ABSTRACT The Lorentzian potential well combines a smooth analytic core with an attractive inverse‐square tail, so different parts of its bound‐state spectrum are governed by different approximation mechanisms. We quantify the crossover from core‐dominated anharmonic states to the discrete‐scale‐invariant threshold spectrum using complementary analytical and numerical approximations. Deep low‐lying states are treated by a strong‐coupling anharmonic expansion including the first inverse‐square‐root correction. For intermediate states, the classical action is evaluated exactly in complete elliptic integrals, yielding an implicit WKB quantization rule and a simple endpoint‐matched elementary surrogate. Near threshold, previously established zero‐energy core‐to‐tail matching provides the parity‐resolved threshold prefactors and the geometric spectrum governed by the quantum‐shifted parameter . The phase unwrapping and branch assignment are verified numerically at three supercritical couplings. A complementary Rayleigh–Ritz treatment gives an exact trigamma expression for a hyperbolic‐secant trial state and a smooth Gaussian‐core/exponential‐tail ansatz with analytic Bessel–Bickley expectation values. The optimized core‐tail ground‐state energy remains within about 0.3% of the numerical result over three orders of magnitude in coupling strength. Numerical spectra and wave functions show quantitatively how the anharmonic, semiclassical, variational, and inverse‐square descriptions take over across the spectrum.
ABSTRACT We study Bose–Einstein condensation of an ideal gas in a solid torus (major radius , minor radius , Dirichlet walls) for – particles and aspect ratios . The low‐energy spectrum is dominated by a soft ladder of states winding around the ring, and this ladder controls the finite‐size thermodynamics. We derive a closed‐form expansion for the ladder energies through eighth order in the thickness. The finite‐ thermodynamics is evaluated exactly in the grand‐canonical and canonical ensembles, and the heat capacity is smooth in both. Measured from the saturation temperature, at which the excited states can hold all particles, the heat‐capacity maximum lies slightly below it in the ball and in the cube but well above it in the torus, by a relative amount decaying as . This dimensional crossover is traced to the soft ring direction by a control computation on a simply connected elongated box. The torus belongs, for every aspect ratio, to the anomalous universality class of condensate statistics; at the saturation temperature its chemical potential lies below the ground‐state energy by an ‐independent amount, leaving a condensate fraction that decays as . A scan over aspect ratios finds no resonant geometry: curvature suppresses spectral shell structure. Quantitative connections to ring‐shaped traps are given.
ABSTRACT The existence of three fermion generations remains one of the unexplained structural features of the Standard Model. This article provides a focused review of an algebraic framework in which an intrinsic family symmetry relates three gauge‐equivalent fermion sectors. The framework is motivated by division algebra and Clifford algebra constructions in which the complex octonions and organise the colour and electromagnetic quantum numbers of one generation. Continuing the Cayley–Dickson sequence, the sedenions provide an intrinsic automorphism structure, while their complexified left‐multiplication operators generate an associative endomorphism algebra isomorphic to . In the formulation, the order‐three family action produces three linearly independent fermion sectors while leaving a single gauge algebra. Extension to incorporates a single gauge algebra, and each generation includes a right‐handed neutrino sterile under the Standard Model gauge interactions. We compare this construction with representative algebraic and family‐symmetry approaches to three generations, including triality‐based proposals, and explain why the family symmetry used here is distinct from standard triality despite the common appearance of and . The framework remains algebraic and representation theoretic rather than a complete dynamical theory; family‐symmetry breaking, realistic fermion masses and mixing, and a dynamical account of the gauge and matter sectors remain open problems.
ABSTRACT The present paper, which is partly a review and turned out to be the corner stone of an entire new development in geometrical deep learning, aims at presenting, in a unified mathematical framework, a complex and articulated lore regarding noncompact symmetric spaces , where is a simple Lie group, whose Lie algebra , is a noncompact real section of a complex simple Lie algebra and is the maximal compact Lie subgroup. All such manifolds are Riemannian normal manifolds , according to Alekseevsky's definition, in the sense that they are metrically equivalent to a solvable Lie group manifold . This identification provides a vision in which, on one side one can derive quite explicit and challenging formulas for the unique distance function between points of the manifold, on the other one, one is able to organize the entire set of the available manifolds in universality classes distinguished by their common Tits–Satake submanifold , and, correspondingly, by their noncompact rank. The members of the class are distinguished by their different paint group , this notion having been established by two of the present authors and their collaborators in the years 2007–2009. In relation with the construction of neural networks, these mathematical structures offer unique possibilities for replacing ad hoc activation functions with the naturally defined nonlinear operations that relate Lie algebras to Lie Groups and vice versa. The paint group invariants offer new tokens both to construct algorithms and inspect (hopefully to control) their working. A conspicuous part of the paper is devoted to the study and systematic construction of parabolic/elliptic discrete subgroups of the Lie groups , in view of discretization and/or tessellations of the space to which data are to be mapped. Furthermore it is shown how the ingredients of special Kähler geometry and the ‐map, well known in the supergravity literature, provide a unified classification scheme of the relevant park of Tits–Satake universality classes with noncompact rank .
We explore the renormalization group-based extension of the model as a potential solution to the current cosmological tensions. In this approach, both the cosmological constant density and Newton's constant are allowed to vary with the energy scale, as a consequence of the remnant effects of massive quantum fields in the low-energy regime. The corresponding cosmological model is consistent with the principles of quantum field theory, based on the covariance of the vacuum effective action, and is characterized by a unique extra parameter . Our analysis yields a best-fit value of , placing the limit at the region of the posterior. This narrow range is consistent with data from CMB (Planck), BAO (DESI), and SN Ia (DES Y5). Our result also alleviates the tension and is consistent with the previously established constraints from large-scale structure. In these kind of models, there is a link between cosmology and particle physics. Our results point to possibility of a new physics, characterized by a mass spectrum lying below the Planck scale but above the values typically associated with Grand Unified Theories (GUTs).
