Representation of the scattering amplitude by a functional integral and quasiclassical asymptotic behavior in quantum mechanics

Quasiclassical asymptotics of the functional integral constructed in [1] for matrix elements of the $S$-matrix in momentum representation is investigated. Quadratic form of the second variation of the action on classical trajectory being degenerated, the problem considered is close to those of the theory of gauge fields and also to the “zeroth mode” problem in soliton models. The central result of the paper is the construction of the diagram technique for calculating quantum corrections to quasiclassical scattering amplitude. As well as in the gauge field theory, a certain additional condition is necessary for constructing the diagram technique and the choice of this condition determines the factor to put in correspondence with the line in the diagram. It is shown that the sum of all diagrams with given number of loops is “gauge invariant” i.e. it does not depend on the choice of the additional condition.