Positive-energy joint spectral input

Assume Countable Choice. Let $\mathcal{H}$ be a nonzero complex Hilbert space with inner product $(\cdot ,\cdot )$ linear in its first argument. Use $p\cdot a:=p^0{a}^0-\mathbf{p}\cdot \mathbf{a}$ on ${\mathbb{R}}^4$, the convention of the proper-orthochronous Poincaré supplier, and put ${\bar{V}}_{+}:=\{p=(p^0,\mathbf{p})\in {\mathbb{R}}^4:p^0\ge \mathrm{\mid }\mathbf{p}\mathrm{\mid }\},V_{+}:=\{b=(b^0,\mathbf{b})\in {\mathbb{R}}^4:b^0>\mathrm{\mid }\mathbf{b}\mathrm{\mid }\}\mathrm{.}$ For estimates only, write $\mathrm{\mid }p{\mathrm{\mid }}_{\mathrm{E}}:=\sqrt{(p^0{)}^2+\mathrm{\mid }\mathbf{p}{\mathrm{\mid }}^2}$.

Let $E$ be a projection-valued measure on the Borel subsets of ${\mathbb{R}}^4$: each $E(S)$ is an orthogonal projection, $E(\mathrm{\varnothing })=0$, $E({\mathbb{R}}^4)=I$, $E(S\cap T)=E(S)E(T)$, and for every disjoint sequence $(S_n)$, $E({\bigcup }_n{S}_n)\xi ={\sum }_{n}E(S_n)\xi$ in norm for each $\xi \in \mathcal{H}$. For a bounded Borel function $g$, write ${\mathrm{\Phi }}_E(g):=\int g(p)dE(p)$ for its bounded spectral integral, defined by uniform simple-function approximation. Let $U:{\mathbb{R}}^4\to \mathcal{B}(\mathcal{H})$ be a strongly continuous unitary representation of translations: $U(0)=I$, $U(a+a^{\mathrm{\prime }})=U(a)U(a^{\mathrm{\prime }})$, every $U(a)$ is a surjective linear isometry, and $a\mapsto U(a)\xi$ is norm-continuous for each $\xi \in \mathcal{H}$. The positive-energy joint spectral input is the additional data and condition $U(a)={\mathrm{\Phi }}_E(p\mapsto e^{ip\cdot a}),E({\mathbb{R}}^4\setminus {\bar{V}}_{+})=0.$

Fix a translation-invariant unit vector $\mathrm{\Omega }\in \mathcal{H}$, so $U(a)\mathrm{\Omega }=\mathrm{\Omega }$ for every $a\in {\mathbb{R}}^4$, and a specified unital $\ast$-subalgebra $\mathfrak{A}\subseteq \mathcal{B}(\mathcal{H})$: it contains $I$ and is closed under complex linear combinations, operator composition, and Hilbert-space adjoints, with these operations inherited from $\mathcal{B}(\mathcal{H})$. Explicitly, $(A+B)\xi =A\xi +B\xi$, $(\lambda A)\xi =\lambda A\xi$, $(AB)\xi =A(B\xi )$ and $I\xi =\xi$; the Hilbert adjoint is characterized by $(A\xi ,\eta )=(\xi ,A^{\ast }\eta )$. Under the already assumed Countable Choice, these bounded adjoints exist and obey the adjoint identities of Hilbert-adjoint identities, so the inherited structure is a unital complex star algebra in the sense of Unital star algebra and positive normalized state. Require $\overline{\{A\mathrm{\Omega }:A\in \mathfrak{A}\}}=\mathcal{H}$; this is the meaning of cyclic vacuum here. If local bounded operators are used, their assignment $O\mapsto \mathfrak{A}(O)\subseteq \mathfrak{A}$ for bounded open $O\subseteq {\mathbb{R}}^4$ is separately specified. This input asserts no locality, isotony, covariance, or field-domain property for that assignment. The PVM and its joint support are input; separate generators or a matrix-coefficient Fourier-support condition do not replace this additional data.

Proper-orthochronous Poincaré action and test pullback

The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)

Hilbert space

Real and complex inner-product spaces and their induced length

Projection valued measure

Bounded borel pvm integral

Fewster and Rejzner, §5.1

Mund, §1.3 and §2.1

The required qft-mc02 reconstruction item proves a matrix-coefficient Fourier-support condition only under its additional moment identities. Its Statement explicitly does not assert joint self-adjoint translation generators or their joint PVM. This definition therefore records a stronger, separate input and does not infer it from that reconstruction result.