Hamiltonians on Phase-Space Noncommutative Laws that are Lipschitz in the Momentum with Linear Growth

In the setting of Plan Jets and Hamiltonians on Square-Integrable Noncommutative Laws: Standing Notation, let $\mathcal{H}:\Sigma^{2}_{2d}\to\mathbb{R}$, with lifts $\mathcal{H}_{M}$ as in Plan Jets and Hamiltonians on Square-Integrable Noncommutative Laws: Standing Notation §lifts; sums of $L^{2}$ $d$-tuples are those of Plan Jets and Hamiltonians on Square-Integrable Noncommutative Laws: Standing Notation §pairing and the $L^{2}$ norm $\lVert\cdot\rVert_{2}$ is that of Tracial W*-Probability Spaces and Square-Integrable Tuples: Standing Notation §tuples. Let $L\ge0$ be real.

¶$\mathcal{H}$ is Lipschitz in the momentum with linear growth, with constant $L$, if for every tracial W*-probability space $(H,M,\Omega)$ and all $L^{2}$ $d$-tuples $X,P,Q$ of $(H,M,\Omega)$,