Ivancevic_Applied-Diff-Geom

Applied Jet Geometry 897This is phrased in terms of the Lie derivatives as follows. Letu G = τ α ∂ α + u A ∂ Abe a general principal vector–field on the product C × E which is projectedonto the vector–field τ = τ α ∂ α on the base X. The corresponding generalprincipal vector–field on the bundle Y readsũ G = ˜τ + u A ∂ A , (5.264)where ˜τ is the canonical lift of τ onto the bundle of geometrical objectsT . A Lagrangian density L is invariant under general isomorphisms of thebundle S iffL j 10 eu GL = 0, (5.265)where the jet lift j 1 0ũ G of the vector–field ũ G takes the coordinate formj 1 0ũ G = j 1 0˜τ − y A α ∂ α u A ∂ α A + u A ∂ A + ̂∂ α u A ∂ α A.There are the topological field theories, besides the gravitation theory,where we can use the condition (5.265) (see subsection 5.11.8 below).5.10 Application: Jets and Hamiltonian Field TheoryRecall that the Hamiltonian counterpart of the classical Lagrangian fieldtheory (see subsection 5.9 above) is the covariant Hamiltonian field theory,in which momenta correspond to derivatives of fields with respect to allworld coordinates. It is well–known that classical Lagrangian and covariantHamiltonian field theories are equivalent in the case of a hyperregularLagrangian, and they are quasi–equivalent if a Lagrangian is almost–regular (see [Sardanashvily (1993); Sardanashvily (1995); Giachetta et. al.(1997); Giachetta et. al. (1999); Mangiarotti and Sardanashvily (2000a);Sardanashvily (2002a)]). Further, in order to quantize covariant Hamiltonianfield theory, one usually attempts to construct and quantize a multisymplecticgeneralization of the Poisson bracket. The path–integral quantizationof covariant Hamiltonian field theory was recently suggested in[Bashkirov and Sardanashvily (2004)].Recall that the symplectic Hamiltonian technique applied to field theoryleads to instantaneous Hamiltonian formalism on an infinite–dimensionalphase–space coordinated by field functions at some instant of time (see [Gotay(1991a)] for the strict mathematical exposition of this formalism). The