Closedness of star products and cohomologies

We first review the introduction of star products in connection with deformations of Poisson brackets and the various cohomologies that are related to them. Then we concentrate on what we have called ``closed star products" and their relations with cyclic cohomology and index theorems. Finally we shall explain how quantum groups, especially in their recent topological form, are in essence examples of star products.Comment: 16 page

A No-Go Theorem for M5-brane Theory

The BLG model for multiple M2-branes motivates an M5-brane theory with a novel gauge symmetry defined by the Nambu-Poisson structure. This Nambu-Poisson gauge symmetry for an M5-brane in large C-field background can be matched, on double dimension reduction, with the Poisson limit of the noncommutative gauge symmetry for a D4-brane in B-field background. Naively, one expects that there should exist a certain deformation of the Nambu-Poisson structure to match with the full noncommutative gauge symmetry including higher order terms. However, We prove the no-go theorem that there is no way to deform the Nambu-Poisson gauge symmetry, even without assuming the existence of a deformation of Nambu-Poisson bracket, to match with the noncommutative gauge symmetry in 4+1 dimensions to all order, regardless of how the double dimension reduction is implemented.Comment: v4: minor modifications

Deformation Quantization of Geometric Quantum Mechanics

Second quantization of a classical nonrelativistic one-particle system as a deformation quantization of the Schrodinger spinless field is considered. Under the assumption that the phase space of the Schrodinger field is $C^{\infty}$, both, the Weyl-Wigner-Moyal and Berezin deformation quantizations are discussed and compared. Then the geometric quantum mechanics is also quantized using the Berezin method under the assumption that the phase space is $CP^{\infty}$ endowed with the Fubini-Study Kahlerian metric. Finally, the Wigner function for an arbitrary particle state and its evolution equation are obtained. As is shown this new "second quantization" leads to essentially different results than the former one. For instance, each state is an eigenstate of the total number particle operator and the corresponding eigenvalue is always ${1 \over \hbar}$.Comment: 27+1 pages, harvmac file, no figure

Comments on the topological open membrane

Just as non-commutative gauge theories arise from quantising open strings in a large magnetic field, non-Abelian two-form gauge theories may conceivably be constructed by quantising open membranes in a large three-form magnetic background. We make some observations that arise in following this strategy, with an emphasis on the relation to the quantisation of volume-preserving diffeomorphisms (vpd). In particular, we construct consistent non-Abelian interactions of a two-form in 3+1 dimensions, based on gauge invariance under vpd.Comment: 6 pages, latex2e, uses revtex

Deformation quantization of linear dissipative systems

A simple pseudo-Hamiltonian formulation is proposed for the linear inhomogeneous systems of ODEs. In contrast to the usual Hamiltonian mechanics, our approach is based on the use of non-stationary Poisson brackets, i.e. corresponding Poisson tensor is allowed to explicitly depend on time. Starting from this pseudo-Hamiltonian formulation we develop a consistent deformation quantization procedure involving a non-stationary star-product $*_t$ and an ``extended'' operator of time derivative $D_t=\partial_t+...$, differentiating the $\ast_t$-product. As in the usual case, the $\ast_t$-algebra of physical observables is shown to admit an essentially unique (time dependent) trace functional $\mathrm{Tr}_t$. Using these ingredients we construct a complete and fully consistent quantum-mechanical description for any linear dynamical system with or without dissipation. The general quantization method is exemplified by the models of damped oscillator and radiating point charge.Comment: 14 pages, typos correcte

On Deformations of n-Lie algebras

The aim of this paper is to review the deformation theory of $n$-Lie algebras. We summarize the 1-parameter formal deformation theory and provide a generalized approach using any unital commutative associative algebra as a deformation base. Moreover, we discuss degenerations and quantization of $n$-Lie algebras.Comment: Proceeding of the conference Dakar's Workshop in honor of Pr Amin Kaidi. arXiv admin note: text overlap with arXiv:hep-th/9602016 by other author

MR of the small bowel with a biphasic oral contrast agent (polyethylene glycol): technical aspects and findings in patients affected by Crohn's disease

