Namespaces
Variants
Actions

Difference between revisions of "Spinor structure"

From Encyclopedia of Mathematics
Jump to: navigation, search
m (tex encoded by computer)
Line 1: Line 1:
''on an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086780/s0867802.png" />-dimensional manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086780/s0867803.png" />, fibration of spin-frames''
+
<!--
 +
s0867802.png
 +
$#A+1 = 66 n = 0
 +
$#C+1 = 66 : ~/encyclopedia/old_files/data/S086/S.0806780 Spinor structure
 +
Automatically converted into TeX, above some diagnostics.
 +
Please remove this comment and the {{TEX|auto}} line below,
 +
if TeX found to be correct.
 +
-->
  
A principal fibre bundle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086780/s0867804.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086780/s0867805.png" /> with structure group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086780/s0867806.png" /> (see [[Spinor group|Spinor group]]), covering some principal fibre bundle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086780/s0867807.png" /> of co-frames with structure group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086780/s0867808.png" />. The latter condition means that there is given a surjective homomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086780/s0867809.png" /> of principal fibre bundles, which is the identity on the base and is compatible with the natural homomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086780/s08678010.png" />. One says that the spinor structure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086780/s08678011.png" /> is subordinate to the Riemannian metric <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086780/s08678012.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086780/s08678013.png" /> defined by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086780/s08678014.png" />. From the point of view of the theory of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086780/s08678015.png" />-structures, a spinor structure is a generalized <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086780/s08678016.png" />-structure with structure group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086780/s08678017.png" /> together with a non-faithful representation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086780/s08678018.png" /> (cf. [[G-structure|<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086780/s08678019.png" />-structure]]).
+
{{TEX|auto}}
 +
{{TEX|done}}
  
In a similar way one defines spinor structures subordinate to pseudo-Riemannian metrics, and spinor structures on complex manifolds subordinate to complex metrics. Necessary and sufficient conditions for the existence of a spinor structure on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086780/s08678020.png" /> consist of the orientability of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086780/s08678021.png" /> and the vanishing of the Stiefel–Whitney class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086780/s08678022.png" />. When these conditions hold, the number of non-isomorphic spinor structures on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086780/s08678023.png" /> subordinate to a given Riemannian metric coincides with the order of the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086780/s08678024.png" /> (see [[#References|[6]]]).
+
''on an  $  n $-
 +
dimensional manifold  $  M $,  
 +
fibration of spin-frames''
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086780/s08678025.png" /> be the complexification of the [[Clifford algebra|Clifford algebra]] of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086780/s08678026.png" /> with quadratic form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086780/s08678027.png" />. Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086780/s08678028.png" /> has an irreducible representation in a space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086780/s08678029.png" /> of dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086780/s08678030.png" />, which defines a representation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086780/s08678031.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086780/s08678032.png" />. Every spinor structure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086780/s08678033.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086780/s08678034.png" /> yields an associated vector bundle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086780/s08678035.png" /> with typical fibre <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086780/s08678036.png" />, called a spinor bundle. The Riemannian connection on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086780/s08678037.png" /> determines in a canonical way a connection on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086780/s08678038.png" />. On the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086780/s08678039.png" /> of smooth sections of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086780/s08678040.png" /> (spinor fields) there acts a linear differential operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086780/s08678041.png" /> of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086780/s08678042.png" />, the Dirac operator, which is locally defined by the formula
+
A principal fibre bundle  $  \widetilde \pi  : \widetilde{P}  \rightarrow M $
 +
over  $  M $
 +
with structure group  $  \mathop{\rm Spin} _ {n} $(
 +
see [[Spinor group|Spinor group]]), covering some principal fibre bundle  $  \pi :  P \rightarrow M $
 +
of co-frames with structure group  $  \mathop{\rm SO} _ {n} $.  
 +
The latter condition means that there is given a surjective homomorphism  $  \kappa : \widetilde{P}  \rightarrow P $
 +
of principal fibre bundles, which is the identity on the base and is compatible with the natural homomorphism  $  \rho :   \mathop{\rm Spin} _ {n} \rightarrow  \mathop{\rm SO} _ {n} $.  
 +
One says that the spinor structure $  ( \widetilde \pi  , \kappa ) $
 +
is subordinate to the Riemannian metric  $  g $
 +
on $  M $
 +
defined by  $  \pi $.  
 +
From the point of view of the theory of $  G $-
 +
structures, a spinor structure is a generalized  $  G $-
 +
structure with structure group  $  G = \mathop{\rm Spin} _ {n} $
 +
together with a non-faithful representation  $  \rho :   \mathop{\rm Spin} _ {n} \rightarrow  \mathop{\rm SO} _ {n} $(
 +
cf. [[G-structure| $  G $-
 +
structure]]).
  
