A Proposal for a Differential Calculus in Quantum Mechanics

In this paper, using the Weyl-Wigner-Moyal formalism for quantum mechanics, we develop a {\it quantum-deformed} exterior calculus on the phase-space of an arbitrary hamiltonian system. Introducing additional bosonic and fermionic coordinates we construct a super-manifold which is closely related to the tangent and cotangent bundle over phase-space. Scalar functions on the super-manifold become equivalent to differential forms on the standard phase-space. The algebra of these functions is equipped with a Moyal super-star product which deforms the pointwise product of the classical tensor calculus. We use the Moyal bracket algebra in order to derive a set of quantum-deformed rules for the exterior derivative, Lie derivative, contraction, and similar operations of the Cartan calculus.Comment: TeX file with phyzzx macro, 43 pages, no figure

Nonlocal Quantum Gravity and the Size of the Universe

Motivated by the conjecture that the cosmological constant problem is solved by strong quantum effects in the infrared we use the exact flow equation of Quantum Einstein Gravity to determine the renormalization group behavior of a class of nonlocal effective actions. They consist of the Einstein-Hilbert term and a general nonlinear function $F_k(V)$ of the Euclidean spacetime volume $V$. For the $V + V \ln V$-invariant the renormalization group running enormously suppresses the value of the renormalized curvature which results from Planck-size parameters specified at the Planck scale. One obtains very large, i.e., almost flat universes without finetuning the cosmological constant. A critical infrared fixed point is found where gravity is scale invariant.Comment: 6 pages, 1 figure, contribution to the proceedings of the 36th International Symposium Ahrenshoop, Berlin, August 26-30, 200