Discrete Dynamics in Nature and Society
Volume 2004 (2004), Issue 1, Pages 205-212

Symmetries, variational principles, and quantum dynamics

J. Manjavidze1,2 and A. Sissakian1

1Joint Institute for Nuclear Research (JINR), Moscow Region, Dubna 141980, Russia
2Institute of Physics, Tbilisi 380077, Georgia

Received 15 October 2003

Copyright © 2004 J. Manjavidze and A. Sissakian. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


We describe the role of symmetries in formation of quantum dynamics. A quantum version of d'Alembert's principle is proposed to take into account the symmetry constrains more exact. It is argued that the time reversibility of quantum process, as the quantum analogy of d'Alembert's principle, makes the measure of the corresponding path integral δ-like. The argument of this δ-function is the sum of all classical forces of the problem under consideration plus the random force of quantum excitations. Such measure establishes the one-to-one correspondence with classical mechanics and, for this reason, allows a free choice of the useful dynamical variables. The analysis shows that choosing the action-angle variables, one may get to the free-from-divergences quantum field theory. Moreover, one can try to get an independence from necessity to extract the degrees of freedom constrained by the symmetry. These properties of new quantization scheme are vitally essential for such theories as the non-Abelian Yang-Mills gauge theory and quantum gravity.