space {#Density operators on Fock space}
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Let $\HH$ denote a Hilbert space, the $1$-particle Hilbert space. Unit vectors in $\HH$ are called “orbitals." Although the notation we will use suggests that $\HH$ has countably infinite Hilbert dimension, this is not required; everything works for $\HH$ of any dimension.

The Hilbert space for finite systems of fermions in $\HH$ is the fermion Fock space over $\HH$, which we shall denote by $\FFF(\HH)$ or simply $\FFF$. Let $\hat{a}^*(f)$ and $\hat{a}(f)$ denote the creation and annihilation operators for $f \in \HH$, defined as bounded operators on the Fock space [@BratteliRobinson].

An $n$-fermion “Slater determinant" wave function can be identified with a Fock space vector $$\label{Slater determinant in Fock space}
       |\Phi\rangle  \equals \hat{a}^*(f_1)\hat{a}^*(f_2) \cdots \hat{a}^*(f_n)|\Omega \rangle$$ where $f_1,\ldots,f_n$ are orthonormal orbitals in $\HH$. We think of the vacuum vector $|\Omega \rangle$ as a $0$-particle Slater determinant wave function.

In this article, we are going to focus on states of many-fermion systems that can be represented by density operators on the Fock space, especially those that represent states of finite average particle number.

Let $\Delta$ be a density operator on $\FFF(\HH)$. The “$1$-particle density matrix" or “$1$-pdm" of $\Delta$ is the bounded operator $\gamma_\Delta$ 