Algebra related to elementary particles of spin 3/2

The algebra generated by the four matricesß μ occurring in the relativistic wave equation of a particle of maximum spinn on the basis of the commutation rules for these matrices obtained previously by one of the authors has been investigated. Auxiliary quantitiesη μ satisfying the equations (5) are introduced. Theseη μ are given as polynomials inß μ . With the help of these, further auxiliary quantitiesξ μ =η μ ß μ are defined. It is shown that for half odd integral spin, theξ ’s andη ’s form two mutually commuting sets of symbols of which theη 's satisfy the same commutation rules as the Dirac matrices. This proves that the algebra in the case of half odd integral spin is the direct product of the Dirac algebra and an associatedξ -algebra. For the special case of maximum spin f the 3/2 theξ -algebra has been studied in detail, and it is shown that this algebra has just three representations of orders 1, 4, 5 such that l2 + 42 + 52 = 42 = rank of the algebra. Explicit representations are given in the non-trivial cases of orders 4 and 5. The 4-row representation of theξ ’s gives a representation of theß ’s of order 16 which is likely to be of importance in connexion with Bhabha’s new theory of the proton.