Difference between revisions of "Pauli matrices"

Revision as of 14:52, 7 June 2020

Certain special constant Hermitian -matrices with complex entries. They were introduced by W. Pauli (1927) to describe spin () and magnetic moment of an electron. His equation describes correctly in the non-relativistic case particles of spin 1/2 (in units ) and can be obtained from the Dirac equation for . In explicit form the Pauli matrices are:

Their eigen values are . The Pauli matrices satisfy the following algebraic relations:

Together with the unit matrix

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 , , 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 around an axis with a directed unit vector , a spinor is transformed according to the formula

From the Pauli matrices one can form the Dirac matrices , :

The real linear combinations of , , , form a four-dimensional subalgebra of the algebra of complex -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 . In this case they are generators of a -dimensional representation of and are denoted by , and . Sometimes it is convenient to use the linear combinations

In certain cases one introduces for a relativistically covariant description of two-component spinor functions instead of the Pauli matrices, matrices related by means of the following identities:

(1)

where the symbol denotes complex conjugation. The matrices satisfy the commutator relations

(2)

where are the components of the metric tensor of the Minkowski space of signature . The formulas (1) and (2) make it possible to generalize the Pauli matrices covariantly to an arbitrary curved space:

where 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=48144

This article was adapted from an original article by V.G. Krechet (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article

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+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:
-p0718601.png
-$#A+1 = 41 n = 0
-$#C+1 = 41 : ~/encyclopedia/old_files/data/P071/P.0701860 Pauli matrices
-Automatically converted into TeX, above some diagnostics.
-Please remove this comment and the {{TEX|auto}} line below,
-if TeX found to be correct.
--->
-{{TEX|auto}}
+<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/p0718606.png" /></td> </tr></table>
-{{TEX|done}}
-Certain special constant Hermitian  $  ( 2 \times 2) $-
+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:
-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|Dirac equation]] for  $  v/c \ll  1 $.
-In explicit form the Pauli matrices are:
-$$
+<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071860/p0718608.png" /></td> </tr></table>
-\sigma _ {1}  =  \left (
-Their eigen values are  $  \pm  1 $.
+<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071860/p0718609.png" /></td> </tr></table>
-The Pauli matrices satisfy the following algebraic relations:
-$$
-\sigma _ {i} \sigma _ {k} + \sigma _ {k} \sigma _ {i}  =  2 \delta _ {ik} ,
-$$
-$$
-\sigma _ {i} \sigma _ {k} - \sigma _ {k} \sigma _ {i}  =  2i \epsilon _ {ikl} \sigma _ {l} .
-$$
 Together with the unit matrix
-$$
+<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/p07186010.png" /></td> </tr></table>
-\sigma _ {0}  =  \left (
-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
-$$
+the Pauli matrices form a complete system of second-order matrices by which an arbitrary linear operator (matrix) of dimension 2 can be expanded. They act on two-component spin functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071860/p07186011.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071860/p07186012.png" />, and are transformed under a rotation of the coordinate system by a linear two-valued representation of the rotation group. Under a rotation by an infinitesimal angle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071860/p07186013.png" /> around an axis with a directed unit vector <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071860/p07186014.png" />, a spinor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071860/p07186015.png" /> is transformed according to the formula
-\psi _ {A}  =  \left [ \sigma _ {0,AB }  +
-\frac{1}{2}
+<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071860/p07186016.png" /></td> </tr></table>
- i \theta ( \sigma \cdot \mathbf n ) _ {AB} \right ] \psi _ {B}  ^  \prime  ,
-$$
-$$
+<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071860/p07186017.png" /></td> </tr></table>
-\sigma \cdot \mathbf n  =  \sigma _ {1} n _ {x} + \sigma _ {2} n _ {y} + \sigma _ {3} n _ {z} .
-$$
-From the Pauli matrices one can form the [[Dirac matrices|Dirac matrices]]  $  \gamma _  \alpha  $,
+From the Pauli matrices one can form the [[Dirac matrices|Dirac matrices]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071860/p07186018.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071860/p07186019.png" />:
-$  \alpha = 0, 1, 2, 3 $:
-$$
+<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/p07186020.png" /></td> </tr></table>
-\gamma _ {0}  =  \left (
-The real linear combinations of  $  \sigma _ {0} $,
+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
-$  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|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
-$$
+<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/p07186032.png" /></td> </tr></table>
-\tau  ^ {+}  =
-\frac{1}{2}
- ( \tau _ {1} + i \tau _ {2} )  = \
-\left (
-In certain cases one introduces for a relativistically covariant description of two-component spinor functions instead of the Pauli matrices, matrices  $  S _  \alpha  $
+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:
-related by means of the following identities:
-$$ \tag{1 }
+<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071860/p07186034.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>
-S _ {0} S _ {0}  ^  \star  + \sigma _ {0}  =  0; \ \
-S _ {i} S _ {0}  ^  \star   =  \sigma _ {i} ,\ \
-i = 1, 2, 3,
-$$
-where the symbol  $  \star $
+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
-denotes complex conjugation. The matrices  $  S _  \alpha  $
-satisfy the commutator relations
-$$ \tag{2 }
+<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071860/p07186037.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>
-S _  \alpha  S _  \beta   ^  \star  + S _  \beta  S _  \alpha   ^  \star   =  2 \eta _ {\alpha , \beta }  ,
-$$
-where  $  \eta _ {\alpha , \beta }  $
+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:
-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:
-$$
+<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071860/p07186040.png" /></td> </tr></table>
-S _  \alpha  S _  \beta   ^  \star  + S _  \beta  S _  \alpha   ^  \star   =  2g _ {\alpha \beta }  ,
-$$
-where  $  g _ {\alpha \beta }  $
+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.
-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>

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Difference between revisions of "Pauli matrices"

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