Admissible Hamiltonians on Unitary Laws

In the setting of Unitary Laws with Free Unitary Noise: Standing Data, with the conventions of Unitary Laws with Free Unitary Noise: Standing Data §spaces, let $T^{\pm}_{\lambda}$ be the set of two-sided tangent vectors to $\mathcal{L}_{d}$ at $\lambda\in\mathcal{L}_{d}$. By The Word Gauge Space Is a Real Hilbert Space, and the Unitary Laws Form a Compact Metric Space in It §inner-product, $E_{d}$ is a real inner product space with norm $\lVert\cdot\rVert_{d}$, so $(x,y)\mapsto\lVert x-y\rVert_{d}$ is a metric on $E_{d}$ by The Norm Metric of a Real Inner Product Space: Triangle Inequalities, Limits and Continuity §metric; and $(\mathcal{L}_{d},d_{\mathcal{L}})$ is a metric space by The Word Gauge Space Is a Real Hilbert Space, and the Unitary Laws Form a Compact Metric Space in It §metric; hence the product metric $\bigl((\mu,p),(\nu,q)\bigr)\mapsto\max\bigl\{d_{\mathcal{L}}(\mu,\nu),\ \lVert p-q\rVert_{d}\bigr\}$ makes $\mathcal{L}_{d}\times E_{d}$ a metric space by claim 1 of The Product Metric is a Metric.

A map $H:\mathcal{L}_{d}\times E_{d}\to\mathbb{R}$ is an admissible Hamiltonian if it has the following two properties.

¶(a) $H$ is continuous on $\mathcal{L}_{d}\times E_{d}$, equipped with the product metric above, with values in the metric space $(\mathbb{R},d_{\mathbb{R}})$.

¶(b) For every $\lambda\in\mathcal{L}_{d}$ and all $p,p'\in E_{d}$ with $\langle p-p',v\rangle_{d}=0$ for every $v\in T^{\pm}_{\lambda}$, one has $H(\lambda,p)=H(\lambda,p')$.