A new study published in the Proceedings of the National Academy of Sciences presents a groundbreaking “big algebra” theorem that could bridge the gap between two distant fields of mathematics: quantum physics and number theory. This breakthrough, developed by Tamás Hausel, professor of mathematics at the Institute of Science and Technology Austria (ISTA), creates a two-way mathematical dictionary that deciphers the most abstract aspects of mathematical symmetry using algebraic geometry.
In this ambitious endeavor, Hausel has proved a new mathematical tool called “big algebras,” which operates at the intersection of symmetry, abstract algebra, and geometry. By recasting sophisticated mathematical information about symmetries in more tangible geometric terms, big algebras can provide unprecedented insights into the hidden depths of symmetry groups.
This innovation holds great promise for consolidating the connection between quantum physics and number theory. In quantum physics, matrices are used to represent complex phenomena, but these matrices are typically non-commutative, posing a problem in algebra and algebraic geometry. Big algebras solve this issue by providing a commutative “mathematical translation” of non-commutative matrix algebras.
Furthermore, Hausel shows that big algebras can reveal relationships between symmetry groups and their Langlands duals, a central concept in the purely mathematical world of number theory. This could have significant applications in number theory, potentially allowing for the connection between the distant worlds of quantum physics and number theory to be strengthened.
The study, titled “Commutative avatars of representations of semisimple Lie groups,” has been published in the Proceedings of the National Academy of Sciences (2024). DOI: 10.1073/pnas.2319341121
Source: https://phys.org/news/2024-09-mathematical-dictionary-quantum-physics-theory.html