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Tuesday, 18 March 2025

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(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.

Quantification of Quantum / Tokyo May 29, 2004 / For the Memory of Hakuba August 23, 2003

 

 

Tokyo January 29, 2008

Sekinan Research Field of Language

www.sekinan.org

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