B₂ Diagrams and Bhabha Equation with Mass Matrix

Abstract

Till now nearly all papers about the Bhabba equation treat equations with a mass term ml. This is merely a matter of mathematical simplicity, because we will see in this work that important physical fields can only be described by a mass term + m 1. The second aim of this publication is to derive a constructive procedure for the calculation of the representations of the Bhabha equa tion and the tensor and spinor field equations, which are equivalent to them. Invariant Wave EquationsBecause a differential equation of any order is being equivalent to a system of first order dif ferential equations the field equation of a free field without subsidary conditions can be written in the form :(1) -ß* + M )ip = 0. The invariance of this equation under infinitesimal Lorentz transformations y)' = (1 + ieSim)rp yields [1]: [Sim, ßj] + ßk (gij Öl -gm] <5?) = 0, [Slm,M ] = 0 (2) (3) (4) (we use the convention: gn = <722 = 933 = -S'oo = -1, Qik = 0 for i = k). The six Sim fulfill [Si mt -+ 9mkSlp + gipSmk -gikSmp -9mpSlk , k, I, m ,p e { 0 ,1 ,2 ,3 } . (5) The Sim can always be choosen in block-diagonal form and then (4) tells us that because of Schur's lemma M is a diagonal matrix. Slm = Sl im Um M = mi m\ mk mkIn order to get the explicit form of the matrices Sim and ßj in a simple manner, one has to make a further assumption. There are two ways to achieve the same result:1) Put ßj = S4], ßi = S41, £744 = -1, 04/ =9j4 = 0, j e { 0,1 ,2 ,3 }. Now(3) has the same structure as (5) and demand ing , $ 4 = -g^Sim = Sim (6) (3), (4), (6) can be written: , = gmk^lp + 9lp®mk ~ gik^mp -gmpSlk, k, I, m ,p e { 0 ,1 ,2 ,3 ,4 } .