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We address the problem of the continuum limit for a system of Hausdorff lattices (namely lattices of isolated points) approximating a topological space $M$. The correct framework is that of projective systems. The projective limit is a universal space from which $M$ can be recovered as a quotient. We dualize the construction to approximate the algebra ${cal C}(M)$ of continuous functions on $M$. In a companion paper we shall extend this analysis to systems of noncommutative lattices (non Hausdorff lattices).
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We consider finite approximations of a topological space $M$ by noncommutative lattices of points. These lattices are structure spaces of noncommutative $C^*$-algebras which in turn approximate the algebra $cc(M)$ of continuous functions on $M$. We show how to recover the space $M$ and the algebra $cc(M)$ from a projective system of noncommutative lattices and an inductive system of noncommutative $C^*$-algebras, respectively.
An approach to calculating approximate solutions to the continuum Schwinger-Dyson equations is outlined, with examples for phi^4 in D=1. This approach is based on the source Galerkin methods developed by Garcia, Guralnik and Lawson. Numerical issues and opportunities for future calculations are also discussed briefly.
We exhibit simple lattice systems, motivated by recently proposed cold atom experiments, whose continuum limits interpolate between real and $p$-adic smoothness as a spectral exponent is varied. A real spatial dimension emerges in the continuum limit if the spectral exponent is negative, while a $p$-adic extra dimension emerges if the spectral exponent is positive. We demonstrate Holder continuity conditions, both in momentum space and in position space, which quantify how smooth or ragged the two-point Greens function is as a function of the spectral exponent. The underlying discrete dynamics of our model is defined in terms of a Gaussian partition function as a classical statistical mechanical lattice model. The couplings between lattice sites are sparse in the sense that as the number of sites becomes large, a vanishing fraction of them couple to one another. This sparseness property is useful for possible experimental realizations of related systems.
We discuss the continuum limit of discrete Dirac operators on the square lattice in $mathbb R^2$ as the mesh size tends to zero. To this end, we propose a natural and simple embedding of $ell^2(mathbb Z_h^d)$ into $L^2(mathbb R^d)$ that enables us to compare the discrete Dirac operators with the continuum Dirac operators in the same Hilbert space $L^2(mathbb R^2)^2$. In particular, we prove strong resolvent convergence. Potentials are assumed to be bounded and uniformly continuous functions on $mathbb R^2$ and allowed to be complex matrix-valued.
We present calculations of certain limits of scheme-independent series expansions for the anomalous dimensions of gauge-invariant fermion bilinear operators and for the derivative of the beta function at an infrared fixed point in SU($N_c$) gauge theories with fermions transforming according to two different representations. We first study a theory with $N_f$ fermions in the fundamental representation and $N_{f}$ fermions in the adjoint or symmetric or antisymmetric rank-2 tensor representation, in the limit $N_c to infty$, $N_f to infty$ with $N_f/N_c$ fixed and finite. We then study the $N_c to infty$ limit of a theory with fermions in the adjoint and rank-2 symmetric or antisymmetric tensor representations.
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