Contents

Overview

To some extent, quantum mechanics and quantum field theory are a deformation of classical mechanics and classical field theory, with the deformation parameterized by Planck's constant $\hbar$. The semiclassical approximation or quasiclassical approximation to quantization/quantum mechanics is the restriction of this deformation to just first order (or some finite order) in $\hbar$.

classical mechanicssemiclassical approximationformal deformation quantizationquantum mechanics
order of Planck's constant $\hbar$$\mathcal{O}(\hbar^0)$$\mathcal{O}(\hbar^1)$$\mathcal{O}(\hbar^n)$$\mathcal{O}(\hbar^\infty)$
statesclassical statesemiclassical statequantum state
observablesclassical observablequantum observable

Applied to path integral quantization, the semiclassical approximation is meant to approximate the path integral $\int_{\phi \in \mathbf{Fields}} D\phi\; F(\phi) e^{iS(\phi)/\hbar}$ by an expansion in $\hbar$ about the critical points of the action functional $S$ (hence the solutions of the Euler-Lagrange equations, hence to the classical trajectories of the system). As usual for the path integral in physics, this often requires work to make precise, but at a heuristic level the idea is famous as the rotating phase approximation?: the idea is that in regions of field-space where $S$ varies fast as measured in units of Planck's constant, the complex phases of the integrand $\exp(i S / \hbar )$ tend to cancel each other in the integral so that substantial contributions to the integral come only from the vicininity of critical points of $S$ (classical trajectories).

But semiclassical approximations can be applied to most other formulations of quantum physics, where they often lead to precise and powerful mathematical tools.

Notably in the Schrödinger picture of quantum evolution, solutions to the Schrödinger equation $i \hbar \frac{d}{d t} \psi = \hat H \psi$ (which characterizes quantum states given by wave functions $\psi$ for Hamiltonian dynamics induced by a Hamilton operator $\hat H$) are usefully considered to first (or any finite) order in $\hbar$. This method, known after (some of) its inventors as the WKB method or similar, amounts to expressing the wave function in the form $\psi = exp(S)$ where $S$ is a slowly varying function and solving the equation for $S$. Globally consistent such solutions to first order lead to what are called Bohr-Sommerfeld quantization conditions. For the formalization of this method in symplectic geometry/geometric quantization see at semiclassical state.

This WKB method makes sense for a more general class of wave equations. For instance in wave optics this yields the short-wavelength limit of the geometrical optics approximation. Here $S$ is called the eikonal?.

Multidimensional generalization of the WKB method appear to be rather nontrivial; they have been pioneered by Victor Maslov who introduced a topological invariant to remove ambiguities of the naive version of the method, called the Maslov index.

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 in gauge theories

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 $1/N^2$.

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

• Victor 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.

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

• Vladimir Arnold, 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)

• Victor Guillemin, Shlomo Sternberg, Geometric asymptotics, AMS 1977, online; Semi-classical analysis, 499 pages, pdf

• 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 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)

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

• Nicole Berline, Ezra Getzler, Michèle 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.

• A. Laptev, I.M. Sigal, Global Fourier integral operators and semiclassical asymptotics, Review of Math. Physics, 12:5 (2000) 749–766 pdf

• Maurice A. de Gosson, Symplectic geometry, Wigner-Weyl-Moyal calculus, and quantum mechanics in phase space, 385 pp. pdf

• Semyon Dyatlov, Semiclassical Lagrangian distributions, pdf; Hoermander–Kashiwara and Maslov indices, pdf

• Shanzhong Sun, Gutzwiller’s semiclassical trace formula and Maslov-type index theory for symplectic paths, arxiv/1608.08294

Borel summability may make sense of the semiclassical expansion to all orders; this approach is sometimes called exact WKB method:

• A. Voros, The return of the quartic oscillator. The complex WKB method, Annales de l’institut Henri Poincaré A39:3, 211-338 (1983) euclid

• Alexander Getmanenko, Dmitry Tamarkin, Microlocal properties of sheaves and complex WKB, arxiv/1111.6325

• Kohei Iwaki, Tomoki Nakanishi, Exact WKB analysis and cluster algebras, J. Phys. A 47 (2014) 474009 arxiv/1401.7094; Exact WKB analysis and cluster algebras II: simple poles, orbifold points, and generalized cluster algebras, arXiv:1409.4641

Relation to quantum integrable systems is in a series of works of Vũ Ngọc, e.g.

• San Vũ Ngọc, Bohr-Sommerfeld conditions for integrable systems with critical manifolds of focus-focus type, Preprint Institut Fourier 433, 1998 15 pdf; Quantum monodromy in integrable systems, Comm. Math. Phys. 203 (1999), no. 2, 465–479 doi

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