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

$p_{\dot {\alpha }}^{\gamma }A_{\gamma \epsilon _{1}\epsilon _{2}\cdots \epsilon _{n}}^{{\dot {\alpha }}_{1}{\dot {\beta }}_{2}\cdots {\dot {\beta }}_{n}}=0\,,\quad p_{\dot {\alpha }}^{\gamma }B_{\gamma \epsilon _{1}\epsilon _{2}\cdots \epsilon _{n}}^{{\dot {\alpha }}_{1}{\dot {\beta }}_{2}\cdots {\dot {\beta }}_{n}}=0$

hold.

See also

• Bargmann–Wigner equations
• Joos–Weinberg equation