In this thesis, we provide a proof of the quantization-reduction problem for general symplectic manifolds. In the final chapter of the thesis, we propose some ideas for future work in this direction.
The motivation for geometric quantization
Geometric prequantization
It is taken as the complement of Γ(L) with respect to this inner product, and the quantization Qf :H → Hof is defined as a classical observable. In order for Q· to satisfy the condition [Qf, Qg] =i~Q{f, g}, the covariant ∇ must be chosen to have curvature.
Geometric quantization
The inner product of ΓF(L) must also be modified: F ∩F can be written as DC for some real subsetD of T M, and the involutivity of F implies the involutivity of D (assuming that dimR(F ∩T M) is constant). For this reason, an inner product (s, t) is determined instead by throwing H(s, t) into M/D and integrating against a suitable measure defined in M/D.
Symplectic reduction and its interaction with geometric quantization
Since geometric quantization is so closely related to the Symplectic structure (M, ω), it is natural to ask whether geometric quantization "nicely" affects the reduction. As far as we know, the first work that deals with reduction in the context of geometric quantization is the paper by Reyman and Semenov-Tian-Shansky1 [RSTS79].
New results in this thesis
Due to the fact that double foliations {J−1(µ)|µ∈ O}and{π−1(a)|a∈π(J−1(O))} are symplectic complements of each other, a one defines a canonical symplectomorphism between J−1R(O)O and π−1Ra(a)×J−1Rµ(µ). The corresponding characteristic foliations (Rh)O,(Rh)µ, and (Rh)a appear to be the horizontal uplifts of the characteristic foliations on the base space.
Limitations of the results
At the end of this chapter, it was noted that the approach has much in common with the original concept of circuit reduction due to Marle [Mar76], [LM87]. Unless otherwise stated, all manifolds and maps discussed in this thesis (including group actions) will be assumed to be smooth.
Lie group and Lie algebra actions
Proper group actions
If H ⊂G is a subgroup of G, the left action H (h, g)7→gh−1 is proper G if and only if H is closed. Since the isotropic group Gx of an action Φ :G×M → M is closed for every x∈ M, the last result and Proposition 2.3.1 show us that G/Gx can be given a smooth structure (namely, what does G→G / Dive Gxa).
Hamiltonian vector fields, Poisson brackets, and symplectic actions
For linear operations G on vector spaces (eg, the joint operation discussed in the next section), the isotropy set of the origin is the entire set G. The right-hand side, being a composition of smooth maps, is smooth , and thus the initial condition of the submanifold for S implies that Φ0:H×S→S is smooth.
Adjoint and coadjoint actions
It is worth noting that in terms of the left action Φg(h) =gand the right action Ψg(h) =hg, we have that. In terms of the above notation, the isotropy groupGµ and the algebraic symmetry of the associative action ong∗ atµ are.
The momentum map
There are several situations in which an equivariant momentum map can be shown to exist. Such an equivariant momentum map is guaranteed to exist in this case as well, provided M is connected.
Notation for projections, inclusions, and restrictions
Then the definition of C is independent of the choice of x∈M,C defines a g∗-valued 1-cocycle on G, and there exists an equivariant momentum map if and only if[C]is trivial inH1(G,g∗ ). From the previous discussion, we already know that an equivariant momentum map J exists, and we just need to establish uniqueness.
Smooth structures on inverse images and their quotients
Transversal mappings
Finally, we will sometimes indicate the restrictions of the symplectic formω to J−1(O),J−1(µ), andπ−1(a) in an analogous way.
Application to the momentum and projection maps
Since J is an immersion, it is in particular transversal to O and so Theorem 2.8.1 (ii) tells us that J−1(O) can be given a smooth structure which makes it an initial submanifold of M and JO makes : J−1(O)→ O an immersion. Since the co-adjoint isotropy group Gµ is closed for every µ, Theorem 2.3.2 guarantees the existence of a smooth structure on the quotient space J−1G(µ).
Relationships between the inverse images
Preservation of submanifold properties under submersions
Properties of immersions, embeddings, and submersions
If S is embedded inM and T is initial inM, then S is embedded inT. Since T is initial inM, it follows that iS,T :S→T is smooth. Since S is initial in M, it follows that f :P →S is smooth. iii) T embedded in M means that TTM ⊂ TT, while S embedded in M means TSM =TS. i) there exists a smooth map f :C→D that makes the following diagram commute:.
Application to quotient spaces under the group action
Foliation reduction
Thens andt are in the same sheet of the foliation N, and sot =ϕ(s) whereϕ is a diffeomorphism consisting of a finite composition of currents exp(Xn)◦exp(Xn−1)◦. Using Tϕ(s)πN◦Tsϕ=Ts(πN◦ϕ) =TsπN, the terms in square brackets project to zero under TtπN, and so must lie in Nt, and in particular the degeneracy directions are forβt.
