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 ) IQ
Clifford algebra C ( V, Q ) = T ( V ) / IQ
Clifford algebra of mod.2 C ( V ) = C + ( V ) ⊕ C - ( V )
Oriented orthonormal basis of V (ei )ni=1
Charity operator C(V) ⊗RC ∋ : = in/2e1…en
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 * : ΛkT *M ⊗RC →Λn-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 : Ωi-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
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