<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/s/s086/s086780/s08678043.png" /></td> </tr></table>
+
In a similar way one defines spinor structures subordinate to pseudo-Riemannian metrics, and spinor structures on complex manifolds subordinate to complex metrics. Necessary and sufficient conditions for the existence of a spinor structure on  $  M $
 +
consist of the orientability of  $  M $
 +
and the vanishing of the Stiefel–Whitney class  $  W _ {2} ( M) $.  
 +
When these conditions hold, the number of non-isomorphic spinor structures on  $  M $
 +
subordinate to a given Riemannian metric coincides with the order of the group  $  H  ^ {1} ( M, \mathbf Z ) $(
 +
see [[#References|[6]]]).
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086780/s08678044.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086780/s08678045.png" />) are the covariant derivatives in the directions of the system of orthonormal vector fields <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086780/s08678046.png" /> and the dot denotes multiplication of spinor fields by vector fields which correspond to the above <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086780/s08678047.png" />-module structure on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086780/s08678048.png" />.
+
Let  $  C $
 +
be the complexification of the [[Clifford algebra|Clifford algebra]] of  $  \mathbf R  ^ {n} $
 +
with quadratic form  $  q= \sum _ {i=} 1  ^ {n} x _ {i}  ^ {2} $.  
 +
Then  $  C $
 +
has an irreducible representation in a space  $  S $
 +
of dimension  $  2 ^ {[ n/2] } $,
 +
which defines a representation of $  \mathop{\rm Spin} _ {n} \subset  C $
 +
in  $  S $.
 +
Every spinor structure  $  \widetilde \pi  $
 +
on  $  M $
 +
yields an associated vector bundle  $  \pi _ {S} : S( M) \rightarrow M $
 +
with typical fibre  $  S $,
 +
called a spinor bundle. The Riemannian connection on  $  M $
 +
determines in a canonical way a connection on  $  \pi _ {S} $.  
 +
On the space  $  \Gamma ( S) $
 +
of smooth sections of  $  \pi _ {S} $(
 +
spinor fields) there acts a linear differential operator  $  D $
 +
of order  $  1 $,
 +
the Dirac operator, which is locally defined by the formula
  
Spinor fields in the kernel of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086780/s08678049.png" /> are sometimes called harmonic spinor fields. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086780/s08678050.png" /> is compact, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086780/s08678051.png" />, and this dimension does not change under conformal deformation of the metric [[#References|[4]]]. If the Riemannian metric on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086780/s08678052.png" /> has positive scalar curvature, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086780/s08678053.png" /> (see [[#References|[4]]], [[#References|[5]]]).
+
$$
 +
Du  = \sum _ { i= } 1 ^ { n }  s _ {i} \cdot \nabla _ {s _ {i}  }
 +
u ,\  u \in \Gamma ( S) ,
 +
$$
  
