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:

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

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

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

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 $:

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

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

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

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)