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In Warren Siegel's Fields page 131 , He says that in order to get rid of the negative norm zero component of the vector wavefunction , we impose the constraint $S^{a}_{b}P_{a}+\omega P_{b}=0$ in analogy to the on-shell constraint $P^2+m^2=0$. I don't see how this can help us get rid of negative probability wavefunctions.

I highly recommend you not to start learning QFT from Siegel's book. His approach is very... non-standard, and sometimes it is impossible to understand what he means. There are many much more accessible textbooks on QFT -- Srednicki, Peskin-Schroeder, Zee, etc.

I don't have time to analyze this problem in details, but it may be shown (as Siegel does in the second chapter of that section), that for the case of, for example, vector particle, this condition leads to the fact that its wave-function obeys Maxwell equations, which are gauge invariant. That invariance allows to gauge unphysical time-like component of the photon away.

I don't see how this can help us get rid of negative probability wavefunctions.

A constraint is an additional equation. Consider this constraint in the rest reference frame (${\bf{P}}=0$). It is imposing a certain value to an otherwise "arbitrary" or "independent" wave function component. If the negative norm component becomes zero due to this condition, it stays "harmless" in the other reference frames due to relativistic invariance of expressions involving this norm.

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