Massive scattering packets and Cook approximant contract

Assume Countable Choice. Use the positive-energy joint spectral input: a nonzero complex Hilbert space $\mathcal{H}$, a strongly continuous translation representation $U:{\mathbb{R}}^4\to \mathcal{B}(\mathcal{H})$, a projection-valued measure $E$ on ${\mathbb{R}}^4$ with $U(a)=\int e^{ip\cdot a}dE(p)$ and $E({\mathbb{R}}^4\setminus {\bar{V}}_{+})=0$, and an invariant unit cyclic vacuum $\mathrm{\Omega }$ for a specified unital complex $\ast$-subalgebra $\mathfrak{A}\subseteq \mathcal{B}(\mathcal{H})$. By the unital star-algebra definition listed below, a unital complex $\ast$-algebra has a complex-bilinear associative product, a specified unit, and a conjugate-linear involution reversing products, squaring to the identity, and fixing the unit; $\mathfrak{A}$ inherits these operations and contains $I$. Cyclicity means $\overline{\{B\mathrm{\Omega }:B\in \mathfrak{A}\}}=\mathcal{H}$. The pairing is $p\cdot a=p^0{a}^0-\mathbf{p}\cdot \mathbf{a}$. Use the shell coordinates and signature of the scalar future mass shell definition: for $m>0$ and $\mathbf{k}\in {\mathbb{R}}^3$, ${\omega }_m(\mathbf{k})=\sqrt{\mathrm{\mid }\mathbf{k}{\mathrm{\mid }}^2+m^2}$ and $p_m(\mathbf{k})=({\omega }_m(\mathbf{k}),\mathbf{k})$. Here $m$ is the positive inverse-length mathematical shell parameter called $\mu$ in that supplier; it is not identified with a measured rest mass. The coordinates have length units, $\mathbf{k}$ and ${\omega }_m$ have inverse-length units, and $\mathbf{k}\mathrm{/}{\omega }_m(\mathbf{k})$ is dimensionless.

${\mathcal{M}}_m=\{(p^0,\mathbf{p}):(p^0{)}^2-\mathrm{\mid }\mathbf{p}{\mathrm{\mid }}^2=m^2,p^0>0\}$ is the positive mass shell; it is closed and hence Borel. Its equivalent parameterization is ${\mathcal{M}}_m=\{p_m(\mathbf{k}):\mathbf{k}\in {\mathbb{R}}^3\}$. It is an isolated massive shell input when $P_m:=E({\mathcal{M}}_m)\mathrm{\ne }0$ and some $0<\delta <m$ satisfies $E(\{p\in {\bar{V}}_{+}:\mathrm{\mid }\sqrt{(p^0{)}^2-\mathrm{\mid }\mathbf{p}{\mathrm{\mid }}^2}-m\mathrm{\mid }<\delta \}\setminus {\mathcal{M}}_m)=0.$ The corresponding one-particle space in this contract is the closed subspace ${\mathcal{H}}_m=P_m\mathcal{H}$. This is an adopted spectral hypothesis and subspace designation; the definition does not infer particle stability from a formal two-point-function pole.

A translation-covariant local net is an assignment $O\mapsto \mathfrak{A}(O)$ of unital $\ast$-subalgebras of $\mathcal{B}(\mathcal{H})$ to bounded open $O\subseteq {\mathbb{R}}^4$, with $\mathfrak{A}(O)\subseteq \mathfrak{A}$, such that $O_1\subseteq O_2$ implies $\mathfrak{A}(O_1)\subseteq \mathfrak{A}(O_2)$, $U(a)\mathfrak{A}(O)U(a{)}^{\ast }=\mathfrak{A}(O+a)$, and $A_1{A}_2=A_2{A}_1$ for every $A_i\in \mathfrak{A}(O_i)$ when the regions are spacelike separated. Spacelike separation means $(x^0-y^0{)}^2-\mathrm{\mid }\mathbf{x}-\mathbf{y}{\mathrm{\mid }}^2<0$ for every $x\in O_1,y\in O_2$. An operator $A$ is local if it belongs to $\mathfrak{A}(O)$ for some bounded open $O$. A bounded operator $A$ is norm-almost-local if, for every integer $N\ge 1$, there is a constant $C_N<\mathrm{\infty }$ and operators $A_R\in \mathfrak{A}(O_R)$ for all $R\ge 1$, where $O_R=\{x:\mathrm{\mid }{x}^0\mathrm{\mid }+\mathrm{\mid }\mathbf{x}\mathrm{\mid }<R\}$, such that $\mathrm{\parallel }A-A_R\mathrm{\parallel }\le C_N(1+R{)}^{-N}$. The domain version of almost locality uses a common dense translation-invariant domain $\mathcal{D}\subseteq \mathcal{H}$ with $\mathrm{\Omega }\in \mathcal{D}$: $A$ and each local approximant $A_R$ act on $\mathcal{D}$, and for every $\xi \in \mathcal{D}$ and integer $N\ge 1$ there is $C_{N,\xi }<\mathrm{\infty }$ such that $\mathrm{\parallel }(A-A_R)\xi \mathrm{\parallel }\le C_{N,\xi }(1+R{)}^{-N}$ for all $R\ge 1$. A local or almost-local operator is an interpolator for $f$ if $A\mathrm{\Omega }$ is defined and the vector below is nonzero.

