We present a non-perturbative computation of the running of the coupling α s in QCD with two flavours of dynamical fermions in the Schrödinger functional scheme. We improve our previous results by a reliable continuum extrapolation. The Λ-parameter characterizing the high-energy running is related to the value of the coupling at low energy in the continuum limit. An estimate of Λr 0 is given using large-volume data with lattice spacings a from 0.07 fm to 0.1 fm. It translates into Λ (2) MS = 245(16)(16) MeV [assuming r 0 = 0.5 fm]. The last step still has to be improved to reduce the uncertainty.

The running of renormalized quark masses is computed in lattice QCD with two flavors of massless O(a) improved Wilson quarks. The regularization and flavor independent factor that relates running quark masses to the renormalization group invariant ones is evaluated in the Schroedinger Functional scheme. Using existing data for the scale r_0 and the pseudoscalar meson masses, we define a reference quark mass in QCD with two degenerate quark flavors. We then compute the renormalization group invariant reference quark mass at three different lattice spacings. Our estimate for the continuum value is converted to the strange quark mass with the help of chiral perturbation theory.Comment: 25 pages, 6 figures; sections 1 and 4 rearranged, minor change to the summary plo

We determine the improvement coefficients b m and b A − b P in quenched lattice QCD for a range of β-values, which is relevant for current large scale simulations. At fixed β, the results are rather sensitive to the precise choices of parameters. We therefore impose improvement conditions at constant renormalized parameters, and the coefficients are then obtained as smooth functions of g 2 0 . Other improvement conditions yield a different functional dependence, but the difference between the coefficients vanishes with a rate proportional to the lattice spacing. We verify this theoretical expectation in a few examples and are therefore confident that O(a) improvement is achieved for physical quantities. As a byproduct of our analysis we also obtain the finite renormalization constant which relates the subtracted bare quark mass to the bare PCAC mass.

We show that the spectral dimension d s of two-dimensional quantum gravity coupled to Gaussian fields is two for all values of the central charge c ≤ 1.The same arguments provide a simple proof of the known result d s = 4/3 for branched polymers.

No theory of four-dimensional quantum gravity exists as yet. In this situation the two-dimensional theory, which can be analyzed by conventional field-theoretical methods, can serve as a toy model for studying some aspects of quantum gravity. It represents one of the rare settings in a quantum-gravitational context where one can calculate quantities truly independent of any background geometry.We review recent progress in our understanding of 2d quantum gravity, and in particular the relation between the Euclidean and Lorentzian sectors of the quantum theory. We show that conventional 2d Euclidean quantum gravity can be obtained from Lorentzian quantum gravity by an analytic continuation only if we allow for spatial topology changes in the latter. Once this is done, one obtains a theory of quantum gravity where space-time is fractal: the intrinsic Hausdorff dimension of usual 2d Euclidean quantum gravity is four, and not two. However, certain aspects of quantum space-time remain two-dimensional, exemplified by the fact that its so-called spectral dimension is equal to two.