Clifford Algebra Note 7 Creation Operator and Annihilation Operator

 Clifford Algebra

 

Note 7

Creation Operator and Annihilation Operator

 

TANAKA Akio

 

1

Manifold     M

Tangent vector bundle of M     TM

Vector field over M = Cross section of TM     X ∈ Γ(M, TM )

Differential map      : M₁ → M_2     * : TM₁ → TM_2     * (V) ∈ T_(x)M₂   V ∈ T_xM₁

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

 

 

Tokyo January 29, 2008

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