Abstract
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.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Baston, R. J.: The algebraic construction of invariant differential operators. D.Phil. thesis, Oxford University 1985
Bateman, H.: The transformation of the electrodynamical equations. Proc. L. M. S.8, 223–264 (1910)
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 János Math. Soc. Gelfand, I. M. (ed), pp. 21–64, London: Hilgar 1975
Boe, B. D., Collingwood, D. H.: A comparison theory for the structure of induced representations. J. Algebra94, 511–545 (1985)
Boe, B. D., Collingwood, D. H.: A comparison theory for the structure of induced representations II. Math. Z.190, 1–11 (1985)
Bott, R.: Homogeneous vector bundles. Ann. Math.66, 203–248 (1957)
Cunningham, E.: The principal of relativity in electrodynamics and an extension thereof. Proc. L. M. S.8, 77–98 (1910)
Dixmier, J.: Enveloping algebras. Amsterdam: North Holland 1977
Dubrovin, B. A., Fomenko, A. T., Novikov, S. P.: Modern geometry, methods and applications. Moscow: Nauka 1979
Eastwood, M. G., Penrose, R., Wells, R. O., Jr.: Cohomology and massless fields. Commun. Math. Phys.78, 305–351 (1981)
Eastwood, M. G., Tod, K. P.: Edth-a differential operator on the sphere. Math. Proc. Camb. Phil. Soc.92, 317–330 (1982)
Eastwood, M. G.: Complexification, twistor theory, and harmonic maps of Riemann surfaces. Bull. A. M. S.11, 317–328 (1984)
Eastwood, M. G.: The generalized Penrose-Ward transform. Math. Proc. Camb. Phil. Soc.97, 165–187 (1985)
Eastwood, M. G.: A duality for homogeneous bundles on twistor space. J. L. M. S.31, 349–356 (1985)
Eastwood, M. G., Singer, M. A.: A conformally invariant Maxwell gauge. Phys. Lett.107A, 73–74 (1985)
Fefferman, C., Graham, C. R.: Conformal invariants. Preprint, Princeton University 1984
Fegan, H. D.: Conformally invariant first order differential operators. Q. J. Math.27, 371–378 (1976)
Harish Chandra: Some applications of the universal enveloping algebra of a semi-simple Lie algebra. Trans. A. M. S.70, 28–99 (1951)
Humphreys, J. E.: Introduction to Lie algebras and representation theory. Grad. Text Math.9, Berlin, Heidelberg, New York: Springer 1972
LeBrun, C. R.: Spaces of complex null geodesics in complex-Riemannian geometry. Trans. A. M. S.278, 209–231 (1983)
Lepowsky, J.: A generalization of the Bernstein-Gelfand-Gelfand resolution. J. Algebra49, 496–511 (1977)
Penrose, R.: Zero-rest-mass fields including gravitation: asymptotic behaviour. Proc. R. Soc. Lond.A248, 159–203 (1965)
Penrose, R.: Twistor algebra. J. Math. Phys.8, 345–366 (1967)
Penrose, R.: The structure of space-time. In: Battelle Rencontres 1967, pp. 121–235. New York: Benjamin 1968
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
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
Penrose, R., Rindler, W.: Spinors and space-time, Vol. I. Cambridge: Cambridge University Press 1984
Penrose, R., Rindler, W.: Spinors and space-time, Vol. II. Cambridge: Cambridge University Press 1986
Spencer, D. C.: Overdetermined systems of linear partial differential equations. Bull. A. M. S.75, 179–239 (1969)
Szekeres, P.: Conformal tensors. Proc. R. Soc. Lond.A304, 113–122 (1968)
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))
Vogan, D. A.: Representations of real reductive Lie groups. Prog. Math.15, Boston Basel Stuttgart: Birkhäuser 1981
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
Author information
Authors and Affiliations
Additional information
Communicated by S.-T. Yau
S.E.R.C. Advanced Fellow and Flinders University Visiting Research Fellow
Rights and permissions
About this article
Cite this article
Eastwood, M.G., Rice, J.W. Conformally invariant differential operators on Minkowski space and their curved analogues. Commun.Math. Phys. 109, 207–228 (1987). https://doi.org/10.1007/BF01215221
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF01215221