Complete Particle Physics

From Quantum Mechanics to Fields

Particle physics combines quantum mechanics, special relativity, and symmetry. Ordinary wave mechanics is not enough because relativistic particle number is not fixed. Quantum field theory treats particles as excitations of fields.

Relativistic Wave Equations

The relativistic energy-momentum relation is

$$E^2=p^2c^2+m^2c^4.$$

Quantizing this relation leads to the Klein-Gordon equation for spin-zero fields. A first-order relativistic equation for spin-one-half particles is the Dirac equation,

$$(i\hbar c\gamma^\mu\partial_\mu-mc^2)\psi=0.$$

Symmetry and Interactions

The Standard Model is organized by gauge symmetry. Its gauge group is

$$SU(3)_C\times SU(2)_L\times U(1)_Y.$$

The gauge fields correspond to the strong, weak, and electromagnetic interactions after electroweak symmetry breaking.

Higgs Mechanism

Gauge symmetry constrains the theory, but observed weak bosons are massive. The Higgs field has a nonzero vacuum expectation value, allowing $W^\pm$ and $Z$ bosons to acquire mass while preserving the consistency of the gauge theory.

Neutrinos

Neutrino flavor states are mixtures of mass states. A simple two-flavor oscillation probability is

$$P_{\alpha\to\beta} =\sin^2(2\theta)\sin^2\left(\frac{\Delta m^2 L}{4E}\right).$$

This shows that neutrinos have nonzero mass differences, extending the minimal original Standard Model picture.