Geometric Quantization  

In General > s.a. canonical quantum theory.
* Idea: A quantization method, generalizing the usual canonical one; The classical phase space doesn't have to be a cotangent bundle, and one has a general framework to talk about different representations (coordinate, momentum, Bargmann).
* History: Souriau prepared the way, studying the phase space as the set of histories, and introducing what later became prequantization; Kostant introduced polarizations and the prequantization scheme.
* Relationships: One is really introducing a complex line bundle over Γ, with a connection ∇ of curvature iΩ (\(\hbar\) = 1).

Prequantization
* Goal: Start with a phase space (Γ, Ω), of dimension 2n, such that Ω has a (local) potential A, and get a pre-quantum Hilbert space of states and a representation of the Poisson algebra of observables thereon.
* Pre-quantum wave functions: Construct the vector space V of charged scalar functions \(\psi_A : \Gamma \to \mathbb C\) such that

ψA+df = exp(i f/\(\hbar\)) ψA

(behavior under gauge transformations); i.e., cross sections of a \(\mathbb C\) line bundle over Γ, associated with the principal U(1) bundle with curvature Ω.
* Pre-quantum Hilbert space: Define on V the gauge-invariant inner product

\(\langle\)ψ, φ\(\rangle\):= ∫Γ ψ*φ dvΩ .

* Covariant derivative: For each potential A, and charged scalar ψ ∈ V, define

∇aψA:= ∂aψA − (i/\(\hbar\)) AaψA .

* Pre-quantum operators: Define a 1-1, linear map from classical observables f to operators Of : V → V by

Of:= −i\(\hbar\) Xfa ∇a + f ;

This preserves the Lie algebra structure, in the sense that [Of, Og] = −i\(\hbar\) O{f, g}, and all operators are symmetric.
@ References: Horváthy pr(80) [and path integrals]; Andersson JFA(87) [in ∞-D]; Kanatchikov gq/00-MG9 [covariant, field theory]; Tuynman JGP(10)mp/03 [super symplectic manifolds]; Zambon & Zhu JGP(07) [two approaches]; Geraci a0911 [intro].

Quantization
* Polarization: Choose some polarization P of Γ.
* Quantum wave functions: Choose the elements of V that satisfy the polarization condition (new input here)

va ∇a ψ = 0 ,    for all va ∈ P .

* Quantum Hilbert space: Choose a (new) inner product on quantum states (the pre-quantum one is ok in the case of a Kähler polarization), and get a Hilbert space \(\cal H\).
* Good observables: Choose as allowed quantum operators those that act within \(\cal H\) (the BKS procedure can be used to construct good operators); This means requiring that the Hamiltonian vector fields preserve \(P\), i.e., "\(X_f^a\nabla_{\!a}\,P\) = 0" or

for all v ∈ P   [Xf , v]a ∈ P ,    or   [Of , va ∇a] ψ = 0 .

@ Polarization: Campbell & Dodson IJTP(79) [projective concepts]; Maraner RPMP(97)qp/98 [dynamical choice].

Examples > s.a. Jacobi Structure.
* Spin system: Phase space Γ = S2; One cannot apply canonical quantization, but one can find a Kähler polarization.
@ Harmonic oscillator: Lim JMP(07) [non-standard complex structure]; Hedrea et al IJGMP(11); Iacob a1607 [harmonic-oscillator-type potentials].
@ Other finite-dimensional systems: Robson JGP(96)ht/94 [particle in Yang-Mills field]; Corichi & Ryan JPA(97)gq/95 [more than one H]; Kimura PTP(98)ht/96 [on a coset space]; Velhinho IJMPA(98)ht/97 [on T2]; Gotay in(02)mp/00; Giachetta et al PLA(02)qp/01 [integrable system]; Sardanashvily IJTP(03) [relativistic Hamiltonian]; Hamilton & Miranda AIF-a0808 [integrable systems with hyperbolic singularities]; Kemp & Veselov a1103 [Dirac magnetic monopole on a unit sphere]; Contreras & Duman EPTcs(17)-a1603 [epistemically restricted theories].
@ T*G: Hall CMP(02)qp/00 [compact G].
@ Field theory: Puta LMP(87) [electromagnetism, prequantization]; Müller JGP(05); Shyam a1304 [canonical gravity]; Clader et al a1309 [applications to Gromov-Witten theory]; Biswas et al IJGMP(15)-a1411 [path space, Klein-Gordon theory]; > s.a. QCD; quantum fields in curved spacetime.

References > s.a. approaches to quantum mechanics; {notes from AA's seminars}; Polymer Representation.
@ General: Van Hove 51; Kostant in(70); Blattner in(74); Qiang JGP(96); Landsman mp/03-proc [functorial point of view].
@ Reviews: Echeverría-Enríquez et al EM(98)mp/99 [foundations]; Ritter mp/02 [short intro]; in Todorov BulgJP(12)-a1206; Carosso a1801 [informal]; Berktav a1902-proc [and Witten's quantum invariant]; Camosso a2012 [intro].
@ Texts: Simms & Woodhouse 76; Śniatycki 80; Hurt 82; Tuynman 85; in Ashtekar 88; Woodhouse 92; Nair a1606-ln; Moshayedi a2010-ln.
@ Inequivalent quantizations: Robson PLB(94)ht; Govaerts & Villanueva IJMPA(00)qp/99 [different bundles]; Lempert & Szőke CMP(14)-a1004.
@ And coherent states: Klauder qp/95; Bartlett et al JPA(02)qp, JPA(02)qp; Florentino et al JFA(05)m.DG/04 [and coherent state transform]; Kirwin JGP(07)m.SG/05.
@ And deformation quantization: Hawkins CMP(00); Nölle a0809, a0903; Duval & Gotay RPMP(12) [deformation of prequantization].
@ Related topics: Chernoff HJ(81) [obstructions]; Guillemin & Sternberg IM(82); Klauder & Onofri IJMPA(89); Batalin & Tyutin NPB(90); Kirwin & Wu CMP(06) [and Fourier transform]; Hochs & Mathai AiM(15)-a1309 [and families of inner products]; Tejero & Vitolo IJGMP(14)-a1408 [geometry of the energy operator]; Camosso JQIS(17)-a1508 [and quantum logic]; Duval a1602 [Souriau's derivation of the Weyl equation].

Variations > s.a. brst; quantum mechanics [geometric approaches]; symplectic structures, types and deformations [& Moyal].
@ Approaches: Fradkin & Linetsky NPB(94) [BFV]; Giachetta et al mp/00 [covariant]; Hawkins m.SG/06 [grupoid approach]; Charles CMP(07) [with metaplectic structure, semiclassical limit]; Aldaya et al IJGMP(11)-a1012 [for non-linear systems]; Patrascu PRD(14)-a1403 [using cohomology groups and the Universal Coefficient Theorem].
@ Similar: Jorjadze JMP(97)ht/96; Isidro qp/01; Abrikosov et al MPLA(03)qp [and Koopman-von Neumann classical mechanics method].
@ With constraints: Ashtekar & Stillerman JMP(86); Blau CQG(88), PLB(88); Batalin & Lavrov TMP(16)-a1505 [second-class constraints].
@ Other extensions: Vaisman JGP(09) [weak-Hamiltonian functions]; Fitzpatrick JGP(11) [for contact manifolds]; Sharatchandra a1503 [torus phase space].

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