Ivancevic_Applied-Diff-Geom

956 Applied Differential Geometry: A Modern Introductionwhere Σ → X is the bundle of gravitational fields h where values of hplay the role of parameter coordinates, besides the familiar world coordinates[Sardanashvily (1992)]. In particular, every spinor bundle S h → Xis isomorphic to the restriction of S → Σ to h(X) ⊂ Σ. Performing thisrestriction, we come to the familiar case of a field model in the presenceof a gravitational field h(x). The feature of the dynamics of field systemson the composite bundle (5.469) lies in the fact that we have the modifiedcovariant differential of fermion fields which depend on derivatives ofgravitational fields h.As a consequence, we get the following covariant derivative of Diracfermion fields in the presence of a gravitational field h(x):˜D α = ∂ α − 1 2 Aabc µ(∂ α h µ c + K µ νλh ν c )I ab , (5.470)A abc µ = 1 2 (ηca h b µ − η cb h a µ),where K is a general linear connection on a world manifold X, 5 η is theMinkowski metric, and I ab = 1 4 [γ a, γ b ] are generators of the spinor Liegroup L s = SL(2, C).The covariant derivative (5.470) has been considered by [Aringazinand Mikhailov (1991); Ponomarev and Obukhov (1982); Tucker and Wang(1995)]. The relation (5.472) correspond to the canonical decomposition ofthe Lie algebra of the general linear group. By the well–known Theorem[Kobayashi and Nomizu (1963/9)], every general linear connection beingprojected onto the Lie algebra of the Lorentz group induces a Lorentz connection.In our opinion, the advantage of the covariant derivative (5.470), consistsin the fact that, being derived in the framework of the gauge gravitationtheory, it may be also applied to the affine–metric gravitation theory and5 The connectioneK ab α = A abc µ (∂αhµ c + Kµ νλh ν c ) (5.471)is not the connectionK k mλ = h k µ(∂ αh µ m + K µ νλh ν m) = K ab α(η am δ k b − η bm δk a )written with respect to the reference frame h a = h a α dxα , but there is the relationeK ab α = 1 2 (Kab α − K ba α). (5.472)If K is a Lorentz connection A h , then the connection e K given by (5.471) is consistentwith K itself.