相关论文: Quaternionic commutations
We study the quantum evolution under the combined action of the exponentials of two not necessarily commuting operators. We consider the limit in which the two evolutions alternate at infinite frequency. This case appears in a plethora of…
This semi-expository paper surveys results concerning three classes of orthogonal polynomials: in one non-hermitian variable, in several isometric non-commuting variables, and in several hermitian non-commuting variables. The emphasis is on…
Application of the noncommutative geometry to several physical models is considered.
New examples of noncommutative 4-spheres are introduced.
We study some counting questions concerning products of positive integers $u_1, \ldots, u_n$ which form a non-zero perfect square, or more generally, a perfect $k$-th power. We obtain an asymptotic formula for the number of such integers of…
Here we follow the basic analysis that is common for real and complex variables and find how it can be applied to a quaternionic variable. Non-commutativity of the quaternion algebra poses obstacles for the usual manipulations; but we show…
In this paper, we present some applications of quaternions and octonions. We present the real matrix representations for complex octonions and some of their properties which can be used in computations where these elements are involved.…
Differential equations with constant and variable coefficients over octonions are investigated. It is found that different types of differential equations over octonions can be resolved. For this purpose non-commutative line integration is…
A systematic theory is introduced for calculating the derivatives of quaternion matrix function with respect to quaternion matrix variables. The proposed methodology is equipped with the matrix product rule and chain rule and it is able to…
We discuss the use of the variational principle within quaternionic quantum mechanics. This is non-trivial because of the non commutative nature of quaternions. We derive the Dirac Lagrangian density corresponding to the two-component Dirac…
Several sets of quaternionic functions are described and studied with respect to hyperholomorphy, addition and (non commutative) multiplication, on open sets of $\mathbb H$. The aim is to get a local function theory.
The quantum quartic oscillator is investigated in order to test the many-body technique of the continuous unitary transformations. The quartic oscillator is sufficiently simple to allow a detailed study and comparison of various…
The quaternionic description of semiconductor single-electron devices is given in the single-electron regime. The conversion scheme of complex value Hamiltonian into a quaternion is formulated for the case of single-electron semiconductor…
Combinatorial aspects of multivariate diagonal invariants of the symmetric group are studied. As a consequence it is proved the existence of a multivariate extension of the classical Robinson-Schensted correspondence. Further byproduct are…
We prove a Nullstellensatz for the ring of polynomial functions in n non-commuting variables over Hamilton's ring of real quaternions. We also characterize the generalized polynomial identities in n variables which hold over the…
We study differential operators, whose coefficients define noncommutative algebras. As algebra of coefficients, we consider crossed products, corresponding to action of a discrete group on a smooth manifold. We give index formulas for…
We give a definition of noncommutative finite-dimensional Euclidean spaces $\mathbb R^n$. We then remind our definition of noncommutative products of Euclidean spaces $\mathbb R^{N_1}$ and $\mathbb R^{N_2}$ which produces noncommutative…
Motivated by a model in quantum computation we study orthogonal sets of integral vectors of the same norm that can be extended with new vectors keeping the norm and the orthogonality. Our approach involves some arithmetic properties of the…
Most of theoretical physics is based on the mathematics of functions of a real or a complex variable; yet we frequently are drawn to try extending our reach to include quaternions. The non-commutativity of the quaternion algebra poses…
We give a nontechnical introduction to the problem of non-uniqueness of star products and describe a covariant resolution of this problem. Some implications (e.g., for noncommutative gravity) and further prospects are discussed.