Friday 20 September 2019

Clifford Algebra Note 1 From Super Space to Quantization


Note 1
From Super Space to Quantization


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.

Tokyo January 10, 2008

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