5 research outputs found
Quantum gravity and the Coulomb potential
We apply a singularity resolution technique utilized in loop quantum gravity
to the polymer representation of quantum mechanics on R with the singular
-1/|x| potential. On an equispaced lattice, the resulting eigenvalue problem is
identical to a finite difference approximation of the Schrodinger equation. We
find numerically that the antisymmetric sector has an energy spectrum that
converges to the usual Coulomb spectrum as the lattice spacing is reduced. For
the symmetric sector, in contrast, the effect of the lattice spacing is similar
to that of a continuum self-adjointness boundary condition at x=0, and its
effect on the ground state is significant even if the spacing is much below the
Bohr radius. Boundary conditions at the singularity thus have a significant
effect on the polymer quantization spectrum even after the singularity has been
regularized.Comment: 10 pages, 5 figures. v2: Minor presentational changes. One data point
added in Table
Background independent quantization and the uncertainty principle
It is shown that polymer quantization leads to a modified uncertainty
principle similar to that obtained from string theory and non-commutative
geometry. When applied to quantum field theory on general background
spacetimes, corrections to the uncertainty principle acquire a metric
dependence. For Friedmann-Robertson-Walker cosmology this translates to a scale
factor dependence which gives a large effect in the early universe.Comment: 6 page