For fermionic fields, the corresponding operator net is a ${\mathbb{Z}}_2$-graded field net: each local algebra splits as $\mathfrak{F}(O)={\mathfrak{F}}_0(O)\oplus {\mathfrak{F}}_1(O)\subseteq \mathcal{B}(\mathcal{H})$ and has a grading automorphism $\gamma$ with $\gamma (A)=(-1{)}^{a}A$ for homogeneous $A\in {\mathfrak{F}}_a(O)$, $a\in \{0,1\}$. Graded locality means $AB=(-1{)}^{ab}BA$ for homogeneous $A\in {\mathfrak{F}}_a(O_1)$ and $B\in {\mathfrak{F}}_b(O_2)$ in spacelike separated regions. The graded net has the same isotony and translation covariance conditions. The local or almost-local interpolators used in that branch, their grading, and the statistics assignment are construction inputs.

${\mathrm{\ell }}_0>0$ denotes the fixed length represented by one numerical unit of the time parameter. A smooth compact momentum packet is a smooth compactly supported function $f\in C_c^{\mathrm{\infty }}({\mathbb{R}}^3;\mathbb{C})$; its support is the closure of $\{\mathbf{k}:f(\mathbf{k})\mathrm{\ne }0\}$. Extend it to the bounded Borel function $\widehat{f}(p^0,\mathbf{p})=f(\mathbf{p})$ on ${\mathbb{R}}^4$ and write ${\mathrm{\Phi }}_E(\widehat{f})=\int \widehat{f}dE$ for its bounded PVM integral. For a bounded interpolating operator $A$, define the one-particle packet vector $\psi (A,f)={\mathrm{\Phi }}_E(\widehat{f})P_{m}A\mathrm{\Omega }\in {\mathcal{H}}_m\mathrm{.}$ The integral is a bounded operator by the bounded Borel PVM integral theorem; it commutes with $P_m$ because both are functions of the same PVM. Its shell support lies in the compact Borel set $S_{m,f}=\{p_m(\mathbf{k}):\mathbf{k}\in \mathrm{supp}f\}$, so the equivalent projection form is $E(S_{m,f}){\mathrm{\Phi }}_E(\widehat{f})A\mathrm{\Omega }$. The velocity support is $V_m(f)=\{\mathbf{k}\mathrm{/}{\omega }_m(\mathbf{k}):\mathbf{k}\in \mathrm{supp}f\}$. Packets have disjoint velocity supports when these sets are pairwise disjoint.

