Generation of Pauli Spin Matrices from Jones Vectors
Abstract
Using the states of polarization of light represented by Jones vectors that belong to a complex linear vector space of one-dimension, algebraic structures are elaborated that are known as dyads or second-order tensors that in this case make up a complex vector space of two dimensions. With these second-order tensors, which can be expressed in a matrix form, sequences of switching relations are constructed with alternating states of light polarization. The sequences of commutation relations, with the property of alternation given by the permutation of the polarization states of light, are presented as linear combinations that generate Pauli spin matrices in a simple way. The polarization states of the Jones vectors used to construct the sequences of the commutation relations of the dyadic forms belong to forms of the circular, left and right, or linear type. The transition from a complex vector space, in which the Jones vectors act, to a complex linear vector space of two dimensions, in which the base of this last space is made up of the unit matrix and the Pauli spin matrices, is feasible through commutation relations using Jones vectors in states of linear and circular polarization.
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