Relativistic Wave Equations - Linear Equations

Linear Equations

Further information: linear differential equation
Particle spin quantum number s Name Equation Typical particles the equation describes
0 Klein–Gordon equation Massless or massive spin-0 particle (such as Higgs bosons).
1/2 Weyl equation Massless spin-1/2 particles.
Dirac equation Massive spin-1/2 particles (such as electrons).
Two-body Dirac equations

Massive spin-1/2 particles (such as electrons).
Majorana equation Massive Majorana particles.
Breit equation Two massive spin-1/2 particles (such as electrons) interacting electromagnetically to first order in perturbation theory.
1 Maxwell equations (in QED) Photons, massless spin-1 particles.
Proca equation Massive spin-1 particle (such as W and Z bosons).
3/2 Rarita–Schwinger equation Massive spin-3/2 particles.
s Bargmann–Wigner equations \begin{align} & (i\hbar \gamma^\mu \partial_\mu + mc)_{\alpha_1 \alpha_1'}\psi_{\alpha'_1 \alpha_2 \alpha_3 \cdots \alpha_{2j}} = 0 \\ & (i\hbar \gamma^\mu \partial_\mu + mc)_{\alpha_2 \alpha_2'}\psi_{\alpha_1 \alpha'_2 \alpha_3 \cdots \alpha_{2j}} = 0 \\ & \qquad \vdots \\ & (i\hbar \gamma^\mu \partial_\mu + mc)_{\alpha_{(2s)} \alpha'_{(2s)}}\psi_{\alpha_1 \alpha_2 \alpha_3 \cdots \alpha'_{2s}} = 0 \\ \end{align}

where ψ is a rank-2s 4-component spinor. The operator (iħγμ∂μ + mc) is a 4×4 matrix (the mc term actually scalar multiplies a 4×4 identity matrix, usually not written for simplicity).

 Free particles of arbitrary spin (bosons and fermions).

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