Classical and Quantum Mechanics with Lie Brackets and Pseudocanonical Transformations

TL;DR

This paper explores the application of Lie brackets to classical and quantum mechanics, especially for systems with constraints, providing detailed calculations of propagators without operators or renormalization.

Contribution

It demonstrates the use of Lie brackets in treating constrained dynamical systems and computes explicit propagators in quantum mechanics using $c$-number techniques.

Findings

Lie brackets effectively handle constrained systems in classical and quantum mechanics.

Exact propagators are derived without operator formalism or renormalization.

The approach simplifies calculations in gauge theories with external magnetic fields.

Abstract

We emphasize the usefulness of the Lie brackets in the context of classical and quantum mechanics. By way of examples we show that many dynamical systems, especially the ones with (gauge) constraints, can equally be treated in their time development with non-canonical variables and Hamiltonians. After a short presentation of the Lie bracket algebra and treating some easier standard problems with the Lie bracket techniques, we concentrate mainly on charged particles with gauge constraint in a constant external magnetic field. Since most of our quantum field theories are meanwhile considered effective, we have purposely treated our final problems with $c$-number instead of field -operator Lagrangians. The van Vleck determinant, which is exact for our problems, is employed to calculate the $c$-number Feynman-Schwinger propagation function. There is no need for operators or renormalization. In particular, the non-relativistic propagator in $2+1$ dimensions and the more complicated one in $3+1$ dimensions are presented in all their glorious detail. On the more editorial side: we have dispensed with numerating the various problems. They are not so much disjoint that they needed an extra title. Also, the article is written in a self-consistent way, meaning one should be able to read it without time-consuming research in textbooks and journals - with a few exceptions, in particular Schwinger's paper [J. Schwinger, Phys. Rev. 82, 664 (1951)], which is the most-cited paper in modern quantum-field-theory physics. Most of the prerequisites for reading the present article can be found in extenso in [W. Dittrich and M. Reuter, Classical and quantum dynamics (Springer, Berlin, Germany, 2016)].