Mathematical Physics and Matrix Representations

Z. Ficek

This chapter illustrates how one can obtain a matrix representation of a state vector, an operator, and an eigenvalue equation in a basis of orthonormal states. It helps to learn that matrix representations are very useful in calculations of the properties of operators, because algebra of matrices is very simple and completely developed. Readers familiar with the algebra of matrices agree that mathematical operations on matrices such as multiplication by a scalar, addition, subtraction, multiplication and diagonalization are simple and easy to perform. In fact, the application of the algebra of matrices to quantum physics results from the direct relations between operators and matrices that any mathematical relation that holds between operators also holds between the corresponding matrices. The eigenvalues and eigenvectors of the matrix representing a given operator  in the basis of orthonormal states are the same as the eigenvalues and eigenvectors of Â.

László Lovász

Boris Khots, Dmitriy Khots

This paper considers spin - j representation of the Lie group SU(2) and its applications to contemporary physics from Observer's Mathematics point of view (see www.mathrelativity.com). This mathematics was introduced by authors based on denial of infinity idea. We proved here the following. Theorem. In Observers Mathematics the probability of spin - j transformation is a homomorphism (representation) of SU(2) to the set of matrix transformations of a linear space of polynomial functions is less than 1. Theorem. The probability of elementary fermions have half-integer spin in Observer's Mathematics is less than 1. The probability of elementary bosons have integer spin in Observer's Mathematics is less than 1.

Mituo Taketani, Masayuki Nagasaki

Richard Healey

Chillemi, Daniel Andres

A mathematical formulation is proposed representing time as a dual matrix structure: physical timeis represented by a matrix T, while its structural or relational time is expressed by the adjoint matrix T†This representation allows for interpreting quantum coupling and decoupling as transformations betweensubspaces of minimal entropy and maximal organization. We prove the equivalence between temporalreversibility and the invertibility of the associated operators.