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Pauli Matrices and the Theory of Representations of the Group of Rotations

Hà ⁄ seyin Önal

Published Jan 1, 2008• 0 citations
OpenAlex

Abstract

It is shown that Pauli Matrixes can be derived from irreducible rotation group representations of the weight =1/2, which in turn based on the system of infinitesimal (elementary) spatial rotations. The last permits to substantiate why Pauli matrixes can be so sufficiently used for modeling of physical rotations.

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The irreducible representations of the three-dimensional rotation group are obtained directly from the irreducible representations of its infinitesimal generators (the spin matrices), parametrized in terms of the rotation angle and the direction of the rotation axis. Expressions are given for the rotation operator exp(iψn ⋅ S) in terms of two different bases of 2j+1 elements for spin j. The results are related to the spectral decomposition of the rotation operator and expressions obtained for spin projection operators along any spatial direction for arbitrary spin.
The orthogonal group admits projective unitary representations which do not derive from true representations, and we describe a fundamental family of such representations. As a consequence there exist quantum systems that change state under a full turn rotation along a given axis (although a second full turn rotation brings them back to the original state). Amazingly, Nature has made essential use of this structure. In order to study the projective representations of the orthogonal and Lorentz groups, it is convenient to replace them by “better versions“; the groups SU(2) and SL(2,C), which are groups of 2 by 2 matrices, and for which projective representations are simply related to true representations. The orthogonal and Lorentz groups are then images of these groups under two-to-one group homomorphisms, and it is these isomorphisms that concentrate the behavior of their projective representations. Finally we describe how the introduction of parity in our theory leads to the discovery of the Dirac matrices.
The proof given by Pauli (Pauli, Relativistic Theory of Elementary Particles'' IIL, 1947, Supplement) for a theorem concerning the connection between the spin and the statistics is based on a consideration of irreducible representations of the tensor and spinor quantities in transformations of a Lorentz group with a determinant equal to unity. A proof is given of the above theorem tn which it is shown that it is enough to restrict oneself to a consideration of the transformation of the quantities with general inversion (inverston of all four co-ordinate axes).