$\rho$ (which is not restricted to a pure state density matrix, but may also correspond to a mixed configuration ${\text{Tr}}{\rho^2} \neq {\text{Tr}}\rho$), the $m$-particle ($m < {\ensuremath{\mathbb N}}$) reduced density matrices can be defined as $$\label{e:yy}
  \rho_m \equiv \frac{1}{(m!)^2}{\ensuremath{|i_1 i_2...i_m\rangle}}{\text{Tr}}\{a_{i_1}a_{i_2}...a_{i_m}
  \rho\,a^\dag_{j_m}a^\dag_{j_{m-1}}...a^\dag_{j_1} \}{\ensuremath{\langlej_1j_{2}...j_m|}}.$$ This is equivalent to Yang’s definition[@yang]. Note that $\rho_m$ is an m-particle operator; and we have simplified all the quantum numbers of the creation operator $a^\dag_{i_k}$ to the label $i_k$. It can then be worked out that Eq.(\[e:yy\]) implies $$\begin{aligned}
\label{e:vev}
  \frac{{\ensuremath{\langlei_1...i_m|}}\rho_m{\ensuremath{|j_1...j_m\rangle}}}{m!({\text{Tr}}\rho_m)} &=&
    \frac{{\ensuremath{\langlei_{1}...i_m{k_{m+1}}...k_n|}}\rho_n{\ensuremath{|j_1...j_mk_{m+1}...k_{n}\rangle}}}{n!({\text{Tr}}\rho_n)} \qquad
    \forall ~ m<n ~,~ 1 < n \leq {{\ensuremath{\mathbb N}}},\\
\text{with} \qquad\frac{\rho_{{\ensuremath{\mathbb N}}}}{{\text{Tr}}\rho_{{\ensuremath{\mathbb N}}}} &=&
\frac{\rho}{{\text{Tr}}\rho}\qquad .\end{aligned}$$ Thus these reduced density matrices are defined by the partial traces of higher particle number density operators.

It is worth emphasizing in quantum field theory Lorentz transformations are implemented [*unitarily*]{} on [*physical states*]{}. Under any Lorentz transformation $\Lambda$, all creation and annihilation operators transform as $U(\Lambda){a^\dag}_i {U^\dag}(\Lambda)$ and $U(\Lambda){a}_i
{U^\dag}(\Lambda)$. It follows that (we also assume a Lorentz invariant vacuum $U(\Lambda){\ensuremath{|0\rangle}}={\ensuremath{|0\rangle}}$) all states obtained through the action of creation and annihilation operators on the vacuum must transform unitarily as ${\ensuremath{|\Psi\rangle}}' = U(\Lambda){\ensuremath{|\Psi\rangle}}$ and ${\ensuremath{\langle\Psi|}}'={\ensuremath{\langle\Psi|}}U^\dag(\Lambda)$. As a consequence, under Lorentz transformations all reduced density matrices defined above 