Factorization of the algebra of particles of half-odd spin

AbstractIt is proved that the matrix algebra for any relativistic wave equation of half-odd integral spin can be factorized as the direct product of a Dirac algebra and another, called a ξ-algebra. The structure and representation of ξ-algebras are studied in detail. The factorization simplifies calculations with particles of spin > ½, because the ξ-algebra contains only one-sixteenth as many elements as the original matrix algebra.

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Algebra related to elementary particles of spin 3/2

Proceedings of the Royal Society of London Series A - Mathematical and Physical Sciences ◽

10.1098/rspa.1946.0083 ◽

1946 ◽

Vol 187 (1010) ◽

pp. 385-397 ◽

Cited By ~ 15

Keyword(s):

Wave Equation ◽

Direct Product ◽

Elementary Particles ◽

Relativistic Wave Equation ◽

Relativistic Wave ◽

Dirac Algebra ◽

Integral Spin ◽

Explicit Representations ◽

Special Case ◽

Maximum Spin

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

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Quaternion treatment of the relativistic wave equation

Proceedings of the Royal Society of London Series A - Mathematical and Physical Sciences ◽

10.1098/rspa.1937.0173 ◽

1937 ◽

Vol 162 (909) ◽

pp. 145-154 ◽

Cited By ~ 21

Keyword(s):

Wave Function ◽

Wave Equation ◽

Linear Functions ◽

Relativistic Wave Equation ◽

Relativistic Wave ◽

Formal Representation ◽

The Matrix ◽

Scalar Products ◽

Relationship Of ◽

The Relationship

A set of matrices can be found which is isomorphic with any linear associative algebra. For the case of quaternions this was first shown by Cayley (1858), but the first formal representation was made by Peirce (1875, 1881). These were two-matrices, and the introduction of the four-row matrices of Dirac and Eddington necessitated the treatment of a wave function as a matrix of one row (as columns). Quaternions have been used by Lanczos (1929) to discuss a different form of wave equation, but here the Dirac form is discussed, the wave function being taken as a quaternion and the four-row matrices being linear functions of a quaternion. Certain advantages are claimed for quaternion methods. The absence of the distinction between outer and scalar products in the matrix notation necessitates special expedients (Eddington 1936). Every matrix is a very simple function of the fundamental Hamiltonian vectors α, β, γ , so that the result of combination is at once evident and depends only on the rules of combination of these vectors. At all stages the relationship of the different quantities to four-space is at once visible. The Dirac-Eddington matrices, the wave equation and its exact solution by Darwin, angular momentum operators, the general and Lorentz transformation, spinors and six-vectors, the current-density four-vector are treated in order to exhibit the working of this method. S and V for scalar and vector products are used. Quaternions are denoted by Clarendon type, and all vectors are in Greek letters.

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