Ivancevic_Applied-Diff-Geom

902 Applied Differential Geometry: A Modern Introductiontonian field theory on a constraint manifold N as a Lagrangian system withthe pull–back Lagrangian L N (5.277).Since a constraint manifold is assumed to be a closed imbedded submanifoldof Π, there exists its open neighborhood U which is a fibre bundle U−→ N. If Π is a fibre bundle π N : Π −→ N over N, it is often convenient toquantize a Lagrangian system on Π with the pull–back Lagrangian π ∗ N L N ,but integration of the corresponding generating functional along the fibresof Π −→ N must be finite.In order to verify this quantization scheme, let us associate to a Lagrangianfield system on Y a covariant Hamiltonian system on Π, then letus quantize this Hamiltonian system and compare this quantization withthat of an original Lagrangian system.5.10.2 Associated Lagrangian and Hamiltonian SystemsIn order to relate classical Lagrangian and covariant Hamiltonian field theories,let us recall that, besides the Euler–Lagrangian equations, a LagrangianL (5.269) also induces the Cartan equations which are given bythe subset(y j µ − yµ)∂ j i α ∂ µ j L = 0, (5.278)∂ i L − d α ∂ α i L + (y j µ − y j µ)∂ i ∂ µ j L = 0, d α = ∂ α + y i α∂ i + y i λµ∂ µ i ,of the repeated jet space J 1 J 1 (X, Y ) coordinated by (x µ , y i , y i λ , yi α, y i λµ ). Asolution of the Cartan equations is a section s of the jet bundle J 1 (X, Y )−→ X whose jet prolongation j 1 s lives in the subset (5.278). Every solutions of the Euler–Lagrangian equations (5.270) defines the solution j 1 s ofthe Cartan equations (5.278). If s is a solution of the Cartan equationsand s = j 1 s, then s is a solution of the Euler–Lagrangian equations. If aLagrangian L is regular, the equations (5.270) and (5.278) are equivalent.Recall that any Lagrangian L (5.269) induces the Legendre map (5.226),i.e.,̂L : J 1 (X, Y ) −→ Π, p α i ◦ ̂L = ∂ α i L, (5.279)over Id Y whose image N L = ̂L(J 1 (X, Y )) is called the Lagrangian constraintspace. A Lagrangian L is said to be hyperregular if the Legendremap (5.279) is a diffeomorphism. A Lagrangian L is called almost–regular if the Lagrangian constraint space is a closed imbedded subbundlei N : N L → Π of the Legendre bundle Π → Y and the surjection