Pauli matrices

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 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 $\gamma _ \alpha$, $\alpha = 0, 1, 2, 3$:

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

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. 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

[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)