User:Maschen/Dirac–Fierz–Pauli equations

Dirac in 1936, and Fierz and Pauli in 1939, built relativistic wave equations from irreducible spinors $A$ and $B$, symmetric in all indices, for a massive particle of spin $n + ½$ for integer $n$ (see Van der Waerden notation for the meaning of the dotted indices):



p_{\gamma\dot{\alpha}}A_{\epsilon_1\epsilon_2\cdots\epsilon_n}^{\dot{\alpha}\dot{\beta}_1\dot{\beta}_2\cdots\dot{\beta}_n} = mcB_{\gamma\epsilon_1\epsilon_2\cdots\epsilon_n}^{\dot{\beta}_1\dot{\beta}_2\cdots\dot{\beta}_n} $$



p^{\gamma\dot{\alpha}}B_{\gamma\epsilon_1\epsilon_2\cdots\epsilon_n}^{\dot{\beta}_1\dot{\beta}_2\cdots\dot{\beta}_n} = mcA_{\epsilon_1\epsilon_2\cdots\epsilon_n}^{\dot{\alpha}\dot{\beta}_1\dot{\beta}_2\cdots\dot{\beta}_n} $$

where $p$ is the momentum as a covariant spinor operator. For $n = 0$, the equations reduce to the coupled Dirac equations and $A$ and $B$ together transform as the original Dirac spinor. Eliminating either $A$ or $B$ shows that $A$ and $B$ each fulfill the Klein–Gordon equation


 * $$ -\hbar^2{\frac {\partial ^2 A }{\partial t^2}}+(\hbar c)^2 \nabla ^2 A =(mc^{2})^2 A \,,$$
 * $$ -\hbar^2{\frac {\partial ^2 B }{\partial t^2}}+(\hbar c)^2 \nabla ^2 B =(mc^{2})^2 B \,,$$

provided the conditions


 * $$ p^\gamma_{\dot{\alpha}} A_{\gamma\epsilon_1\epsilon_2\cdots\epsilon_n}^{\dot{\alpha}_1\dot{\beta}_2\cdots\dot{\beta}_n} = 0 \,,\quad p^\gamma_{\dot{\alpha}} B_{\gamma\epsilon_1\epsilon_2\cdots\epsilon_n}^{\dot{\alpha}_1\dot{\beta}_2\cdots\dot{\beta}_n} = 0 $$

hold.