SCIENCE AN
MIGETAL.pdf

# Its commutator of with π f is d π f ic d f a 1 a f

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Its commutator of with π ( f ) is : [ D ( f )] = - ic (d f a ) 1 a ( f b - f a ) γ 3 M + ( f a - f b ) γ 3 M - ic (d f b ) 1 b , (4.8) where the de Rham exterior differential is d f a = e k ( f a ) θ k and where c ( σ ( k ) ) denotes the Clifford representation of the k-form σ ( k ) : c ( σ i 1 ...i k θ i 1 ... θ i k ) = σ i 1 ...i k γ i 1 ...γ i k . The representation π of (4.7) extends to a -representation of Ω ( A ) by : π ( f 0 d f 1 · · · d f k ) = π ( f 0 )[ D ( f 1 )] · · · [ D ( f k )] . From (4.7) and (4.8) it follows that the element f d g Ω (1) ( A ) is represented by π ( f d g ) = f a - ic (d g a ) 1 a f a ( g b - g a ) γ 3 M + f b ( g a - g b ) γ 3 M f b - ic (d g b ) 1 b , (4.9) 12 These ”generation” indices are not written down explicitely and the ( s ) subscript, fixed once for all, will also be omitted in this section. 22

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A general element F Ω (1) ( A ), given by F α β ( x,y ), is then represented as an operator on H by : π ( F ) = - ic ( σ (1) a ) 1 a σ (0) ab γ 3 M + σ (0) ba γ 3 M - ic ( σ (1) b ) 1 b , (4.10) where the σ ( k ) ’s are differential k-forms given by : σ (1) a ( x ) = e k,y F aa ( x,y ) | y = x θ k x , σ (1) b ( x ) = e k,y F bb ( x,y ) | y = x θ k x , σ (0) ab ( x ) = F ab ( x,y ) | y = x , σ (0) ba ( x ) = F ba ( x,y ) | y = x . (4.11) The representative of a universal 2-form f d g d h will be given by the product of the matrix (4.9) with - ic (d h a ) 1 a ( h b - h a ) γ 3 M + ( h a - h b ) γ 3 M - ic (d h b ) 1 b . The result is : π ( f d g d h ) = π ( f d g d h ) [ aa ] π ( f d g d h ) [ ab ] π ( f d g d h ) [ ba ] π ( f d g d h ) [ bb ] , where π ( f d g d h ) [ aa ] = f a - ic (d g a ) - ic (d h a ) 1 a + f a ( g b - g a )( h a - h b ) M + M, π ( f d g d h ) [ ab ] = f a - ic (d g a )( h b - h a ) - f a ( g b - g a ) - ic (d h b ) γ 3 M + , π ( f d g d h ) [ ba ] = - f b ( g a - g b ) - ic (d h a ) + f b - ic (d g b )( h a - h b ) γ 3 M, π ( f d g d h ) [ bb ] = f b - ic (d g b ) - ic (d h b ) 1 b + f b ( g a - g b )( h b - h a ) MM + . A generic universal two-form G is represented by π ( G ) = π ( G ) [ aa ] π ( G ) [ ab ] π ( G ) [ ba ] π ( G ) [ bb ] , (4.12) 23
with 13 π ( G ) [ aa ] = - c ( ρ (2+0) aaa ) 1 a + ρ (0) aba M + M, π ( G ) [ ab ] = - ic ( ρ (1) ab ) γ 3 M + , π ( G ) [ ba ] = - ic ( ρ (1) ba ) γ 3 M, π ( G ) [ bb ] = - c ( ρ (2+0) bbb ) 1 b + ρ (0) bab MM + , where the differential forms ρ ( k ) ( x ) are given by : ρ (2+0) aaa ( x ) = ρ (2) aaa + ρ (0) aaa , ρ (2+0) bbb ( x ) = ρ (2) bbb + ρ (0) bbb , ρ (2) aaa ( x ) = 1 2 [ e k,y e ,z - e ,y e k,z ] G aaa ( x,y,z ) | y = x,z = x θ k x θ x , ρ (2) bbb ( x ) = 1 2 [ e k,y e ,z - e ,y e k,z ] G bbb ( x,y,z ) | y = x,z = x θ k x θ x , ρ (0) aaa ( x ) = δ kl e k,y e ,z G aaa ( x,y,z ) | y = x,z = x , ˜ ρ (0) bbb ( x ) = δ kl e k,y e ,z G bbb ( x,y,z ) | y = x,z = x , ρ (0) aba ( x ) = G aba ( x,y,z ) | y = x,z = x , ρ (0) bab ( x ) = G bab ( x,y,z ) | y = x,z = x , ρ (1) ab ( x ) = e k,y G aab ( x,y,z ) - e k,z G abb ( x,y,z ) | y = x,z = x θ k x , ρ (1) ba ( x ) = e k,y G bba ( x,y,z ) - e k,z G baa ( x,y,z ) | y = x,z = x θ k x . (4.13) The representation π of Ω ( A ) is a -representation but it is not a differential representation of Ω ( A ) in the sense that G Ker ( π ) = J 0 does not imply π ( d G ) = 0. To obtain a graded differential algebra of operators in H , it is necessary to take the quotient of Ω ( A ) by the graded differential ideal J = J 0 + d J 0 with canonical projection π D : Ω ( A ) Ω D ( A ) = Ω ( A ) J = k =0 Ω ( k ) D ( A ) . (4.14) 13 Here we have used c ( θ k ) c ( θ ) = c ( θ k θ ) + δ k .

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