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| In [[mathematical physics]], '''geometric quantization''' is a mathematical approach to defining a [[Quantum mechanics|quantum theory]] corresponding to a given [[classical theory]]. It attempts to carry out [[Quantization (physics)|quantization]], for which there is [[in general]] no exact recipe, in such a way that certain analogies between the classical theory and the quantum theory remain manifest. For example, the similarity between the Heisenberg equation in the [[Heisenberg picture]] of [[quantum mechanics]] and the [[Hamilton equation]] in classical physics should be built in.
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| One of the earliest attempts at a natural quantization was [[Weyl quantization]], proposed by [[Hermann Weyl]] in 1927. Here, an attempt is made to associate a quantum-mechanical observable (a [[self-adjoint operator]] on a [[Hilbert space]]) with a real-valued function on classical [[phase space]]. The position and momentum in this phase space are mapped to the generators of the [[Heisenberg group]], and the Hilbert space appears as a [[group representation]] of the [[Heisenberg group]]. In 1946, [[Hilbrand J. Groenewold |
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| H. J. Groenewold]] (H.J. Groenewold, "On the Principles of elementary quantum mechanics", ''Physica'','''12''' (1946) pp. 405-460) considered the product of a pair of such observables and asked what the corresponding function would be on the classical phase space. This led him to discover the [[Moyal product|phase-space star-product]] of a pair of functions.
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| More generally, this technique leads to [[deformation quantization]], where the ★-product is taken to be a deformation of the algebra of functions on a [[symplectic manifold]] or [[Poisson manifold]]. However, as a natural quantization scheme (a functor), Weyl's map is not satisfactory. For example, the Weyl map of the classical angular-momentum-squared is not just the quantum angular momentum squared operator, but it further contains a constant term 3ħ<sup>2</sup>/2. (This extra term is actually physically significant, since it accounts for the nonvanishing angular momentum of the ground-state Bohr orbit in the hydrogen atom, cf. {{cite doi|10.1103/PhysRevA.65.022109|noedit}}). As a mere representation change, however, Weyl's map underlies the alternate [[Phase space formulation]] of conventional quantum mechanics.
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| The geometric quantization procedure falls into the following three steps: prequantization, polarization, and metaplectic correction. Prequantization produces a natural Hilbert space together with a quantization procedure for observables that exactly preserves transforms Poisson brackets on the classical side into commutators on the quantum side. Nevertheless, the prequantum Hilbert space is generally understood to be "too big"; see the discussion in Section 22.3 of Hall (2013). The idea is that one should then select a Poisson-commuting set of ''n'' variables on the 2''n''-dimensional phase space and consider functions (or, more properly, sections) that depend only on these ''n'' variables. The ''n'' variables can be either real-valued, resulting in a position-style Hilbert space, or complex-valued, producing something like the [[Segal–Bargmann space]]. A polarization is just a coordinate-independent description of such a choice of ''n'' Poisson-commuting functions; see Section 23.4 of Hall (2013). The metaplectic correction (also known as the half-form correction) is a technical modification of the above procedure that is necessary in the case of real polarizations and often convenient for complex polarizations.
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| * Prequantization of a symplectic manifold <math>(M,\Omega)</math> provides a representation of elements <math>f\in C^\infty(M)</math> of the [[Poisson algebra]] of smooth real functions on <math>M</math> by first order differential operators <math>\widehat f</math> on sections of a complex line [[fiber bundle|bundle]] <math>L\to M</math>. In accordance with the Kostant - Souriau prequantization formula, these operators are expressed via a [[connection (mathematics)|connection]] on <math>L\to M</math> whose [[curvature form]] <math>R</math> obeys the prequantization condition <math>R=i\Omega</math>.
