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Tuesday, 18 March 2025

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 Eare called odd.

2

Super algebra   A·Aj Ai+j   ij  Z2

+, - of elements of Zare 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   ab

a 

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

Supper commutator satisfies super Lie algebra’s next axioms.

a]  + ( -1 )|a||b|b] = 0

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

When [ a] =0, aare called super commutative.

3

Super spaces     E = E+  E-   F = F+  F-

Tensor product EF becomes super space by the next.

(EFE+F⊕ E-F-

(EFE+F⊕ E-F+

4

Super algebra   A, B

Super space    B

Tensor product of super algebra     (a1b1·(a2b2) = (-1)|b1||a2|(a1a2b1b2)

5

Hermitian super space is complex vector space in which E+ and Eare both Hermitian metric.

6

n-dimensional real vector space     V

Inner product of V     Q

Tensor algebra of V     T ( V ) = Tk )

Ideal of V )     Q

Clifford algebra    C ( VQ ) = T ( V ) / I Q

C ( VQ ) satisfies relation  vw wv = -2Q (v) ( 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 ei  ej = - ej  ei.  ( e1, …en    basis of )

Exterior algebra of V     V

9

Family of V’s inner product     Q

Family of Clifford algebra     C ( VQ )

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

www.sekinan.org

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