04/05/2016 20:55
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(M, TM )
Differential map 
: M1 → M2 * : TM1 → TM2 * (V) ∈ T(x)M2 V ∈ TxM1
: M1 → M2 * : TM1 → TM2 * (V) ∈ T(x)M2 V ∈ TxM1
Frame bundle of TM GL (TM)
dim M = 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●(M) →Ω●+1 (M)
3
Vector space V
v ∈ V
exterior product v∧ : ΛV → ΛV
Vector field X
Exterior operator v ( X ) : Ω●(M) →Ω●+1 (M)
4
Vector space V
Dual vector space of V V*
α ∈ V*
Construction ι(α) : ΛV → ΛV
Vector field X
Construction operator ι(X) : Ω●(M) →Ω●-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
Q ( w, w ) = 0
P is Polarization of V ⊗R C .
6
Real vector space V
Linear automorphism of V J
J2 = -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 C ( V ) ⊗R C
C ( V ) ⊗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.
, called annihilation operator, annihilate a particle.
[Note]
Creation operator and annihilation operator are corresponded with the next past work.
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