Holomorphic Factorization for a Quantum Tetrahedron

We provide a holomorphic description of the Hilbert space H(j_1,..,j_n) of SU(2)-invariant tensors (intertwiners) and establish a holomorphically factorized formula for the decomposition of identity in H(j_1,..,j_n). Interestingly, the integration kernel that appears in the decomposition formula turns out to be the n-point function of bulk/boundary dualities of string theory. Our results provide a new interpretation for this quantity as being, in the limit of large conformal dimensions, the exponential of the Kahler potential of the symplectic manifold whose quantization gives H(j_1,..,j_n). For the case n=4, the symplectic manifold in question has the interpretation of the space of "shapes" of a geometric tetrahedron with fixed face areas, and our results provide a description for the quantum tetrahedron in terms of holomorphic coherent states. We describe how the holomorphic intertwiners are related to the usual real ones by computing their overlap. The semi-classical analysis of these overlap coefficients in the case of large spins allows us to obtain an explicit relation between the real and holomorphic description of the space of shapes of the tetrahedron. Our results are of direct relevance for the subjects of loop quantum gravity and spin foams, but also add an interesting new twist to the story of the bulk/boundary correspondence.Comment: 45 pages; published versio

Supercoherent States, Super K\"ahler Geometry and Geometric Quantization

Generalized coherent states provide a means of connecting square integrable representations of a semi-simple Lie group with the symplectic geometry of some of its homogeneous spaces. In the first part of the present work this point of view is extended to the supersymmetric context, through the study of the OSp(2/2) coherent states. These are explicitly constructed starting from the known abstract typical and atypical representations of osp(2/2). Their underlying geometries turn out to be those of supersymplectic OSp(2/2) homogeneous spaces. Moment maps identifying the latter with coadjoint orbits of OSp(2/2) are exhibited via Berezin's symbols. When considered within Rothstein's general paradigm, these results lead to a natural general definition of a super K\"ahler supermanifold, the supergeometry of which is determined in terms of the usual geometry of holomorphic Hermitian vector bundles over K\"ahler manifolds. In particular, the supergeometry of the above orbits is interpreted in terms of the geometry of Einstein-Hermitian vector bundles. In the second part, an extension of the full geometric quantization procedure is applied to the same coadjoint orbits. Thanks to the super K\"ahler character of the latter, this procedure leads to explicit super unitary irreducible representations of OSp(2/2) in super Hilbert spaces of $L^2$ superholomorphic sections of prequantum bundles of the Kostant type. This work lays the foundations of a program aimed at classifying Lie supergroups' coadjoint orbits and their associated irreducible representations, ultimately leading to harmonic superanalysis. For this purpose a set of consistent conventions is exhibited.Comment: 53 pages, AMS-LaTeX (or LaTeX+AMSfonts

Toeplitz Quantization of K\"ahler Manifolds and gl(N)gl(N) N→∞N\to\infty

For general compact K\"ahler manifolds it is shown that both Toeplitz quantization and geometric quantization lead to a well-defined (by operator norm estimates) classical limit. This generalizes earlier results of the authors and Klimek and Lesniewski obtained for the torus and higher genus Riemann surfaces, respectively. We thereby arrive at an approximation of the Poisson algebra by a sequence of finite-dimensional matrix algebras $gl(N)$, $N\to\infty$.Comment: 17 pages, AmsTeX 2.1, Sept. 93 (rev: only typos are corrected

Met expression is an independent prognostic risk factor in patients with oesophageal adenocarcinoma

