Methods of weyl representation of the phase space and canonical transformations. II

Abstract

Thus, we have shown that the solutions (24) and (36) of the basic equation (17) for the symbol of the intertwining operator of general linear canonical transformations are convenient for constructing the Green's functions of quadratic systems and studying group representations; this makes it possible to dispense completely with the technique of coherent states. For example, for a parametrically excited oscillator we have (34), a simple expression for the symbol of the Green's function in terms of fundamental solutions of the classical equation. This fact is not fortuitous: For it is only by virtue of the characteristic property (13) of the Weyl ordering that the invariance group of the symbol, realized by unitary transformations, is identical to the dynamical symmetry groupH(2n, R) λSp(2n, R) of the classical quadratic Hamiltonians. Therefore, it is convenient to use the classification methods for diagonalizing classical systems in the symbol itself. In this way, we have been able to show that the unique δ-functional singularities of the symbol (24) determine the sliding mappings (27)–(28) on invariant subspaces, and we have also obtained the symbols (37)–(41) of the spectral projectors of arbitrary quadratic systems. Note that every irreducible subspace and the projectors acting on it are ultimately determined by the type of some Jordan cell of the matrix of the classical Hamiltonian, and therefore, the theory of canonical Jordan-Arnol'd forms [31] for many-parameter quadratic systems finds a natural quantum-mechanical realization in the formalism we have proposed.