Couplings of a Square-Integrable Plan with a Law: Displacement and Momentum Pairing

In the setting of Plan Jets and Hamiltonians on Square-Integrable Noncommutative Laws: Standing Notation, write $3d=d+d+d$. Affine data, their push-forwards $T_{\#}$ and the marginal data $\mathrm{pr}^{1},\mathrm{pr}^{2}$ are those of Square-Integrable Noncommutative Laws: Standing Notation §affine, and the quadratic moments $\mathrm{m}_{ij}$ are those of Square-Integrable Noncommutative Laws: Standing Notation §moments. Sums, differences, the pairing $\langle\cdot,\cdot\rangle_{2}$ and the $L^{2}$ norm $\lVert\cdot\rVert_{2}$ of $L^{2}$ $d$-tuples are those of Plan Jets and Hamiltonians on Square-Integrable Noncommutative Laws: Standing Notation §pairing; $L^{2}$ tuples, the operations $TZ$, pairs and triples, and laws are those of Tracial W*-Probability Spaces and Square-Integrable Tuples: Standing Notation §tuples. Let $B=(E^{B},0)$ and $U=(E^{U},0)$ be the affine data from $3d$ to $2d$ variables and $C=(E^{C},0)$ the affine datum from $3d$ to $d$ variables given, for $i\in[d]$ and $j\in[3d]$, by

$$E^{B}_{ij}=\begin{cases}1,&j=i,\\0,&j\ne i,\end{cases}\qquad E^{B}_{d+i,j}=\begin{cases}1,&j=d+i,\\0,&j\ne d+i,\end{cases}\qquad E^{C}_{ij}=\begin{cases}1,&j=2d+i,\\0,&j\ne 2d+i,\end{cases}$$ $$E^{U}_{ij}=\begin{cases}1,&j=d+i,\\0,&j\ne d+i,\end{cases}\qquad E^{U}_{d+i,j}=\begin{cases}1,&j=2d+i,\\-1,&j=i,\\0,&\text{otherwise},\end{cases}$$

so that $B(X,P,X')=(X,P)$, $C(X,P,X')=X'$ and $U(X,P,X')=(P,X'-X)$ for all $L^{2}$ $d$-tuples $X,P,X'$ of a tracial W*-probability space.

1. (Displacement and momentum pairing)¶ Let $\gamma\in\Sigma^{2}_{3d}$. A realisation of $\gamma$ is a triple $(X,P,X')$ of $L^{2}$ $d$-tuples of a tracial W*-probability space with $\mathrm{law}(X,P,X')=\gamma$; one exists by Every Square-Integrable Noncommutative Law is the Law of a Square-Integrable Tuple; Realisation of Couplings and of Almost Optimal Pairs §law, applied to $3d$ variables, writing the $L^{2}$ $3d$-tuple it provides as a triple. For every realisation, $\mathrm{law}(P,X'-X)=U_{\#}\gamma$ by Calculus of Laws of Square-Integrable Tuples: Bounded Tuples, the Lipschitz Bound, Moments, Affine Push-Forwards, Couplings and Embeddings §push-forward, so that, by Calculus of Laws of Square-Integrable Tuples: Bounded Tuples, the Lipschitz Bound, Moments, Affine Push-Forwards, Couplings and Embeddings §moments and Sums, Real Multiples and the Pairing of Square-Integrable Tuples in a Tracial W*-Probability Space §pairing,

$$\lVert X'-X\rVert_{2}^{2}=\sum_{i=1}^{d}\mathrm{m}_{d+i,d+i}(U_{\#}\gamma),\qquad\langle P,X'-X\rangle_{2}=\sum_{i=1}^{d}\mathrm{m}_{i,d+i}(U_{\#}\gamma).$$

Hence the reals $s(\gamma)=\lVert X'-X\rVert_{2}$ and $p(\gamma)=\langle P,X'-X\rangle_{2}$ do not depend on the realisation; $s(\gamma)$ is the displacement and $p(\gamma)$ the momentum pairing of $\gamma$.

2. (Couplings with a plan)¶ For $\varpi\in\Sigma^{2}_{2d}$ and $\lambda\in\Sigma^{2}_{d}$, the set of couplings of $\varpi$ with $\lambda$ is

$$\mathcal{C}_{\varpi}(\lambda)=\bigl\{\gamma\in\Sigma^{2}_{3d}:\ B_{\#}\gamma=\varpi\ \text{and}\ C_{\#}\gamma=\lambda\bigr\}.$$

This set is nonempty. Indeed, by Every Square-Integrable Noncommutative Law is the Law of a Square-Integrable Tuple; Realisation of Couplings and of Almost Optimal Pairs §distance (with $\varepsilon=1$) there are $L^{2}$ $d$-tuples $X_{0},Y_{0}$ of a tracial W*-probability space with $\mathrm{law}(X_{0})=\mathrm{pr}^{1}_{\#}\varpi$ and $\mathrm{law}(Y_{0})=\lambda$; then $\gamma_{0}=\mathrm{law}(X_{0},Y_{0})$ satisfies $\mathrm{pr}^{1}_{\#}\gamma_{0}=\mathrm{pr}^{1}_{\#}\varpi$ and $\mathrm{pr}^{2}_{\#}\gamma_{0}=\lambda$ by Every Square-Integrable Noncommutative Law is the Law of a Square-Integrable Tuple; Realisation of Couplings and of Almost Optimal Pairs §coupling. Since $\mathrm{pr}^{1}$ is the datum $F^{d}$ of Gluing Two Square-Integrable Noncommutative Laws along a Common Marginal with $k=d$, Gluing Two Square-Integrable Noncommutative Laws along a Common Marginal §glue, with $k=m=n=d$, $\pi_{1}=\varpi$ and $\pi_{2}=\gamma_{0}$, gives $L^{2}$ $d$-tuples $X,P,X'$ of a tracial W*-probability space with $\mathrm{law}(X,P)=\varpi$ and $\mathrm{law}(X,X')=\gamma_{0}$. Then $\gamma=\mathrm{law}(X,P,X')$ lies in $\mathcal{C}_{\varpi}(\lambda)$: $B_{\#}\gamma=\mathrm{law}(X,P)=\varpi$ and $C_{\#}\gamma=\mathrm{law}(X')=\mathrm{pr}^{2}_{\#}\gamma_{0}=\lambda$ by Calculus of Laws of Square-Integrable Tuples: Bounded Tuples, the Lipschitz Bound, Moments, Affine Push-Forwards, Couplings and Embeddings §push-forward and Every Square-Integrable Noncommutative Law is the Law of a Square-Integrable Tuple; Realisation of Couplings and of Almost Optimal Pairs §coupling.