Bridge to Relativistic Quantum Theory

Special relativity supplies the symmetry the quantum theories assume. This page traces the path from the dispersion relation and Poincaré symmetry to relativistic wave equations and field theory. Convention: $c=1$, $\eta =\mathrm{diag}(+,-,-,-)$.

On-shell condition

The invariant $p^2=m^2$ is the on-shell condition: free quanta live on the mass shell. Promoting $E\to i{\partial }_t$, $\mathbf{p}\to -i\nabla$ turns $E^2={\mathbf{p}}^2+m^2$ into the Klein–Gordon equation $(\square +m^2)\varphi =0$ — see QED historical. Linearizing the dispersion gives the Dirac equation and spin-$\frac{1}{2}$ fields.

Symmetry as postulate

QFT takes the Poincaré group as input: states carry a unitary irrep labeled by mass and spin (Wigner's classification), exactly the SR Casimirs $P^2,W^2$. The QFT postulates start here; the modern derivation builds fields from these reps.

Why fields

Relativistic causality (spacelike events can't communicate) plus quantum superposition force particle number to fluctuate, demanding fields over single wavefunctions. SR's light-cone causal structure becomes microcausality: $[\varphi (x),\varphi (y)]=0$ for spacelike separation.

Continue to QFT preliminaries and group theory.