Phase transitions in spinor quantum gravity on a lattice

We construct a well-defined lattice-regularized quantum theory formulated in terms of fundamental fermion and gauge fields, the same type of degrees of freedom as in the Standard Model. The theory is explicitly invariant under local Lorentz transformations and, in the continuum limit, under diffeomorphisms. It is suitable for describing large nonperturbative and fast-varying fluctuations of metrics. Although the quantum curved space turns out to be on the average flat and smooth owing to the non-compressibility of the fundamental fermions, the low-energy Einstein limit is not automatic: one needs to ensure that composite metrics fluctuations propagate to long distances as compared to the lattice spacing. One way to guarantee this is to stay at a phase transition. We develop a lattice mean field method and find that the theory typically has several phases in the space of the dimensionless coupling constants, separated by the second order phase transition surface. For example, there is a phase with a spontaneous breaking of chiral symmetry. The effective low-energy Lagrangian for the ensuing Goldstone field is explicitly diffeomorphism-invariant. We expect that the Einstein gravitation is achieved at the phase transition. A bonus is that the cosmological constant is probably automatically zero.Comment: 37 pages, 12 figures Discussion of dimensions and of the Berezinsky--Kosterlitz--Thouless phase adde

Cluster ensembles, quantization and the dilogarithm

Cluster ensemble is a pair of positive spaces (X, A) related by a map p: A -> X. It generalizes cluster algebras of Fomin and Zelevinsky, which are related to the A-space. We develope general properties of cluster ensembles, including its group of symmetries - the cluster modular group, and a relation with the motivic dilogarithm. We define a q-deformation of the X-space. Formulate general duality conjectures regarding canonical bases in the cluster ensemble context. We support them by constructing the canonical pairing in the finite type case. Interesting examples of cluster ensembles are provided the higher Teichmuller theory, that is by the pair of moduli spaces corresponding to a split reductive group G and a surface S defined in math.AG/0311149. We suggest that cluster ensembles provide a natural framework for higher quantum Teichmuller theory.Comment: Version 7: Final version. To appear in Ann. Sci. Ecole Normale. Sup. New material in Section 5. 58 pages, 11 picture

Experiment and Theory in Computations of the He Atom Ground State

Extensive variational computations are reported for the ground state energy of the non-relativistic two-electron atom. Several different sets of basis functions were systematically explored, starting with the original scheme of Hylleraas. The most rapid convergence is found with a combination of negative powers and a logarithm of the coordinate s = r_{1}+ r_{2}. At N=3091 terms we pass the previous best calculation (Korobov's 25 decimal accuracy with N=5200 terms) and we stop at N=10257 with E = -2.90372 43770 34119 59831 11592 45194 40444 ... Previous mathematical analysis sought to link the convergence rate of such calculations to specific analytic properties of the functions involved. The application of that theory to this new experimental data leaves a rather frustrating situation, where we seem able to do little more than invoke vague concepts, such as ``flexibility.'' We conclude that theoretical understanding here lags well behind the power of available computing machinery.Comment: 15 page