R. Vasudevan

Kemmer algebra from generalized Clifford elements

Our earlier studies [l, 2, 31 on the generalised Clifford algebra (G.C.il.) formulated by K. Yamazaki [4] 1 e d us to a surprising connection between the generalised Clifford algebra and the unitary groups which describe the internal symmetry of elementary particles. We shall now show that it is possible to obtain the matrices of the Duffin-Kemmer-Petiau [5] (D.K.P.) algebra which enter the space-time description of particles having spin zero or one through a wave equation, known in literature as the D.K.P. equation. Such a derivation of D.K.P. algebra from the generalised Clifford algebra leads us automatically to a method of constructing the elements of the algebra of the orthogonal groups also.

The generalized Clifford algebra and the unitary group

During the past two years following the first formulation of L-matrix theory [I] the Matscience group has been concerned with the generalised Clifford algebra of matrices which are the mth roots of unity. The generalised algebra was discovered by Yamazaki [2] in 1964 and the matrix representations in the lowest dimension were first given by Morris in 1967 [3]. We shall now present some new results on the subject and point out a surprising and unexpected connection with the generators of the special unitary group. It has been established that there are (2n + 1) matrices L, , L, ,..., L2n+l of dimension mn x mn obeying the two generalised Clifford conditions:

A Hierarchy of Idempotent Matrices

We notice that any (3 × 3) antisymmetric matrix

Some new mathematical features in cascade theory

\begin{bmatrix}0 &-\lambda_3 & \lambda_2\\ \lambda_3 & 0 & -\lambda_1\\ -\lambda_2 & \lambda_1 & 0 \end{bmatrix}

MULTIPLE PROCESSES IN ELECTRON-PHOTON CASCADES

where λl, λa, λ3 are pure real or pure imaginary parameters, has the very interesting property

Para-Fermi operators and special unitary algebras

A^3=(\lambda\frac{2}{1} \ +\ \lambda\frac{2}{2}\ \lambda\frac{2}{3})A