Oesophageal adenocarcinoma is an aggressive malignancy with propensity for early lymphatic and haematogenous dissemination. Since conventional TNM staging does not provide accurate prognostic information, novel molecular prognostic markers and potential therapeutic targets are subject of intense research. The aim of the present study was to study the prognostic significance of Met, the hepatic growth factor (HGF) receptor and a possible target for therapy in comparison to cyclooxygenase-2 (COX-2). Tumour sections from 145 consecutive patients undergoing intentionally curative surgery for oesophageal adenocarcinoma were immunohistochemically analysed for Met and COX-2 expression. Clinicopathological data were prospectively collected for all patients. Patients with high Met expression had significantly reduced overall and disease-specific 5-year survival rates (P⩽0.001 and P⩽0.001, respectively) and were more likely to develop distant metastases (P=0.002) and local recurrences (P=0.004) compared to patients with low Met expression. High COX-2 expression tended to be correlated with poor long-term survival but this did not reach statistical significance. Expression of Met was recognised as a significant and independent prognostic factor by stage-specific analysis and multivariate analysis (relative risk=2.3; 95% CI=1.3–4.1). These findings support the importance of Met in oesophageal adenocarcinoma and support the concept of Met tyrosine kinase inhibition as (neo-) adjuvant treatment

Detection of Colorectal Cancer by Serum and Tissue Protein Profiling: A Prospective Study in a Population at Risk

Colorectal cancer (CRC) is the second most common cause of cancer-related death in Europe and its prognosis is largely dependent on stage at diagnosis. Currently, there are no suitable tumour markers for early detection of CRC. In a retrospective study we previously found discriminative CRC serum protein profiles with surface enhanced laser desorption ionisation—time of flight mass spectrometry (SELDI-TOF MS). We now aimed at prospective validation of these profiles. Additionally, we assessed their applicability for follow-up after surgery and investigated tissue protein profiles of patients with CRC and adenomatous polyps (AP). Serum and tissue samples were collected from patients without known malignancy with an indication for colonoscopy and patients with AP and CRC during colonoscopy. Serum samples of controls (CON; n = 359), patients with AP (n = 177) and CRC (n = 73), as well as tissue samples from AP (n = 52) and CRC (n = 47) were analysed as described previously. Peak intensities were compared by non-parametric testing. Discriminative power of differentially expressed proteins was assessed with support vector machines (SVM). We confirmed the decreased serum levels of apolipoprotein C-1 in CRC in the current population. No differences were observed between CON and AP. Apolipoprotein C-I levels did not change significantly within 1 month post-surgery, although a gradual return to normal levels was observed. Several proteins differed between AP and CRC tissue, among which a peak with similar mass as apolipoprotein C-1. This peak was increased in CRC compared to AP. Although we prospectively validated the serum decrease of apolipoprotein C-1 in CRC, serum protein profiles did not yield SVM classifiers with suitable sensitivity and specificity for classification of our patient groups

Projective Fourier Duality and Weyl Quantization

The Weyl-Wigner correspondence prescription, which makes large use of Fourier duality, is reexamined from the point of view of Kac algebras, the most general background for noncommutative Fourier analysis allowing for that property. It is shown how the standard Kac structure has to be extended in order to accommodate the physical requirements. An Abelian and a symmetric projective Kac algebras are shown to provide, in close parallel to the standard case, a new dual framework and a well-defined notion of projective Fourier duality for the group of translations on the plane. The Weyl formula arises naturally as an irreducible component of the duality mapping between these projective algebras.Comment: LaTeX 2.09 with NFSS or AMSLaTeX 1.1. 102Kb, 44 pages, no figures. requires subeqnarray.sty, amssymb.sty, amsfonts.sty. Final version with text improvements and crucial typos correction

Transanal total mesorectal excision: how are we doing so far?

