semiclassical approximation



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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 \hbar𝒪( 0)\mathcal{O}(\hbar^0)𝒪( 1)\mathcal{O}(\hbar^1)𝒪( n)\mathcal{O}(\hbar^n)𝒪( )\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 ϕFieldsDϕF(ϕ)e iS(ϕ)/\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 SS (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 SS varies fast as measured in units of Planck's constant, the complex phases of the integrand exp(iS/)\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 SS (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 iddtψ=H^ψi \hbar \frac{d}{d t} \psi = \hat H \psi (which characterizes quantum states given by wave functions ψ\psi for Hamiltonian dynamics induced by a Hamilton operator H^\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 ψ=exp(S)\psi = exp(S) where SS is a slowly varying function and solving the equation for SS. 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 SS 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 NN-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 21/N^2.

In radiation theory

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.


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

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

For large N-limit compared to semiclassical expansion see

For the semiclassical method in superstring theory see