A spinor structure on a [[Space-time|space-time]] manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086780/s08678054.png" /> (that is, on a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086780/s08678055.png" />-dimensional Lorentz manifold) is defined as a spinor structure subordinate to the Lorentz metric <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086780/s08678056.png" />. The existence of a spinor structure on a non-compact space-time <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086780/s08678057.png" /> is equivalent to the total parallelizability of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086780/s08678058.png" /> (see [[#References|[3]]]). As a module over the spinor group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086780/s08678059.png" />, the spinor space decomposes into the direct sum of two complex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086780/s08678060.png" />-dimensional complexly-conjugate <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086780/s08678061.png" />-modules <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086780/s08678062.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086780/s08678063.png" />. This corresponds to the decomposition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086780/s08678064.png" /> of the spinor bundle, where the tensor product <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086780/s08678065.png" /> is identified with the complexification of the tangent bundle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086780/s08678066.png" />. Spinor fields in space-time that are eigenfunctions of the Dirac operator characterize free fields of particles with spin <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086780/s08678067.png" />, such as electrons.
+
where  $  \nabla _ {s _ {i}  } $(
 +
$  i= 1 \dots n $)
 +
are the covariant derivatives in the directions of the system of orthonormal vector fields  $  s _ {i} $
 +
and the dot denotes multiplication of spinor fields by vector fields which correspond to the above  $  C $-
 +
module structure on  $  S $.
 +
 
 +
Spinor fields in the kernel of  $  D $
 +
are sometimes called harmonic spinor fields. If  $  M $
 +
is compact, then  $  \mathop{\rm dim}  \mathop{\rm ker}  D < \infty $,
 +
and this dimension does not change under conformal deformation of the metric [[#References|[4]]]. If the Riemannian metric on  $  M $
 +
has positive scalar curvature, then  $  \mathop{\rm ker}  D = 0 $(
 +
see [[#References|[4]]], [[#References|[5]]]).
 +
 
 +
A spinor structure on a [[Space-time|space-time]] manifold $  ( M, g) $(
 +
that is, on a $  4 $-
 +
dimensional Lorentz manifold) is defined as a spinor structure subordinate to the Lorentz metric $  g $.  
 +
The existence of a spinor structure on a non-compact space-time $  M $
 +
is equivalent to the total parallelizability of $  M $(
 +
see [[#References|[3]]]). As a module over the spinor group $  \mathop{\rm Spin} ( 1, 3) \approx  \mathop{\rm SL} ( 2, G) $,  
 +
the spinor space decomposes into the direct sum of two complex $  2 $-
 +
dimensional complexly-conjugate $  \mathop{\rm SL} ( 2, G) $-
 +
modules $  {\mathcal C}  ^ {2} $
 +
and $  {\mathcal C}  dot {}  ^ {2} $.  
 +
This corresponds to the decomposition $  S( M)= {\mathcal C}  ^ {2} ( M) \oplus  {\mathcal C}  dot {}  ^ {2} ( M) $
 +
of the spinor bundle, where the tensor product $  {\mathcal C}  ^ {2} ( M) \oplus  {\mathcal C}  dot {}  ^ {2} ( M) $
 +
is identified with the complexification of the tangent bundle $  TM $.  
 +
Spinor fields in space-time that are eigenfunctions of the Dirac operator characterize free fields of particles with spin $  1/2 $,  
 +
such as electrons.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  G. Casanova,  "L'algèbre vectorielle" , Presses Univ. France  (1976)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  R. Penrose,  "The structure of space-time"  C. deWitt (ed.) , ''Batelle Rencontres 1967 Lectures in Math. Physics'' , Benjamin  (1968)  pp. 121–235 (Chapt. VII)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  R. Geroch,  "Spinor structure of space-times in general relativity"  ''J. Math. Phys.'' , '''9'''  (1968)  pp. 1739–1744</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  N. Hitchin,  "Harmonic spinors"  ''Adv. in Math.'' , '''14'''  (1974)  pp. 1–55</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  A. Lichnerowicz,  "Champs spinoriels et propagateurs en rélativité génerale"  ''Bull. Soc. Math. France'' , '''92'''  (1964)  pp. 11–100</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  J. Milnor,  "Spin structure on manifolds"  ''Enseign. Math.'' , '''9'''  (1963)  pp. 198–203</TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top">  R. Penrose,  "The twistor programme"  ''Reports Math. Phys.'' , '''12'''  (1977)  pp. 65–76</TD></TR><TR><TD valign="top">[8]</TD> <TD valign="top">  R.O., jr. Wells,  "Complex manifolds and mathematical physics"  ''Bull. Amer. Math. Soc.'' , '''1'''  (1979)  pp. 296–336</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  G. Casanova,  "L'algèbre vectorielle" , Presses Univ. France  (1976)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  R. Penrose,  "The structure of space-time"  C. deWitt (ed.) , ''Batelle Rencontres 1967 Lectures in Math. Physics'' , Benjamin  (1968)  pp. 121–235 (Chapt. VII)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  R. Geroch,  "Spinor structure of space-times in general relativity"  ''J. Math. Phys.'' , '''9'''  (1968)  pp. 1739–1744</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  N. Hitchin,  "Harmonic spinors"  ''Adv. in Math.'' , '''14'''  (1974)  pp. 1–55</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  A. Lichnerowicz,  "Champs spinoriels et propagateurs en rélativité génerale"  ''Bull. Soc. Math. France'' , '''92'''  (1964)  pp. 11–100</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  J. Milnor,  "Spin structure on manifolds"  ''Enseign. Math.'' , '''9'''  (1963)  pp. 198–203</TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top">  R. Penrose,  "The twistor programme"  ''Reports Math. Phys.'' , '''12'''  (1977)  pp. 65–76</TD></TR><TR><TD valign="top">[8]</TD> <TD valign="top">  R.O., jr. Wells,  "Complex manifolds and mathematical physics"  ''Bull. Amer. Math. Soc.'' , '''1'''  (1979)  pp. 296–336</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
 
