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 ( V )

Ideal of T ( V )     I_Q

Clifford algebra     C ( V, Q ) = T ( V ) / I_Q

Clifford algebra of mod.2     C ( V ) = C ⁺ ( V ) ⊕ C ^- ( V )

Oriented orthonormal basis of V   (e_i )ⁿ_i=1

Charity operator    C(V) ⊗_RC ∋ : = i^n/2e₁…e_n

Exterior product of V     ΛV

Clifford module   ΛV ⊗_RC

Oriented n-dimensional Riemann manifold     M

Tangent vector bundle of M     TM

Bundle of exterior differential of TM     ΛT *M

Complexificated exterior product bundle of M       ΛT *M ⊗_RC

Hodge star operator     * : Λ^kT *M ⊗_RC →Λ^n-kT *M ⊗_RC

2

Cross section space    Γ( M, ΛT *M ) = Ω( M )

Exterior product of Ω( M )     Ωⁱ( M ) =Γ( M, ΛⁱT *M )

Differential form space     Ω( M ) = _I Ωⁱ( M )

Exterior differential     d : Ω^●( M ) →Ω^●+1( M )

Adjoint operator of exterior differential d     d* : Ω^●( M ) →Ω^●-1( M )

de Rham complex    ( Ω( M ), d )

de Rham Cohomology group of M     Hⁱ ( M ) = Hⁱ ( Ω( M ), d ) = ker (d : Ωⁱ( M ) →Ωⁱ⁺¹( M ) ) / Im ( d : Ω^i-1( M ) →Ωⁱ( M ) )

Vector space Hⁱ ( M )

dα= 0  closed form of α

α = dβ exact form of α

Family of forms     [α]

Product of vector space is algebra.     [α₁]∧[α₂] = [α₁∧α₂]

*    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 ΛT *M ⊗_RC

 

[Note]

Exterior differential d 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