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Pauli matrices

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Certain special constant Hermitian -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)

Comments

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)
How to Cite This Entry:
Pauli matrices. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Pauli_matrices&oldid=52212
This article was adapted from an original article by V.G. Krechet (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article