# With the chiral representation γ 1 1 1 and γ 2 i i

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With the chiral representation γ 1 = 0 1 1 0 and γ 2 = 0 i - i 0 , we obtain : J 1( ± ) ψ ( ± ) = a ( ± s ) C 1 K ψ ( ± ) , (5.2) where C 1 = 0 1 α 0 and a ( ± s ) is a complex number. The adjoint of J 1( ± ) is given by : J 1( ± ) + ψ ( ) = a ( ± s ) C t 1 K ψ ( ) . (5.3) Demanding that J 1( ± ) J 1( ± ) + = 1 ( ) , J 1( ± ) + J 1( ± ) = 1 ( ± ) , (5.4) restricts a ( ± s ) be a phase factor. On H 1 = H 1(+) ⊕ H 1( - ) , the antilinear isometry defined by J 1 = 0 J 1( - ) J 1(+) 0 (5.5) obeys J 1 γ k 0 0 γ k = α γ k 0 0 γ k J 1 . 34

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If we require J 1 2 = 1 1 1 , with 1 = ± 1 , (5.6) the phases a ( ± s ) are related by a ( - s ) = α 1 a (+ s ) and J 1( - ) = 1 J 1(+) + . The antilinear mappings J 1( ± ) intertwine with the Dirac operators D ( ± s ) as : J 1( ± ) D ( ± s ) = - α D ( s ) J 1( ± ) . (5.7) On H 1(+) , the Dirac operator is chosen as D 1(+) = D ( s ) , but on H 1( - ) we may choose D ( - s ) up to a sign. Let 1 be another arbitrary sign factor, then the choice 16 D 1( - ) = - α 1 D ( - s ) , (5.8) yields a Dirac operator D 1 = D 1(+) 0 0 D 1( - ) intertwining with J 1 as : J 1 D 1 = 1 D 1 J 1 . (5.9) The representation of A 1 = C ( S 2 ; C ) in H 1 is obtained by taking two copies of the representation in H ( s ) : π 1 ( f ) ψ (+) ψ ( - ) ( x ) = f ( x ) 0 0 f ( x ) ψ (+) ( x ) ψ ( - ) ( x ) . (5.10) In general, a real structure J 1 induces a representation of the opposite algebra A o 1 by : π o 1 ( f ) = J 1 ( π 1 ( f )) + J + 1 , so that the Hilbert space H 1 becomes an A 1 bimodule. Here A 1 is abelian, and with the representation π 1 above, we have π o 1 ( f ) = π 1 ( f ). Since D 1 1 ( f ) = - i κ (+) c ( df ) 0 0 - i κ ( - ) c ( df ) , the first-order condition D 1 1 ( f ) o 1 ( g ) = 0 , (5.11) which is needed to define a connection in bimodules [7], is satisfied. Since J 1( ± ) γ 3 = - γ 3 J 1( ± ) , the chirality in H 1(+) ⊕ H 1( - ) will be taken as χ 1 = γ 3 0 0 γ 3 . (5.12) 16 For notational convenience, we define κ (+) = +1 and κ ( - ) = - α 1 so that D 1( ± ) = κ ( ± ) D ( ± s ) . 35
With this choice : J 1 χ 1 = 1 χ 1 J 1 , with 1 = - 1 . (5.13) The -sign table of Connes [2, 3], corresponding to n = 2, can be satisfied if we choose 1 = - 1 and 1 = +1, but for the moment we shall leave these choices open. The spectral triple T 1 = {A 1 , H 1 , D 1 1 , J 1 } is actually a 0-sphere real spec- tral triple as defined in [2]. For our pragmatic purposes, an S 0 -real spectral triple may be defined as a real spectral triple with an hermitian involution σ 0 commuting with π ( A 1 ) , D 1 1 and anticommuting with J 1 . It is implemented by the decomposition H 1 = H 1(+) ⊕H 1( - ) , which it is given in, by : σ 0 = 1 1(+) 0 0 - 1 1( - ) . (5.14) The doubling of the Hilbert space is justified if we interpret the Pensov spinors of H ( s ) as usual (Euclidean!) Dirac spinors interacting with a mag- netic monopole of strenght s . It seems then natural to consider the (Eu- clidean!) anti-particle fields as Dirac spinors ”seeing” a monopole of strenght - s i.e. as Pensov spinors of H ( - s ) . 5.2 The real discrete spectral triple Proceeding further, as in section 4 , we have to compose the above S 0 -real ”Dirac-Pensov” spectral triple T 1 with a real discrete spectral triple T 2 = {A 2 , H 2 , D 2 2 , J 2 } over the algebra A 2 = C C . The most general finite Hilbert space allowing a A 2 -bimodule structure 17 is given by the direct sum H 2 = α,β C N αβ , (5.15) where α and β vary over { a,b } and where N αβ are integers.

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