Friday, 17 November 2017

Clifford Algebra Note 7 Creation Operator and Annihilation Operator



Note 7
Creation Operator and Annihilation Operator


1
Manifold     M
Tangent vector bundle of M     TM
Vector field over M = Cross section of TM     X  Γ(MTM )
Differential map      M1 → M2     TM TM2     * (V T(x)M2   V  TxM1
Frame bundle of TM     GL (TM)
dim n     GL (n)
Representation space of arbitrary representation ρ in GL (n)     E
Tensor bundle of M = Associated bundle      ε GL (TM) ×ρ E
Exterior algebra    Λ(Rn)*
Exterior differential bundle     ΛT*M GL (TM) ×ρ Λ(Rn)*
2
Space of cross section    Γ(M, ΛT*M )
Space of differential form    Ω(M)
Ωi(M) = Γ(M, ΛiT*M )
Exterior differential     d : Ω(MΩ+1 (M)
3
Vector space     V
 V
exterior product     v : Λ→ ΛV
Vector field     X
Exterior operator     v ( X ) : Ω(MΩ+1 (M)
4
Vector space     V
Dual vector space of V     V*
α ∈ V*
Construction     ι(α) : Λ→ ΛV
Vector field     X
Construction operator    ι(X) : Ω(MΩ-1 (M)
5
Complex vector space     V R C
Complex subspace of V R C     P
V R C  P  
Inner product     Q
w  P
ww ) = 0
P is Polarization of V R .
6
Real vector space    V
Linear automorphism of V     J
J= -1
J     Complex structure of V
7
P’s exterior algebra    ΛP
Spinor space    S =ΛP
Spinor module ( Complex Clifford module )     S = S+  S-
Complex Clifford module     R C
R C = End ( S ) = S+  S-
8
From upper 2, 3 and 5, elements of P, called creation operator, create a particle and elements of , called annihilation operator, annihilate a particle.

[Note]
Creation operator and annihilation operator are corresponded with the next past work.


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