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Register a new userRandom Aharonov-Bohm vortices and some exact families of integrals: Part III
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Stephane Ouvry
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As a sequel to [1] and [2], I present some recent progress on Bessel integrals $int_0^{infty}{rmd u}; uK_0(u)^{n}$, $int_0^{infty}{rmd u}; u^{3}K_0(u)^{n}$, ... where the power of the integration variable is odd and where $n$, the Bessel weight, is a positive integer. Some of these integrals for weights n=3 and n=4 are known to be intimately related to the zeta numbers zeta(2) and zeta(3). Starting from a Feynman diagram inspired representation in terms of n dimensional multiple integrals on an infinite domain, one shows how to partially integrate to n-2 dimensional multiple integrals on a finite domain. In this process the Bessel integrals are shown to be periods. Interestingly enough, these reduced multiple integrals can be considered in parallel with some simple integral representations of zeta numbers. One also generalizes the construction of [2] on a particular sum of double nested Bessel integrals to a whole family of double nested integrals. Finally a strong PSLQ numerical evidence is shown to support a surprisingly simple expression of zeta(5) as a linear combination with rational coefficients of Bessel integrals of weight n= 8.
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At 6th order in perturbation theory, the random magnetic impurity problem at second order in impurity density narrows down to the evaluation of a single Feynman diagram with maximal impurity line crossing. This diagram can be rewritten as a sum of ordinary integrals and nested double integrals of products of the modified Bessel functions $K_{ u}$ and $I_{ u}$, with $ u=0,1$. That sum, in turn, is shown to be a linear combination with rational coefficients of $(2^5-1)zeta(5)$, $int_0^{infty} u K_0(u)^6 du$ and $int_0^{infty} u^3 K_0(u)^6 du$. Unlike what happens at lower orders, these two integrals are not linear combinations with rational coefficients of Euler sums, even though they appear in combination with $zeta(5)$. On the other hand, any integral $int_0^{infty} u^{n+1} K_0(u)^p (uK_1(u))^q du$ with weight $p+q=6$ and an even $n$ is shown to be a linear combination with rational coefficients of the above two integrals and 1, a result that can be easily generalized to any weight $p+q=k$. A matrix recurrence relation in $n$ is built for such integrals. The initial conditions are such that the asymptotic behavior is determined by the smallest eigenvalue of the transition matrix.
We show that the Aharonov-Bohm effect finds a natural description in the setting of QFT on curved spacetimes in terms of superselection sectors of local observables. The extension of the analysis of superselection sectors from Minkowski spacetime to an arbitrary globally hyperbolic spacetime unveils the presence of a new quantum number labeling charged superselection sectors. In the present paper we show that this topological quantum number amounts to the presence of a background flat potential which rules the behaviour of charges when transported along paths as in the Aharonov-Bohm effect. To confirm these abstract results we quantize the Dirac field in presence of a background flat potential and show that the Aharonov-Bohm phase gives an irreducible representation of the fundamental group of the spacetime labeling the charged sectors of the Dirac field. We also show that non-Abelian generalizations of this effect are possible only on space-times with a non-Abelian fundamental group.
The Aharonov-Bohm effect is the prime example of a zero-field-strength configuration where a non-trivial vector potential acquires physical significance, a typical quantum mechanical effect. We consider an extension of the traditional A-B problem, by studying a two-dimensional medium filled with many point-like vortices. Systems like this might be present within a Type II superconducting layer in the presence of a strong magnetic field perpendicular to the layer, and have been studied in different limits. We construct an explicit solution for the wave function of a scalar particle moving within one such layer when the vortices occupy the sites of a square lattice and have all the same strength, equal to half of the flux quantum. From this construction we infer some general characteristics of the spectrum, including the conclusion that such a flux array produces a repulsive barrier to an incident low-energy charged particle, so that the penetration probability decays exponentially with distance from the edge.
We study the Hamiltonian describing two anyons moving in a plane in presence of an external magnetic field and identify a one-parameter family of self-adjoint realizations of the corresponding Schr{o}dinger operator. We also discuss the associated model describing a quantum particle immersed in a magnetic field with a local Aharonov-Bohm singularity. For a special class of magnetic potentials, we provide a complete classification of all possible self-adjoint extensions.
We present a summary of recent and older results on Bessel integrals and their relation with zeta numbers.
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