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| * By polarization is meant an integrable maximal [[distribution (differential geometry)|distribution]] <math>T</math> on <math>M</math> such that <math>\Omega(v,v')=0</math> for all <math>v,v'\in T</math>. Integrable means <math>[v,v']\in\Gamma(T)</math> for <math>v,v'\in\Gamma(T)</math> (sections of ''T''). The quantum algebra <math>A_M</math> of a symplectic manifold <math>M</math> consists of the operators <math>\widehat f</math> of functions <math>f</math> whose [[Hamiltonian vector field]]s <math>X_f</math> satisfiy the condition <math>[X_f,T]\subset T</math>.
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| * In accordance with the metaplectic correction, elements of the quantum algebra <math>A_M</math> act in the [[inner product space|pre-Hilbert space]] of [[half-form]]s with values in the prequantization Line bundle on a symplectic manifold <math>M</math>. The quantization is simply
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| *:<math>f\mapsto f\cdot +i\hbar^{1/2}\mathcal{L}_{X_f}</math>
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| :where <math>\mathcal{L}_X</math> is the Lie derivative of a half-form with respect to a vector field ''X''. See Section 23.6 of Hall (2013) for further discussion.
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| Geometric quantization of Poisson manifolds and symplectic foliations also is developed. For instance, this is the case of [[integrable system|partially integrable]] and [[superintegrable Hamiltonian system|superintegrable]] Hamiltonian systems and [[non-autonomous mechanics]].
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| ==See also==
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| * [[Half-form]]
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| * [[Lagrangian foliation]]
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| * [[Kirillov orbit method]]
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| == References ==
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| *{{cite book
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| | author = J. Śniatycki
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| | year = 1980
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| | title = Geometric Quantization and Quantum Mechanics
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| | publisher = Springer
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| | isbn = 0-387-90469-7
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| | url =
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| }}
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| *{{cite book
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| | author = N.M.J. Woodhouse
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| | year = 1991
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| | title = Geometric Quantization
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| | publisher = Clarendon Press
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| | isbn = 0-19-853673-9
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| | url =
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| }}
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| *{{cite book
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| | author = B.C. Hall
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| | year = 2013
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| | title = Quantum Theory for Mathematicians
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| | publisher = Springer
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| }}
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| *{{cite book
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| | author = I. Vaisman
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| | year = 1991
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| | title = Lectures on the Geometry of Poisson Manifolds
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| | publisher = Birkhauser
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| | isbn = 978-3-7643-5016-1
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| | url =
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| }}
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| *{{cite book
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| | author = G. Giachetta, L. Mangiarotti, [[Gennadi Sardanashvily|G. Sardanashvily]]
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| | year = 2005
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| | title = Geometric and Algebraic Topological Methods in Quantum Mechanics
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| | publisher = World Scientific
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| | isbn = 981-256-129-3
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| | url =
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| }}
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| *{{cite book
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| | author = K.Kong Wan
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| | year = 2006
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| | title = From Micro to Macro Quantum Systems, (A Unified Formalism with Superselection Rules and Its Applications)
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| | publisher = World Scientific
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| | isbn = 978-1-86094-625-7
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| | url =
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| }}
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| ==External links==
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| * [http://arxiv.org/abs/math-ph/0208008 William Ritter's review of Geometric Quantization] presents a general framework for all problems in [[physics]] and fits geometric quantization into this framework
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| *[http://math.ucr.edu/home/baez/quantization.html John Baez's review of Geometric Quantization], by [[John Baez]] is short and pedagogical
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| *[http://www.blau.itp.unibe.ch/lecturesGQ.ps.gz Matthias Blau's primer on Geometric Quantization], one of the very few good primers (ps format only)
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| * A. Echeverria-Enriquez, M. Munoz-Lecanda, N. Roman-Roy, Mathematical foundations of geometric quantization, [http://arxiv.org/abs/math-ph/9904008 arXiv: math-ph/9904008.]
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| * [[Gennadi Sardanashvily|G. Sardanashvily]], Geometric quantization of symplectic foliations, [http://xxx.lanl.gov/abs/math/0110196 arXiv: math-ph/0110196.]
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| [[Category:Symplectic geometry]]
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| [[Category:Mathematical quantization]]
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| [[Category:Functional analysis]]
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