geometric quantization on coadjoint orbits
DESCRIPTION
Geometric Quantization on coadjoint orbitsTRANSCRIPT
Coadjoint Orbits Geometric Quantization
Workshop on Diffeology etcAix en Provence, France- June 25-26-27, 2014Geometric Quantization On Coadjoint Orbits
Hassan Jolany, University of Lille1
University of Lille1
Geometric Quantization on coadjoint orbits
Coadjoint Orbits Geometric Quantization
List of sections
1 Coadjoint Orbits
2 Geometric Quantization
University of Lille1
Geometric Quantization on coadjoint orbits
Coadjoint Orbits Geometric Quantization
Coadjoint Orbits
Let G be a compact semisimple Lie group with Lie algebra g.Group G acts on dual of Lie algebra g∗ with coadjointrepresentation Ad∗ : G × g∗ → g∗ by convention:⟨
Ad∗gµ,X⟩
=⟨µ,Adg−1X
⟩where µ ∈ g∗, g ∈ G ,X ∈ g.
Definition
The subset Oµ = Ad∗gµ; g ∈ G of g∗ is called a coadjoint orbitof G through µ ∈ g∗
Coadjoint orbit through µ ∈ g∗ can be written Oµ ∼= G/Gµ,which the stabilizer subgroup Gµ can be written as
Gµ = g ∈ G : Ad∗gµ = µUniversity of Lille1
Geometric Quantization on coadjoint orbits
Coadjoint Orbits Geometric Quantization
Some Examples of Co-Adjoint Orbits
University of Lille1
Geometric Quantization on coadjoint orbits
Coadjoint Orbits Geometric Quantization
Some Examples of Co-Adjoint Orbits
University of Lille1
Geometric Quantization on coadjoint orbits
Coadjoint Orbits Geometric Quantization
Complexified of Lie Group
Definition
If G is a Lie group, a universal complexification is given by acomplex Lie group GC and a continuous homomorphismϕ : G → GC with the universal property that, if f : G → H is anarbitrary continuous homomorphism into a complex Lie group H,then there is a unique complex analytic homomorphismF : GC → H such that f = F oϕ.
For a classical Real Lie group G , The complexification of Liegroup GC can be defined as
GC := expg + igLet G = U(n), then GC = GL(n,C) or Let K = SU(n), thenKC = SL(n,C)
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Geometric Quantization on coadjoint orbits
Coadjoint Orbits Geometric Quantization
Complexified of Lie Group
Theorem
Let G be a compact and connected Lie group, then
GC ∼= G n g∗ ∼= T ∗G
which this decomposition known as Polar Decomposition.
Every coadjoint orbit Oµ can be written as
Oµ ∼= GC/P ∼= T ∗G/P
which P is the Parabolic subgroup.
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Geometric Quantization on coadjoint orbits
Coadjoint Orbits Geometric Quantization
Volume of a coadjoint orbits
Kirillov’s Character formula:Kirillov’s famous formula says that the characters of the irreducibleunitary representations of a compact Lie group G can be writtenby following form
χλ(exp(x)) =1
p(x)
∫Oλ+ρ
e2πi<µ,x>dµOλ+ρ(µ)
where p is a certain function on lie algebra g and it can be writtenas
p(x) = det1/2 sinh(ad(x/2))
ad(x/2)
and µ is the highest weight
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Geometric Quantization on coadjoint orbits
Coadjoint Orbits Geometric Quantization
Volume of a coadjoint orbit
Theorem
The Volume of a Coadjoint Orbit through µ is
Vol(Oµ) =∏α∈R+
< α, µ >
< α, ρ >
where ρ = 12
∑α∈R+ α
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Geometric Quantization on coadjoint orbits
Coadjoint Orbits Geometric Quantization
Measure on coadjoint orbits
Explicit formula for measure of coadjoint orbits:For µ ∈ g∗, we define a skew-symmetric bilinear form bµ onTeGµOµ ∼= g/gµ by
bµ(u, v) =< µ, [u, v ] >
where for u, v ∈ g, we write u, v ∈ g/gµ.So, the dual form of bµ is the bilinear form βµ on(g/gµ)∗ = g⊥µ = g.µ corresponding to bµ under this isomorphism
βµ(u.µ, v .µ) = bµ(u, v)
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Geometric Quantization on coadjoint orbits
Coadjoint Orbits Geometric Quantization
Measure on coadjoint orbits
Now choose a basis v1, v2, ..., vm for g/gµ and µ1, µ2, ..., µm be thedual basis for (g/gµ)∗ (where m is the maximal dimension of anorbit in g∗)then
1
(m/2)!bm/2µ = Pf < µ, [vi , vj ] > µ1 ∧ ... ∧ µm
and1
(m/2)!βm/2µ = Pf−1 < µ, [vi , vj ] > v1 ∧ ... ∧ vm
where Pf(aij) denotes the Pfaffian of a skew-symmetric matrix (aij)Now define,
ωµ =1
(m/2)!βm/2µ
So integration of ωµ gives measure on Oµ up to the factor (2π)m/2
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Geometric Quantization on coadjoint orbits
Coadjoint Orbits Geometric Quantization
Geometric properties of coadjoint orbits
Every coadjoint orbit is symplectic manifold. The symplecticform on Oλ is given by
ωλ(ad∗X , ad∗Y ) = 〈λ, [X ,Y ]〉. This is obviously anti-symmetric and non-degenerate andalso closed 2-form.Every coadjoint orbit has Kaehler and Hyper-Kaehlerstructure.Coadjont orbits are simply connected.
