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.