Total momentum multipliers on transverse symmetric Fock sectors

[Explicit transverse tensor embedding] The measurable matrices $\mathrm{\Pi }(k)$ are orthogonal projections with norm at most one. Hence $M_{\mathrm{\Pi }}$ is a bounded orthogonal projection on $G$, with range exactly $\mathfrak{h}$, because $\mathrm{\Pi }(k)f(k)=f(k)$ is equivalent to $k\cdot f(k)=0$ off the null point $0$. In particular $\mathfrak{h}$ is closed and Hilbert. The inclusion $\mathfrak{h}\to G$ induces an isometric inclusion of every finite tensor power: its elementary-tensor inner products agree by [F1], and density extends that isometry to the completion. Its closed range is the range of $(M_{\mathrm{\Pi }}{)}^{\otimes n}$, since the latter projection maps the dense elementary span onto elementary transverse tensors. Iterating [F1] with dimensions $3,3(n-1)$ and finite components gives a unitary ${\mathrm{\Theta }}_n:G^{\otimes n}\to L_n$ with the stated product formula. Under it $(M_{\mathrm{\Pi }}{)}^{\otimes n}$ becomes multiplication by $Q_n(k)=\mathrm{\Pi }(k_1)\otimes \cdots \otimes \mathrm{\Pi }(k_n)$: both bounded maps agree on all elementary tensors, hence on their dense span. No undocumented product-measure or tensor-subspace identification is used.

[Symmetry includes polarization permutations] Define the unitary ${\tau }_{\pi }$ on $({\mathbb{C}}^3{)}^{\otimes n}$ by the same tensor-factor permutation as in [F2]. The conjugated ambient permutation is $(C_{\pi }u)(k_1,\dots ,k_n)={\tau }_{\pi }u(k_{\widehat{\pi }(1)},\dots ,k_{\widehat{\pi }(n)})$: substitution into the elementary product formula verifies this identity, and the bounded maps agree everywhere by density. Its average $S_n=(n!{)}^{-1}{\sum }_{\pi }{C}_{\pi }$ is the orthogonal symmetry projection. The tensor projection $(M_{\mathrm{\Pi }}{)}^{\otimes n}$ commutes with every ambient permutation, as is verified on elementary tensors, so $M_{Q_n}$ commutes with $S_n$. Their product $R_n=M_{Q_n}{S}_n$ is an orthogonal projection, whose range is exactly the image of the symmetric transverse tensor sector. Thus $K_n=\mathrm{ran}{R}_n$ and $I_n={\mathrm{\Theta }}_n$ restricted to that sector is an isometry onto $K_n$. Every bounded scalar multiplier $b(p_{j,n})$ commutes with $M_{Q_n}$ pointwise and with $C_{\pi }$, since $p_{j,n}$ is invariant under permutations; therefore it commutes with $R_n$ and preserves $K_n$.

[Sector PVM and regularity] The scalar function $p=p_{j,n}$ is real, measurable and finite on ${\mathbb{R}}^{3n}$. By [F3], multiplication by ${\mathbf{1}}_{p^{-1}(B)}$ is its spectral projection in each scalar component of $L_n$; the finite component sum is a regular PVM there. It commutes with $R_n$ by step 2.1. Its restriction to $K_n$, transported by $I_n$, is therefore an orthogonal projection for every $B$, with total projection $I$, product law and strong countable additivity inherited from the ambient PVM. Its scalar measure at $u$ is exactly ${\int }_{p^{-1}(B)}\mathrm{\parallel }{I}_{n}u{\mathrm{\parallel }}^2$, the sum of its finite component measures, so it is regular and is the push-forward of $\mathrm{\parallel }{I}_{n}u{\mathrm{\parallel }}^2{d}^{3n}k$. This defines the sector PVM $E_{j,n}$. For $n=0$, the indicated atom at zero is a regular PVM on $\mathbb{C}$.

[Exact signed sector domain and self-adjointness] For the PVM of step 3.1, push-forward integration gives $\int t^{2}d{E}_{j,n,u}(t)=\int \mathrm{\mid }p(k){\mathrm{\mid }}^2\mathrm{\parallel }{I}_{n}u(k){\mathrm{\parallel }}^2{d}^{3n}k$: it holds first for indicators and simple nonnegative functions and then for increasing nonnegative simple approximations. By [F4], $\int td{E}_{j,n}(t)$ is densely defined and self-adjoint on exactly the displayed multiplier domain. For bounded Borel $b$, indicator/simple-function approximation and the norm estimate of [F4] show that $b(E_{j,n})$ is the restriction of multiplication by $b(p)$, transported by $I_n$. In particular the defining truncations for the identity function act as $p{\mathbf{1}}_{\mathrm{\mid }p\mathrm{\mid }\le L}{I}_{n}u$. If $p{I}_{n}u\in L_n$, these converge to $p{I}_{n}u$ in norm, because the integrable squared tail ${\int }_{\mathrm{\mid }p\mathrm{\mid }>L}\mathrm{\mid }p{\mathrm{\mid }}^2\mathrm{\parallel }{I}_{n}u{\mathrm{\parallel }}^2$ tends to zero. Every truncation lies in the closed $K_n$, so its limit lies there also. Thus the PVM integral is exactly $P_{j,n}$ with the stated action and domain. Spectral uniqueness [F4] identifies $E_{j,n}$ as its spectral PVM; $P_{j,0}=0$ is the corresponding vacuum operator. The sign of $p$ played no role in this squared-integrability argument.