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  H. Baum,  "Spin-Strukturen und Dirac-Operatoren über pseudoriemannschen Mannigfaltigkeiten" , Teubner  (1981)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  C.T.J. Dodson,  "Categories, bundles, and spacetime topology" , Kluwer  (1988)  pp. Chapt. V, §3</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  H. Baum,  "Spin-Strukturen und Dirac-Operatoren über pseudoriemannschen Mannigfaltigkeiten" , Teubner  (1981)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  C.T.J. Dodson,  "Categories, bundles, and spacetime topology" , Kluwer  (1988)  pp. Chapt. V, §3</TD></TR></table>

Revision as of 08:22, 6 June 2020


on an $ n $- dimensional manifold $ M $, fibration of spin-frames

A principal fibre bundle $ \widetilde \pi : \widetilde{P} \rightarrow M $ over $ M $ with structure group $ \mathop{\rm Spin} _ {n} $( see Spinor group), covering some principal fibre bundle $ \pi : P \rightarrow M $ of co-frames with structure group $ \mathop{\rm SO} _ {n} $. The latter condition means that there is given a surjective homomorphism $ \kappa : \widetilde{P} \rightarrow P $ of principal fibre bundles, which is the identity on the base and is compatible with the natural homomorphism $ \rho : \mathop{\rm Spin} _ {n} \rightarrow \mathop{\rm SO} _ {n} $. One says that the spinor structure $ ( \widetilde \pi , \kappa ) $ is subordinate to the Riemannian metric $ g $ on $ M $ defined by $ \pi $. From the point of view of the theory of $ G $- structures, a spinor structure is a generalized $ G $- structure with structure group $ G = \mathop{\rm Spin} _ {n} $ together with a non-faithful representation $ \rho : \mathop{\rm Spin} _ {n} \rightarrow \mathop{\rm SO} _ {n} $( cf. $ G $- structure).

In a similar way one defines spinor structures subordinate to pseudo-Riemannian metrics, and spinor structures on complex manifolds subordinate to complex metrics. Necessary and sufficient conditions for the existence of a spinor structure on $ M $ consist of the orientability of $ M $ and the vanishing of the Stiefel–Whitney class $ W _ {2} ( M) $. When these conditions hold, the number of non-isomorphic spinor structures on $ M $ subordinate to a given Riemannian metric coincides with the order of the group $ H ^ {1} ( M, \mathbf Z ) $( see [6]).

