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Recently I was reading about 5-component field $(\varphi , \psi_{\mu})$, for which $$ \hat {p}^{\mu} \varphi = mc\psi^{\mu}, \quad \hat {p}_{\mu}\psi^{\mu} = mc\varphi . $$ This field refers to the spin-zero representation of the Lorentz group and obeys the Klein-Gordon massive equation (so it reproduces Poincare group irreducible representations).

What physical system is described by it?

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It might help if you provided context or references for your issue. As it stands, for non vanishing m, which is necessary if you are talking about spin and not helicity, the first equation is a definition of the auxiliary field function $$ \psi^\mu \equiv \frac{\hat{p}^\mu}{mc} ~\phi $$ which makes the second equation the K-G equation for φ. I'm not sure what physical system you are seeking... Any physical system involving a spineless field is also likely to involve its gradient. The "5-array" you wrote is, of course, not a Lorentz vector.

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