Geometry and field theory in multi-fractional spacetime

We construct a theory of fields living on continuous geometries with fractional Hausdorff and spectral dimensions, focussing on a flat background analogous to Minkowski spacetime. After reviewing the properties of fractional spaces with fixed dimension, presented in a companion paper, we generalize to a multi-fractional scenario inspired by multi-fractal geometry, where the dimension changes with the scale. This is related to the renormalization group properties of fractional field theories, illustrated by the example of a scalar field. Depending on the symmetries of the Lagrangian, one can define two models. In one of them, the effective dimension flows from 2 in the ultraviolet (UV) and geometry constrains the infrared limit to be four-dimensional. At the UV critical value, the model is rendered power-counting renormalizable. However, this is not the most fundamental regime. Compelling arguments of fractal geometry require an extension of the fractional action measure to complex order. In doing so, we obtain a hierarchy of scales characterizing different geometric regimes. At very small scales, discrete symmetries emerge and the notion of a continuous spacetime begins to blur, until one reaches a fundamental scale and an ultra-microscopic fractal structure. This fine hierarchy of geometries has implications for non-commutative theories and discrete quantum gravity. In the latter case, the present model can be viewed as a top-down realization of a quantum-discrete to classical-continuum transition.Comment: 1+82 pages, 1 figure, 2 tables. v2-3: discussions clarified and improved (especially section 4.5), typos corrected, references added; v4: further typos correcte

Comparative study between intravenous urography and renal scintigraphy with DMSA for the diagnosis of renal scars in children with vesicoureteral reflux

PURPOSE: To assess the value of intravenous urography (IVU) in detecting and grading the renal scar, comparing its results with those of scintigraphy with dimercaptosuccinic acid (DMSA). MATERIALS AND METHODS: The study included 43 children investigated by DMSA and IVU, who had vesicoureteral reflux diagnosed and classified through voiding cystourethrography. RESULTS: Among the kidneys with reflux, there was agreement between the results of DMSA and IVU concerning the presence and the absence of scars in 82.4% of the cases. Based on the results obtained, IVU would have a sensitivity of 66.6%, specificity of 94.4%; accuracy of 82.5%; positive predictive value (PPV) of 90% and negative predictive value (NPV) of 79%, when compared with DMSA results. Our data also confirm the close relation between the reflux grade and the presence of renal scar, since 75% of the kidneys with grade IV and V reflux presented scars. In relation to the grading of nephropathy, in 78% of patients the classification of the scar by both methods was identical. The highest disagreement was verified in the group with segmental scar on DMSA, where 41.6% of the kidneys were classified as normal on IVU. CONCLUSION: The data obtained confirm that the scintigraphy with DMSA is essential in the investigation of patients with renal scar, and cannot be replaced by IVU, due to its low sensitivity and lower ability of satisfactory grading