The associated scalar positive-energy packet solution, with fixed Fourier normalization, is $u_{m,f}^{\eta }(t,\mathbf{x})=(2\pi {)}^{-3}{\int }_{{\mathbb{R}}^3}f(\mathbf{k})e^{i\mathbf{k}\cdot \mathbf{x}-i\eta {\mathrm{\ell }}_{0}t{\omega }_m(\mathbf{k})}{d}^3\mathbf{k}\mathrm{.}$ For each fixed $t$, this is smooth in $\mathbf{x}$ and rapidly decreasing together with all spatial derivatives, meaning that for every $\alpha =({\alpha }_1,{\alpha }_2,{\alpha }_3)\in {\mathbb{N}}_0^3$ and integer $N\ge 0$, where ${\mathrm{\partial }}_{\mathbf{x}}^{\alpha }:={\prod }_{j=1}^3{\mathrm{\partial }}_{x_j}^{{\alpha }_j}$, ${\sup }_{\mathbf{x}}(1+\mathrm{\mid }\mathbf{x}\mathrm{\mid }{)}^N\mathrm{\mid }{\mathrm{\partial }}_{\mathbf{x}}^{\alpha }{u}_{m,f}^{\eta }(t,\mathbf{x})\mathrm{\mid }<\mathrm{\infty }$. Indeed, the momentum integrand and all its derivatives are smooth and compactly supported, so repeated integration by parts in $\mathbf{k}$ gives these bounds. Let ${\alpha }_a(A)=U(a)AU(a{)}^{\ast }$ for $a\in {\mathbb{R}}^4$. For bounded local or almost-local $A$, define the candidate smeared operator by the strong operator integral $A_t^{\eta }(f)=\mathrm{s}\text{\hspace{0.17em}⁣}\text{\hspace{0.17em}⁣}{\int }_{{\mathbb{R}}^3}{u}_{m,f}^{\eta }(t,\mathbf{x}){\alpha }_{(\eta {\mathrm{\ell }}_{0}t,\mathbf{x})}(A)d^3\mathbf{x}\mathrm{.}$ The rapid-decay estimate with $N=4$ is integrable on ${\mathbb{R}}^3$, since in spherical coordinates ${\int }_0^{\mathrm{\infty }}{r}^2(1+r{)}^{-4}dr<\mathrm{\infty }$. For $\xi \in \mathcal{H}$ put $g_{\xi }(\mathbf{x}):=u_{m,f}^{\eta }(t,\mathbf{x})U(\eta {\mathrm{\ell }}_{0}t,\mathbf{x})AU(\eta {\mathrm{\ell }}_{0}t,\mathbf{x}{)}^{\ast }\xi \mathrm{.}$ Strong continuity of $U$ and boundedness of $A$ make $g_{\xi }$ continuous, with $\mathrm{\parallel }{g}_{\xi }(\mathbf{x})\mathrm{\parallel }\le \mathrm{\mid }{u}_{m,f}^{\eta }(t,\mathbf{x})\mathrm{\mid }\mathrm{\parallel }A\mathrm{\parallel }\mathrm{\parallel }\xi \mathrm{\parallel }\mathrm{.}$ It is strongly measurable: on each $[-n,n{)}^3$, partition into finitely many half-open cubes of side $1\mathrm{/}n$, use the value of $g_{\xi }$ at each cube's lower corner, and set the simple function to zero outside the box. These simple functions converge pointwise to $g_{\xi }$. The displayed majorant is integrable; the Bochner-integrability criterion therefore gives a Bochner integral for each $g_{\xi }$. By the Bochner-integrable-function definition listed below, this integral is the norm limit of integrals of integrable simple approximants. Define $A_t^{\eta }(f)\xi :=\int g_{\xi }(\mathbf{x})d^3\mathbf{x}$. Linearity and the Bochner integral norm inequality give $\mathrm{\parallel }{A}_t^{\eta }(f)\xi \mathrm{\parallel }\le \mathrm{\parallel }A\mathrm{\parallel }\mathrm{\parallel }\xi \mathrm{\parallel }\mathrm{\parallel }{u}_{m,f}^{\eta }(t,\cdot ){\mathrm{\parallel }}_{L^1},$ so this strong integral defines a bounded operator and $\mathrm{\parallel }{A}_t^{\eta }(f)\mathrm{\parallel }\le \mathrm{\parallel }A\mathrm{\parallel }\mathrm{\parallel }{u}_{m,f}^{\eta }(t,\cdot ){\mathrm{\parallel }}_{L^1}\mathrm{.}$

