Fractional Fourier transform and geometric quantization

Generalized Fourier transformation between the position and the momentum representation of a quantum state is constructed in a coordinate independent way. The only ingredient of this construction is the symplectic (canonical) geometry of the phase-space: no linear structure is necessary. It is shown that the “fractional Fourier transform” provides a simple example of this construction. As an application of this technique we show that for any linear Hamiltonian system, its quantum dynamics can be obtained exactly as the lift of the corresponding classical dynamics by means of the above transformation. Moreover, it can be deduced from the free quantum evolution. This way new, unknown symmetries of the Schrödinger equation can be constructed. It is also argued that the above construction defines in a natural way a connection in the bundle of quantum states, with the base space describing all their possible representations. The non-flatness of this connection would be responsible for the non-existence of a quantum representation of the complete algebra of classical observables.     

Singular unitarity in “quantization commutes with reduction”

Let M$M$ be a connected compact quantizable Kähler manifold equipped with a Hamiltonian action of a connected compact Lie group G$G$. Let M//G=?⁻¹(0)/G=_M0$M//G={?}^{-1}\left(0\right)/G=M_0$ be the symplectic quotient at value 0 of the moment map ?$?$. The space _M0$M_0$ may in general not be smooth. It is known that, as vector spaces, there is a natural isomorphism between the quantum Hilbert space over _M0$M_0$ and the G$G$-invariant subspace of the quantum Hilbert space over M$M$. In this paper, without any regularity assumption on the quotient _M0$M_0$, we discuss the relation between the inner products of these two quantum Hilbert spaces under the above natural isomorphism; we establish asymptotic unitarity to leading order in Planck’s constant of a modified map of the above isomorphism under a “metaplectic correction” of the two quantum Hilbert spaces.     

Semiclassical Almost Isometry

Let (M , ω , J) be a compact and connected polarized Hodge manifold, an isodrastic leaf of half-weighted Bohr–Sommerfeld Lagrangian submanifolds. We study the relation between the Weinstein symplectic structure of and the asymptotics of the the pull-back of the Fubini–Study form under the projectivization of the so-called BPU maps on .     

Geometric quantization of weak-Hamiltonian functions

The paper presents an extension of the geometric quantization procedure to integrable, big-isotropic structures. We obtain a generalization of the cohomology integrality condition, we discuss geometric structures on the total space of the corresponding principal circle bundle and we extend the notion of polarization.     

Geometric quantization of an OSp(1/2) coadjoint orbit

In this Letter, we show how the complete geometric quantization extends to specific supersymplectic supermanifolds. More precisely, we extend this procedure to OSp(1/2)-coadjoint orbits, which are graded extensions of elliptic Sp(2, )-coadjoint orbits. Our approach exploits results obtained in a previous work, where the notion of a super-Kähler supermanifold was defined, and the former orbits were shown to be nontrivial examples of such a notion. As their underlying Kähler manifolds, these supermanifolds carry a natural (super-Kähler) polarization, a crucial notion that was so far lacking. Geometric quantization leads here to a nontrivial representation of osp(1/2), which is realized in a space of square integrable holomorphic sections of a super-Hermitian complex line bundle sheaf-with-connection over the homogenous space OSp(1/2)/U(1).     

Deformation quantization of nonholonomic almost Kähler models and Einstein gravity

Nonholonomic distributions and adapted frame structures on (pseudo) Riemannian manifolds of even dimension are employed to build structures equivalent to almost Kähler geometry and which allows to perform a Fedosov-like quantization of gravity. The nonlinear connection formalism that was formally elaborated for Lagrange and Finsler geometry is implemented in classical and quantum Einstein gravity.     

Coherent states in geometric quantization

In this paper we study overcomplete systems of coherent states associated to compact integral symplectic manifolds by geometric quantization. Our main goals are to give a systematic treatment of the construction of such systems and to collect some recent results. We begin by recalling the basic constructions of geometric quantization in both the Kähler and non-Kähler cases. We then study the reproducing kernels associated to the quantum Hilbert spaces and use them to define symplectic coherent states. The rest of the paper is dedicated to the properties of symplectic coherent states and the corresponding Berezin–Toeplitz quantization. Specifically, we study overcompleteness, symplectic analogues of the basic properties of Bargmann’s weighted analytic function spaces, and the ‘maximally classical’ behavior of symplectic coherent states. We also find explicit formulas for symplectic coherent states on compact Riemann surfaces.     

Deformation quantization of the Pais–Uhlenbeck fourth order oscillator

We analyze the quantization of the Pais–Uhlenbeck fourth order oscillator within the framework of deformation quantization. Our approach exploits the Noether symmetries of the system by proposing integrals of motion as the variables to obtain a solution to the ?$?$-genvalue equation, namely the Wigner function. We also obtain, by means of a quantum canonical transformation the wave function associated to the Schrödinger equation of the system. We show that unitary evolution of the system is guaranteed by means of the quantum canonical transformation and via the properties of the constructed Wigner function, even in the so called equal frequency limit of the model, in agreement with recent results.     

