···(1/N(·))∏(((-1)^(1/Zq)-e^(-Pi i/Zq))/2sin(Pi/Zq)) = O(Erf(≈)) + ∑((-1)^((4Zq+1)/2)-(-1)^((4Zq-1)/2))/2···bmatt.org

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[   ] Nonlocal scalar QFT func. int., basis func. rep. and S.C.E.Perhaps, the key conceptual hardness we had to overcome was how to "embed" a collection of countable lattices into larger universe of "lattice-like objects", as re-scaling limits when size of lattices grows to uncountable infinity. More precisely, from our prior publication doi.org/10.3390/particles2010009, we propose a duality (due to Efimov) between discrete lattices and suitable lattice paths, which appear as nonlocal QFT with "positive driftings" of standard Gaussian random walk; the sort of encoding of discrete lattices as planar continuous random functions to make sense of (weak) convergence (radius) of random discrete lattices just as random functions in the sense of (weak) convergence with respect to the QFT (norm) spaces L^(p>=0).

We make two cases: the 1-norm case and the 0-norm case. The 1-norm case is explicitly defined by a new symbol for product integral in renormalization over p >= 1. Given that the subspace of L^infinity defined by the space of convergent subsequences is isomorphic to the space of real-valued continuous functions on countably infinite compact set of real line equipped with the sup norm, and there is natural isomorphism between finite discrete lattice with lattice paths and compact label subset of L^1, the (weak) convergence of countably infinite random discrete lattices can be treated as just (weak) convergence of countably infinite random compact sets, where compact subsets of L^(p>=1) are equipped with Hausdorff distance arising from the usual norm on QFT spaces L^(p>=1). The 0-norm case is explicitly defined by a new notion of replica functional Taylor series for 0-metric and smoothed Dirac delta-functions in the p --> zero limit for the QFT spaces L^(p>0).

Although we do not prove the natural isomorphism (left as elementary analysis task), both embedding and labeling (for continuous functions and compact subset, resp.) can be an artifact if the lattices do not naturally have such structure; in particular, lattice-valued stochastic processes, where intermittent embedding or labeling must be kept updated as process evolves. Moreover, a topology inherited from supremum norm or Hausdorff metric may be too strong for some functions. As a result, more intrinsic ways of characterizing what is meant by "lattice-like object" must be sought.
2019-08-15 17:38 446.5K  
[   ] S-matrix of nonlocal scalar QFT in basis func. rep.We fundamentally obtain mathematically rigorous nonlocal Quantum Field Theory (QFT), namely: Arbitrary spacetime, entire and explicit analytical nonlocal form factors for problem of divergences in transition amplitude and statistical physics model dualities that arise in Euclidean QFT, as well as various nonlocal variational function models that appear in Functional (nonperturbative, exact) Renormalization Group (FRG), light hadrons low-energy physics a.k.a nonlocal quark theory, Quantum Chromodynamics (QCD), quark confinement, high-energy scalar theory and Quantum Electrodynamics (QED), supersymmetric non-Abelian QFT, integrable QFT, and the Theory of Critical Phenomena (TCP).   2019-02-19 17:38 1.1M  
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