= Polarized fermionic Fock space
{title2=$\mathcal F_P=\Lambda(PH)\,\widehat\otimes\,\Lambda(\overline{(I-P)H})$}
An orthogonal projection $P$ splits a one-particle <Hilbert space> into empty and filled modes. Take completed exterior Fock spaces of empty particles and of holes in the filled modes, with the graded tensor product enforcing the <canonical anticommutation relations> between both factors. The vacuum is killed by particle annihilation in $PH$ and by particle creation in $(I-P)H$. Only finite particle-hole excitations are needed for a dense basis. Normal ordering subtracts the filled-vacuum expectation from bilinear observables.
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