Theorem
Let G be a Lie group, and Φ : T ∗G → g∗ be a moment map and
ζ ∈ g∗ then the symplectic quotient Φ−1(ζ)Gζ
is coadjoint orbit
through ζ, i.e., Oζ .
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Geometric Quantization on coadjoint orbits
Coadjoint Orbits Geometric Quantization
Here, by previous theorem we see that the symplectic form oncotangent bunlde of a Lie group is related to Kostant KirillovSouriau symplectic structure of the coadjoint orbits in thedual of a Lie algebra as follows
Gi //
π
T ∗G
Oζand we have
π∗ωOζ= i∗ωT∗G
Theorem of Patrick Iglesias-Zemmour:
Theorem
Every connected Hausdorff symplectic manifold is isomorphic to acoadjoint orbit of its group of hamiltonian diffeomorphisms.
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Geometric Quantization on coadjoint orbits
Coadjoint Orbits Geometric Quantization
Geometric PDE on coadjoint orbits
Following theorem known as M. Stenzel’s theorem: We denote thecomplexified of a coadjoint orbit as
OCζ := GC/GC
ζ
where ζ ∈ g∗
Theorem
Let G be a compact, connected and semisimple, Lie group. Thereexists a G invariant, real analytic, strictly plurisubharmonicfunction of complexified coadjoint orbit ρ : OC
ζ → [0,∞) such thatu =√ρ satisfies the Monge-Ampere equation,
(∂∂u)n = 0
where n = dimOλUniversity of Lille1
Geometric Quantization on coadjoint orbits
Coadjoint Orbits Geometric Quantization
Geometric Quantization
Now, we define the geometric quantization by some axioms whichare compatible with physical view.
Geometric quantization we associate to a symplectic manifold(M, ω) a Hilbert space H, and one associates to smoothfunctions f : M → R skew-adjoint operators Of : H → H.Paul Dirac introduced in his doctoral thesis, the ”method ofclassical analogy”for quantization which is now known asDirac axioms as follows.
1] Poisson bracket of functions passes to commutator of operators:
Of ,g = [Of ,Og ]
2] Linearity condition must holds ,Oλ1f +λ2g = λ1Of + λ2Og forλ1, λ2 ∈ C3] Normalization condition must holds: 1 7→ i .I (Which I is identityoperator and i =
√−1)
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Geometric Quantization on coadjoint orbits
Coadjoint Orbits Geometric Quantization
Dirac Principal for Geometric quantization
Before to establish the axiom 4], we need to following definition.
Definition
Let (M, ω), be a symplectic manifold. A set of smooth functionsfj is said to be a complete set of classical observables if and onlyif every other function g such that fi , g = 0 for all fj, isconstant. Also we say that a family of operators is complete if itacts irreducibly on H
4] Minimality condition must holds: Any complete family offunctions passes to a complete family of operators. Moreover, if Gbe a group acting on (M, ω) by symplectomorphisms and on H byunitary transformations. If the G -action on M is trnsitive, then itsaction on H must be irreducible.
