Quantum and Classical Mechanics with Connected Graphs.
The dynamics of a nonrelativistic spinless N-particle system with time-dependent, smooth, scalar interactions is investigated. If the system also couples to an external vector field, the generalized Wigner-Kirkwood W-K expansion of the quantum propagator <x(VBAR)U(t,s)(VBAR)y> is obtained from a large-mass expansion of the higher-order 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 time-independent Hamiltonians H the W-K 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 W-K and WKB methods, three complete integrals of the classical Hamilton-Jacobi 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 two-point boundary conditions, for sufficiently short time displacements t-s.
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- Physics: General