โ Scribed by I. Bialynicki-Birula
No coin nor oath required. For personal study only.
The purpose of the present paper is to elucidate the relationship betweeI\ the time dependence of quantum operators in the Heisenberg picture and the time dependence of the corresponding dynamical variables in the underlying classical theory. This problem is studied in the nonrelativistic particle mechanics and in the field theory. It is shown how the operator solutions of the quantum equations of motion are related to the corresponding solutions of the classical equations of motion. An explicit formula is given, which expresses quantum operators in the Heisenberg picture in terms of their classical counterparts. This formula is particularly useful in. the study of the classical limit of the quantum theory. The dependence on /j of the matrix elements of. the coordinate operators and the field operators is explicitly given, which enables one to study the quantum corrections to the classical theory in all orders. Coherent states of the quantum system play an essential role in the formalism. * Supported in part by the U. S. Atomic Energy Commission under Contract No. AT-30-1-3829.
252 * The difference between these rules and Dyson's rules was discussed in the footnote 21 of Ref. [l].
๐ SIMILAR VOLUMES
From the assumption that the collective Hamiltonian be invariant under the orthogonal group O(A -1, iR) it is concluded that classical collective dynamics can be formulated on a symplectic manifold. This manifold is shown to be a coset space of the symplectic group Y/2(6, [R) of dimension 12, 16 or