nLab
semiclassical approximation

Overview

Semiclassical or quasiclassical approximation to quantum mechanics (including infinitely many degrees of freedom, e.g. QFT) in most general sense is the series expansion of the transition amplitudes in Planck constant. However more specifically one considers the Feynman path integral $\int D\varphi F(\varphi )e^{\mathrm{iS}(\varphi )/h}$ and develops it by a stationary phase method about the critical points of the action functional $S$ (the critical points of $S$ correspond to the solutions of the Euler-Lagrange equations, hence to the classical trajectories of the system). Namely $S$ is assumed to be large and change fast in comparison to the Planck constant, so the increments of $i\pi$ in the phase of the exponent appear very often leading to mutual cancellation, except in the regions close to the critical points.

Semiclassical method in more traditional Schroedinger wave equation approach due to Wentzel, Kramers and Brillouin (WKB or WKBJ method, see wikipedia), amounts to the expression of the wave function in the form $\varphi =\exp (S)$ where $S$ is a slowly varying function and solving the equation for $S$. This makes sense for a more general class of wave equations, and in wave optics this short-wave limit is the approximation of geometrical optics where $S$ is called the eikonal. Globally consistent solutions in first order lead to so-called Bohr-Sommerfeld quantization conditions. Multidimensional generalization of WKB method appear to be rather nontrivial; it has been pioneered by V. Maslov who introduced a topological invariant to remove ambiguities of the naive version of the method (Maslov index).

Semiclassical approximation and equivariant localization

In some special cases (most often in the presence of supersymmetry) the main contribution (the first term in expansion) amounts to the true result; the quantum correction sometimes leads however to an overall scalar factor. This is the case of so-called localization (related directly in some cases to the equivariant localization in cohomology and Lefshetz-type fixed point formulas). Most of well known examples of integrable systems and TQFTs lead to localization.

Large N-limit

The large N limit of gauge theories, which is of importance in collective field theory and in the study of relation between gauge and string theories is formally very similar to semiclassical expansion, where the role of Planck constant is played by $1/N^2$.

Terminology

In the theory of radiation there is a different meaning of semiclassical treatment: one considers particles in a sorrounding electromagnetic field and the particles are treated as in finite-dimensional quantum mechanics, with the electromagnetic field as an external classical field coupled to the particles via an interaction term.

Literature

• M.V. Fedoryuk, Semi-classical approximation, Springer Online Enc. of Math.

• Sean Bates, Alan Weinstein, Lectures on the geometry of quantization, pdf

• V. P. Maslov, Stationary-phase method for Feynman’s continual integral, Theoret. and Math. Phys., 2:1 (1970), 21–25; Russian original: ТМФ, 2:1 (1970), 30–35 pdf.

• V. P. Maslov, Theory of perturbations and asymptotic methods (Russian), Izdat. Moskov. Gos. Univ. 1965.

• V. I. Arnol’d, Characteristic class entering in quantization conditions, Funct. Anal. its Appl. 1967, 1:1, 1–13, doi (В. И. Арнольд, “О характеристическом классе, входящем в условия квантования”, Функц. анализ и его прил., 1:1 (1967), 1–14, pdf)

• V. Guillemin, S. Sternberg, Geometric asymptotics, AMS 1977, online

• A. S. Mishchenko, B. Yu. Sternin, V. E. Shatalov, Lagrangian manifolds and the canonical operator method, Nauka, Moscow, 1978. (in Russian). English transl.: Lagrangian manifolds and the Maslov operator, Springer, Berlin, 1990.

• Richard J. Szabo, Equivariant cohomology and localization of path integrals, Lecture Notes in Physics, N.S. Monographs 63. Springer 2000. xii+315 pp. (early version: Equivariant localization of path integrals, hep-th/9608068)

• M. F. Atiyah, Circular symmetry and stationary phase approximation, Asterisque 131 (1985) 43–59

• N. Berline, Ezra Getzler, M. Vergne, Heat kernels and Dirac operators, Grundlehren 298, Springer 1992, “Text Edition” 2003.

• Albert Schwarz, Oleg Zaboronsky, Supersymmetry and localization, Comm. Math. Phys. 183, 2 (1997), 463-476, euclid

• Albert Schwarz, Semiclassical approximation in Batalin-Vilkovisky formalism, Comm. Math. Phys. 158 (1993), no. 2, 373–396, euclid.

For large N-limit compared to semiclassical expansion see

• L. G. Yaffe, Large N limits as classical mechanics, Rev. Mod. Phys. 54, 407–435 (1982), pdf

For the semiclassical method in superstring theory see

• J. Maldacena, G. Moore, N. Seiberg, D. Shih, Exact vs. semiclassical target space of the minimal string, hep-th/0408039

• K. Hori, A. Iqbal, C. Vafa, D-Branes and mirror symmetry, hep-th/0005247