The foliation-reduced symplectic manifolds
We will demonstrate in Section 2.15 that foliation reduction on (J−1R(O)O, ωRO) corresponds to the circuit reduction image ([MMO+07]) of symplectic reduction. Since ωRO, ωµR and ωaR are obtained by "quoting out" the same fibersR, the latter two are limitations of the first.
The reduced group action, projection, and momentum map
Also since we have a G-action on the reduced symplectic manifold (J−1R(O)O , ωOR), it is logical to ask whether there exists a corresponding momentum map, and whether it is equivalent with respect to the G-action . In terms of the notation for individual R-leaves, (i) the reduced G-action satisfies g· R(µ,a)=R(Ad∗g−1µ,a); ii) the reduced projection satisfies πOR(R(µ,a)) =a;. iii) the reduced momentum map satisfies JOR(R(μ,a)) =μ. i) The action catch∈Gon the leaf space is.
The canonical symplectomorphism
The inverse function theorem implies that φOR is a local diffeomorphism, and part (i) implies that it is a global diffeomorphism. iii) First we note that since TR(ν,b).
Relationship to orbit reduction
- Notation for left modules and their quotients
- Connections and curvature
- Connection-invariant Hermitian forms
- Equivalence classes of bundle-connection pairs with given curvature
Properties (i) and (ii) ensure that the result is independent of the lift or point. Moreover, for M connected unequal bundle connection pairs ( ˙L, α) with the same curvatureΩα= Ω are characterized by elements of the sign group π1(M)∗ = Hom(π1(M),U(1)) of the basic group of M.
The prequantization procedure
- The prequantization of classical observables
- Geometric interpretation of the prequantized observables
- Closedness of the lifted vector fields under the Lie bracket
- The lifted Lie group and Lie algebra actions
In terms of the original divisions∈Γ(L), the right-hand side is just Qfs=−i~∇Xfs+f s, and Uft corresponds to the exponentiation operator. The construction of the reduced U(1) bundle (RL˙hµ)µ detailed in this chapter corresponds to that proposed in [RSTS79].
Properties of the universal cover of a compact semisimple Lie group
Let R=TR denote the tangent distribution to the generalized foliation R ⊂T M, and Rh⊂TL˙ its horizontal lift on ˙L. By the above convention, let RO denote the restriction of R to J−1(O), (Rh)O the restriction of Rh to ˙LO, and similarly for Rµ, (Rh)µ, Ra and (Rh)a.
Foliation reduction of the prequantum data
- The characteristic distributions of the restricted connections
- Consistency of foliation reduction and the notion of admissibility
- A characterization of admissible momenta
- Foliation reduction of the bundles
- The bundle structure of the reduced spaces
- The reduced connections and their curvatures
The freedom of the action follows from the fact that the horizontal Gμ action on ˙Lμ covers the Gμ action on J−1(μ), and the freedom of the Gμ action on J−1(μ). Examination of the proof of Theorem 4.3.6 makes it clear that the existence of a horizontal Gµ action on the beam ˙L(a,µ) passing over R(a,µ) = π−1(a)∩J− 1 lies (μ) is sufficient to guarantee the existence of the characterχ−~iμ, which in turn guarantees the existence of the horizontal action on all of J−1(μ).
The reduced lifted group action
If we apply ˙ΣO to both sides of the above equation, and take advantage of the fact that ˙ΣO is both U(1) and G equivariant, we get e. This compatibility allows the polarization to be dropped to the reduced space, and guarantees a one-to-one correspondence between covariant constant sections of the unreduced and reduced spaces.
The use of polarizations in quantization
A polarization on coadjoint orbits
The structure of simple Lie algebras
Let ad-stabk(ζ) denote the stabilizer ofζ∈kunder the additional action, ad-stabk(ζ) ={ξ∈k|adξ(ζ) = 0}. also called the centralizer of ζ in k), and ad∗-stabk(µ) denotes the stabilizer of µ ∈ k∗ under the coadjoint action.
Construction of the polarization
Therefore adξ, ξ∈g, is skew symmetric with respect to some inner product of g, and can therefore be represented on an orthogonal basis by a skew symmetric matrix Aξ. Since the space of vector fields iRonπ−1(U) forms a C∞(π−1(U)) module with basisXi, it follows from the properties of the Jacobi-Lie bracket that it is involutive.
Connection with the root space decomposition
If T is a maximal torus of G (i.e. a maximal commutatively connected subgroup of G), with corresponding Lie algebra, then h=tC is a Cartan subalgebra ork=gC, and the corresponding root space decomposition is. It is a standard theorem that every element of G is contained in some maximal torus T, which implies that every element of g is contained in some maximal commutative (real) subalgebra.
