Causal Kernel Research Articles

Causal localizations of the massive scalar boson

The positive operator-valued localizations (POL) of a massive scalar boson are constructed, and a characterization and structural analyses of their kernels are obtained. In the focus of this article are the causal features of the POL. There is the well-known causal time evolution (CT). Recently a POL by Terno and Moretti, which is a kinematical deformation of the Newton–Wigner localization (NWL) and belongs to the here fully analyzed class of finite POL, is shown by V. Moretti to comply with CT. A further POL with CT treated here, which is in the same class, is the only one being the trace of a projection-valued localization (like NWL) with CT. Causality imposes a condition CC, which implies CT but is more restrictive than CT. Extending Moretti’s method it is shown rigorously that the POL of the class introduced by Petzold et al. satisfy CC. Their kernels are called causal kernels, of which a rather detailed description is achieved. One the way there the case of one spatial dimension is solved completely. This case is instructive. In particular it directed Petzold et al. and subsequently Henning, Wolf to find their basic one-parameter family Kr\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$K_r$$\\end{document} of causal kernels. The causal kernels are, up to a fixed energy factor, normalized positive definite Lorentz invariant kernels. A full characterization of the latter is attained due to their close relation to the zonal spherical functions on the Lorentz group. Finally these considerations discharge into the main result that K3/2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$K_{3/2}$$\\end{document} is the absolute maximum, viz. |K|≤K3/2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$|K|\\le K_{3/2}$$\\end{document} for all causal kernels K.

An improved time-marching formulation based on an explicit time integration method for dynamic analysis of non-viscous damping systems

An improved time-marching formulation based on an explicit method [1] is proposed for non-viscous damping systems. In this method, an improved integral approximation scheme is developed to improve the convolution solution accuracy and is applicable to any causal kernel function. The mathematical derivation and calculation procedure based on the explicit method are formulated for non-viscous damping systems. For comparison, a composite explicit method proposed by Noh and Bathe is also extended for non-viscous damping systems. The proposed explicit method shows more desirable stability and accuracy properties than other competitive explicit methods. Numerical simulations of some representative examples demonstrate the proposed time-marching procedure is efficient for dynamic analysis of non-viscous damped systems.

Nonlocal Gravity: Fundamental Tetrads and Constitutive Relations

Nonlocal gravity (NLG) is a classical nonlocal generalization of Einstein’s theory of gravitation based on a certain analogy with the nonlocal electrodynamics of media. The history dependence enters NLG through a constitutive relation involving a causal kernel that should ultimately be determined via observational data. The purpose of this paper is to reformulate nonlocal gravity such that the nonlocal aspect of the constitutive relation directly connects measurable quantities as in the nonlocal electrodynamics of media. The resulting constitutive relation turns out to coincide with the recent suggestion of Puetzfeld, Obukhov, and Hehl. With the new constitutive relation of NLG, it is possible to show that de Sitter spacetime is not a solution of NLG.

A time integration method based on the weak form Galerkin method for non-viscous damping systems

A time integration method based on the weak form Galerkin method is proposed for non-viscous damping systems with a convolution damping model. The non-viscous damping model takes the complete velocity history into account through a convolution. This generic time integration method is developed for any causal kernel functions. The displacements is approximated by using Lagrangian interpolation function in one time elements. The weak form of the differential equation is computed by using Galerkin method. The weak form Galerkin method computational procedure of dynamic responses of non-viscous damping systems is given in detail. Numerical examples are provided to illustrate the accuracy and the efficiency of the integration by comparison with other numerical methods.

Determinism and randomness in quantum dynamics

A class of models is considered where a finite or infinite quantum dynamical system in a stationary state is probed by sequences of observations acting on a specified finite subsystem. The whole set of such experiments is described by a time-ordered (causal) and stationary quantum correlation kernel. It is shown that any such kernel can be decomposed in a unique way into a convex combination of two kernels, here called the regular and singular components. A singular system has a strong deterministic property, the predictability of the future from the knowledge of the past is limited only by the inevitable indeterminism of quantum measurements. Furthermore, in this case the full set of correlation functions of arbitrary time order, and hence the dynamical system itself, is determined by the causal kernel. Finite systems and infinite systems satisfying the KMS condition at finite temperature are of this type. In the regular case the dynamics contains a shift, there is a genuine asymptotic randomness and the dynamical system cannot be reconstructed in a unique way from the causal kernel. Non-trivial quantum Markov processes are shown to belong to this class.

Complexified de Sitter space: Analytic causal kernels and Källen-Lehmann-type representation

Global analyticity properties of functions associated with causal kernels on de Sitter space are considered. These properties extend in a reasonable way those implied by the general framework of quantum field theory in complex Minkowski space. Mathematical results of J. Faraut, G.A. Viano and J. Bros (motivated in particular by complex angular momentum analysis in field theory) find here new applications.

Two Qualms about Functionalist Marxism

InKarl Marx's Theory of History: A Defence(1978), G. A. Cohen has developed a distinctively functionalist interpretation of historical materialism. In this paper I outline Cohen's novel reconstruction of Marx and subject it to two independent internal criticisms. I first argue that explanations cannot conform to Cohen's functionalist model. I then suggest that even if there could be explanations having the structure he has proposed, they would fail to be helpful in illuminating the causal kernel of Marx's theory. Finally I sketch a neglected but more promising approach to clarifying historical materialism, due to Geoffrey Hellman.

Defect-induced tunnelling and the conductivity of strongly disordered systems

For a model describing the motion of a zero-temperature electron gas in an environment causing a fluctuating potential and a fluctuating effective mass, a theory is developed which allows for an approximate calculation of the particle density propagation self-consistently with momentum relaxation and defect-induced tunnelling processes. All correlation functions of the system are expressed in terms of two causal kernels for which non-linear equations are derived and solved. A detailed discussion of the influence of the defect-induced tunnelling on the mobility, in particular on mobility edges, and on the Fermi glass stiffness is given and an anomalous high-frequency tail of the dynamical conductivity is predicted.