5,292 research outputs found
Closure of Constraints for Plane Gravity Waves
The metric for gravitational plane waves has very high symmetry (two
spacelike commuting Killing vectors). For this high symmetry, a simple
renormalization of the lapse function is found which allows the constraint
algebra for canonical quantum gravity to close; also, the vector constraint has
the correct form to generate spatial diffeomorphisms. A measure is constructed
which respects the reality conditions, but does not yet respect the invariances
of the theory.Comment: 19 pages; LaTe
Partitions with fixed differences between largest and smallest parts
We study the number of partitions of with difference between
largest and smallest parts. Our main result is an explicit formula for the
generating function . Somewhat
surprisingly, is a rational function for ; equivalently,
is a quasipolynomial in for fixed . Our result generalizes to
partitions with an arbitrary number of specified distances.Comment: 5 page
Closed-Flux Solutions to the Constraints for Plane Gravity Waves
The metric for plane gravitational waves is quantized within the Hamiltonian
framework, using a Dirac constraint quantization and the self-dual field
variables proposed by Ashtekar. The z axis (direction of travel of the waves)
is taken to be the entire real line rather than the torus (manifold
coordinatized by (z,t) is RxR rather than x R). Solutions to the
constraints proposed in a previous paper involve open-ended flux lines running
along the entire z axis, rather than closed loops of flux; consequently, these
solutions are annihilated by the Gauss constraint at interior points of the z
axis, but not at the two boundary points. The solutions studied in the present
paper are based on closed flux loops and satisfy the Gauss constraint for all
z.Comment: 18 pages; LaTe
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