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Register a new userFinite Approximations of Physical Models over Local Fields
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We show that the Schrodinger operator associated with a physical system over a local field can be approximated in a very strong sense by finite Schrodinger operators. Some striking numerical results are included at the end of the article.
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We give a stochastic proof of the finite approximability of a class of Schru007fodinger operators over a local field, thereby completing a program of establishing in a non-Archimedean setting corresponding results and methods from the Archimedean (real) setting. A key ingredient of our proof is to show that Brownian motion over a local field can be obtained as a limit of random walks over finite grids. Also, we prove a Feynman-Kac formula for the finite systems, and show that the propagator at the finite level converges to the propagator at the infinite level.
We strengthen a result of two of us on the existence of effective interactions for discretised continuous-spin models. We also point out that such an interaction cannot exist at very low temperatures. Moreover, we compare two ways of discretising continuous-spin models, and show that, except for very low temperatures, they behave similarly in two dimensions. We also discuss some possibilities in higher dimensions.
We prove the split property for any finite helicity free quantum fields. Finite helicity Poincare representations extend to the conformal group and the conformal covariance plays an essential role in the argument. The split property is ensured by the trace class condition: Tr (exp(-s L_0)) is finite for all s>0 where L_0 is the conformal Hamiltonian of the Mobius covariant restriction of the net on the time axis. We extend the argument for the scalar case presented in [7]. We provide the direct sum decomposition into irreducible representations of the conformal extension of any helicity-h representation to the subgroup of transformations fixing the time axis. Our analysis provides new relations among finite helicity representations and suggests a new construction for representations and free quantum fields with non-zero helicity.
If the $ell$-adic cohomology of a projective smooth variety, defined over a local field $K$ with finite residue field $k$, is supported in codimension $ge 1$, then every model over the ring of integers of $K$ has a $k$-rational point. For $K$ a $p$-adic field, this is math/0405318, Theorem 1.1. If the model $sX$ is regular, one has a congruence $|sX(k)|equiv 1 $ modulo $|k|$ for the number of $k$-rational points 0704.1273, Theorem 1.1. The congruence is violated if one drops the regularity assumption.
We consider one-dimensional long-range spin models (usually called Dyson models), consisting of Ising ferromagnets with slowly decaying long-range pair potentials of the form $frac{1}{|i-j|^{alpha}}$ mainly focusing on the range of slow decays $1 < alpha leq 2$. We describe two recent results, one about renormalization and one about the effect of external fields at low temperature. The first result states that a decimated long-range Gibbs measure in one dimension becomes non-Gibbsian, in the same vein as comparable results in higher dimensions for short-range models. The second result addresses the behaviour of such models under inhomogeneous fields, in particular external fields which decay to zero polynomially as $(|i|+1)^{- gamma}$. We study how the critical decay power of the field, $gamma$, for which the phase transition persists and the decay power $alpha$ of the Dyson model compare, extending recent results for short-range models on lattices and on trees. We also briefly point out some analogies between these results.
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