The Unreasonable Effectiveness of Mathematics in Physics (Wigner I): A Group-Theoretic Model for Fundamental Particles and Forces

This talk explores how the particles and forces observed in nature arise from the group-theoretic framework of the Standard Model of physics, namely SU(3) × SU(2) × U(1). The aim is to illustrate how mathematics does not merely describe physical reality in a suggestive way, but in a remarkably precise and predictive one.

Beginning with SU(2), we will build a quantum vector space that incorporates superposition and entanglement, thereby distinguishing it from the classical setting associated with SO(2). Using the structural properties and representation-theoretic rules of SU(n), we will then construct eigenvector spaces and group generators that model both particles and the mediators of their interactions.

By expanding the framework through the Kronecker product, we arrive at the full Standard Model, where additional particles and force mediators emerge together with their characteristic charges and interaction laws. The resulting “zoo” of particles and interactions, when elevated to covariant operator-valued quantum fields and constrained by an Euler–Lagrange formalism, yields the most successful qualitative and quantitative theory currently available for known particles, forces, and measurements.

As this talk comes near the end of the semester, it is intended to be informal, interactive, and accessible, inviting conversation across mathematics and physics.