Ivancevic_Applied-Diff-Geom

946 Applied Differential Geometry: A Modern IntroductionMoreover, we can restrict ourselves to connections of the following type.Every principal connection B on P induces the connection Γ B (5.431) onC such thatΓ B ◦ B = S ◦ j 1 B,Γ Bmµλ = 1 2 [cm nlk n αk l µ + ∂ µ B m α + ∂ α B m µ − c m nl(k n µB l α + k n αB l µ)] − Γ β µλ (Bm β − k m β ).For all these connections, the following Hamiltonian formsH B = p µλm dk m µ ∧ ω α − p µλm Γ Bmµλ ω − ˜H Y M ω, (5.432)˜H Y M = ε24 amn Gg µν g λβ p [µλ]mp [νβ]n |g| −1/2 ,are associated with the Lagrangian density L Y M and constitute a completefamily. The corresponding Hamiltonian equations for sections r of Π −→ Xread∂ α p µλm = −c n lmkνp l n[µν] + c n mlBνp l (µν)n − Γ µ λν p(λν) m , (5.433)∂ α kµ m + ∂ µ kα m m= 2Γ B(µλ) , (5.434)plus the equation (5.426). The equations (5.426) and (5.433) restricted tothe constraint space (5.425) are the familiar Yang–Mills equations. DifferentHamiltonian forms (5.432) lead to the different equations (5.434). Theequation (5.434) is independent of canonical momenta and plays the role ofthe gauge–type condition. Its solution is k(x) = B.Let A be a solution of the Yang–Mills equations. There exists the Hamiltonianform H B=A (5.432) such that r A = ̂L Y M ◦ A is a solution of thecorresponding Hamiltonian equations (5.433), (5.434) and (5.426) on theconstraint space (5.425).On the solution r A , the curvature of the connection Γ A is reduced toR m λαµ = 1 2 (∂ αF m αµ − c m qnk q αF n αµ − Γ β αλ F m βµ − Γ β µλ F m αβ) =12 [(∂ αF m λµ − c m qnk q αF n λµ − Γ β λα F m µβ) − (∂ µ F m λα − c m qnk q µF n λα − Γ β λµ F m αβ)]where F = F ◦ A is the strength of A. If we setthen we haveS α µ = p [αλ]m ∂αµ m ˜H Y M = ε22 √ |g| amn Gg µν g αβ p [αλ]S α µ = 1 2 p[αλ] F m µα, ˜HY M = 1 2 Sα α.mp [βν]n ,