Pauli matrices

Certain special constant Hermitian $(2\times 2)$-matrices with complex entries. They were introduced by W. Pauli (1927) to describe spin ($\overrightarrow{s}=(\hslash /2)\overrightarrow{\sigma }$) and magnetic moment $(\overrightarrow{\mu }=(e\hslash /2mc)\overrightarrow{\sigma })$ of an electron. His equation describes correctly in the non-relativistic case particles of spin 1/2 (in units $\hslash$) 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:

$${\sigma }_i{\sigma }_k+{\sigma }_k{\sigma }_i=2{\delta }_{ik},$$

$${\sigma }_i{\sigma }_k-{\sigma }_k{\sigma }_i=2i{ϵ}_{ikl}{\sigma }_l.$$

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

$${\psi }_A=[{\sigma }_{0,AB}+\frac{1}{2}i\theta (\sigma \cdot \mathbf{n}{)}_{AB}]{\psi }_B^{\prime },$$

$$\sigma \cdot \mathbf{n}={\sigma }_1{n}_x+{\sigma }_2{n}_y+{\sigma }_3{n}_z.$$

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 $\mathrm{SU}(2)$. In this case they are generators of a $2$-dimensional representation of $\mathrm{SU}(2)$ 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:

$$\begin{array}{}\text{(1)}& S_0{S}_0^{\star }+{\sigma }_0=0;S_i{S}_0^{\star }={\sigma }_i,i=1,2,3,\end{array}$$

where the symbol $\star$ denotes complex conjugation. The matrices $S_{\alpha }$ satisfy the commutator relations

$$\begin{array}{}\text{(2)}& S_{\alpha }{S}_{\beta }^{\star }+S_{\beta }{S}_{\alpha }^{\star }=2{\eta }_{\alpha ,\beta },\end{array}$$

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.

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