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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