Unitary Laws with Free Unitary Noise: Standing Data

1. (Spaces)¶ The conventions of The Real Numbers: Standing Notation and Background are in force. $d\in\mathbb{N}$, where $\mathbb{N}$ is the set of natural numbers. $\mathcal{L}_{d}$ is the set of unitary laws of $d$-tuples; $E_{d}$ is the gauge space with inner product $\langle\cdot,\cdot\rangle_{d}$ and norm $\lVert\cdot\rVert_{d}$, a real inner product space by The Word Gauge Space Is a Real Hilbert Space, and the Unitary Laws Form a Compact Metric Space in It §inner-product and a real Hilbert space by The Word Gauge Space Is a Real Hilbert Space, and the Unitary Laws Form a Compact Metric Space in It §hilbert; $\lVert\cdot\rVert_{d,1}$ is the length-weighted gauge; $\iota_{d}:\mathcal{L}_{d}\to E_{d}$ is the embedding and $d_{\mathcal{L}}$ the gauge distance, a metric on $\mathcal{L}_{d}$ by The Word Gauge Space Is a Real Hilbert Space, and the Unitary Laws Form a Compact Metric Space in It §metric. Continuity, upper and lower semicontinuity, local maxima, local minima and the upper and lower semicontinuous envelopes of real functions on $\mathcal{L}_{d}$ refer to the metric space $(\mathcal{L}_{d},d_{\mathcal{L}})$, relative to $\mathcal{L}_{d}$.

2. (Test functions)¶ For a map $F:E_{d}\to\mathbb{R}$, $F\circ\iota_{d}$ is the map $\lambda\mapsto F(\iota_{d}(\lambda))$ on $\mathcal{L}_{d}$. $C^{1}(E_{d})$ is the class of $C^{1}$ functions on the open set $E_{d}$ of the real Hilbert space $E_{d}$, in the setting of Real Hilbert Spaces: Standing Notation and Background, with gradient $DF(x)\in E_{d}$ at $x\in E_{d}$.

3. (Noise)¶ $\Theta$ is the free unitary heat generator and, for $\lambda\in\mathcal{L}_{d}$, $\widehat{\Theta}\lambda$ is the restriction of $\Theta\lambda$ to the set of cyclically reduced words, an element of $E_{d}$ by The Free Unitary Heat Generator: Bound, Continuity and Strict Dissipation in the Length-Weighted Gauge §bound. $T^{+}_{\lambda}$ and $T^{\pm}_{\lambda}$ are the sets of inward and two-sided tangent vectors to $\mathcal{L}_{d}$ at $\lambda$.

4. (Constants)¶ $\rho$ and $\beta$ are real numbers with $0<\rho$ and $0\le\beta$.

5. (Source)¶ $f:\mathcal{L}_{d}\to\mathbb{R}$ is a map, the source, that is continuous on the metric space $(\mathcal{L}_{d},d_{\mathcal{L}})$.

6. (Background)¶ The results Unitary Laws: Adjoints, the Bound One, and Reduction to Cyclically Reduced Words, in particular Unitary Laws: Adjoints, the Bound One, and Reduction to Cyclically Reduced Words §nonempty, by which $\mathcal{L}_{d}$ is nonempty, The Word Gauge Space Is a Real Hilbert Space, and the Unitary Laws Form a Compact Metric Space in It, The Free Unitary Heat Generator: Bound, Continuity and Strict Dissipation in the Length-Weighted Gauge, Gradients at Relative Extrema on the Unitary Law Space: Signs on Inward Tangent Vectors and Vanishing on Two-Sided Ones and The Free Unitary Heat Generator Points into the Space of Unitary Laws are in force.