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:
![](https://www.encyclopediaofmath.org/legacyimages/p/p071/p071860/p07186034.png) | (1) |
where the symbol
denotes complex conjugation. The matrices
satisfy the commutator relations
![](https://www.encyclopediaofmath.org/legacyimages/p/p071/p071860/p07186037.png) | (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.
References
[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) |
References
[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. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Pauli_matrices&oldid=49519
This article was adapted from an original article by V.G. Krechet (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098.
See original article