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 *ψ*_{A}: Γ → \(\mathbb C\) 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} ∇_{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 a1603-conf [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.

@ __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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2017