A polarization compatible with foliation reduction
F is a generalized polarization in the sense that it is not necessarily smooth, nor does it necessarily satisfy dimR(T M ∩F) = constant. For a given coadjoint orbit O, let QJ−1(O)/G be an arbitrary polarization on J−1G(O) and let P be the polarization on O described in the previous section.
Admissibility and covariantly constant sections
The reduced polarizations
The relationship between covariant sections on the reduced and unreduced bundles . 79
- Equivalence of the unreduced and reduced representations
- The correspondence between polarized sections and functions on the group . 80
- Irreducibility of the representation
- Application to the cotangent bundle of a Lie group
- The symplectomorphism between the reduced spaces revisited
- The lifted construction
- The polarization isomorphism
We are now able to demonstrate the irreducibility of the G representation on Γe FRa(LaR). This fact motivates the following construction of the canonical symplectomorphism, which will be "lifted" to a U(1)-bundle-connection isomorphism in Section 6.1.2.
The lifted dynamics and group action under the decomposition
- The relation between unreduced and reduced flows
- Group invariance and decomposition
- The lifted dynamics under the decomposition
- The lifted group action under the decomposition
Recall the discussion from Section 3.2.2: the elevated flux (˙ψtH)O in ˙LO is generated by the vector field AOH. The raised flow in (RL˙hO)O decomposes in the identification times a flow in (RL˙hµ)µ under decomposition.
Decomposition of the space of covariantly constant sections
We denote the corresponding isomorphism of sections using the same symbol for convenience Eqa,µ0 : Γ (LaRLµR)−→Γ LOR. The discussion of the previous section shows us that through the decomposition (˙Eqa,µ0)−1, the action G and the elevated flow ˙e ψHtO.
Commutativity of quantization and reduction
The discussion of the Borel-Weil theorem in section 5.8 tells us that ΓFRa(LaR)'q0 (H~iµ)∗ (whereq0∈L˙aRlies above R(µ,a)). It is seen that the factorization of the space of polarized sections coming from quantum reduction (Appendix C) is caused by a corresponding factorization on the reduced U(1) beams, which in turn gives rise to the canonical symplectomorphism J−1R(O)O ' covers' π−1R(a)a ×J−1R(μ)μ of the basic manifold.
The isomorphism group as a Lie group central extension of the isotropy group
The lifted Hamiltonian vector fields as a Lie algebra central extension of the Hamil-
Relationship between the central extensions
This is exactly the one-parameter subgroup generated by X = Xf (ie, the Hamiltonian flux φtf off). We now have a one-parameter subgroup Isom( ˙L, α ), which, according to part (i), is generated by the vector field Af for some ∈C∞(M).
The exterior tensor product on line bundles
The connection on the exterior product and its curvature
The induced covariant derivative on sections of the exterior tensor product of line
The symmetry defines a projective representation of G onto H, which can be raised to a proper representation U :Ge→U(H) of the universalGeofG. The isometry Kλ : ImPλ −→(Hλ)∗⊗(Hλ⊗ H)Ge depends implicitly on the choice of the maximal torus He ⊂G, mimicking the q0-dependence of the decomposition (Eqa,µ0 )−1 : ΓFO.
Complex structures
We do this by taking advantage of the fact that ΓFRa(LaR)'R(µ,a) (H~iµ)∗, and by considering the space of G-invariant covariant constant sections of (Le bR)∗LaR. For strictly positive complex structures, the non-degenerate bilinear form g(X, Y) := ω(X,JFY) (which can easily be shown to be symmetric based on properties of the complex structure) defines a Riemann metric, called the K¨ahler metricon (M, ω) corresponding toJF.
The complexified group action
The involutivity of F implies that M can be given the structure of a complex manifold in such a way that JF corresponds to a holomorphic complex structure described in any system of complex coordinates zα=xα+iyα angle. This defines the structure of an infinitesimalgag⊕ig-action on M that potentiates into aGfC-action.
Lifting of the complexified action
C×(˙LC,C) of equivariant functions on ˙LC (from equivariant extension in one direction, restriction in the other), and clearly constant covariant sections of Lgo to constant covariant sections of LC. Then for any Hamiltonian vector field Xf we have (Xf−iJFXf)hs˙= 0, or equivalently (JFXf)hs˙=i(Xfhs).
The complex structure on the reduced group orbit
Therefore, the complex structure JFRa on π−1Ra(a) is just that induced by the usual complex structure on GC/Pµ− under the diffeomorphism π−1R(a)a 'R(µ,a) GC. Looking back at the definition of the GC-action on π−1Ra(a), we see that it simply corresponds to the natural left GC-action on GC/Pµ− under the above diffeomorphism.
Schur’s Lemma
Proceedings of the Symposium held at the University of Bonn, July Lecture Notes in Mathematics, vol. Harris, Principles of Algebraic Geometry, Wiley Classics Library, John Wiley & Sons Inc., New York, 1994, reprint of the 1978 original.
