Symmetry means that physical laws remain unchanged after operations such as mirror reflection or rotation. By introducing mathematical group theory, Wigner treated these symmetry operations with precise equations. He showed that conservation laws like energy or electric charge follow directly from symmetry. His framework organized nuclear energy levels and particle "quantum numbers," making experimental data easier to interpret. The concept now underpins technologies such as lasers and semiconductors, which are key to today’s information society. Even the momentum-conservation rule you learn in school is actually one example of symmetry at work.

Wigner employed Lie groups and representation theory to systematize conserved quantities associated with symmetry operations of the quantum-mechanical Hamiltonian. The SU(2) rotation group representations became indispensable for classifying nuclear spin and orbital angular momentum, while the Wigner–Eckart theorem greatly simplified transition-probability calculations. He also analyzed discrete symmetries such as time reversal and parity, laying the groundwork for later discoveries of parity violation in weak interactions. By introducing the isospin concept for nucleons (protons and neutrons), he clarified level degeneracies and selection rules. These results influenced pre-Standard-Model quantum field theories and the group-theoretic classification of particles (e.g., the Eightfold Way), still essential for modern high-energy data analysis.

Wigner formulated symmetry operations in quantum systems as unitary or anti-unitary operators and, via Wigner’s theorem, proved the preservation of transition probabilities. His group-theoretical approach spans particle classification with irreducible representations of the Poincaré group, SU(4) supermultiplets in nuclear structure, and early studies of conformal symmetry. The unified treatment of discrete symmetries, including parity and time reversal, provided the mathematical footing for the rigorous CPT theorem. He expressed selection rules in covariant perturbation theory through Wigner 3j and 6j symbols, establishing universal algorithms for angular-momentum coupling. These tools enable applications ranging from nuclear magnetic dipole transitions and the Mössbauer effect to baryon spectroscopy.