Realising a Square-Integrable Field of the GNS Space of a Bounded Law next to Its Positions: Uniqueness and the Joint Law with a Momentum

In the setting of Plan Jets and Hamiltonians on Square-Integrable Noncommutative Laws: Standing Notation, for a law $\gamma\in\Sigma_{k}$ ($k=d$ or $k=2d$) let $(\mathcal{H}_{\gamma},\mathcal{M}_{\gamma},\Omega_{\gamma})$ be the tracial W*-probability space of The Tracial Algebra of a Noncommutative Law is a Tracial W*-Probability Space: the W*-Closure of the Left Multiplications §w-star, with conjugation $J_{\gamma}$. $\Sigma_{d,r}$ is the set of laws with norm bound $r$ of Noncommutative Laws, Couplings and the Wasserstein Distance: Standing Notation §laws, and $\kappa_{d}$, $\kappa_{2d}$ are the canonical maps of Square-Integrable Noncommutative Laws: Standing Notation §laws. $L^{2}$ tuples, their pairs and triples and their laws are those of Tracial W*-Probability Spaces and Square-Integrable Tuples: Standing Notation §tuples; sums, real multiples, the pairing $\langle\cdot,\cdot\rangle_{2}$ and the 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.

Data. Let $r>0$ be real, let $\lambda\in\Sigma_{d,r}$, and let $\zeta=(\zeta_{1},\dots,\zeta_{d})$ be an $L^{2}$ $d$-tuple of $(\mathcal{H}_{\lambda},\mathcal{M}_{\lambda},\Omega_{\lambda})$. Write

$$X_{\lambda}=(\widehat{x_{1}},\dots,\widehat{x_{d}})$$

for the classes of the variables $x_{1},\dots,x_{d}$ in $\mathcal{H}_{\lambda}$; it is an $L^{2}$ $d$-tuple of $(\mathcal{H}_{\lambda},\mathcal{M}_{\lambda},\Omega_{\lambda})$ since $J_{\lambda}\widehat{x_{i}}=\widehat{x_{i}^{*}}=\widehat{x_{i}}$ by Left and Right Multiplication Operators on the Complex GNS Space of a Noncommutative Law: Boundedness, Algebra Rules, Adjoints, Commutation, the Vacuum and the Conjugation §conjugation and Noncommutative Polynomials Form a Unital Complex Algebra with Involution: Linear Extension from Monomials, Products, Adjoints and Self-Adjoint Parts §adjoint. Let $(H,M,\Omega)$ be a tracial W*-probability space as in Tracial W*-Probability Spaces and Square-Integrable Tuples: Standing Notation §spaces, and let $X$, $Q$ and $Q'$ be $L^{2}$ $d$-tuples of it with $\mathrm{law}(X)=\kappa_{d}(\lambda)$.

1. (Uniqueness)¶ If $\mathrm{law}(X,Q)=\mathrm{law}(X_{\lambda},\zeta)$ and $\mathrm{law}(X,Q')=\mathrm{law}(X_{\lambda},\zeta)$, then $Q=Q'$.

2. (Joint law with a momentum)¶ Assume $\mathrm{law}(X,Q)=\mathrm{law}(X_{\lambda},\zeta)$. Let $\pi$ be a bounded plan at $\lambda$, let $X_{\pi},P_{\pi}$ be its positions and momenta and $V^{1}_{\pi}\zeta$ the field carried to $\pi$, all $L^{2}$ $d$-tuples of $(\mathcal{H}_{\pi},\mathcal{M}_{\pi},\Omega_{\pi})$, and let $P$ be an $L^{2}$ $d$-tuple of $(H,M,\Omega)$ with $\mathrm{law}(X,P)=\kappa_{2d}(\pi)$. Then

$$\mathrm{law}(X,P,Q)=\mathrm{law}\bigl(X_{\pi},P_{\pi},V^{1}_{\pi}\zeta\bigr).$$

¶Consequently $\mathrm{law}(X,P+tQ)=\pi\oplus t\zeta$, the shift of $\pi$ by $t\zeta$, for every real $t$. ¶Moreover $\langle Q,P\rangle_{2}=\mathcal{J}(\zeta,\pi)$, the plan pairing.

3. (Norm)¶ If $\mathrm{law}(X,Q)=\mathrm{law}(X_{\lambda},\zeta)$, then $\lVert Q\rVert_{2}=\lVert\zeta\rVert_{2}$.