With the stated translation and Fourier conventions, its mass-shell component on $\mathrm{\Omega }$ is exactly $\psi (A,f)$. In fact, invariance of $\mathrm{\Omega }$ gives ${\alpha }_{(\eta {\mathrm{\ell }}_{0}t,\mathbf{x})}(A)\mathrm{\Omega }=U(\eta {\mathrm{\ell }}_{0}t,\mathbf{x})A\mathrm{\Omega }$. For $\xi =A\mathrm{\Omega }$ and any $y\in \mathcal{H}$, the bounded linear functional $v\mapsto (v,y)$ passes through the Bochner integral by its simple-function definition. The spectral representation of $U$, the polarization formula $E_{\xi ,y}=\frac{1}{4}{\sum }_{k=0}^3{i}^k{E}_{\xi +i^{k}y}$, and Fubini's theorem yield $(A_t^{\eta }(f)\mathrm{\Omega },y)={\int }_{{\mathbb{R}}^4}G(p)d{E}_{\xi ,y}(p)=({\mathrm{\Phi }}_E(G)\xi ,y),G(p):={\int }_{{\mathbb{R}}^3}{u}_{m,f}^{\eta }(t,\mathbf{x})e^{i\eta {\mathrm{\ell }}_{0}t{p}^0-i\mathbf{p}\cdot \mathbf{x}}{d}^3\mathbf{x}\mathrm{.}$ For each $z=\xi +i^{k}y$, use the finite positive scalar measure $E_z$ from the scalar and complex PVM-measures lemma listed below, which has total mass $\mathrm{\parallel }z{\mathrm{\parallel }}^2$, and apply the sigma-finite Fubini theorem listed below to Lebesgue measure on ${\mathbb{R}}^3$ and the finite positive Borel measure $E_z$ on ${\mathbb{R}}^4$. The product integrand $F(\mathbf{x},p):=u_{m,f}^{\eta }(t,\mathbf{x})e^{i\eta {\mathrm{\ell }}_{0}t{p}^0-i\mathbf{p}\cdot \mathbf{x}}$ is continuous and has absolute integral $\mathrm{\parallel }{u}_{m,f}^{\eta }(t,\cdot ){\mathrm{\parallel }}_{L^1}\mathrm{\parallel }z{\mathrm{\parallel }}^2<\mathrm{\infty }$. Thus $G$ is bounded, and dominated convergence with majorant $\mathrm{\mid }{u}_{m,f}^{\eta }(t,\cdot )\mathrm{\mid }$ makes it continuous, hence Borel. Fourier inversion in the convention of the supplier listed under Definition references, after rescaling $\mathbf{k}=2\pi \mathit{\xi }$, gives ${\int }_{{\mathbb{R}}^3}{u}_{m,f}^{\eta }(t,\mathbf{x})e^{-i\mathbf{p}\cdot \mathbf{x}}{d}^3\mathbf{x}=f(\mathbf{p})e^{-i\eta {\mathrm{\ell }}_{0}t{\omega }_m(\mathbf{p})}\mathrm{.}$ Hence $G(p)=e^{i\eta {\mathrm{\ell }}_{0}t{p}^0}f(\mathbf{p})e^{-i\eta {\mathrm{\ell }}_{0}t{\omega }_m(\mathbf{p})}$; on ${\mathcal{M}}_m$ this is $f(\mathbf{p})=\widehat{f}(p)$. The PVM product identity $E(M){\mathrm{\Phi }}_E(G)={\mathrm{\Phi }}_E({\mathbf{1}}_{M}G)$ follows first for simple functions and then by uniform approximation. Since ${\mathbf{1}}_{M}G={\mathbf{1}}_M\widehat{f}$, $M={\mathcal{M}}_m$, and $P_m=E({\mathcal{M}}_m)$, the same identity gives $P_m{A}_t^{\eta }(f)\mathrm{\Omega }={\mathrm{\Phi }}_E(\widehat{f})P_{m}A\mathrm{\Omega }\mathrm{.}$ Convergence of the full vector $A_t^{\eta }(f)\mathrm{\Omega }$ to that component is a separate construction obligation. For a domain-valued variant, the corresponding integrals and all products must instead be defined on the common invariant domain. Use the completed Hilbert tensor powers and their unitary permutation action: $S_n$ is the permutation group of $\{1,\dots ,n\}$ and $U_{\pi }(v_1\otimes \cdots \otimes v_n)=v_{{\pi }^{-1}(1)}\otimes \cdots \otimes v_{{\pi }^{-1}(n)}$ on elementary tensors. The sign $\mathrm{sgn}(\pi )$ is $+1$ for an even permutation and $-1$ for an odd permutation. The orthogonal symmetrizing and antisymmetrizing projections are $P_{s,n}=n{!}^{-1}{\sum }_{\pi \in S_n}{U}_{\pi }$ and $P_{a,n}=n{!}^{-1}{\sum }_{\pi \in S_n}\mathrm{sgn}(\pi )U_{\pi }$. For $\sigma \in \{s,a\}$, the Fock space ${\mathcal{F}}_{\sigma }({\mathcal{H}}_m)$ is the square-summable direct sum of the ranges of these projections for $n\ge 1$, together with vacuum sector $\mathbb{C}$ at $n=0$; use the componentwise second-variable-linear inner product and vacuum conventions of the previously defined QFT Fock spaces, with $P_{\sigma ,n}=P_{s,n}$ for $\sigma =s$ and $P_{\sigma ,n}=P_{a,n}$ for $\sigma =a$. The finite packet domain ${\mathcal{D}}_{\sigma }$ is the linear span of the vacuum and the $P_{\sigma ,n}$-projections of elementary tensors built from $\psi (A_j,f_j)$ with pairwise disjoint velocity supports. The selected value of $\sigma$ and, for fermionic fields, any required graded locality are construction inputs.