PURPOSE: To report our experience using MR of the small bowel with polyethylene glycol (PEG) solution as an oral contrast agent in a population of adults and children with known Crohn's disease. MATERIALS AND METHODS: 40 patients (29 males; 11 females), 15 adults (age range 24-52 years) and 25 children (age range 5-17 years), with known Crohn's disease, underwent MR of the small bowel using a supeconductive 1.5 T magnet, and polyethylene glycol solution as an oral contrast agent. The fixed amount of contrast agent was 750-1000 ml for adults and 10 ml/kg of body weight for children. The Crohn's Disease Activity Index (CDAI) was available in all patients. Our study protocol included the acquisition of T2-weighted half-Fourier single-shot turbo spin-echo (HASTE) sequences and true fast imaging in the steady-state precession (true-FISP) sequences, followed by the acquisition of "spoiled" 2D gradient echo T1-weighted sequences with fat suppression (FLASH, fast low-angle shot) or alternatively "spoiled" 3D (VIBE, volume interpolated breath-hold examination), acquired 70 seconds after intravenous administration of gadopentetate dimeglumine (Gd-DTPA) (0,1 mmol/kg). A specific MR score was created and calculated for each patient and was compared by means of the Spearman rank with CDAI. RESULTS: In all patients no significant side effects were observed and the MR examination was well tolerated even by paediatric patients. In all cases MR showed a small bowel wall thickening (> 4 mm) in the terminal ileum, with lumen stenosis in 26 patients. In 3 cases pathological segments proximal to the terminal ileum were observed and in another 3 cases caecal involvement was visible. The MR examination was able to show abnormalities of perivisceral fat tissue in 15 patients, mesenteric lymphadenopathy in 1 patient and abdominal abscess in 1 case. The Spearman rank showed a statistically significant correlation between CDAI and the MR score (r = 0.91, P = 0,0001). DISCUSSION: MR using PEG as an oral contrast agent could be considered a test of great interest in the evaluation of the small bowel in patients suspected of having Crohn's disease in that it is easily reproducible, well tolerated even by paediatric patients and it provides useful information about the localisation, extension and activity of inflammatory disease without the use of ionising radiation

Contrast-enhanced magnetic resonance imaging (MRI) of the distal small bowel in children with Chron's disease

JPGN 2002; vol 34, No. 4, pg 446, No.

Contrast enhanced magnetic resonance imaging of the terminal ileum in children with Crohn’s disease

Background and aims: Recently, magnetic resonance imaging (MRI) has been introduced in the diagnosis of patients with inflammatory bowel disease (IBD). However, it is still rarely reported in paediatric IBD. We studied the diagnostic value of gadolinium enhanced MRI in revealing inflammation of the distal ileum in children with Crohn’s disease (CD) and in differentiating them from patients with other inflammatory diseases of the gut. MRI was performed using a polyethylene glycol (PEG) solution as oral contrast agent to distend the small bowel (CE-PEG-MRI). Subjects and methods: Seventy five consecutive patients (median age 13.6 years, range 8–17) with suspected CD underwent ileocolonoscopy with biopsy and CE-PEG-MRI. CD activity was measured by the paediatric Crohn’s disease activity index (PCDAI). CE-PEG-MRI was evaluated with an overall score calculated, taking into account both wall thickness and contrast enhancement. Results: Active CD with distal ileitis was diagnosed in 26 cases, active ulcerative colitis (UC) in 18, and spondyloarthropathy and indeterminate ileocolitis in 11; 20 children served as controls. In all CD patients, CE-PEG-MRI revealed a marked ileal involvement with increased wall thickness and parietal contrast enhancement and showed a high concordance with endoscopy and histology, whereas the test was negative in all controls. Of the 18 UC patients, CE-PEG-MRI was negative in 15 and showed a mild parietal contrast enhancement of the terminal ileum in only three of seven patients with backwash ileitis. Among the group of spondyloarthropathy patients, six had mucosal erosions and five mild superficial ileitis: CE-PEG-MRI was negative in four and revealed only mild parietal contrast enhancement of the ileal wall in seven. CE-PEG-MRI did not show an increase in wall thickness of the distal ileum in any of the UC or spondyloarthropathy patients. The sensitivity and specificity of CE-PEG-MRI related to the presence of erosive ileitis, as documented by endoscopy, were 84% and 100%, respectively. In addition, the test correlated markedly with endoscopy and histology in the entire population (r=0.94; r=0.95, respectively) as well as with the PCDAI in CD patients (r=0.91). Conclusions: In children with active CD, CE-PEG-MRI is a very sensitive and specific test for the detection of distal ileitis and for differentiation from other inflammatory diseases of the gut. The test could also be useful for the firstline diagnostic approach in children with suspected CD. The high correlation of CE-PEG-MRI with ileal endoscopy and histology as well as with PCDAI makes this test of great interest for future studies as a tool for monitoring the clinical course and the effect of therapy in CD patients