arXiv subjects
Publications and source records attributed to Christian Rakotonirina.
Within the framework of quantum theory, we review the Chrestenson operator, the Weyl operator basis, and the Kronecker-Pauli operator basis in $d$-dimensional Hilbert spaces using Dirac notation, where $d$ is a prime integer strictly greater than 2. We establish a new algebraic relation connecting these operators and present the cases $d=3$ and $d=5$ as illustrative examples.
We construct two sets of representations of the Dirac equation. They transform themselves from one to another in multiplying by the tensor commutation matrix (TCM) $2\otimes 2$. The Gauss matrix can lead us to the Cholesky decomposition. The TCMs can be used in order to obtain different transforms of some matrix equations to linear matrix equations of the form $AX=B$. A TCM $n\otimes n$ is expressed in terms of the $n\otimes n$-Gell-Mann matrices. In order to generalize this expression to which of TCMs $n\otimes p$, we introduced what we call rectangle Gell-Mann matrices. The electric charge operator (ECO) for eight leptons and quarks of the Standard Model (SM) of a same generation proposed by Zenczykowski is expressed in terms of the TCM $2\otimes 2$. An ECO including all the fermions of the SM is constructed in terms of TCM $4\otimes 4$. Then the eigenvalues of a TCM take sense.
In this paper, some tensor commutation matrices are expressed in termes of the generalized Pauli matrices by tensor products of the Pauli matrices.
We show that two tensor permutation matrices permutate tensor product of rectangle matrices. Some examples, in the particular case of tensor commutation matrices, for studying some linear matrix equations are given.
Electric Charges operator (ECO) in phase space formulation, proposed by Zenczykowski, is expressed in terms of a swap operator, in some expressions for possible physical interpretations. An expression of an ECO in terms of a swap operator makes sense to the eigenvalues of the swap operator. An ECO including all the fermions of the standard model (SM) is constructed.
The Dirac-Siddharth Equation has been constructed from the Siddharth hamiltonian by quantization of the energy and momentum in Pauli algebra. We have solved this equation by using tensor product of matrices.
The Hamilton-Jacobi method which can be used for solving this equation has been presented. The solution of the equation suggests that there exist some second order linear ordinary differential equations whose resolution can be done by means of characteristic equation.
In this paper, we prove that if the matrix of the linear system is symetric, the Cholesky decomposition can be obtained from the Gauss elimination method without pivoting, without proving that the matrix of the system is positive definite.
We call rectangle Gell-Mann matrices rectangle matrices which make generalization of the expression of a tensor commutation matrix $n \otimes n$ in terms of tensor products of square Gell-Mann matrices.