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 2*n*, 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

*ψ*_{A+df}
= exp(i* f*/\(\hbar\)) *ψ*_{A}

(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*
d*v*_{Ω} .

* __Covariant derivative__:
For each potential *A*, and charged scalar
*ψ* ∈ *V*, define

∇_{a}*ψ*_{A}:=
∂_{a}*ψ*_{A} − (i/\(\hbar\))
*A*_{a}*ψ*_{A} .

* __Pre-quantum operators__:
Define a 1-1, linear map from classical observables *f* to operators
*O*_{f} : *V* → *V* by

*O*_{f}:=
−i\(\hbar\) *X*_{f}^{a}
∇_{a} + *f* ;

This preserves the Lie algebra structure, in the sense that
[*O*_{f},
*O*_{g}]
= −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)

*v*^{a} ∇_{a}
*ψ* = 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 \(\cal H\).

* __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* [*X*_{f} ,
*v*]^{a} ∈ *P* ,
or [*O*_{f },
*v*^{a }∇_{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 = S^{2}; 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 T^{2}];
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

@

**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].

@ __Texts__: Simms & Woodhouse 76;
Śniatycki 80;
Hurt 82;
Tuynman 85;
in Ashtekar 88;
Woodhouse 92;
Nair a1606-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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