Tuesday 6 January 2015

Clifford Algebra Note 4 Dirac Operator From 3-dimensional Spinor Group to 4-dimensional Spin Manifold / January 20, 2008


Clifford Algebra

Note 4
Dirac Operator
From 3-dimensional Spinor Group to 4-dimensional Spin Manifold

TANAKA Akio

1 <Clifford algebra>
has inner product.
Orthonormal basis of the inner product space     e1, …, en
Algebra generated from e1, …, ehas next relations.
eiej = -ejei  (i  )
(e)2 = -1   (i = 1, …, n )       (1)
The algebra is called n-dimensional Clifford algebra, expressed by Cln.
Clhas vector space generated from ei1eik  against i1 < …<ik
≤ k ≤ n. When k = 0, ei1eik = 1.
2 <Dirac operator>
Differential operator defined over open set of n
γ1  +  +γn  
γiγj = -γjγi  (i  )
(γ)2 = -1   (i = 1, …, n )
D becomes Dirac operator.
3 <Representation space>
Clis presentation space of Cln , for  Cln’s vector space generated from ei1eik  against i1 < …<ik
4 <Exterior algebra>
At (1), now relation is changed to (e)2 = 0   (i = 1, …, n )      (2)
New elation is called exterior algebra, abbreviated to ex.
Exterior algebra’s product is expressed by wedge product .
Vector space of exterior algebra is generated from
ei1eik .     (3)
≤ k ≤ n. When k = 0, ei1eik = 1.
Now 0 ≤ k ≤ n. When k = 0, ei1eik = 1.
Vector subspace generated from (3) against fixed k is expressed by k.
5 <Differential form>
Basis of is expressed by dx1, …, dxn.
k valued function on ex is expressed by the next,
α = αi1,...,αk dxi1, …, dxin.    (4)
(4 ) is called k-dimensional differential form.
6 <exterior differentiation operator, associated operator>
All of k-dimensional differential forms is expressed by Ωk .
Next operators are given against Ωk .
Exterior differentiation operator     d : ΩΩk+1
Associated operator     d* : ΩΩk-1
7 <Spinor group>
Rotation group of 3-dimensional Euclid space     SO ( 3 )
SO ( 3 ) is homeomorphic with 3-dimensional sphere Sthat is called spinor group.
n-dimensional spinor group is expressed by Spin ( n ).
Spinor group has two 2-dimensional complex expression S±.
Sis called plus 2-dimensional spinor.                                                   
Sis called minus 2-dimensional spinor.
8 <Spinor representation>
By Sand SS is expressed to the next.
S S-
9 <Riemann manifold>
Euclid space     2l
Dirac operator is expressed to the next by generating element e.
D = er      (5)
When Euclid space is lifted to oriented Riemann manifold, the condition of 2-dimensional Stiefel Whitney class is defined .
The condition is the next.
w2(TM ) = 0
TM is tangent bundle.
w is vector bundle ξ’s base space B’s Zcoefficient’s cohomology group’s element.
w(ξ Hi Z)  i = 0, 1, 2, …
10 <Spin Riemann manifold >
2l-dimensional spin Riemann manifold     M
Dirac operator     D
Spinor field that satisfies Ds = 0 is called harmonic spinor.
Space given by harmonic spinor     H
From S = S+ + S-
Decomposition H = H+  H-
dim H- dim Hbecomes topological invariant.
The invariant is called index D.
11 <Seiberg-Witten equation>
Oriented compact4-dimensional spin manifold     M
Complex linear bundle over M     L
( 1 ) connection of L is fixed.
Plus spinor bundle     S+
Section of S L     
Seiberg-Witten equation is defined by the next.
DA = 0,  FA+ = [ ∅ ∅- ]+      (6)
Here
DA = 0   Dirac equation
FA+ = 2+      
0 +2S+  S+
0   0-dimensional differential form
2+  Self-dual 2-dimensional differential form  

Tokyo January 20, 2008
Sekinan Research Field of Language

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