View source for Pauli matrices

Certain special constant Hermitian <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071860/p0718601.png" />-matrices with complex entries. They were introduced by W. Pauli (1927) to describe spin (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071860/p0718602.png" />) and magnetic moment <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071860/p0718603.png" /> of an electron. His equation describes correctly in the non-relativistic case particles of spin 1/2 (in units <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071860/p0718604.png" />) and can be obtained from the [[Dirac equation|Dirac equation]] for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071860/p0718605.png" />. In explicit form the Pauli matrices are:

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Their eigen values are <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071860/p0718607.png" />. The Pauli matrices satisfy the following algebraic relations:

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Together with the unit matrix

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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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071860/p07186011.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071860/p07186012.png" />, 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071860/p07186013.png" /> around an axis with a directed unit vector <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071860/p07186014.png" />, a spinor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071860/p07186015.png" /> is transformed according to the formula

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071860/p07186016.png" /></td> </tr></table>

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From the Pauli matrices one can form the [[Dirac matrices|Dirac matrices]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071860/p07186018.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071860/p07186019.png" />:

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The real linear combinations of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071860/p07186021.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071860/p07186022.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071860/p07186023.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071860/p07186024.png" /> form a four-dimensional subalgebra of the algebra of complex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071860/p07186025.png" />-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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071860/p07186026.png" />. In this case they are generators of a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071860/p07186027.png" />-dimensional representation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071860/p07186028.png" /> and are denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071860/p07186029.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071860/p07186030.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071860/p07186031.png" />. Sometimes it is convenient to use the linear combinations

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In certain cases one introduces for a relativistically covariant description of two-component spinor functions instead of the Pauli matrices, matrices <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071860/p07186033.png" /> related by means of the following identities:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071860/p07186034.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>

where the symbol <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071860/p07186035.png" /> denotes complex conjugation. The matrices <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071860/p07186036.png" /> satisfy the commutator relations

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071860/p07186037.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071860/p07186038.png" /> are the components of the metric tensor of the Minkowski space of signature <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071860/p07186039.png" />. The formulas (1) and (2) make it possible to generalize the Pauli matrices covariantly to an arbitrary curved space:

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where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071860/p07186041.png" /> 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>