. B 199 (1987), 69; Bergshoeff, E., Salam, A., Sezgin, E. and Tanii, Y.: Singletons, higher spin massless states and the supermembrane, Phys. Lett. B 205 (1988), 237.
Flato, M. and Frønsdal, C.: One massless particle equals two Dirac singletons, Lett. Math. Phys. 2 (1978), 421.
Nicolai, H.: Representations of supersymmetry in anti-de Sitter space, In: B. de Wit, P. Fayet, P. van Nieuwenhuizen (eds), Supersymmetry and Supergravity '84, World Scientific, Singapore, 1984.
Ferrara, S., Frønsdal, C. and Zaffaroni, A.: On N = 8 supergravity on AdS(5) and N = 4 superconformal Yang-Mills theory, Nuclear Phys. B 532 (1998), 153, hep-th/9802203; Ferrara, S. and Zaffaroni, A.: Bulk gauge fields in AdS supergravity and supersingletons, hep-th/9807090.
Coddington, E. A. and Levinson, N.: Theory of Ordinary Differential Equations, McGraw-Hill, Englewood Cliffs, 1955.
Birrell, N. D. and Davies, P. C. W.: Quantum Fields in Curved Space, Cambridge Univ. Press, 1982.
Bargmann, V.: Irreducible unitary representations of the Lorentz group, Ann. Math. 48 (1947), 568.
Flato, M. and Frønsdal, C.:Quantum field theory of singletons. TheRac, J. Math. Phys. 22 (1981), 1100.
Frønsdal, C.: Dirac supermultiplet, Phys.Rev. D 26 (1982), 1988.
Flato, M. and Frønsdal, C.: The singleton dipole, Comm. Math. Phys. 108 (1987), 469.
Flato, M. and Frønsdal, C.: Interacting singletons, Lett. Math. Phys. 44 (1998), 249, hep-th/9803013.
Araki, H.: Indecomposable representations with invariant inner product. A theory of the Gupta-Bleuler triplet, Comm. Math. Phys. 97 (1985), 149.
Laoues, M.: Some properties of massless particles in arbitrary dimension, Rev. Math. Phys. 10 (1998), 1079.
Prudnikov, A. P., Brychkov, Yu. A. and Marichev, O. I.: Integraly i ryady. Specialnye funktsii, Nauka, Moscow, 1983. English translation: Integrals and Series. Vol. 2: Special Functions; Vol. 3: More Special Functions, Gordon & Breach, New York, 1988.
Starinets, A.: math-ph/9809014.
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Starinets, A. Singleton Field Theory and Flato–Frønsdal Dipole Equation. Letters in Mathematical Physics 50, 283–300 (1999). https://doi.org/10.1023/A:1007644223085
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DOI: https://doi.org/10.1023/A:1007644223085