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
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.
[Note]
Creation
operator and annihilation operator are corresponded with the next past work.
Quantification
of Quantum / Tokyo May 29, 2004 / For the Memory of Hakuba August 23, 2003