Haag–Ruelle Scattering-State Construction

Haag–Ruelle theory constructs incoming and outgoing particles from the local operators of an interacting theory. An isolated massive shell supplies stable one-particle vectors, and separated velocities let locality control products of propagating packets. Here we prove norm convergence for every finite product in a neutral bosonic vacuum sector, derive the two-particle Fock inner product, and apply the construction to a massive weak-coupling $P(\phi)_2$ model. Equality of the scattering space with the full Hilbert space is a separate completeness problem.

Required background. Jost points, edge-of-the-wedge, and locality supplies spacelike commutation; clustering, vacuum uniqueness, and mass-gap implications supplies decay and spectral separation; and particles and one-particle subspaces supplies the stable shell.

Helpful background. LSZ reduction: poles, residues, and stable external states gives the physical scattering bridge.

The massive vacuum sector

Work on Minkowski space in $1+d$ dimensions, $d\geq1$ , with a net of bounded local operators $\mathcal A(O)$ on a Hilbert space $\mathcal H$ . Assume isotony, spacelike commutativity, and covariance under a strongly continuous positive-energy Poincaré representation. Its translations are $U(t,\mathbf x)=e^{iHt-i\mathbf P\cdot\mathbf x}$ , and $E(\Delta)$ is their joint spectral measure. Let the invariant vacuum $\Omega$ be unique and cyclic for the quasilocal algebra. A convenient sufficient spectral hypothesis is

\begin{gathered} \operatorname{Sp}(H,\mathbf P) \subset\{0\}\cup H_m^+\cup G_M,\qquad 0<m<M,\\ H_m^+=\{(\omega_{\mathbf p},\mathbf p): \omega_{\mathbf p}=\sqrt{\mathbf p^2+m^2}\},\qquad G_M=\{(p^0,\mathbf p):p^0\geq\sqrt{\mathbf p^2+M^2}\},\\ P_1=E(H_m^+)\neq0,\qquad E(\{0\})=|\Omega\rangle\langle\Omega|. \end{gathered}

Thus the shell has a gap both from the vacuum and from the higher spectrum. We select a neutral scalar bosonic species reached by local observables. The proof below uses ordinary commutators; charged sectors and fermionic fields require the corresponding localization and graded-locality framework. The chapter’s dependency map places this spectral input before the scattering construction.

Spectral smearing and propagating packets

Choose $A\in\mathcal A(O)$ with nonzero overlap onto a compact patch of $\mathcal H_1=P_1\mathcal H$ . Let $g$ be a Schwartz spacetime function whose Fourier transform is smooth and compactly supported in a small positive-energy neighborhood $K$ of that patch, such that

K\cap\operatorname{Sp}(H,\mathbf P)\subset H_m^+, \qquad B=\int d^{1+d}x\,g(x)U(x)AU(x)^*.

The integral can first be defined weakly and gives a bounded operator. With the global Fourier convention, vacuum invariance gives the exact identity

B\Omega=\widetilde g(H,\mathbf P)A\Omega\in\mathcal H_1, \qquad B^*\Omega=0.

More generally, $B$ has energy–momentum transfer in $K$ : $B E(\Delta)=E(\Delta+K)B E(\Delta)$ . This says what the smearing filters, including when $B$ acts on a nonvacuum vector.

$B$ is almost local: for every $N$ it has local approximants $B_R$ in a double cone of radius $O(R)$ with $\|B-B_R\|\leq C_N(1+R)^{-N}$ . To see this, truncate the integral to spacetime distances at most $R$ and use the Schwartz tail of $g$ . Translations of $B$ are smooth in operator norm: each derivative transfers to a derivative of $g$ , which is again Schwartz. All these derivatives remain almost local. The relevant operator class and spectral-transfer property are stated in Dybalski and Gérard 2012, §2.2, Definitions 2.1–2.3 and Eqs. (2.4)–(2.5), PDF.

