Pauli Matrices

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High Quality Content by WIKIPEDIA articles! High Quality Content by WIKIPEDIA articles! In physics, the Pauli matrices are a set of 2 × 2 complex Hermitian and unitary matrices (See also ref.[1]). Usually indicated by the Greek letter 'sigma' (¿), they are occasionally denoted with a 'tau' (¿) when used in connection with isospin symmetries. They are: sigma_1 = sigma_x = begin{pmatrix} 0&1 1&0 end{pmatrix} sigma_2 = sigma_y = begin{pmatrix} 0&-i i&0 end{pmatrix} sigma_3 = sigma_z = begin{pmatrix} 1&0 0&-1 end{pmatrix}. The name refers to Wolfgang Pauli. The real (hence also, complex) subalgebra generated by the ¿i (that is, the set of real or complex linear combinations of all the elements which can be built up as products of Pauli matrices) is the full set M2(C) of complex 2×2 matrices. The ¿i can also be seen as generating the real Clifford algebra of the real quadratic form with signature (3,0): this shows that this Clifford algebra C¿3,0(R) is isomorphic to M2(C), with the Pauli matrices providing an explicit isomorphism. (In particular, the Pauli matrices define a faithful representation of the real Clifford algebra C¿3,0(R) on the complex vector space C2 of dimension 2.)

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