The Shift of a Bounded Plan by a Self-Adjoint Field

In the setting of Plan Jets and Hamiltonians on Square-Integrable Noncommutative Laws: Standing Notation, for a law $\lambda\in\Sigma_{k}$ ($k=d$ or $k=2d$) let $(\mathcal{H}_{\lambda},\mathcal{M}_{\lambda},\Omega_{\lambda})$ 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 $J_{\lambda}$.

Data. Let $\mu\in\Sigma_{d}$, let $\pi$ be a bounded plan at $\mu$, let $\zeta=(\zeta_{1},\dots,\zeta_{d})$ be an $L^{2}$ $d$-tuple of $(\mathcal{H}_{\mu},\mathcal{M}_{\mu},\Omega_{\mu})$, and let $t$ be real.

Positions and momenta of the plan.¶ The classes of the variables in $\mathcal{H}_{\pi}$,

$$X_{\pi}=(\widehat{x_{1}},\dots,\widehat{x_{d}}),\qquad P_{\pi}=(\widehat{x_{d+1}},\dots,\widehat{x_{2d}}),$$

are $L^{2}$ $d$-tuples of $(\mathcal{H}_{\pi},\mathcal{M}_{\pi},\Omega_{\pi})$, since $J_{\pi}\widehat{x_{i}}=\widehat{x_{i}^{*}}=\widehat{x_{i}}$ 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.

The field carried to the plan.¶ Let $V^{1}_{\pi}:\mathcal{H}_{\mu}\to\mathcal{H}_{\pi}$ be the marginal isometry. Since $V^{1}_{\pi}J_{\mu}=J_{\pi}V^{1}_{\pi}$ by Marginals of a Noncommutative Law: the Isometry of GNS Spaces, the Trace-Preserving Embedding of Tracial Algebras and the Conditional Expectation §isometry, $J_{\pi}V^{1}_{\pi}\zeta_{j}=V^{1}_{\pi}J_{\mu}\zeta_{j}=V^{1}_{\pi}\zeta_{j}$ for every $j\in[d]$, so $V^{1}_{\pi}\zeta=(V^{1}_{\pi}\zeta_{1},\dots,V^{1}_{\pi}\zeta_{d})$ is an $L^{2}$ $d$-tuple of $(\mathcal{H}_{\pi},\mathcal{M}_{\pi},\Omega_{\pi})$. Hence the sum and real multiple $P_{\pi}+tV^{1}_{\pi}\zeta$ is an $L^{2}$ $d$-tuple, and the pair $(X_{\pi},P_{\pi}+tV^{1}_{\pi}\zeta)$ is an $L^{2}$ $2d$-tuple, with a law in $\Sigma^{2}_{2d}$.

¶Definition. The shift of $\pi$ by $t\zeta$ is the $L^{2}$ law

$$\pi\oplus t\zeta=\mathrm{law}\bigl(X_{\pi},\,P_{\pi}+t\,V^{1}_{\pi}\zeta\bigr)\in\Sigma^{2}_{2d}.$$