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Conformally invariant differential operators on Minkowski space and their curved analogues. (English) Zbl 0659.53047

This article describes the construction of a natural family of conformally invariant differential operators on a four-dimensional (pseudo-) Riemannian manifold. Included in this family are the usual massless field equations for arbitrary helicity but there are many more besides. The article begins by classifying the invariant operators on flat space. This is a fairly straightforward task in representation theory best solved through the theory of Verma modules. The method generates conformally invariant operators in the curved case by means of Penrose’s local twistor transport.

MSC:

53C50 Global differential geometry of Lorentz manifolds, manifolds with indefinite metrics
58J70 Invariance and symmetry properties for PDEs on manifolds
83C05 Einstein’s equations (general structure, canonical formalism, Cauchy problems)
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[1] Baston, R. J.: The algebraic construction of invariant differential operators. D.Phil. thesis, Oxford University 1985 · Zbl 0562.90101
[2] Bateman, H.: The transformation of the electrodynamical equations. Proc. L. M. S.8, 223-264 (1910) · JFM 41.0942.03 · doi:10.1112/plms/s2-8.1.223
[3] Bernstein, I. N., Gelfand, I. M., Gelfand, S. I.: Differential operators on principal affine spaces and investigation of g-modules. In: Lie groups and their representations, Summer School of Bolyai J?nos Math. Soc. Gelfand, I. M. (ed), pp. 21-64, London: Hilgar 1975
[4] Boe, B. D., Collingwood, D. H.: A comparison theory for the structure of induced representations. J. Algebra94, 511-545 (1985) · Zbl 0606.17007 · doi:10.1016/0021-8693(85)90197-8
[5] Boe, B. D., Collingwood, D. H.: A comparison theory for the structure of induced representations II. Math. Z.190, 1-11 (1985) · Zbl 0562.17003 · doi:10.1007/BF01159158
[6] Bott, R.: Homogeneous vector bundles. Ann. Math.66, 203-248 (1957) · Zbl 0094.35701 · doi:10.2307/1969996
[7] Cunningham, E.: The principal of relativity in electrodynamics and an extension thereof. Proc. L. M. S.8, 77-98 (1910) · doi:10.1112/plms/s2-8.1.77
[8] Dixmier, J.: Enveloping algebras. Amsterdam: North Holland 1977 · Zbl 0346.17010
[9] Dubrovin, B. A., Fomenko, A. T., Novikov, S. P.: Modern geometry, methods and applications. Moscow: Nauka 1979 · Zbl 0601.53001
[10] Eastwood, M. G., Penrose, R., Wells, R. O., Jr.: Cohomology and massless fields. Commun. Math. Phys.78, 305-351 (1981) · Zbl 0465.58031 · doi:10.1007/BF01942327
[11] Eastwood, M. G., Tod, K. P.: Edth-a differential operator on the sphere. Math. Proc. Camb. Phil. Soc.92, 317-330 (1982) · Zbl 0511.53026 · doi:10.1017/S0305004100059971
[12] Eastwood, M. G.: Complexification, twistor theory, and harmonic maps of Riemann surfaces. Bull. A. M. S.11, 317-328 (1984) · Zbl 0556.53049 · doi:10.1090/S0273-0979-1984-15294-7
[13] Eastwood, M. G.: The generalized Penrose-Ward transform. Math. Proc. Camb. Phil. Soc.97, 165-187 (1985) · Zbl 0581.32035 · doi:10.1017/S030500410006271X
[14] Eastwood, M. G.: A duality for homogeneous bundles on twistor space. J. L. M. S.31, 349-356 (1985) · Zbl 0534.14008
[15] Eastwood, M. G., Singer, M. A.: A conformally invariant Maxwell gauge. Phys. Lett.107A, 73-74 (1985) · Zbl 1177.83074
[16] Fefferman, C., Graham, C. R.: Conformal invariants. Preprint, Princeton University 1984 · Zbl 0602.53007
[17] Fegan, H. D.: Conformally invariant first order differential operators. Q. J. Math.27, 371-378 (1976) · Zbl 0334.58016 · doi:10.1093/qmath/27.3.371
[18] Harish Chandra: Some applications of the universal enveloping algebra of a semi-simple Lie algebra. Trans. A. M. S.70, 28-99 (1951) · Zbl 0042.12701 · doi:10.1090/S0002-9947-1951-0044515-0
[19] Humphreys, J. E.: Introduction to Lie algebras and representation theory. Grad. Text Math.9, Berlin, Heidelberg, New York: Springer 1972 · Zbl 0254.17004
[20] LeBrun, C. R.: Spaces of complex null geodesics in complex-Riemannian geometry. Trans. A. M. S.278, 209-231 (1983) · Zbl 0562.53018 · doi:10.1090/S0002-9947-1983-0697071-9
[21] Lepowsky, J.: A generalization of the Bernstein-Gelfand-Gelfand resolution. J. Algebra49, 496-511 (1977) · Zbl 0381.17006 · doi:10.1016/0021-8693(77)90254-X
[22] Penrose, R.: Zero-rest-mass fields including gravitation: asymptotic behaviour. Proc. R. Soc. Lond.A248, 159-203 (1965) · Zbl 0129.41202
[23] Penrose, R.: Twistor algebra. J. Math. Phys.8, 345-366 (1967) · Zbl 0163.22602 · doi:10.1063/1.1705200
[24] Penrose, R.: The structure of space-time. In: Battelle Rencontres 1967, pp. 121-235. New York: Benjamin 1968 · Zbl 0174.55901
[25] Penrose, R.: Twistor theory, its aims and achievements. In: Quantum gravity: an Oxford Symposium. Isham, C. J., Penrose, R., Sciama, D. W. (eds.). pp. 267-407. Oxford: Clarendon Press 1975
[26] Penrose, R., Ward, R. S.: Twistors for flat and curved space-time. In: General relativity and gravitation, Vol. II Held, A. (ed.). pp. 283-328. New York, London: Plenum Press 1980
[27] Penrose, R., Rindler, W.: Spinors and space-time, Vol. I. Cambridge: Cambridge University Press 1984 · Zbl 0538.53024
[28] Penrose, R., Rindler, W.: Spinors and space-time, Vol. II. Cambridge: Cambridge University Press 1986 · Zbl 0591.53002
[29] Spencer, D. C.: Overdetermined systems of linear partial differential equations. Bull. A. M. S.75, 179-239 (1969) · Zbl 0185.33801 · doi:10.1090/S0002-9904-1969-12129-4
[30] Szekeres, P.: Conformal tensors. Proc. R. Soc. Lond.A304, 113-122 (1968) · Zbl 0159.23903
[31] Verma, D.-N.: Structure of certain induced representations of complex semisimple Lie algebras. Ph.D. thesis, Yale University 1966 (cf. Bull. A. M. S.74, 160-166 (1968)) · Zbl 0157.07604
[32] Vogan, D. A.: Representations of real reductive Lie groups. Prog. Math.15, Boston Basel Stuttgart: Birkh?user 1981 · Zbl 0469.22012
[33] Wells, R. O., Jr.: The conformally invariant Laplacian and the instanton vanishing theorem. In: Seminar on differential geometry. Yau, S.-T. (ed.). pp. 483-498. Ann. Math. Stud.102, Princeton: Princeton University Press 1982
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