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 E+ are called odd.
2
Super algebra Ai ·Aj = Ai+j i, 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 ) = 
Tk ( V )
Tk ( 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 ei 
ej = - ej ei. ( e1, …en basis of V )
ej = - ej ei. ( e1, …en basis of V )
Exterior algebra of V 
V
V
9
Family of V’s inner product 
Q
Q
Family of Clifford algebra C ( V, Q )
Q )
10
When is seemed to be Planck constant, C ( V ) is represented as quantization of V.
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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