Quantum and Classical Mechanics with Connected Graphs.
Abstract
The dynamics of a nonrelativistic spinless Nparticle system with timedependent, smooth, scalar interactions is investigated. If the system also couples to an external vector field, the generalized WignerKirkwood WK expansion of the quantum propagator <x(VBAR)U(t,s)(VBAR)y> is obtained from a largemass expansion of the higherorder WKB approximation. This procedure illuminates the structural interconnection between these two semiclassical approximations. In the case of zero vector potential, a complete formal exponential representation of the propagator is obtained in terms of connected simple graphs. For timeindependent Hamiltonians H the WK expansion of the heat kernel <x(VBAR)e('(beta)H)(VBAR)y> is recovered. The practical efficiency of this expansion is illustrated by computing quantum corrections to some correlation functions of statistical mechanics. By comparing the WK and WKB methods, three complete integrals of the classical HamiltonJacobi equation for H = p('2)/2m + (nu)(x,t) are constructed. These integrals include Hamilton's principal function S, and they are analytic functions of inverse mass in a punctured disc about the origin. The coefficients of their Laurent expansions may be expressed using gradient structures associated with tree graphs. Jacobi's theorem is used to induce unique classical paths satisfying twopoint boundary conditions, for sufficiently short time displacements ts.
 Publication:

Ph.D. Thesis
 Pub Date:
 1986
 Bibcode:
 1986PhDT........76M
 Keywords:

 Physics: General