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The analogy between determinantal point processes (DPPs) and free fermioniccalculi is well-known. We point out that, from the perspective of freefermionic algebras, Pfaffian point processes (PfPPs) naturally emerge, and showthat a positive contraction acting on a "doubled" one-particle space with anadditional structure defines a unique PfPP. Recently, Olshanski inverted thedirection from free fermions to DPPs, proposed a scheme to construct afermionic state from a quasi-invariant probability measure, and introduced thenotion of perfectness of a probability measure. We propose a method to checkthe perfectness and show that Schur measures are perfect as long as they arequasi-invariant under the action of the symmetric group. We also studyconditional measures for PfPPs associated with projection operators.Consequently, we show that the conditional measures are again PfPPs associatedwith projection operators onto subspaces explicitly described.

Pfaffian Point Processes from Free Fermion Algebras: Perfectness and Conditional Measures