For $\widetilde f\in C_c^\infty(\mathbb R^d)$ define a positive-energy Klein–Gordon packet and its bounded approximant by

\begin{aligned} f_t(\mathbf x) &=\int\frac{d^d\mathbf p}{(2\pi)^d} \widetilde f(\mathbf p) e^{-i\omega_{\mathbf p}t+i\mathbf p\cdot\mathbf x},\\ B_t(f)&=\int d^d\mathbf x\,f_t(\mathbf x) U(t,\mathbf x)BU(t,\mathbf x)^*. \end{aligned}

Here $\widetilde f$ is a spatial Fourier multiplier. If one instead writes the packet with the invariant measure $d^d\mathbf p/[(2\pi)^d2\omega_{\mathbf p}]$ , its amplitude must be $2\omega_{\mathbf p}\widetilde f$ to describe the same packet. No unspecified normalization or extra Klein–Gordon derivative is needed in the definition above. The spectral theorem gives

\begin{aligned} B_t(f)\Omega &=e^{it[H-\omega(\mathbf P)]}\widetilde f(\mathbf P)B\Omega\\ &=\widetilde f(\mathbf P)B\Omega\equiv\psi, \qquad \frac{d}{dt}B_t(f)\Omega=0. \end{aligned}

The equality is exact because $B\Omega$ lies entirely on the isolated shell. The smooth compact momentum support also confines every finite product applied to the vacuum to a bounded energy–momentum region, though no unbounded-operator domain is needed for these bounded approximants.

Locality gives the convergence estimate

The velocity support of a packet is the compact set

\Gamma(f)=\left\{\nabla\omega(\mathbf p) =\frac{\mathbf p}{\omega_{\mathbf p}}: \mathbf p\in\operatorname{supp}\widetilde f\right\}.

Consider packets with pairwise disjoint velocity supports. Choose smooth velocity cutoffs $\chi_j$ equal to one near $\Gamma(f_j)$ , with mutually disjoint compact supports. Their support separation has a positive lower bound $\delta$ .

Packet tails. The momentum phase has gradient $\mathbf x-t\nabla\omega(\mathbf p)$ . Outside the region where $\chi_j(\mathbf x/t)=1$ , it has no stationary point. Repeated integration by parts in $\mathbf p$ then gives, for every $N$ ,

\left\|[1-\chi_j(\mathbf x/t)]f_{j,t}\right\|_{L^1} \leq C_N(1+|t|)^{-N}.

The same estimate holds for $\partial_t f_{j,t}$ , since its momentum amplitude is $-i\omega\widetilde f_j$ with the same support. Inside a velocity cutoff the spatial volume is $O(|t|^d)$ and the Fourier integral has a uniform pointwise bound. Together with the tail estimate this gives the sufficient, though nonoptimal, bounds

\|B_{j,t}(f_j)\|+\|\partial_t B_{j,t}(f_j)\| \leq C(1+|t|)^d.

Commutators. In the cutoff regions, the equal-time centers satisfy $|\mathbf x-\mathbf y|\geq\delta|t|$ . Replace both almost-local operators by local approximants of radius $c|t|$ , with $c$ small compared with $\delta$ . Their translated double cones are spacelike separated, so their commutator vanishes. Errors from this replacement and from the packet tails decay faster than any inverse power. Polynomial factors from the spatial integrations can be absorbed by choosing a larger tail exponent. Consequently,

\|[B_{j,t}(f_j),B_{k,t}(f_k)]\| +\|[\partial_tB_{j,t}(f_j),B_{k,t}(f_k)]\| \leq C_N(1+|t|)^{-N}

for every $N$ and $j\neq k$ . The estimate also holds with an operator replaced by its adjoint. Norm smoothness of $B_j$ is essential here: differentiating an approximant produces both a differentiated packet and a translated derivative of $B_j$ , and both obey the preceding estimates.

Cook’s argument. Write $B_j(t)=B_{j,t}(f_j)$ and $\Psi_t=B_1(t)\cdots B_n(t)\Omega$ . Differentiate and commute each $\dot B_j(t)$ to the right. Its term at the vacuum vanishes because $\dot B_j(t)\Omega=0$ . Each remaining term has one commutator $[\dot B_j(t),B_k(t)]$ and $n-2$ ordinary factors. For $n\geq2$ ,

\|\dot\Psi_t\| \leq C_N(1+|t|)^{(n-2)d-N}.