Equivariant quantization of orbifolds

Equivariant quantization is a new theory that highlights the role of symmetries in the relationship between classical and quantum dynamical systems. These symmetries are also one of the reasons for the recent interest in quantization of singular spaces, orbifolds, stratified spaces, etc. In this work, we prove the existence of an equivariant quantization for orbifolds. Our construction combines an appropriate desingularization of any Riemannian orbifold by a foliated smooth manifold, with the foliated equivariant quantization that we built in Poncin et al. (2009) [19]. Further, we suggest definitions of the common geometric objects on orbifolds, which capture the nature of these spaces and guarantee, together with the properties of the mentioned foliated resolution, the needed correspondences between singular objects of the orbifold and the respective foliated objects of its desingularization.     

New Spin Generalisation for Long Range Interaction Models

We study new interactions between degrees of freedom for Calogero, Sutherland and confined Calogero spin models. These interactions are encoded by the generators of the Lie algebra so(N) or sp(N). We find the symmetry algebras of these new models: the half-loop algebra based on so(N) or sp(N) for the Calogero models and the Yangian of so(N) or sp(N) for the two types of other models. Surprisingly, these symmetry occur only for a specific value of the coupling constant.Dedicated to my PhD supervisor and friend D. Arnaudon.     

A first approximation for quantization of singular spaces

Many mathematical models of physical phenomena that have been proposed in recent years require more general spaces than manifolds. When taking into account the symmetry group of the model, we get a reduced model on the (singular) orbit space of the symmetry group action. We investigate quantization of singular spaces obtained as leaf closure spaces of regular Riemannian foliations on compact manifolds. These contain the orbit spaces of compact group actions and orbifolds. Our method uses foliation theory as a desingularization technique for such singular spaces. A quantization procedure on the orbit space of the symmetry group–that commutes with reduction–can be obtained from constructions which combine different geometries associated with foliations and new techniques originated in Equivariant Quantization. The present paper contains the first of two steps needed to achieve these just detailed goals.     

Goldfish Geodesics and Hamiltonian Reduction of Matrix Dynamics

We describe the Hamiltonian reduction of a time-dependent real-symmetric N×N matrix system to free vector dynamics, and also provide a geodesic interpretation of Ruijsenaars–Schneider systems. The simplest of the latter, the goldfish equation, is found to represent a flat-space geodesic in curvilinear coordinates.      

The vacuum and lightcone quantization of interaction Hamiltonians

Let denote the conformally invariant neutral free scalar field on ×S ⁿ. The naive lightcone Hamiltonian for a ^p interaction is given by c^p, where C denotes a lightcone in ×S ⁿ, and the Wick power is relative to the free vacuum. We show that this sesquilinear form annihilates the free vacuum if n3 is odd, p>2, and p(n–1)0 mod 4.     

Differential geometry of the Fermat quartic and theta functions

The universal curve over a finite cover of the moduli space of elliptic curves with level four structure is embedded in CP³ as the Fermat quartic and is parametrized via the four Jacobi theta functions. Constructions from completely integrable systems have shown the importance of looking at the curvature of certain spaces and here we compute sectional curvatures. For our computations, we choose the ambient Fubini-Study metric of CP³. We also derive several theta identities which arise from the quartic’s holomorphic two-form.     

The cohomology of the classical and quantum Weyl CCR in curved spaces

An analysis is presented of the cohomological underpinnings for the Weyl group of the canonical commutation relations on manifolds of constant negative curvature. Several uniqueness results are obtained leading from purely classical considerations to the group associated with the systems of imprimitivity of the orthodox approach to quantum mechanics.     

Canonical quantization and the spectral action; a nice example

We study the canonical quantization of the theory given by Chamseddine–Connes spectral action on a particular finite spectral triple with algebra _M2(C)⊕C$M_2\left(\mathbb{C}\right)\oplus \mathbb{C}$. We define a quantization of the natural distance associated with this noncommutative space and show that the quantum distance operator has a discrete spectrum. We also show that it would be the same for any other geometric quantity. Finally we propose a physical Hilbert space for the quantum theory. This spectral triple had been previously considered by Rovelli as a toy model, but with a different action which was not gauge invariant. The results are similar in the two cases, but the gauge invariance of the spectral action manifests itself by the presence of a non-trivial degeneracy structure for our distance operator.     

Differential Galois obstructions for non-commutative integrability

We show that if a holomorphic Hamiltonian system is holomorphically integrable in the non-commutative sense in a neighbourhood of a non-equilibrium phase curve which is located at a regular level of the first integrals, then the identity component of the differential Galois group of the variational equations along the phase curve is Abelian. Thus necessary conditions for the commutative and non-commutative integrability given by the differential Galois approach are the same.