Pauli Matrices

The Pauli matrices are a set of three 2 × 2 complex matrices which are Hermitian and unitary. 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:

These matrices were used by, then named after, the Austrian-born physicist Wolfgang Pauli (1900–1958), in his 1925 study of spin in quantum mechanics.

Each Pauli matrix is Hermitian, and together with the identity matrix I (sometimes considered the zeroth Pauli matrix ), the Pauli matrices (being multiplied by real coefficients) span the full vector space of 2x2 Hermitian matrices. In the language of quantum mechanics, hermitian matrices are observables, so the Pauli matrices span the space of observables of the 2-dimensional complex Hilbert space. In the context of Pauli's work, is the observable corresponding to spin along the kth coordinate axis in three-dimensional Euclidean space ℝ3.

The Pauli matrices (after multiplication by i to make them anti-hermitian), also generate transformations in the sense of Lie algebras: the matrices form a basis for, which exponentiates to the spin group SU(2), and for the identical Lie algebra, which exponentiates to the Lie group SO(3) of rotations of 3-dimensional space. Moreover, the algebra generated by the three matrices is isomorphic to the Clifford algebra of ℝ3, called the algebra of physical space.