The action principle in statistical mechanics

H. A. Cohen, M. Z. Shaharir

Anindya Biswas

Enrico Santamato

A gauge-invariant average action principle is presented that permits the formulation of a completely classical theory which is proved to be equivalent to Klein-Gordon quantum mechanics. In this approach, here called geometric quantum mechanics, the particle motion as well as the space-time geometry are determined simultaneously from the same average action principle. Quantum effects are proved to be related to space-time affine connections rather than to space-time metric tensor components. In this way geometric quantum mechanics is made compatible with axiomatic approaches to both space-time structure and probability calculus.

Jean-Claude Zambrini

Within the framework of stochastic calculus of variations for time-symmetric semimartingales X(t,ω), we consider two different stochastic versions of Maupertuis’ least action principle, in Lagrangian and Hamiltonian terms. The general results are applied to classical statistical mechanics, where they coincide with those of classical calculus of variations, and to Nelson’s stochastic mechanics, an approach to quantum mechanics where a time-symmetric semimartingale represents the position of a particle and the dynamics is expressed by a stochastic version of Hamilton’s principle of least action. Some historic examples of old quantum theory are discussed.

Marco Baldovin, Giacomo Gradenigo, Angelo Vulpiani +1 more

Oliver Bühler

This book provides a rapid overview of the basic methods and concepts in mechanics for beginning Ph.D. students and advanced undergraduates in applied mathematics or related fields. It is based on a graduate course given in 2006-07 at the Courant Institute of Mathematical Sciences. Among other topics, the book introduces Newton's law, action principles, Hamilton-Jacobi theory, geometric wave theory, analytical and numerical statistical mechanics, discrete and continuous quantum mechanics, and quantum path-integral methods. The focus is on fundamental mathematical methods that provide connectio