% miko:type: complete
% miko:title: Complete Particle Physics
% miko:summary: A standard introductory path through relativistic waves, symmetries, the Standard Model, and neutrino mixing.
% miko:tags: particle
% miko:difficulty: undergrad
% miko:related: standard-model, dirac-equation, klein-gordon-equation, feynman-diagrams, higgs-mechanism, neutrino-oscillation, qcd-asymptotic-freedom, commutator
% miko:keywords: Standard Model, gauge symmetry, Higgs, neutrino, QCD

\documentclass{article}
\usepackage{amsmath}
\title{Complete Particle Physics}
\begin{document}
\maketitle

\section{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.

\section{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,

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

\section{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.

\section{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.

\section{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.

\end{document}
