[工程科技]A completely solvable model with an infinite number of Dirac observables for a real sector.pdf

A completely solvable model with an infinite number of Dirac observables for a real sector of complexified Ashtekar gravity T. Thiemann∗ Institute for Theoretical Physics, RWTH Aachen, 3 D-52074 Aachen, Germany 9 9 1 Preprint PITHA 93-33, August 93 v o N 1 Abstract 1 We introduce a reduced model for a real sector of complexified Ashtekar 1 gravity that does not correspond to a subset of Einstein’s gravity but for which v 1 the programme of canonical quantization can be carried out completely, both, 2 via the reduced phase space approach or along the lines of the algebraic quan- 0 tization programme. 1 1 This model stands in a certain correspondence to the frequently treated cylin- 3 drically symmetric waves. 9 / In contrast to other models that have been looked at up to now in terms of c the new variables the reduced phase space is infinite dimensional while the q - scalar constraint is genuinely bilinear in the momenta. r g The infinite number of Dirac observables can be expressed in compact and : explicit form in terms of the original phase space variables. v i They turn out, as expected, to be non-local and form naturally a set of count- X able cardinality. r a 1 Introduction Ashtekar’s variables ([1]) simplify the algebraic structure of the constrai