The real covariant dirac operator matter 20 in this

Info icon This preview shows pages 44–48. Sign up to view the full content.

(5.47) 44
Image of page 44

Info icon This preview has intentionally blurred sections. Sign up to view the full version.

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
Image of page 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
Image of page 46

Info icon This preview has intentionally blurred sections. Sign up to view the full version.

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 .
Image of page 47
Image of page 48
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

What students are saying

  • Left Quote Icon

    As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

    Student Picture

    Kiran Temple University Fox School of Business ‘17, Course Hero Intern

  • Left Quote Icon

    I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

    Student Picture

    Dana University of Pennsylvania ‘17, Course Hero Intern

  • Left Quote Icon

    The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

    Student Picture

    Jill Tulane University ‘16, Course Hero Intern