SOLUTIONS OF THE LIOUVILLE EQUATION BY THE TECHNIQUE OF STATIONARY PHASE IN INFINITELY MANY DIMENSIONS
Abstract
The Feynman path integral has been presented in past years as a formulation of the propagator for the Schrodinger equation in quantum mechanics in terms of an heuristic infinite-dimensional integral and the path along which classical motion occurs has been exhibited in terms of the Feynman path integral by letting Planck's constant tend zero. These results have been made mathematically rigorous recently and investigated by S. A. Albevario and R. J. Hoegh-Krohn {Oscillatory integrals and the method of stationary phase in infinitely many dimensions, with applications to the classical limit of quantum mechanics I, Inventiones mathematicae 40, 59-106 (1977)}. From their results it is possible to develop a mathematically rigorous scheme within the space of tempered distributions for passing from a solution of a modified form of the Schrodinger equation to a solution of the Liouville equation by letting the Planck's contant term tend to zero. This is done by first proving a mathematically rigorous form of the classical limit in physics and then by constructing maps into the Hilbert space of paths that make it possible to actually compute the function space integrals necessary to invoke the stationary phase calculation. The solutions of the Liouville equation derived in this way are exhibited in closed form as tempered distributions expressed in terms of a Fredholm determinant involving the potential energy functional and the initial physical distribution of the particles of the system. These functionals are evaluated along the classical path of motion and the physical distribution of the particles is then seen to propogate along the classical path. A simple demonstration of the scheme is given by considering the one-dimensional problem for a gas whose potential function is V(x) = cos x.
Degree
Ph.D.
Subject Area
Mathematics
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