Gauge symmetries, Noether's theorem, and the Standard Model
Noether's theorem reveals the deep connection between symmetries and conservation laws: every continuous symmetry of a physical system corresponds to a conserved quantity. Rotational symmetry gives conservation of angular momentum, translational symmetry gives conservation of momentum, and time-translation symmetry gives conservation of energy.
The Standard Model of particle physics is built on the gauge symmetry group SU(3) × SU(2) × U(1). The Higgs mechanism breaks SU(2) × U(1) down to U(1)ₑₘ, giving mass to the W and Z bosons. Every force in nature corresponds to a Lie group acting as a gauge symmetry.
A particle moves in a potential field. Toggle symmetries on and off to see how conservation laws are affected. When rotational symmetry is present, angular momentum remains constant — the “charge meter” stays steady. Break the symmetry, and the conserved quantity fluctuates. This one-to-one correspondence between symmetries and conservation laws is one of the deepest results in theoretical physics.
Key insight: Noether's theorem works in both directions: given a symmetry you can construct the conserved charge, and given a conserved charge you can reconstruct the symmetry. The Lie algebra structure determines the Poisson bracket of charges.
A quantum wavefunction ψ(x) is a section of a U(1) bundle over spacetime.Global phase symmetry (ψ → eⁱθψ) is trivial, but local gauge symmetry (ψ → eⁱθ(x)ψ) requires introducing a gauge field Aμ (the electromagnetic potential) to keep the physics invariant. Adjust the local phase and see how the gauge field compensates to preserve |ψ|².
The Standard Model organizes all known particles by their representations under three gauge groups: SU(3) for the strong force (color), SU(2) for the weak force (isospin), and U(1) for hypercharge. Click any particle to see its quantum numbers. The “symmetry breaking” slider shows how the Higgs mechanism breaks SU(2) × U(1) down to electromagnetic U(1).
Gauge theory is the physics of connections on principal bundles. A U(1) bundle over a 2D base space looks like circles sitting over each point. Draw curves on the base and watch parallel transport carry the phase around the loop. If the connection has curvature (= electromagnetic field), the transported phase picks up a holonomy — a net rotation that depends on the enclosed flux. This connects directly to the differential geometry module.