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\section*{semiclassical approximation}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{physics}{}\paragraph*{{Physics}}\label{physics}

[[!include physicscontents]]

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{overview}{Overview}\dotfill \pageref*{overview} \linebreak
\noindent\hyperlink{related_phenomena}{Related phenomena}\dotfill \pageref*{related_phenomena} \linebreak
\noindent\hyperlink{equivariant_localization}{Equivariant localization}\dotfill \pageref*{equivariant_localization} \linebreak
\noindent\hyperlink{large_limit_in_gauge_theories}{Large $N$-limit in gauge theories}\dotfill \pageref*{large_limit_in_gauge_theories} \linebreak
\noindent\hyperlink{in_radiation_theory}{In radiation theory}\dotfill \pageref*{in_radiation_theory} \linebreak
\noindent\hyperlink{literature}{Literature}\dotfill \pageref*{literature} \linebreak


\hypertarget{overview}{}\subsection*{{Overview}}\label{overview}

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

[[!include classical-to-quantum notions - table]]

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 \emph{[[rotating phase approximation]]}: the idea is that in regions of [[field (physics)|field]]-space where $S$ varies fast as measured in units of [[Planck's constant]], the [[complex number|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 mechanics|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 \textbf{[[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 leaf|Bohr-Sommerfeld quantization conditions]]. For the formalization of this method in [[symplectic geometry]]/[[geometric quantization]] see at \emph{[[semiclassical state]]}.

This [[WKB method]] makes sense for a more general class of [[wave equations]]. For instance in [[wave mechanics|wave]] [[optics]] this yields the short-[[wavelength]] limit of the [[geometrical optics]] approximation. Here $S$ is called the \textbf{[[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 \emph{[[Maslov index]]}.

\hypertarget{related_phenomena}{}\subsection*{{Related phenomena}}\label{related_phenomena}

\hypertarget{equivariant_localization}{}\subsubsection*{{Equivariant localization}}\label{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.

\hypertarget{large_limit_in_gauge_theories}{}\subsubsection*{{Large $N$-limit in gauge theories}}\label{large_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 $1/N^2$.

\hypertarget{in_radiation_theory}{}\subsubsection*{{In radiation theory}}\label{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.

\hypertarget{literature}{}\subsection*{{Literature}}\label{literature}

\begin{itemize}%
\item M.V. Fedoryuk, \emph{Semi-classical approximation}, Springer \href{http://eom.springer.de/S/s083990.htm}{Online} Enc. of Math.


\item Sean Bates, Alan Weinstein, \emph{Lectures on the geometry of quantization}, \href{http://www.math.berkeley.edu/~alanw/GofQ.pdf}{pdf}


\item [[Victor Maslov]], \emph{Stationary-phase method for Feynman's continual integral}, Theoret. and Math. Phys., 2:1 (1970), 21--25; Russian original: , 2:1 (1970), 30--35 \href{http://www.mathnet.ru/php/getFT.phtml?jrnid=tmf&paperid=3986&what=fullt&option_lang=rus}{pdf}.


\item [[Victor Maslov]], \emph{Theory of perturbations and asymptotic methods} (Russian), Izdat. Moskov. Gos. Univ. 1965.


\item [[Vladimir Arnold]], \emph{Characteristic class entering in quantization conditions}, Funct. Anal. its Appl. 1967, 1:1, 1--13, \href{http://dx.doi.org/10.1007/BF01075861}{doi} (. . , ``  ,    '', .    ., 1:1 (1967), 1--14, \href{http://www.mathnet.ru/php/getFT.phtml?jrnid=faa&paperid=2802&what=fullt&option_lang=rus}{pdf})


\item [[Victor Guillemin]], [[Shlomo Sternberg]], \emph{Geometric asymptotics}, AMS 1977, \href{http://www.ams.org/online_bks/surv14}{online}; \emph{Semi-classical analysis}, 499 pages, \href{http://www-math.mit.edu/~vwg/semistart.pdf}{pdf}


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


\item [[Richard Szabo]], \emph{Equivariant cohomology and localization of path integrals}, Lecture Notes in Physics, N.S. Monographs \textbf{63}. Springer 2000. xii+315 pp. (early version: \emph{Equivariant localization of path integrals}, \href{http://arxiv.org/abs/hep-th/9608068}{hep-th/9608068})


\item [[Michael Atiyah]], \emph{Circular symmetry and stationary phase approximation}, Asterisque \textbf{131} (1985) 43--59


\item [[Nicole Berline]], [[Ezra Getzler]], [[Michèle Vergne]], \emph{Heat kernels and Dirac operators}, Grundlehren \textbf{298}, Springer 1992, ``Text Edition'' 2003.


