Clifford Algebra Note 1 From Super Space to Quantization

 

Clifford Algebra

Note 1

From Super Space to Quantization

TANAKA Akio

1

Super space   Vector space that has Z₂ grading  E = E⁺ ⊕ E^-

Elements of E⁺ are called even.

Elements of E⁺ are called odd.

2

Super algebra   Aⁱ ·A^j = A^i+ji, j ∈ Z₂

+, - of elements of Z₂ are expressed by 0, 1.

3

Element of super algebra A = A⁺ ⊕ A^-      a

|a| = 0  a ∈ A⁺

|a| = 1  a ∈ A^-

Two elements of super algebra   a, b

a, b ∈ 

Super commutator    [ a, b ] = ab – ( -1 )^|a||b|ba

Supper commutator satisfies super Lie algebra’s next axioms.

[ a, b ]  + ( -1 )^|a||b|[ b, a ] = 0

[ a, [ b,c] ] = [ [a, b ], c ] + ( -1 )^|a||b|[ b, [ a, c] ]

When [ a, b ] =0, a, b are called super commutative.

3

Super spaces     E = E⁺ ⊕ E^-   F = F⁺ ⊕ F^-

Tensor product E⊗F becomes super space by the next.

(E⊗F) ⁺ = E⁺⊗F⁺ ⊕ E^-⊗F^-

(E⊗F) ^- = E⁺⊗F^- ⊕ E^-⊗F⁺

4

Super algebra   A, B

Super space   A ⊗ B

Tensor product of super algebra     (a₁⊗b₁) ·(a₂⊗b₂) = (-1)^|b1||a2|(a₁a₂⊗b₁b₂)

5

Hermitian super space is complex vector space in which E⁺ and E^- are both Hermitian metric.

6

n-dimensional real vector space     V

Inner product of V     Q

Tensor algebra of V     T ( V ) = T^k ( V )

Ideal of T ( V )     I _Q

Clifford algebra    C ( V, Q ) = T ( V ) / I _Q

C ( V, Q ) satisfies relation  vw + wv = -2Q (v, w ) ( v, w ∈ V )

7

Super module     E = E⁺ ⊕ E^-

Clifford module    C⁺(V)·⊂

                 C^-(V)·⊂

8

Exterior product space’s n-dimensional vector space V over field has direct sum and product e_i  e_j = - e_j  e_i.  ( e₁, …e_n    basis of V )

Exterior algebra of V     V

9

Family of V’s inner product     Q

Family of Clifford algebra     C ( V, Q )

10

When  is seemed to be Planck constant, C ( V ) is represented as quantization of V.

[Reference]

Quantization of language and property of quantum are considerable from Clifford algebra’s quantization.

Refer to the next.

Property of Language / Tokyo May 21, 2004

Tokyo January 10, 2008

Sekinan Research Field of Language