The correspondence between the particle and the wave aspects of the meson and the photon

Abstract

In the Kemmer formulation of the meson equation certain $\Gamma _{k}^{\ast}$ matrices are introduced. They are determined completely by the condition that for a Lorentz transformation $\Gamma _{k}^{\ast}\psi $ transforms as a vector. The $\Gamma ^{\ast}$'s, together with another matrix $\beta $ (defined in terms of them), turn out to be very convenient for discussing the properties of the meson matrices $\beta _{k}$. With their help, a formula for the spur of a product of any number of $\beta _{k}$'s is derived both for the 10- and the 5-row representations. Also they enable one to bring out, without making use of an explicit representation, the complete equivalence of the Kemmer formulation with the vector and scalar meson equations in the usual tensor form. Besides, the use of the $\Gamma ^{\ast}$-matrices now permits the introduction of nuclear interaction in the Kemmer formulation in an elegant way. The extension of the 'particle' formulation to the case of a particle of zero rest-mass is discussed. Finally, a specially convenient representation of the $\beta _{k}$'s is given both for the case of spin 1 and of spin 0. The $\Gamma ^{\ast}$'s and the $\beta $ are explicitly given for this particular representation.