\item [[Albert Schwarz]], Oleg Zaboronsky, \emph{Supersymmetry and localization}, Comm. Math. Phys. \textbf{183}, 2 (1997), 463-476, \href{http://projecteuclid.org/euclid.cmp/1158328185}{euclid}


\item [[Albert Schwarz]], \emph{Semiclassical approximation in [[Batalin-Vilkovisky quantization|Batalin-Vilkovisky formalism]]}, Comm. Math. Phys. \textbf{158} (1993), no. 2, 373--396, \href{http://projecteuclid.org/euclid.cmp/1104254246}{euclid}.


\item A. Laptev, I.M. Sigal, \emph{Global Fourier integral operators and semiclassical asymptotics}, Review of Math. Physics, \textbf{12}:5 (2000) 749--766 \href{http://www.math.kth.se/~laptev/Research/Papers/LSig.pdf}{pdf}


\item Maurice A. de Gosson, \emph{Symplectic geometry, Wigner-Weyl-Moyal calculus, and quantum mechanics in phase space}, 385 pp. \href{http://opus.kobv.de/ubp/volltexte/2009/3021/pdf/2006_06.pdf}{pdf}


\item Semyon Dyatlov, \emph{Semiclassical Lagrangian distributions}, \href{http://math.mit.edu/~dyatlov/files/2012/hlagrangians.pdf}{pdf}; \emph{Hoermander--Kashiwara and Maslov indices}, \href{http://math.berkeley.edu/~dyatlov/files/2009/maslov.pdf}{pdf}


\item Shanzhong Sun, \emph{Gutzwiller's semiclassical trace formula and Maslov-type index theory for symplectic paths}, \href{https://arxiv.org/abs/1608.08294}{arxiv/1608.08294}



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

\begin{itemize}%
\item A. Voros, \emph{The return of the quartic oscillator. The complex WKB method}, Annales de l'institut Henri Poincar\'e{} A39:3, 211-338 (1983) \href{http://eudml.org/doc/76217}{euclid}


\item Alexander Getmanenko, Dmitry Tamarkin, \emph{Microlocal properties of sheaves and complex WKB}, \href{http://arxiv.org/abs/1111.6325}{arxiv/1111.6325}


\item Kohei Iwaki, Tomoki Nakanishi, \emph{Exact WKB analysis and cluster algebras}, J. Phys. A 47 (2014) 474009 \href{http://arxiv.org/abs/1401.7094}{arxiv/1401.7094}; \emph{Exact WKB analysis and cluster algebras II: simple poles, orbifold points, and generalized cluster algebras}, \href{http://arxiv.org/abs/1409.4641}{arXiv:1409.4641}



\end{itemize}
Relation to quantum [[integrable system]]s is in a series of works of V Ngc, e.g.

\begin{itemize}%
\item San V Ngc, \emph{Bohr-Sommerfeld conditions for integrable systems with critical manifolds of focus-focus type}, Preprint Institut Fourier 433, 1998 15 \href{http://perso.univ-rennes1.fr/san.vu-ngoc/articles/focus-tout.pdf}{pdf}; \emph{Quantum monodromy in integrable systems}, Comm. Math. Phys. 203 (1999), no. 2, 465--479 \href{http://dx.doi.org/10.1007/s002200050621}{doi}

\end{itemize}
For large N-limit compared to semiclassical expansion see

\begin{itemize}%
\item L. G. Yaffe, \emph{Large N limits as classical mechanics}, Rev. Mod. Phys. \textbf{54}, 407--435 (1982), \href{http://rmp.aps.org/pdf/RMP/v54/i2/p407_1}{pdf}

\end{itemize}
For the semiclassical method in [[superstring theory]] see

\begin{itemize}%
\item J. Maldacena, G. Moore, N. Seiberg, D. Shih, \emph{Exact vs. semiclassical target space of the minimal string}, \href{http://arxiv.org/abs/hep-th/0408039}{hep-th/0408039}


\item K. Hori, A. Iqbal, C. Vafa, \emph{D-Branes and mirror symmetry}, \href{http://arxiv.org/abs/hep-th/0005247}{hep-th/0005247}



\end{itemize}
[[!redirects quasiclassical approximation]] [[!redirects quasi-classical approximation]] [[!redirects semiclassical expansion]] [[!redirects semiclassical analysis]][[!redirects semi-classical analysis]]

[[!redirects semiclassical quantization]]

[[!redirects semiclassical limit]] [[!redirects semiclassical limits]]



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