Related papers: Spencer Operator and Applications: From Continuum …
I review the various algebraic foundations of quantum mechanics. They have been suggested since the birth of this theory till up to last year. They are the following ones: Heisenberg-Born-Jordan (1925), Weyl (1928), Dirac (1930), von…
We discuss two distinct aspects in supersymmetric quantum mechanics. First, we introduce a new class of operators A and $\bar{A}$ in terms of anticommutators between the momentum operator and N+1 arbitrary superpotentials. We show that…
A long standing problem of Gian-Carlo Rota for associative algebras is the classification of all linear operators that can be defined on them. In the 1970s, there were only a few known operators, for example, the derivative operator, the…
Motivated by the sharp contrast between classical and quantum physics as probability theories, in these lecture notes I introduce the basic notions of operator algebras that are relevant for the algebraic approach to quantum physics.…
The theory of Non-Relativistic Quantum Mechanics was created (or discovered) back in the 1920's mainly by Schr\"odinger and Heisenberg, but it is fair enough to say that a more modern and unified approach to the subject was introduced by…
The history of the quadratic stochastic operators can be traced back to work of S.Bernshtein (1924). During more than 80 years this theory developed and many papers were published. In recent years it has again become of interest in…
Operator splitting schemes have been successfully used in computational sciences to reduce complex problems into a series of simpler subproblems. Since 1950s, these schemes have been widely used to solve problems in PDE and control.…
We apply to operator algebra theory a monotone selection principle which apparently escaped attention (of operator algebra theorists) so far. This principle relates to the basic order theoretic characterisation of von Neumann algebras given…
The purpose of this note is to show that the construction of the C*-algebra for the space-time uncertainty relations which was introduced by Doplicher, Fredenhagen and Roberts fits comfortably into the strict deformation quantization…
This paper explains the fundamental relation between Jacobi structures and the classical Spencer operator coming from the theory of PDEs so as to provide a direct and geometric approach to the integrability of Jacobi structures. It uses…
This paper is mostly a collection of ideas already published by various authors, some of them even a long time ago. Its intention is to bring the reader to know some rather unknown papers of different fields that merit interest and to show…
Motivated by our attempt to recast Cartan's work on Lie pseudogroups in a more global and modern language, we are brought back to the question of understanding the linearization of multiplicative forms on groupoids and the corresponding…
The main achievements of Pseudo-Hermitian Quantum Mechanics and its distinction with the indefinite-metric quantum theories are reviewed. The issue of the non-uniqueness of the metric operator and its consequences for defining the…
A few physicists have recently constructed the generating compatibility conditions (CC) of the Killing operator for the Minkowski (M) , Schwarzschild (S) and Kerr (K) metrics. They discovered second order CC, well known for M, but also…
The present manuscript was written in 1994 and was not published. It addresses the form that the quantum-mechanical current density must take in mesoscopic treatments of semiconductor heterostructures, in which the electron dispersion…
The purpose of this paper is to present for the first time an elementary summary of a few recent results obtained through the application of the formal theory of partial differential equations and Lie pseudogroups in order to revisit the…
We introduce a symmetric operad whose algebras are the Operator Product Expansion (OPE) Algebras of quantum fields. There is a natural classical limit for the algebras over this operad and they are commutative associative algebras with…
Several quantum systems have been used in the last few years to extend supersymmetry. In this paper we show all this systems fit into the picture of what we call "Number Operator Algebras".
We generalize the notion of involutivity to systems of differential equations of different orders and show that the classical results due to Guillemin and Quillen relating involutivity, restrictions, characteristics and characteristicity,…
The singular value decomposition going with many problems in medical imaging, non-destructive testing, geophysics, is of central importance. Unfortunately the effective numerical determination of the singular functions in question is a very…