A holomorphic representation of the Jacobi algebra

A representation of the Jacobi algebra $\mathfrak{h}_1\rtimes \mathfrak{su}(1,1)$ by first order differential operators with polynomial coefficients on the manifold $\mathbb{C}\times \mathcal{D}_1$ is presented. The Hilbert space of holomorphic functions on which the holomorphic first order differential operators with polynomials coefficients act is constructed.Comment: 34 pages, corrected typos in accord with the printed version and the Errata in Rev. Math. Phys. Vol. 24, No. 10 (2012) 1292001 (2 pages) DOI: 10.1142/S0129055X12920018, references update

Drag in a resonantly driven polariton fluid

We study the linear response of a coherently driven polariton fluid in the pump-only configuration scattering against a point-like defect and evaluate analytically the drag force exerted by the fluid on the defect. When the system is excited near the bottom of the lower polariton dispersion, the sign of the interaction-renormalised pump detuning classifies the collective excitation spectra into three different categories (Ciuti and Carusotto 2005 Phys. Status Solidi b 242 2224): linear for zero, diffusive-like for positive and gapped for negative detuning. We show that both cases of zero and positive detuning share a qualitatively similar crossover of the drag force from the subsonic to the supersonic regime as a function of the fluid velocity, with a critical velocity given by the speed of sound found for the linear regime. In contrast, for gapped spectra, we find that the critical velocity exceeds the speed of sound. In all cases, the residual drag force in the subcritical regime depends on the polariton lifetime only. Also, well below the critical velocity, the drag force varies linearly with the polariton lifetime, in agreement with previous work (Cancellieri et al 2010 Phys. Rev. B 82 224512), where the drag was determined numerically for a finite-size defect

A convenient coordinatization of Siegel-Jacobi domains

We determine the homogeneous K\"ahler diffeomorphism $FC$ which expresses the K\"ahler two-form on the Siegel-Jacobi ball \mc{D}^J_n=\C^n\times \mc{D}_n as the sum of the K\"ahler two-form on \C^n and the one on the Siegel ball \mc{D}_n. The classical motion and quantum evolution on \mc{D}^J_n determined by a hermitian linear Hamiltonian in the generators of the Jacobi group G^J_n=H_n\rtimes\text{Sp}(n,\R)_{\C} are described by a matrix Riccati equation on \mc{D}_n and a linear first order differential equation in z\in\C^n, with coefficients depending also on W\in\mc{D}_n. $H_n$ denotes the $(2n+1)$-dimensional Heisenberg group. The system of linear differential equations attached to the matrix Riccati equation is a linear Hamiltonian system on \mc{D}_n. When the transform $FC:(\eta,W)\rightarrow (z,W)$ is applied, the first order differential equation in the variable \eta=(\un-W\bar{W})^{-1}(z+W\bar{z})\in\C^n becomes decoupled from the motion on the Siegel ball. Similar considerations are presented for the Siegel-Jacobi upper half plane \mc{X}^J_n=\C^n\times\mc{X}_n, where \mc{X}_n denotes the Siegel upper half plane.Comment: 32 pages, corrected typos, Latex, amsart, AMS font

Vlasov moment flows and geodesics on the Jacobi group

By using the moment algebra of the Vlasov kinetic equation, we characterize the integrable Bloch-Iserles system on symmetric matrices (arXiv:math-ph/0512093) as a geodesic flow on the Jacobi group. We analyze the corresponding Lie-Poisson structure by presenting a momentum map, which both untangles the bracket structure and produces particle-type solutions that are inherited from the Vlasov-like interpretation. Moreover, we show how the Vlasov moments associated to Bloch-Iserles dynamics correspond to particular subgroup inclusions into a group central extension (first discovered in arXiv:math/0410100), which in turn underlies Vlasov kinetic theory. In the most general case of Bloch-Iserles dynamics, a generalization of the Jacobi group also emerges naturally.Comment: 45 page

Identification of Berezin-Toeplitz deformation quantization

We give a complete identification of the deformation quantization which was obtained from the Berezin-Toeplitz quantization on an arbitrary compact Kaehler manifold. The deformation quantization with the opposite star-product proves to be a differential deformation quantization with separation of variables whose classifying form is explicitly calculated. Its characteristic class (which classifies star-products up to equivalence) is obtained. The proof is based on the microlocal description of the Szegoe kernel of a strictly pseudoconvex domain given by Boutet de Monvel and Sjoestrand.Comment: 26 page

Isospin equilibration processes and dipolar signals: Coherent cluster production

The total dipolar signal related to multi-break-up processes induced on the system 48Ca + 27Al at 40 MeV/nucleon has been investigated with the CHIMERA multi-detector. Experimental data related to semi-peripheral collisions are shown and compared with CoMD-III calculations. The strong connection between the dipolar signal as obtained from the detected fragments and the dynamics of the isospin equilibration processes is also shortly discussed

The Geometry of Quantum Mechanics

A recent notion in theoretical physics is that not all quantum theories arise from quantising a classical system. Also, a given quantum model may possess more than just one classical limit. These facts find strong evidence in string duality and M-theory, and it has been suggested that they should also have a counterpart in quantum mechanics. In view of these developments we propose "dequantisation", a mechanism to render a quantum theory classical. Specifically, we present a geometric procedure to "dequantise" a given quantum mechanics (regardless of its classical origin, if any) to possibly different classical limits, whose quantisation gives back the original quantum theory. The standard classical limit $\hbar\to 0$ arises as a particular case of our approach.Comment: 15 pages, LaTe