Related papers: Jordan Form and Quantum Tomography
The research presented in this article concerns the stroboscopic approach to quantum tomography, which is an area of science where quantum Physics and linear algebra overlap. In this article we introduce the algebraic structure of the…
There have been several propositions for a geometric and essentially non-linear formulation of quantum mechanics. From a purely mathematical point of view, the point of view of Jordan algebra theory might give new strength to such…
We define a general notion of partially ordered Jordan algebra (over a partially ordered ring), and we show that the Jordan geometry associated to such a Jordan algebra admits a natural invariant partial cyclic order, whose intervals are…
The question of matrix similarity is a classical one in linear algebra. For a field $\mathbb{F}$ and some positive integer $n \in \mathbb{N}$, one may consider the following problems: 1. Given two matrices $A, B \in \mathrm{GL}(n,…
This is a transcription of a conference proceedings from 1985. It reviews the Jordan algebra formulation of quantum mechanics. A possible novelty is the discussion of time evolution; the associator takes over the role of $i$ times the…
Let k be an algebraically closed field of characteristic p \ge 0. We shall consider the problem of finding out a Jordan canonical form of J(\alpha,s) \otimes_{k} J(\beta,t), where J(\alpha,s) means the Jordan block with eigenvalue \alpha…
We establish a natural correspondence between (the equivalence classes of) cubic solutions of an eiconal type equation and (the isomorphy classes of) cubic Jordan algebras.
Using geometric approach we formulate quantum theory in terms of Jordan algebras. We analyze the notion of (quasi)particle (=elementary excitation of translation-invariant stationary state) and the scattering of (quasi)particles in this…
Over a field of characteristic $0$ we give a concrete, computation--ready description of Jordan algebra structures and their low--order deformation theory. The Jordan identity is quartic in the elements and cubic in the multiplication, and…
I explore several related routes to deriving the Jordan-algebraic structure of finite-dimensional quantum theory from more transparent operational or physical principles, mainly involving ideas about the symmetries of, and the correlations…
The quantum deformation of the Jordanian twist F_qJ for the standard quantum Borel algebra U_q(B) is constructed. It gives the family U_qJ(B) of quantum algebras depending on parameters x and h. In a generic point these algebras represent…
The classification, up to isomorphism, of two-dimensional (not necessarily commutative) Jordan algebras over algebraically closed fields and $\mathbb{R}$ is presented in terms of their matrices of structure constants.
We introduce some basic notions and results for quaternionic linear operators analogous to those for complex linear operators. Our main result is to prove the additive and multiplicative Jordan-Chevalley decompositions for quaternionic…
In this note we mainly study the fine Jordan-Chevalley decomposition: a refinement of the classical Jordan-Chevalley decomposition of a matrix and we pay a particular attention to the field of the coefficients of the matrix. Moreover we…
A representation of finite-dimensional probabilistic models in terms of formally real Jordan algebras is obtained, in a strikingly easy way, from simple assumptions. This provides a framework in which real, complex and quaternionic quantum…
This paper suveys some recent algebraic developments in two parameter Quantum deformations and their Nonstandard (or Jordanian) counterparts. In particular, we discuss the contraction procedure and the quantum group homomorphisms associated…
Various reconstructions of finite-dimensional quantum mechanics result in a formally real Jordan algebra A and a last step remains to conclude that A is the self-adjoint part of a C*-algebra. Using a quantum logical setting, it is shown…
A square matrix $A$ has the usual Jordan canonical form that describes the structure of $A$ via eigenvalues and the corresponding Jordan blocks. If $A$ is a linear relation in a finite-dimensional linear space ${\mathfrak H}$ (i.e., $A$ is…
The nonlinear dynamics of a system with periodic structure can be analyzed using a square matrix. We show that because the special property of the square matrix constructed for nonlinear dynamics, we can reduce the dimension of the matrix…
Classical and quantum perturbations can be described in terms of marginal distribution functions in the framework of tomographic cosmology. In particular, the so called Radon transformation and the mode-parametric quantum oscillator…