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Tuesday, 18 March 2025

Clifford Algebra Note 3 Anti-self-dual Form

 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

www.sekinan.org

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