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

Singular projective varieties and quantization

By the quantization condition compact quantizable Kaehler manifolds can be embedded into projective space. In this way they become projective varieties. The quantum Hilbert space of the Berezin-Toeplitz quantization (and of the geometric quantization) is the projective coordinate ring of the embedded manifold. This allows for generalization to the case of singular varieties. The set-up is explained in the first part of the contribution. The second part of the contribution is of tutorial nature. Necessary notions, concepts, and results of algebraic geometry appearing in this approach to quantization are explained. In particular, the notions of projective varieties, embeddings, singularities, and quotients appearing in geometric invariant theory are recalled.Comment: 21 pages, 3 figure

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

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

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

Disparity in association of obesity measures with ankle and brachial systolic blood pressures in Europeans and South Asians

Obesity causes increases in brachial systolic-blood-pressures (SBP), risks of type 2 diabetes (T2DM) and cardiovascular diseases (CVD). Brachial and ankle SBPs have differential relationship with T2DM and CVD. Our objective was to study the relationship of obesity measures with brachial and ankle SBPs. A population of 1098 adults (South Asians n = 699; 41.70% male and 58.3% female) were recruited over 5 years from primary care practices in England. Their four limbs SBPs were measured using Doppler machine and body-mass-index (BMI) and waist-to-height-ratio (WHtR) calculated. Linear regressions were performed between SBPs and obesity measures, after adjustments for sex, age, ethnicity, T2DM and CVD. The mean age of all participants was 51.3 (SD = 17.2), European was 57.7 (SD 17.2) and South Asian was 47.8 (SD = 16.1). The left posterior tibial [Beta = 1.179, P = 4.559 × 10−15] and the right posterior tibial SBP [Beta = 1.178, P = 1.114 × 10−13] most significantly associated with the BMI. In South Asians, although the left brachial [Beta = 25.775, P = 0.032] and right brachial SBP [Beta = 22.792, P = 0.045] were associated to the WHtR, the left posterior tibial SBP [Beta = 39.894, P = 0.023], association was the strongest. For the first time, we have demonstrated that ankle SBPs had significant association with generalised obesity than brachial systolic blood pressures (SBP), irrespective of ethnicity. However, with respect to visceral obesity, the association with ankle SBP was more significant in South Asians compared to Europeans