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We define torsion T and curvature R of the spin connection as the following 2forms, i i Tjk = ([Dj , Gk ] − [Dk , Gj ]) , Rjk = [Dj , Dk ] . 5. The spin connection is torsion-free. 15). Proof. 18) yields that [Dj , Gk ] = [Dj , gkl Gl ] = (∂j gkl ) Gl − gkl Γljm Gm = Γm jk Gm and thus, using that the Levi-Civita connection is torsion-free, i − Γm Tjk = (Γm kj ) Gm = 0 . 2 jk 26 1. 18), we can rewrite the covariant derivative as a spin derivative, Gl ∇k ul = [Dk , Gl ul ] . 7) by i ul = [Dj , [Dk , Gl ul ]] − [Dk , [Dj , Gl ul ]] Gi Rjkl = [[Dj , Dk ], Gl ul ] = −2i [Rjk , Gl ul ] .
14). In general, however, it does not satisfy ˜ = 0, which does not seem to be what we want. 16) should coincide. This yields a condition for the operator B, which can be characterized as follows. We demand that Xs∨ B Xs∨ = s∨ B Xs∨ = Xs∨ B s∨ . 17) ∨ ∨ ∗ Since the operator s∨ m=0 is odd, we have Xs = s X . 17) yields the condition X B = B X. 17). e. X ∗ Y = Y X = Y ) and that X is composed of chiral projectors (which implies that X ∗ ∂/ = ∂/X). Def. 2. The Dirac operator is called causality compatible with X if X ∗ (i∂/ + B − mY ) = (i∂/ + B − mY ) X .
Another approach is to work with the so-called Hadamard states [H, Wa2]. The disadvantage of this approach is that the states of the quantum field no longer have a particle interpretation. In other words, the notions of “particle” and “anti-particle” depend on the local observer, and therefore also the notion of the Dirac sea loses its universal meaning. 2 by showing that the Dirac sea can indeed be introduced as a global object of space-time, even in the presence of a general interaction. 32 2.