Ivancevic_Applied-Diff-Geom

962 Applied Differential Geometry: A Modern Introduction(5.477) relative to the connection on the bundle S h , given byA h = dx α ⊗ [∂ α + (ÃB α + A Ba µ∂ α h µ a)∂ B ].In order to construct the differential ˜D (5.479) on J 1 (X, S) in explicitform, let us consider the principal connection on the bundle LX Σ which isgiven by the local connection formà = (Ãab µdx µ + A abc µdσ µ c ) ⊗ I ab , (5.481)à ab µ = 1 2 Kν λµσ α c (η ca σ b ν − η cb σ a ν),A abc µ = 1 2 (ηca σ b µ − η cb σ a µ), (5.482)where K is a general linear connection on T X and (5.482) correspondsto the canonical left–invariant connection on the bundle GL + (4, R) −→GL + (4, R)/L.Therefore, the differential ˜D relative to the connection (5.481) reads˜D = dx α ⊗ [∂ α − 1 2 Aabc µ(σ µ cλ + Kµ νλσ ν c )I abAB ψ B ∂ A ]. (5.483)Given a section h, the connection à (5.481) is reduced to the Lorentzconnection ˜K (5.471) on L h X, and the differential (5.483) leads to thecovariant derivatives of fermion fields (5.470). We will use the differential(5.483) in order to construct a Lagrangian density of Dirac fermion fields.Their Lagrangian density is defined on the configuration space J 1 (X, S ⊕S + ) coordinated by (x µ , σ µ a, ψ A , ψ + A , σµ aλ , ψA α , ψ + Aλ). It readsL ψ = { i 2 [ψ+ A (γ0 γ α ) A B(ψ B α − 1 2 Aabc µ(σ µ cλ + Kµ νλσ ν c )I abBC ψ C )− (ψ + Aλ − 1 2 Aabc µ(σ µ cλ + Kµ νλσ ν c )ψ + C I+ abCA )(γ 0 γ α ) A Bψ B ] (5.484)− mψ + A (γ0 ) A Bψ B }σ −1 ω,where γ µ = σ µ aγ a , and σ = det(σ µ a), while ψ + A (γ0 ) A Bψ B is the Lorentz–invariant fibre metric in the bundle S ⊕ S ∗ [Crawford (1991)].One can show that∂L ψ∂K µ νλ+ ∂L ψ∂K µ λν= 0.Hence, the Lagrangian density (5.484) depends on the torsion of the generallinear connection K only. In particular, it follows that, if K is the