LINEAR FORM OF CANONICAL GRAVITY

Recent work in the literature has shown that general relativity can be formulated in terms of a jet bundle which, in local coordinates, has five entries: local coordinates on Lorentzian space-time, tetrads, con nection one-forms, multivelocities corresponding to the tetrads and mu ltivelocities corresponding to the connection one-forms. The derivativ es of the Lagrangian with respect to the latter class of multivelociti es give rise to a set of multimomenta which naturally occur in the con straint equations. Interestingly, all the constraint equations of gene ral relativity are linear in terms of this class of multimomenta. This construction has been then extended to complex general relativity, wh ere Lorentzian space-time is replaced by a four-complex-dimensional co mplex-Riemannian manifold. One then finds a holomorphic theory where t he familiar constraint equations are replaced by a set of equations Li near in the holomorphic multimomenta, providing such multimomenta vani sh on a family of two-complex-dimensional surfaces. In quantum gravity , the problem arises to quantize a real or a holomorphic theory on the extended space where the multimomenta can be defined.