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If I consider one single Dirac electron in momentum representation, I use the wavefunction $u(p)e^{-ipx}$, however if I consider an one-particle state in the Fock space I use $|p\rangle$. Should it not be same?

Obviously the Dirac 1-particle wavefunction is a bispinor, and probably $|p\rangle$ is not a spinor. But could it not be spinor?

For a 2-particle wavefunction $|p,k\rangle$, I would use $$\frac{1}{\sqrt2}(u_1(p)u_2(k)e^{ipx_1+ikx_2} - u_2(k)u_1(p)e^{ipx_2 + ikx_1})$$ or something similar. I regret my limited way of expressing correctly. Certainly there is the problem if I consider instead of a half-spin particle a scalar particle, then I would have to build my multi-particles state out scalar wave function instead of spinor wave functions. May be my understanding of the Fock space is incomplete.

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