For such a finite packet family $(A_j,f_j{)}_{j=1}^n$, a candidate Haag–Ruelle approximant is ${\mathrm{\Psi }}_t^{\eta }=A_{1,t}^{\eta }(f_1)\cdots A_{n,t}^{\eta }(f_n)\mathrm{\Omega }$, where $t\ge 1$ is the dimensionless positive-time parameter, physical coordinate time is $\eta {\mathrm{\ell }}_{0}t$, and $\eta \in \{+1,-1\}$ labels outgoing ($+1$) or incoming ($-1$) scattering. The operators $A_{j,t}^{\eta }(f_j)$ act on a common dense domain $\mathcal{D}$ invariant under translations and these operators, with $\mathrm{\Omega }\in \mathcal{D}$; every displayed product is required to be defined there. Bounded operator families are the special case $\mathcal{D}=\mathcal{H}$. The operator families are supplied by the chosen construction. The one-particle limit condition is $A_{j,t}^{\eta }(f_j)\mathrm{\Omega }\to \psi (A_j,f_j)$ in norm as $t\to \mathrm{\infty }$. On the zero-particle vacuum label $1\in \mathbb{C}$, set ${\mathrm{\Psi }}_t^{\eta }(1)=\mathrm{\Omega }$ for every $t$. For each $f\in {\mathcal{D}}_{\sigma }$, the construction supplies a linear map ${\mathrm{\Psi }}_t^{\eta }:{\mathcal{D}}_{\sigma }\to \mathcal{H}$ whose values on elementary packet tensors are these operator products; the assignment must be well-defined independently of a tensor presentation. The Cook condition is that $t\mapsto {\mathrm{\Psi }}_t^{\eta }(f)$ is norm-$C^1$ on $[1,\mathrm{\infty })$ and, for constants $C_{\eta ,f}\ge 0$ and ${ϵ}_{\eta ,f}>0$, $\parallel \frac{d}{dt}{\mathrm{\Psi }}_t^{\eta }(f)\parallel \le C_{\eta ,f}(1+t{)}^{-1-{ϵ}_{\eta ,f}}(t\ge 1)\mathrm{.}$ Norm-$C^1$ means that the norm derivative exists for each $t>1$, has the corresponding one-sided derivative at $t=1$, and is norm-continuous on $[1,\mathrm{\infty })$. The parameter $t$ is the dimensionless numerical time in the fixed coordinate unit; changing that unit changes $C_{\eta ,f}$ but not integrability. A construction must also establish any required convergence of approximant inner products to the specified ${\mathcal{F}}_{\sigma }({\mathcal{H}}_m)$ inner products. The joint spectral input states cyclicity but not uniqueness of the invariant vacuum; any vacuum uniqueness or clustering hypothesis needed for this convergence must be stated or proved separately.

This is a contract for a proof, not an existence theorem. It does not derive the Cook bound, the one-particle limits, or the Fock inner-product limits from locality and shell isolation. An actual construction must also record uniform bounds on the vector approximants; those bounds are distinct from bounds on the operator factors $A_{j,t}^{\eta }(f_j)$.

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

Unital star algebra and positive normalized state

Positive-energy joint spectral input

Scalar future mass shell, invariant measure and test restriction

Schwartz/distribution spaces and Hilbert Fock conventions

Bounded borel pvm integral

Strongly measurable Banach-valued function

Bochner-integrable function

Bochner integrability criterion

Dominated convergence

Bochner integral norm inequality

Scalar and complex measures from a pvm

Fubini's theorem for L^1 functions on a sigma-finite product

Fourier inversion on Schwartz space

Charged quantum electrodynamics is outside this isolated massive charged-shell contract: massless photons and infraparticle sectors can invalidate its shell hypothesis. An infraparticle here means a charged excitation whose mass distribution has no isolated mass-shell eigen-subspace because of arbitrarily soft massless radiation. Lehmann–Symanzik–Zimmermann reduction (LSZ) is a separate route from distributional asymptotic fields and pole residues; this contract does not supply those fields, residues, or their domain arguments. Haag's no-go theorem concerns an exact unitary interaction-picture identification under its own hypotheses; it is a separate theorem obligation and is not a conclusion about finite-regulator Dyson evolution here.