Physical laws manifest themselves through the amalgamation of mathematical symbols, numbers, functions, geometries, and relationships. These intricate combinations unfold within a mathematical model devised to capture and represent the ``objective reality'' of the system under examination. In this symbiotic relationship between physics and mathematics, the language of mathematics becomes a powerful tool for describing and predicting the behavior of the physical world. The language used and the associated concepts are in a perpetual state of evolution, mirroring the ongoing expansion of the phenomena accessible to our scientific understanding. In this contribution, written in honor of Richard Kerner, we delve into fundamental, at times seemingly elementary, elements of the mathematical language inherent to the physical sciences, guided by the overarching principles of symmetry and group theory. Our focus turns to the captivating realm of spheres, those strikingly symmetric entities that manifest prominently within our geometric landscape. By exploring the interplay between mathematical abstraction and the tangible beauty of symmetry, we seek to deepen our understanding of the underlying structures that govern our interpretation of the physical world.

An explanation of the origin of dark matter is suggested in this work. The argument is based on symmetry considerations about the concept of mass. In the Wigner's view, the rest mass and the spin of a free elementary particle in flat space-time are the two invariants that characterize the associated unitary irreducible representation of the Poincar\'e group. The Poincar\'e group has two and only two deformations with maximal symmetry. They describe respectively the de Sitter (dS) and Anti de Sitter (AdS) kinematic symmetries. Analogously to their shared flat space-time limit, two invariants, spin and energy scale for de Sitter and rest energy for Anti de Sitter, characterize the unitary irreducible representation associated with dS and AdS elementary systems. While the dS energy scale is a simple deformation of the Poincaré rest energy and so has a purely mass nature, AdS rest energy is the sum of a purely mass component and a kind of zero-point energy derived from the curvature. An analysis based on recent estimates on the chemical freeze-out temperature marking in Early Universe the phase transition quark-gluon plasma epoch to the hadron epoch supports the guess that dark matter energy might originate from an effective AdS curvature energy.

Covariant integral quantization with rotational symmetry SO(3) is established for the quantum motion on this group manifold, or, alternatively, for Gabor signal analysis on this group. We revisit the action of the related (non-unitary) Weyl-Gabor operator on the Hilbert space of square integrable functions on SO(3) and disclose a set of various properties. By selecting weight functions on the corresponding discrete-continuous phase space, one derives related coherent states and Wigner transform.

In the same way as the realization of some of the famous gedanken experiments imagined by the founding fathers of quantum mechanics has recently led to the current renewal of the interpretation of quantum physics, it seems that the most recent progresses of observational astrophysics can be interpreted as the realization of some cosmological gedanken experiments such as the removal from the universe of the whole visible matter or the cosmic time travel leading to a new cosmological standard model. This standard model involves two dark components of the universe, dark energy and dark matter. Whereas dark energy is usually associated with the cosmological constant, we propose to explain dark matter as a pure QCD effect, namely a gluonic Bose Einstein condensate, following the transition from the quark gluon plasma phase to the colorless hadronic phase. Our approach not only allows us to assume a ratio Dark/Visible equal to 11/2 but also provides gluons and (anti-)quarks with an extra mass of vibrational nature. Such an interpretation would comfort the idea that, apart from the violation of the matter/antimatter symmetry satisfying the Sakharov’s conditions, the reconciliation of particle physics and cosmology needs not the recourse to any ad hoc fields, particles or hidden variables.

Any quantization maps linearly functions on a phase space to symmetric operators in a Hilbert space. Covariant integral quantization combines operator-valued measure with symmetry group of the phase space. Covariant means that the quantization map intertwines classical (geometric operation) and quantum (unitary transformations) symmetries. Integral means that we use all ressources of integral calculus, in order to implement the method when we apply it to singular functions, or distributions, for which the integral calculus is an essential ingredient. In this paper we emphasize the deep connection between Fourier transform and covariant integral quantization when the Weyl-Heisenberg and affine groups are involved. We show with our generalisations of the Wigner-Weyl transform that many properties of the Weyl integral quantization, commonly viewed as optimal, are actually shared by a large family of integral quantizations.

Currently, there is no widely accepted consensus regarding a consistent thermodynamic framework within the special relativity paradigm. However, by postulating that the inverse temperature 4-vector, denoted as β, is future-directed and timelike, intriguing insights emerge. Specifically, it is demonstrated that the q-dependent Tsallis distribution can be conceptualized as a de-Sitterian deformation of the relativistic Maxwell-Jüttner distribution. In this context, the curvature of the de Sitter space-time is characterized by Λ/3, where Λ represents the cosmological constant within the ΛCDM standard model for Cosmology. For a simple gas composed of particles with proper mass m, and within the framework of quantum statistical de Sitterian considerations, the Tsallis parameter q exhibits a dependence on the cosmological constant given by q=1+ℓcΛ/n where ℓc=ℏ/mc is the Compton length of the particle and n is a positive numerical factor. This formulation establishes a novel connection between the Tsallis distribution, quantum statistics, and the cosmological constant, shedding light on the intricate interplay between relativistic thermodynamics and fundamental cosmological parameters.