Representações da equação de Dirac em 1+1 dimensões

Abstract

In this paper a general classification of representations of the Dirac equation in 1+1 dimensions is described. There are considered such representations in which every Dirac matrix β, α is associated with a unique Pauli matrix. The classification includes 24 representations among which six representations are main and others are ones with sign modified Pauli matrices. The unitary transformations between all the representations are determined in the explicit form. The study of structure of the set of transformations results in conclusions that all the representations are equivalent, so that anyone of the representations can be obtained from any other by a unitary transformation. It is established that the set of the transformations forms a non-abelian group in respect of matrix product in equivalence classes, which are defined through indistinguishability of transformations, which differ by a phase factor. The group possesses one non-trivial subgroup, which in its turn includes two non-trivial subgroups. The most general form of the Dirac matrices in 1+1 dimensions determined by three arbitrary parameters is presented. Isomorphism of the couple of the Dirac matrices to the couple of orthogonal unitary vectors is demonstrated. Applications of the results obtained in the paper are presented.