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Geometric Quantization on coadjoint orbits
Coadjoint Orbits Geometric Quantization
Geometric Quantization
Now, we again recall the pre-quantization Line bundle informal language. In fact, we have two important method forGeometric Quantization;
1)Using line bundle. More precisely, In geometric quantizationwe construct the Hilbert space H as a subspace of the spaceof sections of a line bundle L on a symplectic manifold M.
2)Without using line bundle: Using Mpc -structure instead ofline bundle. One of advantage of this construction is betterbehaved of physical view but definig it is not so easy.
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Geometric Quantization on coadjoint orbits
Coadjoint Orbits Geometric Quantization
Mpc Quantization
The use of Mpc structures clarifies and extends the traditionalKonstant scheme of geometric in a number of ways.
In fact, prequantized Mpc structures generalize the combinedtraditional data consisting of a prequantum line bundle and ametaplectic structure and are used in constructing andcomparing representations of poisson algebras on symplecticmanifolds.
Also, Mpc structures exists on any symplectic manifold.
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Geometric Quantization on coadjoint orbits
Coadjoint Orbits Geometric Quantization
Mpc- Group
Definition
Let (V ,Ω) be a real symplectic vector space and fix an irreducibleunitary projective representation W of V on a Hilbert space Hsuch that
x , y ∈ V ⇒W (x)W (y) = exp 1
2i~Ω(x , y)W (x + y)
. If g ∈ Sp(V ,Ω), so that g is a linear automorphism of V withg∗Ω = Ω. So, by uniqueness theorem of Stone and von Neumannthere exists a unitary operator U on H such thatv ∈ V ⇒W (gv) = UW (v)U−1. The group of all such operatorsU as g ranges over the symplectic group Sp(V ,Ω) is denoted byMpc(V ,Ω)
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Geometric Quantization on coadjoint orbits
Coadjoint Orbits Geometric Quantization
Mpc- Group
We have a central short exact sequence
1→ U(1)→ Mpc(V ,Ω)σ→ Sp(V ,Ω)→ 1
where σ sends U to g .
Mpc(V ,Ω) supports a unique unitary characterη : Mpc(V ,Ω)→ U(1) such that η(λ) = λ2 wheneverλ ∈ U(1).
The kernel of η is a connected double cover of Sp(V ,Ω)called metaplectic group Mp(V , ω)
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Geometric Quantization on coadjoint orbits
Coadjoint Orbits Geometric Quantization
Mpc- Structure
Definition
Let (E , ω) be a real symplectic vector bundle over the manifold M.If the rank of E equals the dimension of V then we may model(E , ω) on (V ,Ω) and define the symplectic frame bundleSp(E , ω) = Sp(E ) to be the principal Sp(V ,Ω) bundle over Mwhose fibre over m ∈ M consists of all linear isomorphismsb : V → Em such that b∗ωm = Ω
Now, we are ready to define Mpc -structure.
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Geometric Quantization on coadjoint orbits
Coadjoint Orbits Geometric Quantization
Mpc- Structure
Definition
By an Mpc -structure on (E , ω) we mean a principal Mpc(V ,Ω)bundle π : P → M with a fibre-preserving map φ : P → Sp(E , ω)such that the group actions are compatible:
φ(p.g) = φ(p).σ(g), ∀p ∈ P, g ∈ Mpc(V ,Ω)
where σ : Mpc(V )→ Sp(V )
Mpc -structures for (E , ω) are parametrized by H2(M,Z).
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Geometric Quantization on coadjoint orbits
Coadjoint Orbits Geometric Quantization
Mpc- Quantization
Definition
Let (M, ω) be a symplectic manifold. A prequantized Mpc
structure for (M, ω) is a pair (P, γ) with P an Mpc structure for(TM, ω) and γ an Mpc -invariant u(1)-valued one-form on P suchthat: if z ∈ mpc(V ,Ω) = Lie(Mpc(V ,Ω)) determines thefundamental vector field z on P then
γ(z) =1
2η∗z
anddγ = (1/i~)π∗ω
where π : P → M is the bundle projection. We say that (M, ω) isquantizable iff it admits prequantized Mpc structures.
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Geometric Quantization on coadjoint orbits
Coadjoint Orbits Geometric Quantization
Mpc- Quantization
Theorem
(M, ω) is quantizable if and only if
[ω]− 1
2c1(TM)
be an integral cohomology class.
