Ivancevic_Applied-Diff-Geom

Applied Jet Geometry 899where a, b and c are functions of world coordinates x µ and field variablesy i . Note that, in the framework of perturbative quantum field theory, anyLagrangian is split into the sum of a quadratic Lagrangian (5.268) and aninteraction term quantized as a perturbation.For example, let the Lagrangian (5.268) be hyperregular, i.e., the matrixfunction a is nondegenerate. Then there exists a unique associated Hamiltoniansystem whose Hamiltonian H is quadratic in momenta p µ i , and sois the Lagrangian L H (5.266). If the matrix function a is positive–definiteon an Euclidean space–time, the generating functional (5.267) is a Gaussianintegral of momenta p µ i (x). Integrating Z with respect to pµ i (x), onerestarts the generating functional of quantum field theory with the originalLagrangian L (5.268). We extend this result to field theories withalmost–regular Lagrangians L (5.268), e.g., Yang–Mills gauge theory. Thekey point is that, though such a Lagrangian L induces constraints and admitsdifferent associated Hamiltonians H, all the Lagrangians L H coincideon the constraint manifold, and we have a unique constrained Hamiltoniansystem which is quasi–equivalent to the original Lagrangian one [Giachettaet. al. (1997)].5.10.1 Covariant Hamiltonian Field SystemsTo develop the covariant Hamiltonian field theory suitable for path–integral quantization, we start by following the geometrical formulationof classical field theory (see [Sardanashvily (1993); Sardanashvily (1995);Giachetta et. al. (1997); Mangiarotti and Sardanashvily (2000a); Sardanashvily(2002a)]), in which classical fields are represented by sections offibre bundles. Let Y → X be a smooth fibre bundle provided with bundlecoordinates (x µ , y i ). Recall from subsection 5.9 above, that the configurationspace of Lagrangian field theory on Y is the 1–jet space J 1 (X, Y ) ofY . It is equipped with the bundle coordinates (x µ , y i , yµ) i compatible withthe composite fibrationJ 1 (X, Y ) π1 0−→ Yπ−→ X.Any section s : X −→ Y of a fibre bundle Y → X is prolonged to thesection j 1 s : X −→ J 1 (X, Y ) of the jet bundle J 1 (X, Y ) → X, such thaty i µ ◦ j 1 s = ∂ µ s i .Also, recall that the first–order Lagrangian is defined as a horizontal