Pauli matrices

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In physics, the Pauli matrices are a set of 2 × 2 complex Hermitian and unitary matrices.[1] Usually indicated by the Greek letter "sigma" (σ), they are occasionally denoted with a "tau" (τ) when used in connection with isospin symmetries. They are:

The name refers to Wolfgang Pauli.

The real (hence also, complex) subalgebra generated by the σi (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 σi 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 Cℓ3,0(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 Cℓ3,0(R) on the complex vector space C2 of dimension 2.)

Contents

Algebraic properties

where I is the identity matrix, i.e. the matrices are involutory.

From above we can deduce that the eigenvalues of each σi are ±1.

  • Together with the identity matrix I (which is sometimes written as σ0), the Pauli matrices form an orthogonal basis, in the sense of Hilbert-Schmidt, for the real Hilbert space of 2 × 2 complex Hermitian matrices, or the complex Hilbert space of all 2 × 2 matrices.

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