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 , with a connection of curvature i ( = 1).

Prequantization
* Goal: Start with a phase space (, ), of dimension 2n, such that 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 _A: → C such that

_A+df = exp(i f/) _A

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

,:= _Gamma * dv_Omega .

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

_aA:= _aA – (i/) A_aA .

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

O_f:= –i X_f^a _a + f ;

This preserves the Lie algebra structure, in the sense that [O_f, O_g] = –i O_{{f, g}}, and all operators are symmetric.
@ References: Horvathy pr(85) [and path integrals]; Andersson JFA(87) [in -D]; Kanatchikov gq/00-MG9 [covariant, field theory]; Tuynman mp/03 [super symplectic manifolds]; Zambon & Zhu JGP(07) [two approaches].

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

v^a _a = 0 ,    for all v^a 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 .
* Good observables: Choose as allowed quantum operators those that act within (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 _a P = 0" or

for all v P   [X_f ,v]^a P ,   or   [O_f , v^a _a] = 0 .

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

Examples > s.a. Jacobi Structure.
* Spin system: Phase space = S²; One cannot apply canonical quantization, but one can find a Kähler polarization.
@ Finite-dimensional systems: Robson 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 mp/00-in; Giachetta et al PLA(02)qp/01 [integrable system]; Sardanashvily IJTP(03) [relativistic Hamiltonian]; Lim JMP(07) [harmonic oscillator, non-standard complex structure].
@ T*G: Hall CMP(02)qp/00 [compact G].
@ Field theory: Puta LMP(87) [em, prequantization]; Müller JGP(05); > s.a. QCD, quantum fields in curved spacetime.

References > s.a. approaches to quantum mechanics; {notes from AA's seminars}.
@ General: Van Hove 51; Kostant in(70); Blattner in(74); Qiang JGP(96); Landsman mp/03-in [functorial point of view]; Hawkins CMP(00) [and deformation quantization].
@ Reviews: Echeverría-Enríquez et al EM(98)mp/99 [foundations]; Ritter mp/02 [short intro].
@ Texts: Simms & Woodhouse 76; Sniatycki 80; Hurt 82; Tuynman 85; in Ashtekar 88; Woodhouse 92.
@ Inequivalent quantizations: Robson PLB(94)ht; Govaerts & Villanueva IJMPA(00)qp/99 [different bundles].
@ 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.
@ 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].

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].
@ 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).

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Send feedback and suggestions to bombelli at olemiss.edu – Modified 28 jun 2008