View source for Pauli matrices

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Certain special constant Hermitian  $  ( 2 \times 2) $-matrices with complex entries. They were introduced by W. Pauli (1927) to describe spin ($  \vec{s} = (\hbar /2) \vec{\sigma} $) 
and magnetic moment  $  ( \vec{\mu} = ( e \hbar /2mc) \vec{\sigma} ) $
of an electron. His equation describes correctly in the non-relativistic case particles of spin 1/2 (in units  $  \hbar $) 
and can be obtained from the [[Dirac equation|Dirac equation]] for  $  v/c \ll  1 $. 
In explicit form the Pauli matrices are:

$$ 
\sigma _ {1}  =  \left ( 
\begin{array}{cc}
 0  & 1  \\
 1  & 0  \\
\end{array}
 \right ) ; \ \ 
\sigma _ {2}  =  \left ( 
\begin{array}{cc}
 0  &- i  \\
 i  & 0  \\
\end{array}
 \right ) ; \ \ 
\sigma _ {3}  =  \left ( 
\begin{array}{cc}
 1  & 0  \\
 0  &- 1  \\
\end{array}
 \right ) .
$$

Their eigen values are  $  \pm  1 $. 
The Pauli matrices satisfy the following algebraic relations:

$$ 
\sigma _ {i} \sigma _ {k} + \sigma _ {k} \sigma _ {i}  =  2 \delta _ {ik} ,
$$

$$ 
\sigma _ {i} \sigma _ {k} - \sigma _ {k} \sigma _ {i}  =  2i \epsilon _ {ikl} \sigma _ {l} .
$$

Together with the unit matrix

$$ 
\sigma _ {0}  =  \left ( 
\begin{array}{cc}
 1  & 0  \\
 0  & 1  \\
\end{array}
 \right )
$$

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  $  \psi _ {A} $, 
$  A = 1, 2 $, 
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  $  \theta $
around an axis with a directed unit vector  $  \mathbf n $, 
a spinor  $  \psi _ {A} $
is transformed according to the formula

$$ 
\psi _ {A}  =  \left [ \sigma _ {0,AB }  +

\frac{1}{2}
 i \theta ( \sigma \cdot \mathbf n ) _ {AB} \right ] \psi _ {B}  ^  \prime  ,
$$

$$ 
\sigma \cdot \mathbf n  =  \sigma _ {1} n _ {x} + \sigma _ {2} n _ {y} + \sigma _ {3} n _ {z} .
$$

From the Pauli matrices one can form the [[Dirac matrices|Dirac matrices]]  $  \gamma _  \alpha  $, 
$  \alpha = 0, 1, 2, 3 $:

$$ 
\gamma _ {0}  =  \left ( 
\begin{array}{cc}
\sigma _ {0}  & 0  \\
 0  &-
\sigma _ {0}  \\
\end{array}
 \right ) ; \ \ 
\gamma _ {k}  =  \left ( 
\begin{array}{cc}
 0  &\sigma _ {k}  \\
- \sigma _ {k}  & 0  \\
\end{array}
 \right ) ; \ \ 
k = 1, 2, 3.
$$

The real linear combinations of  $  \sigma _ {0} $, 
$  i \sigma _ {1} $, 
$  i \sigma _ {2} $, 
$  i \sigma _ {3} $
form a four-dimensional subalgebra of the algebra of complex  $  ( 2 \times 2) $-matrices (under matrix multiplication) that is isomorphic to the simplest system of hypercomplex numbers, the quaternions, cf. [[Quaternion|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  $   \mathop{\rm SU} ( 2) $. 
In this case they are generators of a  $  2 $-dimensional representation of  $   \mathop{\rm SU} ( 2) $
and are denoted by  $  \tau _ {1} $, 
$  \tau _ {2} $
and  $  \tau _ {3} $. 
Sometimes it is convenient to use the linear combinations

$$ 
\tau  ^ {+}  =  
\frac{1}{2}
 ( \tau _ {1} + i \tau _ {2} )  = \ 
\left ( 
\begin{array}{cc}
 0  & 1  \\
 0  & 0  \\
\end{array}
 \right ) ; \ \ 
\tau  ^ {-}  =  
\frac{1}{2}
 ( \tau _ {1} - i \tau _ {2} )  = \ 
\left ( 
\begin{array}{cc}
 0  & 0  \\
 1  & 0  \\
\end{array}
 \right ) .
$$

In certain cases one introduces for a relativistically covariant description of two-component spinor functions instead of the Pauli matrices, matrices  $  S _  \alpha  $
related by means of the following identities:

$$ \tag{1 }
S _ {0} S _ {0}  ^  \star  + \sigma _ {0}  =  0; \ \ 
S _ {i} S _ {0}  ^  \star   =  \sigma _ {i} ,\ \ 
i = 1, 2, 3,
$$

where the symbol  $  \star $
denotes complex conjugation. The matrices  $  S _  \alpha  $
satisfy the commutator relations

$$ \tag{2 }
S _  \alpha  S _  \beta   ^  \star  + S _  \beta  S _  \alpha   ^  \star   =  2 \eta _ {\alpha , \beta }  ,
$$

where  $  \eta _ {\alpha , \beta }  $
are the components of the metric tensor of the Minkowski space of signature  $  + 2 $. 
The formulas (1) and (2) make it possible to generalize the Pauli matrices covariantly to an arbitrary curved space:

$$ 
S _  \alpha  S _  \beta   ^  \star  + S _  \beta  S _  \alpha   ^  \star   =  2g _ {\alpha \beta }  ,
$$

where  $  g _ {\alpha \beta }  $
are the components of the metric tensor of the curved space.

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

====Comments====

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  W. Pauli,   "Zur Quantenmechanik des magnetischen Elektrons"  ''Z. Phys.'' , '''43''' :  601</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  W. Pauli (ed.) , ''Handbuch der Physik'' , '''24''' , Springer  (1933)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  R.M. Wald,   "General relativity" , Univ. Chicago Press  (1984)  pp. Chapt. 4</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  Y. Choquet-Bruhat,   C. DeWitt-Morette,   M. Dillard-Bleick,   "Analysis, manifolds and physics" , North-Holland  (1982)  (Translated from French)</TD></TR></table>