Applications of Four-Vectors in Particle Physics

Special Relativity requires a mathematical framework that treats space and time as components of a single four-dimensional structure. This paper examines the four-vector formalism as that framework, tracing its use from the derivation of the Lorentz transformations through applications in modern particle physics. This paper begins with the failure of Galilean relativity to accommodate Maxwell’s prediction of a constant speed of light, and derives the Lorentz transformations that resolve this conflict. Time dilation is illustrated using the atmospheric muon experiment, which confirms the effect experimentally. Four-vectors are then introduced as quantities whose invariant length under Lorentz transformations makes them suited to describing energy, momentum, and spacetime intervals across reference frames. Using this formalism, this paper analyzes Compton scattering, showing how conservation of four-momentum produces the scattering formula and supports the particle interpretation of light. The paper extends to Feynman diagrams and the Mandelstam variables s, t, and u, which provide a Lorentz-invariant description of scattering processes used throughout collider physics. Each application illustrates the same underlying principle: four-vector algebra converts calculations that would otherwise depend on the observer’s frame into results that hold universally. The paper closes by connecting this formalism to its role in contemporary high-energy physics, where scattering amplitudes and cross sections are computed directly in terms of invariant four-vector quantities.