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In this paper we show how semiclassical pseudodifferential operators together with Getzler’s rescaling can be used to give a systematic proof of the Atiyah-Singer index theorem using the resolvent or the heat kernel.
A quaternionic version of Quantum Mechanics is constructed using the Schwinger’s formulation based on measurements and a Variational Principle. Commutation relations and evolution equations are provided, and the results are compared with other formulations.
Although intrinsic spin is usually viewed as a purely quantum property with no classical analog, we present evidence that fermion spin has a classical origin rooted in the geometry of three-dimensional physical space. Our approach to the quantum/classical interface is based on a formulation of...
Let Q be a symmetric bilinear form on with corank r , rank p + q and signature type ( p , q ), p resp. q denoting positive resp. negative dimensions. We consider the degenerate spin group Spin ( Q ) = Spin ( p , q , r ) in the sense of Crumeyrolle and prove that this group is isomorphic to the...
This paper uses an advanced geometric language for the development of theory, concepts, and computer algorithms in the domain of robot vision. Traditionally researchers use Gibb’s vector calculus, matrix algebra, tensor calculus or quaternions. In this article, the authors utilize a modern...
We formulate and sketch a solution method for the boundary value problem of electromagnetic radiation in a vacuum with a given gravitational background, in terms of Clifford Analysis over a pseudo-Riemannian manifold with signature (1, 3). The general solution for the electromagnetic field,...
The Atiyah-Bott-Shapiro classification of the irreducible Clifford algebra is used to derive general properties of the minimal representations of the 1 D N -Extended Supersymmetry algebra (the Z 2 -graded symmetry algebra of the Supersymmetric Quantum Mechanics) linearly realized on a finite...
The aim of this paper is to review the formalism of noncommutativity using canonical parametrization theory. In the first part we present the formalism for the case of Quantum Mechanics, and we show that using this approach and an appropriate basis we can get the noncommutativity expressed in...
We present a new theoretical framework for multidimensional image processing using Clifford algebras. The aim of the paper is to detect edges by computing the first fundamental form of a surface associated to an image. We propose to construct this metric in the Clifford bundles setting. A nD...
In this paper we define directional quaternionic Hilbert operators on the three–dimensional space . We consider functions in the kernel of the Cauchy-Riemann operator a variant of the Cauchy. Fueter operator. This choice is motivated by the strict relation existing between this type of...
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