[The Fock PVM and push-forward sum] For each Borel $B$, define $E_j(B)={⨁}_{n\ge 0}{E}_{j,n}(B)$. It is a bounded orthogonal projection with the product law and total projection $I$. Strong countable additivity follows sectorwise and then on the full sum: first retain finitely many sectors, where convergence is already known, and bound the omitted squared tail by ${\sum }_{n>N}\mathrm{\parallel }{\psi }_n{\mathrm{\parallel }}^2$, which tends to zero uniformly in the partial union, using [F5]. Its scalar measure at $\psi$ is [step 3.1, step 4.1, F3, F5, algebra] $E_{j,\psi }(B)={\mathbf{1}}_B(0)\mathrm{\mid }{\psi }_0{\mathrm{\mid }}^2+{\sum }_{n\ge 1}{\int }_{p_{j,n}^{-1}(B)}\mathrm{\parallel }{I}_n{\psi }_n(k){\mathrm{\parallel }}^2{d}^{3n}k\mathrm{.}$ This is a finite Borel measure of mass $\mathrm{\parallel }\psi {\mathrm{\parallel }}^2$. It is regular: given $ϵ>0$, choose finitely many sector measures leaving total tail mass below $ϵ\mathrm{/}2$; approximate the set from inside by compact sets for those finitely many regular measures and take their finite union, or from outside by open sets and take their finite intersection, allocating the remaining error among those measures. The vacuum measure is supported on the compact singleton $\{0\}$ and is regular. Thus $E_j$ is a regular PVM and the scalar push-forward formula in (b) holds.

[Total operator, exact domain and self-adjointness] For the scalar measure of step 5.1, nonnegative simple approximation and monotone convergence give [step 3.1, step 4.1, step 5.1, F4, F5, algebra] ${\int }_{\mathbb{R}}{t}^{2}d{E}_{j,\psi }(t)={\sum }_{n\ge 1}{\int }_{{\mathbb{R}}^{3n}}\mathrm{\mid }{p}_{j,n}(k){\mathrm{\mid }}^2\mathrm{\parallel }{I}_n{\psi }_n(k){\mathrm{\parallel }}^2{d}^{3n}k={\sum }_{n\ge 1}\mathrm{\parallel }{P}_{j,n}{\psi }_n{\mathrm{\parallel }}^2,$ with the vacuum contribution zero; the monotone convergence step uses Monotone convergence for the integral. By [F4], $A_j:=\int td{E}_j(t)$ is self-adjoint on the domain where this second moment is finite, exactly the domain $D(P_j)$ in (b). For bounded Borel functions, the PVM integral acts sectorwise: this holds for indicators by the definition of $E_j$, for simple functions by linearity, and for bounded Borel functions by bounded approximation and the norm estimate in [F4]. In particular the truncations $t{\mathbf{1}}_{\{\mathrm{\mid }t\mathrm{\mid }\le L\}}(E_j)$ act on sector $n$ as the corresponding truncations of $P_{j,n}$. On $D(P_j)$ these converge to $P_{j,n}{\psi }_n$ in each sector, and their squared total error is bounded by ${\sum }_n\mathrm{\parallel }{P}_{j,n}{\psi }_n{\mathrm{\parallel }}^2$; truncating the sector tail and then taking $L\to \mathrm{\infty }$ in the finitely many retained sectors proves convergence in $F(\mathfrak{h})$. Hence $A_j\psi =(P_{j,n}{\psi }_n{)}_n=P_j\psi$ on precisely $D(P_j)$. This proves self-adjointness of the displayed total operator directly, with the vacuum sector $P_{j,0}=0$ included, and identifies $E_j$ as its spectral PVM by uniqueness.

[Vacuum, one-particle restriction and scope] The vacuum has only its zeroth component, where $P_{j,0}=0$, so it belongs to $D(P_j)$ and $P_j\mathrm{\Omega }=0$. For $n=1$, ${\mathrm{\Theta }}_1=I_G$ and the symmetric projection is the identity, so the sector construction gives exactly the maximal restricted multiplier $f\mapsto \mathrm{\hslash }{k}_{j}f$ on $\mathfrak{h}$. Steps 1.1–6.1 establish all tensor embeddings, sector domains, self-adjointness and PVM formulas without importing any physical radiation postulate. CC and AC are the declared hypotheses of the actual mathematical suppliers; no positivity, dispersion relation, commutation between distinct unbounded observables or physical law was used.