The Gaussian family attains the position-momentum bound in the massive model

Given: The postulates P1 and P2 and the model P5 read through The massive spin-s particle model of the framework: state space, operators, units and frame readings with $r=1$; the Axiom of Choice and the Axiom of Countable Choice; a width $\sigma >0$; the action constant $\mathrm{\hslash }>0$; the vector ${\psi }_{\sigma }$; the momentum transform $U$; a coordinate $j$ or $k$.

Heat kernel. For $t>0$ the function $\mathrm{\Gamma }(x,t)=(4\pi t{)}^{-n\mathrm{/}2}{e}^{-\mathrm{\mid }x{\mathrm{\mid }}^2\mathrm{/}(4t)}$ on ${\mathbb{R}}^n$ is in $C^{\mathrm{\infty }}$, satisfies ${\int }_{{\mathbb{R}}^n}\mathrm{\Gamma }(x,t)dx=1$ (The heat kernel on ${\mathbb{R}}^n$ and its causal extension, Normalisation, parabolic scaling, heat equation and derivative bounds for the heat kernel), and has absolutely integrable first and second moments with ${\int }_{{\mathbb{R}}^n}{x}_i\mathrm{\Gamma }(x,t)dx=0$ and ${\int }_{{\mathbb{R}}^n}{x}_i{x}_j\mathrm{\Gamma }(x,t)dx=2t{\delta }_{ij}$ (First and second Gaussian heat-kernel moments).

Gaussian Fourier transform and momentum transform. With the $2\pi$-normalized unitary Plancherel transform one has ${\mathcal{F}}_2(e^{-\pi t\mathrm{\mid }x{\mathrm{\mid }}^2})(\xi )=t^{-n\mathrm{/}2}{e}^{-\pi \mathrm{\mid }\xi {\mathrm{\mid }}^2\mathrm{/}t}$ for $t>0$ (Euclidean Gaussian transform with the 2π normalization); the model's momentum transform satisfies $U=D_{\mathrm{\hslash }}{\mathcal{F}}_2$ with $D_{\mathrm{\hslash }}g(p)=(2\pi \mathrm{\hslash }{)}^{-3\mathrm{/}2}g(p\mathrm{/}(2\pi \mathrm{\hslash }))$ and $(U\phi )(p)=(2\pi \mathrm{\hslash }{)}^{-3\mathrm{/}2}\int e^{-ip\cdot x\mathrm{/}\mathrm{\hslash }}\phi (x)dx$ for $\phi \in L^1\cap L^2$ (Momentum Fourier Jacobian: the unitary momentum representation clauses (a),(c), The momentum representation of the componentwise kinematics clause (a)).

Outcome laws of the model. For a unit class $\psi$ the position law of $Q_j$ is ${\mu }_{\psi }^{Q_j}(B)={\int }_{x_j^{-1}(B)}\mathrm{\mid }\psi (x){\mathrm{\mid }}^{2}dx$ and the momentum law of $P_k$ is ${\mu }_{\psi }^{P_k}(B)={\int }_{\{p_k\in B\}}\mathrm{\mid }(U\psi )(p){\mathrm{\mid }}^{2}dp$ (Position and momentum outcome laws of the massive spin-s model clauses (a) and (c), Assigned sharp observables and the outcome law of the framework clause (b)).

Moments and spreads. For a unit vector in the domain of the operator the mean and variance of the outcome law exist and equal the quadratic-form mean and the squared norm of the centered vector (Mean, second moment and variance of an assigned observable in a preparation clauses (a) and (b), Preparation variance and spread of an assigned sharp observable clauses (b) and (c)).

Position-momentum relation. For every unit $\psi \in D(Q_j)\cap D(P_j)$ one has ${\mathrm{\Delta }}_{\psi }{Q}_j{\mathrm{\Delta }}_{\psi }{P}_j\ge \mathrm{\hslash }\mathrm{/}2$, with equality if and only if the centered vectors are linearly dependent and their pairing has vanishing real part (The position-momentum preparation relation of the massive spin-s model clauses (a) and (c)).

Schwartz functions and weak derivatives. A smooth function $f:{\mathbb{R}}^3\to \mathbb{C}$ lies in $\mathcal{S}({\mathbb{R}}^3;\mathbb{C})$ exactly when all seminorms ${\sup }_x\mathrm{\mid }{x}^{\alpha }{\mathrm{\partial }}^{\beta }f(x)\mathrm{\mid }$ are finite (Schwartz space and its seminorms). A $C^1$ function has its classical partial derivatives as locally integrable weak derivatives (Classical derivatives agree with weak derivatives). When those derivatives are in $L^2$, their classes give the model's momentum-domain membership. We distinguish a smooth function from its almost-everywhere $L^2$ class, using the same symbol when passing to that class.

Product integration. Integrals of nonnegative measurable functions on product spaces may be iterated (Tonelli's theorem for nonnegative measurable functions on a sigma-finite product), and ${\int }_{\mathbb{R}}{e}^{-u^2}du=\sqrt{\pi }$ (The Gaussian integral ${\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{e}^{-x^2}dx=\sqrt{\pi }$).

Choice in the spectral laws and moments. The Axiom of Choice is the spectral theorem's stated hypothesis for constructing the spectral projection-valued measures of the self-adjoint operators; the moment suppliers also state the Axiom of Choice and Countable Choice. Thus the Born-law and moment steps below are conditional on these assumptions. The integrations and Gaussian Fourier calculation use the separate measure and Fourier suppliers [F1], [F2] and [F7].