A Canonical Analysis of the First Order EinsteinHilbert Action.pdfVIP

UWO -TH- 07/17 A Canonical Analysis of the First Order Einstein-Hilbert Action R. N. Ghalati∗ and D. G. C. McKeon† Department of Applied Mathematics, University of Western Ontario, London, N6A 5B7 Canada 7 0 (Dated: February 2, 2008) 0 2 Abstract v o N The Dirac constraint formalism is applied to the d (d 2) dimensional Einstein-Hilbert action 6 when written in first order form, using the metric density and affine connection as independent 1 fields. Field equations not involving time derivatives are not used to eliminate fields. Primary, ] c q secondary and tertiary constraints arise, leaving d(d − 3) degrees of freedom in phase space. The - r Poisson Bracket algebra of these constraints is given. g [ 1 v 3 4 5 2 . 1 1 7 0 : v i X r a ∗Electronic address: rnowbakh@uwo.ca †Electronic address: dgmckeo2@uwo.ca 1 A full understanding of the Einstein-Hilbert action which could possibly lead to a consis- tent quantization of gravity is contingent upon understanding its canonical structure. Any canonical analysis depends on the choice of geometrical quantities that are taken to be the fundamental fields. Here we follow refs. [1-7] and work with the metric and affine connection; in particular we treat them as being independent as in refs. [3,7]. We differ from these papers however, as we do not use all the equations of motion which do not contain time derivatives to eliminate any fields. This is because some of these equations turn out to be secondary first class constraints that lead to furthe