arXiv subjects
Publications and source records attributed to C. Duval.
We extend projectively equivariant quantization and symbol calculus to symbols of pseudo-differential operators. An explicit expression in terms of hypergeometric functions with noncommutative arguments is given. Some examples are worked out, one of them yielding a quantum length element on $S^3$.
Taking advantage of the two-parameter central extension of the planar Galilei group, we construct a non relativistic particle model in the plane. Owing to the extra structure, the coordinates do not commute. Our model can be viewed as the non-relativistic counterpart of the relativistic anyon considered before by Jackiw and Nair. For a particle moving in a magnetic field perpendicular to the plane, the two parameters combine with the magnetic field to provide an effective mass. For vanishing effective mass the phase space admits a two-dimensional reduction, which represents the condensation to collective ``Hall'' motions and justifies the rule called ``Peierls substitution''. Quantization yields the wave functions proposed by Laughlin to describe the Fractional Quantum Hall Effect.
Let $(M,g)$ be a pseudo-Riemannian manifold and $F_λ(M)$ the space of densities of degree $λ$ on $M$. We study the space $D^2_{λ,μ}(M)$ of second-order differential operators from $F_λ(M)$ to $F_μ(M)$. If $(M,g)$ is conformally flat with signature $p-q$, then $D^2_{λ,μ}(M)$ is viewed as a module over the group of conformal transformations of $M$. We prove that, for almost all values of $μ-λ$, the $O(p+1,q+1)$-modules $D^2_{λ,μ}(M)$ and the space of symbols (i.e., of second-order polynomials on $T^*M$) are canonically isomorphic. This yields a conformally equivariant quantization for quadratic Hamiltonians. We furthermore show that this quantization map extends to arbitrary pseudo-Riemannian manifolds and depends only on the conformal class $[g]$ of the metric. As an example, the quantization of the geodesic flow yields a novel conformally equivariant Laplace operator on half-densities, as well as the well-known Yamabe Laplacian. We also recover in this framework the multi-dimensional Schwarzian derivative of conformal transformations.
This article is a survey of recent work of the authors developing a new approach to quantization based on the equivariance with respect to some Lie group of symmetries. Examples are provided by conformal and projective differential geometry: given a smooth manifold M endowed with a flat conformal/projective structure, we establish a canonical isomorphism between the space of symmetric contravariant tensor fields on M and the space of differential operators on M. This leads to a notion of conformally/projectively invariant star-product on $T^*M$.
We prove the existence and the uniqueness of a conformally equivariant symbol calculus and quantization on any conformally flat pseudo-Riemannian manifold $(M,\rg)$. In other words, we establish a canonical isomorphism between the spaces of polynomials on $T^*M$ and of differential operators on tensor densities over $M$, both viewed as modules over the Lie algebra $\so(p+1,q+1)$ where $p+q=\dim(M)$. This quantization exists for generic values of the weights of the tensor densities and compute the critical values of the weights yielding obstructions to the existence of such an isomorphism. In the particular case of half-densities, we obtain a conformally invariant star-product.
We relate in this note the classical Schwarzian derivative to the curvature of time-like curves in Lorentz surfaces of constant curvature.
We investigate, in some details, symplectic equivalence between several conformal classes of Lorentz metrics on the hyperboloid of one sheet $H^{1,1} \cong T \times T - Δ$ and affine coadjoint orbits of the group $Diff_+(Δ)$ of orientation preserving diffeomorphisms of $Δ\cong T$ with its natural projective structure. This will allow for generalizations, namely, to the case of arbitrary projective structures on null infinity.
It is shown that the non-relativistic `Dirac' equation of Lévy-Leblond, we used recently to describe a spin $1/2$ field interacting non-relativistically with a Chern-Simons gauge field, can be obtained by lightlike reduction from $3+1$ dimensions. This allows us to prove that the system is Schrödinger symmetric. A spinor representation of the Schrödinger group is presented. Static, self-dual solutions, describing spinor vortices are given and shown to be the non-relativistic limits of the fermionic vortices found by Cho et al. The construction is extended to external harmonic and uniform magnetic fields.
The non-relativistic `Dirac' equation of Lévy-Leblond is used to describe a spin {\small 1/2} particle interacting with a Chern-Simons gauge field. Static, purely magnetic, self-dual spinor vortices are constructed. The solution can be `exported' to a uniform magnetic background field.
The space of linear differential operators on a smooth manifold $M$ has a natural one-parameter family of $Diff(M)$ (and $Vect(M)$)-module structures, defined by their action on the space of tensor-densities. It is shown that, in the case of second order differential operators, the $Vect(M)$-module structures are equivalent for any degree of tensor-densities except for three critical values: $\{0,{1\over 2},1\}$. Second order analogue of the Lie derivative appears as an intertwining operator between the spaces of second order differential operators on tensor-densities.
The construction and the symmetries of Chern-Simons vortices in harmonic and uniform magnetic force backgrounds found by Ezawa, Hotta and Iwazaki, and by Jackiw and Pi are generalized using the non-relativistic Kaluza-Klein-type framework presented in our previous paper. All Schrodinger-symmetric backgrounds are determined.
The vanishing of the anomaly in the recent example of Nappi and Witten, constructed from the Wess-Zumino-Witten model based on a certain non-semisimple group, follows from a more general result valid for gravitational waves. The construction of the metric is explained.}
The non-relativistic conformal symmetry found by Jackiw and Pi for the coupled Chern-Simons and gauged nonlinear Schrödinger equations in the plane is derived in a non-relativistic Kaluza-Klein framework.