Pauli matrices

Certain special constant Hermitian -matrices with complex entries. They were introduced by W. Pauli (1927) to describe spin ( ) and magnetic moment of an electron. His equation describes correctly in the non-relativistic case particles of spin 1/2 (in units ) and can be obtained from the Dirac equation for . In explicit form the Pauli matrices are: Their eigen values are . The Pauli matrices satisfy the following algebraic relations:  Together with the unit matrix the Pauli matrices form a complete system of second-order matrices by which an arbitrary linear operator (matrix) of dimension 2 can be expanded. They act on two-component spin functions , , and are transformed under a rotation of the coordinate system by a linear two-valued representation of the rotation group. Under a rotation by an infinitesimal angle around an axis with a directed unit vector , a spinor is transformed according to the formula  From the Pauli matrices one can form the Dirac matrices , : The real linear combinations of , , , form a four-dimensional subalgebra of the algebra of complex -matrices (under matrix multiplication) that is isomorphic to the simplest system of hypercomplex numbers, the quaternions, cf. Quaternion. They are used whenever an elementary particle has a discrete parameter taking only two values, for example, to describe an isospin nucleon (a proton-neutron). Quite generally, the Pauli matrices are used not only to describe isotopic space, but also in the formalism of the group of inner symmetries . In this case they are generators of a -dimensional representation of and are denoted by , and . Sometimes it is convenient to use the linear combinations In certain cases one introduces for a relativistically covariant description of two-component spinor functions instead of the Pauli matrices, matrices related by means of the following identities: (1)

where the symbol denotes complex conjugation. The matrices satisfy the commutator relations (2)

where are the components of the metric tensor of the Minkowski space of signature . The formulas (1) and (2) make it possible to generalize the Pauli matrices covariantly to an arbitrary curved space: where are the components of the metric tensor of the curved space.