Pauli matrices

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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:


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


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.


[1] W. Pauli, , Works on quantum theory , 1–2 , Moscow (1975–1977) (In Russian; translated from German)
[2] N.F. Nelina, "Physics of elementary particles" , Moscow (1977) (In Russian)
[3] D. Bril, J.A. Wheeler, , The latest problems on gravitation , Moscow (1961) pp. 381–427 (In Russian)



[a1] W. Pauli, "Zur Quantenmechanik des magnetischen Elektrons" Z. Phys. , 43 : 601
[a2] W. Pauli (ed.) , Handbuch der Physik , 24 , Springer (1933)
[a3] R.M. Wald, "General relativity" , Univ. Chicago Press (1984) pp. Chapt. 4
[a4] Y. Choquet-Bruhat, C. DeWitt-Morette, M. Dillard-Bleick, "Analysis, manifolds and physics" , North-Holland (1982) (Translated from French)
How to Cite This Entry:
Pauli matrices. V.G. Krechet (originator), Encyclopedia of Mathematics. URL:
This text originally appeared in Encyclopedia of Mathematics - ISBN 1402006098