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# The real covariant dirac operator matter 20 in this

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(5.47) 44

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5.5 The ”Real” covariant Dirac operator Matter 20 in this ”real spectral triple” approach is represented by states of the covariant Hilbert space H Cov = M ⊗ A H ⊗ A M * . Fot the S 0 -real spectral triple H = H (+) ⊕ H ( - ) so that also H Cov splits in a sum of ”particle” and ”antiparticle” Hilbert spaces H Cov = H (+ p ) ⊕ H ( - p ) where each H ( ± p ) = M ⊗ A H ( ± ) A M * has typical elements Ψ ( ± p . The projective module M = P A 2 and its dual M * were examined in section 3 and in appendix A . In bases { E i } and { E j } of the free modules A 2 and A 2 * , we may represent a state of A 2 A H ( ± ) A A 2 * as E i A Ψ ( ± ) i j A E j . It is a state Ψ ( ± p of H ( ± p ) if Ψ ( ± ) i j ∈ H ( ± ) obeys Ψ ( ± ) i j = π ( ± ) ( P i k ) π o ( ± ) ( P j ) Ψ ( ± ) k . (5.48) Since P i k,a ( x ) = δ i k and P i k,b ( x ) = | ν ( x ) i ν ( x ) | k (cf. section 3 ) the vector Ψ ( ± ) i j ∈ H ( ± ) is represented by the column vector Ψ ( ± ) i j = (( ψ ( ± ) aa )) i j | ψ ( ± ) ab i ν | j | ν i ψ ( ± ) ba | j | ν i ψ ( ± ) bb ν | j , (5.49) where (( ψ ( ± ) aa )) is a quadruplet and ψ ( ± ) bb = ν | (( ψ ( ± ) bb )) | ν a singlet of Pensov spinor fields of spin weight ± s , while | ψ ( ± ) ab = (( ψ ( ± ) ab )) | ν , re- spectively ψ ( ± ) ba | = ν | (( ψ ( ± ) ba )), are doublets of Pensov spinors of weight ± s - n/ 2, respectively ± + n/ 2. The covariant real structure J Cov , as defined in appendix B , acts on Ψ Cov as ( J Cov Ψ Cov ) ( ± ) i j = χ δ i ¯ C 1 K (( ψ ( ) aa )) k δ ¯ kj δ i ¯ C 1 K ψ ( ) ba | ν | j | ν i C 1 K| ψ ( ) ab k δ ¯ kj -| ν i C 1 K ψ ( ) bb ν | j . (5.50) 20 In this section we consider case 2.a) only. At the end the final result for the covariant Dirac operator in case 2.b) will also be given 45
The covariant Dirac operator, defined in (B.10), is also block diagonal : D = D (+ p ) ⊕ D ( - p ) and is given by : D ( ± p ) ( E i P i k ) A ψ ( ± ) k A ( P j E j ) = E i A π ( ± ) (( A )) i k ψ ( ± ) k j A E j + E i A π ( ± ) ( P i k ) π o ( ± ) ( P j ) D ( ± ) ψ ( ± ) k A E j + E i A π o ( ± ) (( A )) j ψ ( ± ) i A E j , (5.51) where (( A )) is the 2 × 2 matrix of universal one-forms given in (3.6). It is represented by the matrix valued differential one- and zero-forms given in terms of (( α a )), α b , | Φ ab and Φ ba | , defined in (4.22) by : (( A a )) = - i γ r (( α a,r )) , (( A b )) = - i γ r α b,r | ν ν | , (( A ab )) = | Φ ab ν | γ 3 , (( A ba )) = | ν Φ ba | γ 3 . (5.52) We obtain : π ( ± ) ((( A ))) = (( A a )) 1 N 0 0 0 0 (( A a )) 1 N 0 (( A ab )) A + 0 0 (( A b )) 1 N 0 0 (( A ba )) A 0 (( A b )) 1 N . (5.53) The π o ( ± ) representative of (( A )) is computed as : π o ( ± ) ((( A ))) = (( A a )) 1 N 0 0 0 0 (( A b )) 1 N 0 0 0 0 (( A a )) 1 N - (( A ba )) B + 0 0 - (( A ab )) B (( A b )) 1 N . (5.54) 46

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Substituting (5.53) and (5.54) in (5.51), we obtain : (( D Ψ Cov ( ± ) aa )) = D 1( ± ) (( ψ ( ± ) aa )) - i γ r (( α a,r )) , (( ψ ( ± ) aa )) , |D Ψ Cov ( ± ) ab = D ( - n/ 2) 1( ± ) | ψ ( ± ) ab + | η ab γ 3 A + ψ ( ± ) bb - i γ r (( α a,r )) | ψ ( ± ) ab - | ψ ( ± ) ab α b,r ; D Ψ Cov ( ± ) ba | = D (+ n/ 2) 1( ± ) ψ ( ± ) ba | + η ba | γ 3 B + ψ ( ± ) bb - i γ r α b,r ψ ( ± ) ba | - ψ ( ± ) ba | (( α a,r )) , D Ψ Cov ( ± ) bb = D 1( ± ) ψ ( ± ) bb + γ 3 A η ba | ψ ( ± ) ab + γ 3 B ψ ( ± ) ba | η ab .
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