Aim This subgroup analysis of a prospective multicentre cohort study aims to compare postoperative morbidity between transanal total mesorectal excision (TaTME) and laparoscopic total mesorectal excision (LaTME). Method The study was designed as a subgroup analysis of a prospective multicentre cohort study. Patients undergoing TaTME or LaTME for rectal cancer were selected. All patients were followed up until the first visit to the outpatient clinic after hospital discharge. Postoperative complications were classified according to the Clavien–Dindo classification and the comprehensive complication index (CCI). Propensity score matching was performed. Results In total, 220 patients were selected from the overall prospective multicentre cohort study. After propensity score matching, 48 patients from each group were compared. The median tumour height for TaTME was 10.0 cm (6.0–10.8) and for LaTME was 9.5 cm (7.0–12.0) (P = 0.459). The duration of surgery and anaesthesia were both significantly longer for TaTME (221 vs 180 min, P < 0.001, and 264 vs 217 min, P < 0.001). TaTME was not converted to laparotomy whilst surgery in five patients undergoing LaTME was converted to laparotomy (0.0% vs 10.4%, P = 0.056). No statistically significant differences were observed for Clavien–Dindo classification, CCI, readmissions, reoperations and mortality. Conclusion The study showed that TaTME is a safe and feasible approach for rectal cancer resection. This new technique obtained similar postoperative morbidity to LaTME

Cotangent bundle quantization: Entangling of metric and magnetic field

For manifolds $\cal M$ of noncompact type endowed with an affine connection (for example, the Levi-Civita connection) and a closed 2-form (magnetic field) we define a Hilbert algebra structure in the space $L^2(T^*\cal M)$ and construct an irreducible representation of this algebra in $L^2(\cal M)$. This algebra is automatically extended to polynomial in momenta functions and distributions. Under some natural conditions this algebra is unique. The non-commutative product over $T^*\cal M$ is given by an explicit integral formula. This product is exact (not formal) and is expressed in invariant geometrical terms. Our analysis reveals this product has a front, which is described in terms of geodesic triangles in $\cal M$. The quantization of $\delta$-functions induces a family of symplectic reflections in $T^*\cal M$ and generates a magneto-geodesic connection $\Gamma$ on $T^*\cal M$. This symplectic connection entangles, on the phase space level, the original affine structure on $\cal M$ and the magnetic field. In the classical approximation, the $\hbar^2$-part of the quantum product contains the Ricci curvature of $\Gamma$ and a magneto-geodesic coupling tensor.Comment: Latex, 38 pages, 5 figures, minor correction

Long-term survival after hyperthermic intraperitoneal chemotherapy using mitomycin C or oxaliplatin in colorectal cancer patients with synchronous peritoneal metastases:A nationwide comparative study

Objectives: In the Netherlands, limited variability exists in performance of cytoreductive surgery and hyperthermic intraperitoneal chemotherapy (CRS-HIPEC) among centers treating colorectal peritoneal metastases (PM), except for the intraperitoneal drug administration. This offers a unique opportunity to investigate any disparities in survival between the two most frequently used HIPEC regimens worldwide: mitomycin C (MMC) and oxaliplatin.Methods: This was a comparative, population-based cohort study of all Dutch patients diagnosed with synchronous colorectal PM who underwent CRS-HIPEC between 2014 and 2017. They were retrieved from the Netherlands Cancer Registry. Main outcome was overall survival (OS). The effect of the intraperitoneal drug on OS was investigated using multivariable Cox regression analysis.Results: In total, 297 patients treated between 2014 and 2017 were included. Among them, 177 (59.6%) received MMC and 120 (40.4%) received oxaliplatin. Only primary tumor location was different between the two groups: more left-sided colon in the Oxaliplatin group (47.5% vs. 33.3%, respectively, p=0.048). The 1-, 2- and 3-year OS were 84.6% vs. 85.8%, 61.6% vs. 63.9% and 44.7% vs. 53.5% in patients treated with MMC and oxaliplatin, respectively. Median OS was 30.7 months in the MMC group vs. 46.6 months in the oxaliplatin group (p=0.181). In multivariable analysis, no influence of intraperitoneal drug on survival was observed (adjusted HR 0.77 [0.53-1.13]).Conclusions: Long-term survival between patients treated with either MMC or oxaliplatin during CRS-HIPEC was not significantly different. (C) 2020 Elsevier Ltd, BASO similar to The Association for Cancer Surgery, and the European Society of Surgical Oncology. All rights reserved.</p