Theorem
Inequivalent prequantized Mpc structures are parametrized by
H1(M; U(1))
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Geometric Quantization on coadjoint orbits
Coadjoint Orbits Geometric Quantization
Mpc- Quantization on cotangent bundle of Lie groups,T ∗G
Let G be a Lie group with Lie algebra g . Take, Z = T ∗G . ModleT ∗G on the symplectic vector space V = g⊕ g∗ with symplecticform
Ω((ξ, φ), (η, ψ)) = φ(η)− ψ(ξ)
The standard left action of G on the cotangent bundle T ∗G isHamiltonian , with equivariant moment map
J : T ∗G → g∗
withJ(αg ) = g .α
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Geometric Quantization on coadjoint orbits
Coadjoint Orbits Geometric Quantization
Mpc- Quantization on cotangent bundle of Lie groups,T ∗G
A canonical global section b : T ∗G → Sp(T ∗G ) of the symplecticframe bundle Sp(T ∗G ) is defined by
bαg (ξ, φ) = Λ∗
(√2(g−1.ξ)g ,
1√2
(g−1.φ− (g−1.ξ).α)α
)where Λ : G × g∗ → T ∗G sends (g , α) to αg an where dots signifyadjoint and coadjoint action.
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Geometric Quantization on coadjoint orbits
Coadjoint Orbits Geometric Quantization
Mpc- Quantization on cotangent bundle of Lie groups,T ∗G
The Mpc structure is the product π : T ∗G ×Mpc(V )→ T ∗Gwith projection T ∗G ×Mpc(V )→ Sp(T ∗G ) determined bysending (z , I ) to bz ; the prequantum form δ is given by
δ =1
i~π∗θ +
1
2η∗ε
where θ is the canonical one-form on T ∗G and ε is the flatconnection in π : T ∗G ×Mpc(V )→ T ∗G and η is the unitarycharacter of Mpc(V ) restricting to U(1) as the squaring map.The passage of prequantized Mpc structures to Marsden Weinsteinredced phase spaces here gives to us prequantized Mpc structureson coadjoint orbits.(because Marsden Weinstein redced phasespaces of T ∗G is the coadjoint orbit Oλ )
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Geometric Quantization on coadjoint orbits
Coadjoint Orbits Geometric Quantization
Mpc- Quantization on coadjoint orbits
For φ ∈ g∗ we denote Oφ = G .φ ⊂ g∗ its coadjoint orbit we canidentify Oφ ∼= G/Gφ, which Gφ is coadjoint stabilizer with Liealgebra gφ, and also we denote the identity component of Gφ withG 0φ .
Now we explain Robinson and J. H. Rawnsley’s quantization: Weknow that the vanishing holonomy generalizes the Keller Maslovcorrected Bohr sommerfeld rule. So we give the following theoremfor quantization on coadjoint orbits
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Geometric Quantization on coadjoint orbits
Coadjoint Orbits Geometric Quantization
Theorem
Mpc - Quantization on coadjoint orbits: For the coadjoint orbitOφ∼= G/Gφ, the vanishing holonomy condition amounts to the
following requirement:That the adjoint isotropy representation Ad : G 0
φ → Sp(g/h), given
by Adh(ξ + gφ) = h.ξ + gφ for h ∈ G 0φ and ξ ∈ g should lift to a
homomorphism τ : G 0φ → Mpc(g/h) with the property that
(ηoτ)∗ = − 2i~φ
Now here we give a connection between two different quantizationson coadjoint orbits with line bundles and without line bundle,
Theorem
Hassan Jolany’s Theorem: Let the coadjoint orbit Oµ withpre-quantum line bundle (G ×Gµ C)⊗2 be quantizable then thecoadjoint orbit Oµ is Mpc quantizable.
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Geometric Quantization on coadjoint orbits
Coadjoint Orbits Geometric Quantization
Theorem
Hassan Jolany’s theorem: A coadjoint orbit (Oφ, ωKKS) isMpc -quantizable if and only if
[ω]− 1
4
∑α∈R+
α
belongs to a lattice Zk
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Geometric Quantization on coadjoint orbits
Coadjoint Orbits Geometric Quantization
Thanks a lot for your attentionEND
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Geometric Quantization on coadjoint orbits