On the geometry of the spin-statistics connection in quantum mechanics

The Spin-Statistics theorem states that the statistics of a system of identical particles is determined by their spin: Particles of integer spin are Bosons (i.e. obey Bose-Einstein statistics), whereas particles of half-integer spin are Fermions (i.e. obey Fermi-Dirac statistics). Since the original proof by Fierz and Pauli, it has been known that the connection between Spin and Statistics follows from the general principles of relativistic Quantum Field Theory. In spite of this, there are different approaches to Spin-Statistics and it is not clear whether the theorem holds under assumptions that are different, and even less restrictive, than the usual ones (e.g. Lorentz-covariance). Additionally, in Quantum Mechanics there is a deep relation between indistinguishabilty and the geometry of the configuration space. This is clearly illustrated by Gibbs' paradox. Therefore, for many years efforts have been made in order to find a geometric proof of the connection between Spin and Statistics. Recently, various proposals have been put forward, in which an attempt is made to derive the Spin-Statistics connection from assumptions different from the ones used in the relativistic, quantum field theoretic proofs. Among these, there is the one due to Berry and Robbins (BR), based on the postulation of a certain single-valuedness condition, that has caused a renewed interest in the problem. In the present thesis, we consider the problem of indistinguishability in Quantum Mechanics from a geometric-algebraic point of view. An approach is developed to study configuration spaces Q having a finite fundamental group, that allows us to describe different geometric structures of Q in terms of spaces of functions on the universal cover of Q. In particular, it is shown that the space of complex continuous functions over the universal cover of Q admits a decomposition into C(Q)-submodules, labelled by the irreducible representations of the fundamental group of Q, that can be interpreted as the spaces of sections of certain flat vector bundles over Q. With this technique, various results pertaining to the problem of quantum indistinguishability are reproduced in a clear and systematic way. Our method is also used in order to give a global formulation of the BR construction. As a result of this analysis, it is found that the single-valuedness condition of BR is inconsistent. Additionally, a proposal aiming at establishing the Fermi-Bose alternative, within our approach, is made.

Deep geometry of subduction below the Andean belt of Colombia as revealed by seismic tomography

In this study new tomographic models of Colombia were calculated. I used the seismicity recorded by the Colombian seismic network during the period 2006-2009. In this time period, the improvement of the seismic network yields more stable hypocentral results with respect to older data set and allows to compute new 3D Vp and Vp/Vs models. The final dataset consists of 10813 P- and 8614 S-arrival times associated to 1405 earthquakes. Tests with synthetic data and resolution analysis indicate that velocity models are well constrained in central, western and southwestern Colombia to a depth of 160 km; the resolution is poor in the northern Colombia and close to Venezuela due to a lack of seismic stations and seismicity. The tomographic models and the relocated seismicity indicate the existence of E-SE subducting Nazca lithosphere beneath central and southern Colombia. The North-South changes in Wadati-Benioff zone, Vp & Vp/Vs pattern and volcanism, show that the downgoing plate is segmented by slab tears E-W directed, suggesting the presence of three sectors. Earthquakes in the northernmost sector represent most of the Colombian seimicity and concentrated on 100-170 km depth interval, beneath the Eastern Cordillera. Here a massive dehydration is inferred, resulting from a delay in the eclogitization of a thickened oceanic crust in a flat-subduction geometry. In this sector a cluster of intermediate-depth seismicity (Bucaramanga Nest) is present beneath the elbow of the Eastern Cordillera, interpreted as the result of massive and highly localized dehydration phenomenon caused by a hyper-hydrous oceanic crust. The central and southern sectors, although different in Vp pattern show, conversely, a continuous, steep and more homogeneous Wadati-Benioff zone with overlying volcanic areas. Here a "normalthickened" oceanic crust is inferred, allowing for a gradual and continuous metamorphic reactions to take place with depth, enabling the fluid migration towards the mantle wedge.

Geometry of BV quantization and Mathai-Quillen formalism

Il formalismo Mathai-Quillen (MQ) è un metodo per costruire la classe di Thom di un fibrato vettoriale attraverso una forma differenziale di profilo Gaussiano. Lo scopo di questa tesi è quello di formulare una nuova rappresentazione della classe di Thom usando aspetti geometrici della quantizzazione Batalin-Vilkovisky (BV). Nella prima parte del lavoro vengono riassunti i formalismi BV e MQ entrambi nel caso finito dimensionale. Infine sfrutteremo la trasformata di Fourier “odd" considerando la forma MQ come una funzione definita su un opportuno spazio graduato.

The geometry of Lagrangian fibres

If the generic fibre f−1(c) of a Lagrangian fibration f : X → B on a complex Poisson– variety X is smooth, compact, and connected, it is isomorphic to the compactification of a complex abelian Lie–group. For affine Lagrangian fibres it is not clear what the structure of the fibre is. Adler and van Moerbeke developed a strategy to prove that the generic fibre of a Lagrangian fibration is isomorphic to the affine part of an abelian variety.rnWe extend their strategy to verify that the generic fibre of a given Lagrangian fibration is the affine part of a (C∗)r–extension of an abelian variety. This strategy turned out to be successful for all examples we studied. Additionally we studied examples of Lagrangian fibrations that have the affine part of a ramified cyclic cover of an abelian variety as generic fibre. We obtained an embedding in a Lagrangian fibration that has the affine part of a C∗–extension of an abelian variety as generic fibre. This embedding is not an embedding in the category of Lagrangian fibrations. The C∗–quotient of the new Lagrangian fibration defines in a natural way a deformation of the cyclic quotient of the original Lagrangian fibration.

Operator variability using different polishing methods and surface geometry of a nanohybrid composite

This study evaluated the operator variability of different finishing and polishing techniques. After placing 120 composite restorations (Tetric EvoCeram) in plexiglassmolds, the surface of the specimens was roughened in a standardized manner. Twelve operators with different experience levels polished the specimens using the following finishing/polishing procedures: method 1 (40 ?m diamond [40D], 15 ?m diamond [15D], 42 ?m silicon carbide polisher [42S], 6 ?m silicon carbide polisher [6S] and Occlubrush [O]); method 2 (40D, 42S, 6S and O); method 3 (40D, 42S, 6S and PoGo); method 4 (40D, 42S and PoGo) and method 5 (40D, 42S and O). The mean surface roughness (Ra) was measured with a profilometer. Differences between the methods were analyzed with non-parametric ANOVA and pairwise Wilcoxon signed rank tests (?=0.05). All the restorations were qualitatively assessed using SEM. Methods 3 and 4 showed the best polishing results and method 5 demonstrated the poorest. Method 5 was also most dependent on the skills of the operator. Except for method 5, all of the tested procedures reached a clinically acceptable surface polish of Ra?0.2 ?m. Polishing procedures can be simplified without increasing variability between operators and without jeopardizing polishing results.

Rotational integral geometry of tensor valuations

We derive a new rotational Crofton formula for Minkowski tensors. In special cases, this formula gives (1) the rotational average of Minkowski tensors defined on linear subspaces and (2) the functional defined on linear subspaces with rotational average equal to a Minkowski tensor. Earlier results obtained for intrinsic volumes appear now as special cases.