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This paper describes an approach to quantum electrodynamics (QED) in curved space–time obtained by considering infinite-dimensional algebra bundles associated to a natural principal bundle Q associated with any locally conformally flat space–time, with typical fibers including the Fock space and a space of fermionic multiparticle states which forms a Grassmann algebra. Both these algebras are direct sums of generalized Hilbert spaces. The requirement of K covariance associated with the geometry of space–time, where K is the structure group of Q , leads to the consideration of K intertwining operators between various spaces. Scattering processes are associated with such operators and are encoded in an algebra of kernels. Intertwining kernels can be generated using K covariant matrix-valued measures. Feynman propagators, fermion loops and the electron self-energy can be given well-defined interpretations as such measures. Divergence-free calculations in QED can be carried out by computing the spectra of these measures and kernels (a process called spectral regularization). As an example of the approach the precise Uehling potential function for the H atom is calculated without requiring renormalization from which the Uehling contribution to the Lamb shift can be calculated exactly. [ABSTRACT FROM AUTHOR]

We consider the sequences of matrix bi-orthogonal polynomials with respect to the bilinear forms · , · R ^ and · , · L ^ ⟨ P (z 1) , Q (z 2) ⟩ R ^ = ∫ T × T P (z 1) † L (z 1) d μ (z 1 , z 2) Q (z 2) , P , Q ∈ L p × p [ z ] P (z 1) , Q (z 2) L ^ = ∫ T × T P (z 1) L (z 1) d μ (z 1 , z 2) Q (z 2) † , where μ (z 1 , z 2) is a matrix of bi-variate measures supported on T × T , with T the unit circle, L p × p [ z ] is the set of matrix Laurent polynomials of size p × p and L(z) is a special polynomial in L p × p [ z ] . A connection formula between the sequences of matrix Laurent bi-orthogonal polynomials with respect to · , · R ^ , (resp. · , · L ^ ) and the sequence of matrix Laurent bi-orthogonal polynomials with respect to d μ (z 1 , z 2) is given. [ABSTRACT FROM AUTHOR]

In this work we consider open quantum random walks on the non-negative integers. By considering orthogonal matrix polynomials we are able to describe transition probability expressions for classes of walks via a matrix version of the Karlin-McGregor formula. We focus on absorbing boundary conditions and, for simpler classes of examples, we consider path counting and the corresponding combinatorial tools. A non-commutative version of the gambler's ruin is studied by obtaining the probability of reaching a certain fortune and the mean time to reach a fortune or ruin in terms of generating functions. In the case of the Hadamard coin, a counting technique for boundary restricted paths in a lattice is also presented. We discuss an open quantum version of Foster's Theorem for the expected return time together with applications. [ABSTRACT FROM AUTHOR]

In this paper we begin a study of free analysis in the quaternionic setting, and consider Boolean convolution for quaternion-valued measures. To this end we also study Boolean convolution for matrix-valued complex measures, also proving Boolean infinite divisibility and central limit theorems for these measures. Moreover we prove an integral representation for quaternionic Carathéodory and Herglotz functions. [ABSTRACT FROM AUTHOR]

Abstract: Let be a matrix-valued measure with the essential spectrum a single interval and countably many point masses outside of it. Under the assumption that the absolutely continuous part of satisfies Szegő’s condition and the point masses satisfy a Blaschke-type condition, we obtain the asymptotic behavior of the orthonormal polynomials on and off the support of the measure. The result generalizes the scalar analogue of Peherstorfer–Yuditskii (2001) and the matrix-valued result of Aptekarev–Nikishin (1983) , which handles only a finite number of mass points. [Copyright &y& Elsevier]

We survey the analytic theory of matrix orthogonal polynomials. [ABSTRACT FROM AUTHOR]

Abstract: A Bochner-integral formulation of Jensen''s inequality is presented for Hermitian matrix-valued functions and measures. [Copyright &y& Elsevier]