Department of Physics,
Sub-Department of Quantum Statistics and Field Theory,
Moscow State University,
Vorobievy Gory, 119899, Moscow,  RUSSIA
E-mail: shvedov@qs.phys.msu.su

Symmetries in semiclassical mechanics

Semiclassical approximation is widely used in quantum mechanics and field theory. Many different approaches to constructing semiclassical theories (functional integral approach, operator approach etc.) have been developed. However, the most suitable for mathematical justification approach is based on direct substitution of the wave function to the quantum evolution equation.

Historically, the WKB wave function was the first substitution that obeyed the Schrodinger equation in the semiclassical approxiamtion. Now, many different semiclassical substitutions (arised in the Maslov theory of wave packets, in the Maslov theory of Lagrangian manifolds with complex germs etc)  to the Schrodinger equation are known. The simplest one is the wave packet function which is specified by classical variables (numbers: S - phase, Pi - momenta, Qi - coordinates) and quantum function f specifying  a shape of the wave packet.

Generally, a semiclassical theory may be viewed as follows. A semiclassical state is a point on the space of a bundle ("semiclassical bundle"), which can be denoted as (X,f), where X is a classical state (for ordinary quantum mechanics, X=(S,P,Q)) and f is a quantum state in the given classical background. Set of all {X} may be considered as a base of the bundle, while f belongs to a Hilbert space FX, which  may be generally X-dependent.

If the quantum theory model is symmetric under a Lie group, the corresponding semiclassical theory should be also symmetric. This means that the symmetry Lie group acts on the semiclassical bundle; an automorphism of the bundle corresponds to each element of the Lie group and the group property is satisfied.

When one interests whether the quantum theory is symmetric under a Lie group, provided that the corresponding classical theory is symmetric, a first step to solve the problem is to investigate the corresponding semiclassical theory. Qunatum anomalies may be investigated even in the semiclassical level.

The purpose of the talk is to investigate the properties of the semiclassical mechanics symmetric under Lie groups. The following problems are to be discussed:
    (a) infinitesimal properties; correspondence between Lie groups and algebras in the semiclassical mechanics;
    (b) properties of semiclassical gauge theories;
    (c) applications to quantum field theory.


  1. O.Yu. Shvedov. On Maslov canonical operator in abstract spaces. Mat.zametki, 1999, vol.65, N3.
  2. O.Yu. Shvedov. On Maslov complex germ in abstract spaces. Mat. sbornik, 1999, vol.190, N10.
  3. O.Yu. Shvedov. Semiclssical symmetries, math-ph/0109016, Annals Phys. 296:51-89, 2002.
  4. O.Yu. Shvedov. Renormalization of Poincare transformations in Hamiltonian semiclassical field theory, hep-th/0109142, J. Math. Phys. 43: 1809-1843, 2002.
  5. O.Yu. Shvedov. Semiclassical mechanics of constrained systems, hep-th/0111265 (short version is published in Teor. Mat. Fiz, 2003).
  6. O.Yu. Shvedov. Group transformations of semiclassical gauge systems. math-ph/0112062 (short version is published in  Mat. zametki, 2003, v.73, N3).