Mathematics offers an astonishing bridge between the infinitely small world of particles and the cosmic immensity. Researchers are exploring how abstract geometric shapes can describe both collisions in particle accelerators and the evolution of the Universe since the Big Bang. This innovative approach could unify previously disjointed physical domains.
The study led by Claudia Fevola and Anna-Laura Sattelberger, published in
Notices of the American Mathematical Society, focuses on positive geometry. Inspired by work in theoretical physics, this branch of mathematics represents interactions between particles as volumes in high-dimensional spaces.
Representation of an amplituhedron.
Image Wikimedia.
For example, the amplituhedron, a geometric object introduced in 2013, allows for simpler calculation of scattering amplitudes, which determine the probabilities of events like proton collisions.
These mathematical tools find direct applications in cosmology, such as modeling correlations in the cosmic microwave background, the first light of the Universe. Scientists can trace back to the physical laws that governed the first moments after the Big Bang, thus offering a window into the origin of everything around us.
The methodology combines algebra, geometry, and combinatorics. Feynman integrals, used to describe quantum processes, are related to generalized Euler integrals. These objects are studied through topological properties, reflecting physical concepts.
This work is part of a growing international effort to bring mathematics and fundamental physics closer together. The authors emphasize that positive geometry, although recent, could revolutionize our understanding of nature at all scales.
Going further: positive geometry and fundamental physics
Positive geometry is an emerging mathematical discipline that defines spaces where all coordinates are positive or zero. It generalizes concepts like simplices and convex polytopes to model physical phenomena.
In particle physics, it allows representing scattering amplitudes – key quantities for predicting experimental outcomes – as volumes in high-dimensional spaces. This approach avoids certain traditional calculations and offers an intuitive geometric interpretation.
In cosmology, similar objects called cosmological polytopes encode statistical correlations in the cosmic microwave background. These structures help understand how primordial inhomogeneities evolved to form the galaxies and galaxy clusters observed today.
The unifying potential of positive geometry lies in its ability to describe very different physical systems with the same mathematical formalism, thus connecting quantum physics and general relativity.