The Lie algebra s o (N) and the Duffin‐Kemmer‐Petiau ring

An explicit expression is given for the unit element E of the ring generated by the Duffin‐Kemmer‐Petiau (DKP) operators βμ. The relation of E to the unit operator I (unit matrix in a matrix representation) is also derived. It is pointed out that one must be careful to distinguish E from I. Bhabha's observation that one may use the irreducible representations (irreps) of the Lie algebra s o (5) to obtain the irreps of the Dirac, DKP, and other algebras is given a concise and general setting in terms of a relation between the Lie algebra s o (n + 1) and a family of semisimple operator rings. We emphasize that for the case n + 1 = 5 this means that there is an underlying relationship between the physical DKP and Dirac algebras and wave equations.

[1]  E. Fischbach,et al.  Resolution of the SU(3) Puzzle:Γ(η→γγ)Γ(π0→γγ) , 1972 .

[2]  E. Fischbach,et al.  The Association of the Sakata-Taketani (Feshbach-Villars) Field with the Kemmer Field, under Symmetry Breaking , 1972 .

[3]  E. Fischbach,et al.  K-l-3 form-factors and the scaling behavior of spin-0 fields , 1971 .

[4]  E. Fischbach,et al.  RESOLUTION OF THE DISCREPANCY IN THE DETERMINATION OF THE CABIBBO ANGLE theta/sub V/ FROM K/sub l3/ AND NUCLEAR $beta$ DECAYS. , 1971 .

[5]  I. Gyuk,et al.  WAVE EQUATIONS WITH COMPACT AND NONCOMPACT SPECTRUM. , 1971 .

[6]  P. Chandrasekaran,et al.  Kemmer algebra from generalized Clifford elements , 1969 .

[7]  M. Wong.  Representations of the Orthogonal Group. I. Lowering and Raising Operators of the Orthogonal Group and Matrix Elements of the Generators , 1967 .

[8]  K. Hecht,et al.  Lowering and Raising Operators for the Orthogonal Group in the Chain O(n) ⊃ O(n − 1) ⊃ … , and their Graphs , 1967 .

[9]  P. Roman,et al.  Theory of elementary particles , 1960 .

[10]  Homi Jehangir Bhabha,et al.  On the Postulational Basis of the Theory of Elementary Particles , 1949 .

[11]  N. Kemmer,et al.  The algebra of meson matrices , 1943, Mathematical Proceedings of the Cambridge Philosophical Society.

[12]  B. S. M. Rao.  Commutation rules for matrices related to particles of higher spins , 1942 .

[13]  N. Kemmer.  The Particle Aspect of Meson Theory , 1939 .

[14]  R. Duffin.  On The Characteristic Matrices of Covariant Systems , 1938 .

[15]  H. Weyl,et al.  Spinors in n Dimensions , 1935 .