Let $ C $ be the complexification of the Clifford algebra of $ \mathbf R ^ {n} $ with quadratic form $ q= \sum _ {i=} 1 ^ {n} x _ {i} ^ {2} $. Then $ C $ has an irreducible representation in a space $ S $ of dimension $ 2 ^ {[ n/2] } $, which defines a representation of $ \mathop{\rm Spin} _ {n} \subset C $ in $ S $. Every spinor structure $ \widetilde \pi $ on $ M $ yields an associated vector bundle $ \pi _ {S} : S( M) \rightarrow M $ with typical fibre $ S $, called a spinor bundle. The Riemannian connection on $ M $ determines in a canonical way a connection on $ \pi _ {S} $. On the space $ \Gamma ( S) $ of smooth sections of $ \pi _ {S} $( spinor fields) there acts a linear differential operator $ D $ of order $ 1 $, the Dirac operator, which is locally defined by the formula

$$ Du = \sum _ { i= } 1 ^ { n } s _ {i} \cdot \nabla _ {s _ {i} } u ,\ u \in \Gamma ( S) , $$

where $ \nabla _ {s _ {i} } $( $ i= 1 \dots n $) are the covariant derivatives in the directions of the system of orthonormal vector fields $ s _ {i} $ and the dot denotes multiplication of spinor fields by vector fields which correspond to the above $ C $- module structure on $ S $.

Spinor fields in the kernel of $ D $ are sometimes called harmonic spinor fields. If $ M $ is compact, then $ \mathop{\rm dim} \mathop{\rm ker} D < \infty $, and this dimension does not change under conformal deformation of the metric [4]. If the Riemannian metric on $ M $ has positive scalar curvature, then $ \mathop{\rm ker} D = 0 $( see [4], [5]).

A spinor structure on a space-time manifold $ ( M, g) $( that is, on a $ 4 $- dimensional Lorentz manifold) is defined as a spinor structure subordinate to the Lorentz metric $ g $. The existence of a spinor structure on a non-compact space-time $ M $ is equivalent to the total parallelizability of $ M $( see [3]). As a module over the spinor group $ \mathop{\rm Spin} ( 1, 3) \approx \mathop{\rm SL} ( 2, G) $, the spinor space decomposes into the direct sum of two complex $ 2 $- dimensional complexly-conjugate $ \mathop{\rm SL} ( 2, G) $- modules $ {\mathcal C} ^ {2} $ and $ {\mathcal C} dot {} ^ {2} $. This corresponds to the decomposition $ S( M)= {\mathcal C} ^ {2} ( M) \oplus {\mathcal C} dot {} ^ {2} ( M) $ of the spinor bundle, where the tensor product $ {\mathcal C} ^ {2} ( M) \oplus {\mathcal C} dot {} ^ {2} ( M) $ is identified with the complexification of the tangent bundle $ TM $. Spinor fields in space-time that are eigenfunctions of the Dirac operator characterize free fields of particles with spin $ 1/2 $, such as electrons.

References

[1] G. Casanova, "L'algèbre vectorielle" , Presses Univ. France (1976)
[2] R. Penrose, "The structure of space-time" C. deWitt (ed.) , Batelle Rencontres 1967 Lectures in Math. Physics , Benjamin (1968) pp. 121–235 (Chapt. VII)
[3] R. Geroch, "Spinor structure of space-times in general relativity" J. Math. Phys. , 9 (1968) pp. 1739–1744
[4] N. Hitchin, "Harmonic spinors" Adv. in Math. , 14 (1974) pp. 1–55
[5] A. Lichnerowicz, "Champs spinoriels et propagateurs en rélativité génerale" Bull. Soc. Math. France , 92 (1964) pp. 11–100
[6] J. Milnor, "Spin structure on manifolds" Enseign. Math. , 9 (1963) pp. 198–203
[7] R. Penrose, "The twistor programme" Reports Math. Phys. , 12 (1977) pp. 65–76
[8] R.O., jr. Wells, "Complex manifolds and mathematical physics" Bull. Amer. Math. Soc. , 1 (1979) pp. 296–336

Comments

References

[a1] H. Baum, "Spin-Strukturen und Dirac-Operatoren über pseudoriemannschen Mannigfaltigkeiten" , Teubner (1981)
[a2] C.T.J. Dodson, "Categories, bundles, and spacetime topology" , Kluwer (1988) pp. Chapt. V, §3
How to Cite This Entry:
Spinor structure. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Spinor_structure&oldid=33893
This article was adapted from an original article by D.V. Alekseevskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article