Tuesday, 6 January 2015

Clifford Algebra Note 3 Anti-self-dual Form / January 15, 2008


Clifford Algebra

Note 3
Anti-self-dual Form

TANAKA Akio

1
Oriented Euclid vector space     V
Complexification of V     V C
Non-negatively definitive value inner product of V     Q
Tensor algebra of V     T ( )
Ideal of T ( )     IQ
Clifford algebra     C ( VQ ) = T ( ) / IQ
Clifford algebra of mod.2     V ) =  C - )
Oriented orthonormal basis of V   (e)ni=1
Charity operator    C(VR : = in/2e1en
Exterior product of V     ΛV
Clifford module   ΛRC
Oriented n-dimensional Riemann manifold     M
Tangent vector bundle of M     TM
Bundle of exterior differential of TM     Λ*M
Complexificated exterior product bundle of M       Λ*RC
Hodge star operator     * : Λk*RC →Λn-k*RC
2
Cross section space    Γ( M, Λ*) = Ω( )
Exterior product of Ω( )     Ωi) =Γ( M, Λi*)
Differential form space     Ω( ) = Ωi)
Exterior differential     : Ω) →Ω●+1)
Adjoint operator of exterior differential d     d* : Ω) →Ω●-1)
de Rham complex    ( Ω( ), d )
de Rham Cohomology group of M     Hi M ) = Hi ( Ω( ), d ) = ker (: Ωi) →Ωi+1) ) / Im ( : Ωi-1) →Ωi) )
Vector space Hi M )
= 0  closed form of α
α dβ exact form of α
Family of forms     [α]
Product of vector space is algebra.     [α1][α2] = [α1α2]
*    Dirac operator      d + d*
Differential form that satisfies *α = α     Self-dual form    
Differential form that satisfies *α = - α     Anti-self-dual form
3
4-multiple dimensional oriented compact Riemann manifold     M
Signature operator     Operator d + d* over Clifford algebra Λ*RC

[Note]
Exterior differential and adjoint operator of exterior differential d* are corresponded with the concept of <orbit table> that is described in the paper, Quantum Theory for Language.
More details, refer to <24> and <#10> in the next paper.
Quantum Theory for Language Synopsis / Tokyo January 15, 2004
On history of Quantum Theory for Language, refer to the next.
Invitation to Quantum Theory for Language / Data arranged at Tokyo January 24, 2005

Tokyo January 15, 2008
Sekinan Research Field of Language

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