Choose $N>(n-2)d+1$ . For $t_2>t_1\geq1$ ,

\|\Psi_{t_2}-\Psi_{t_1}\| \leq\int_{t_1}^{t_2}\|\dot\Psi_s\|\,ds \longrightarrow0\qquad(t_1\to\infty).

Completeness of $\mathcal H$ therefore gives $\Psi^{\rm out}=\lim_{t\to+\infty}\Psi_t$ in Hilbert-space norm. The estimate in $|t|$ proves the incoming limit at $t\to-\infty$ as well. For $n=1$ the approximant on the vacuum is already constant. This is a vector limit, and implies neither operator-norm convergence nor strong-operator convergence on all of $\mathcal H$ . An independent two-particle formulation is Dybalski and Gérard 2012, §6.2, Lemma 6.4 and Theorem 6.5(1), PDF.

If one replaces an interpolator by another with the same $\psi$ and the same separated velocity neighborhood, their difference $C(t)$ annihilates $\Omega$ . Commuting $C(t)$ to the right leaves only rapidly decreasing commutators multiplied by polynomially bounded factors. The scattering limit is unchanged. Reordering the factors is justified by the same commutator bound.

Two-particle inner products

Let $\Psi^{\rm out}$ arise from $B_1(t)B_2(t)\Omega$ and let $\Phi^{\rm out}$ arise from $C_1(t)C_2(t)\Omega$ , with separated velocities within each pair. Write $\psi_j=B_j(t)\Omega$ and $\varphi_j=C_j(t)\Omega$ . Then

\begin{aligned} \langle\Phi^{\rm out},\Psi^{\rm out}\rangle ={}&\langle\varphi_1,\psi_1\rangle \langle\varphi_2,\psi_2\rangle\\ &+\langle\varphi_1,\psi_2\rangle \langle\varphi_2,\psi_1\rangle. \end{aligned}

There are two ingredients beyond convergence. First, refine the spacetime smearing, without changing the one-particle vectors, so that each transfer support lies in a sufficiently thin neighborhood of its compact shell patch. A difference of two distinct positive-energy momenta on the same massive shell is spacelike. Compactness and the vacuum gap therefore let us choose these neighborhoods so that their differences meet the physical spectrum only at the vacuum. The spectral-transfer rule then yields

C_i(t)^*B_j(t)\Omega =\langle\varphi_i,\psi_j\rangle\Omega, \qquad C_i(t)^*\Omega=0.

Second, a nested commutator such as $[[C_1(t)^*,B_1(t)],B_2(t)]$ decreases rapidly. To justify this when the velocity support of $C_1$ meets both other supports, split its momentum amplitude with a smooth partition into a part separated from $\Gamma(f_1)$ and a part separated from $\Gamma(f_2)$ . Such a partition exists because those two compact sets are disjoint. Apply the pairwise commutator estimate to each part and use the Jacobi identity. This supplies the nested estimate without assuming separation between the two different scattering states.

Now expand $C_1^*B_1B_2\Omega$ by moving $C_1^*$ to the right. The vacuum-annihilating term vanishes, the two single contractions give the displayed pairings after multiplication by $\langle\Omega,C_2^*(\cdot)\rangle$ , and the remaining nested commutator tends to zero even after the polynomial norm bound for $C_2$ . This proves the formula. The same contraction argument is recorded in Dybalski and Gérard 2012, Theorem 6.5(2), Eqs. (6.8)–(6.10), PDF.

The formula proves full independence of the two-particle limit from interpolating operators: two constructions with the same ordered one-particle vectors have a difference of squared norm zero. It also fixes the normalization. With

\psi_1\otimes_s\psi_2 =\frac{\psi_1\otimes\psi_2+\psi_2\otimes\psi_1}{\sqrt2},

the scattering map preserves the symmetric tensor-product inner product. Smooth separated packets span a dense subspace of the two-particle space: the one-particle spectral measure is absolutely continuous on the massive shell, so the equal-momentum diagonal has product measure zero. Vacuum cyclicity supplies the dense local-interpolator vectors. The isometry consequently extends by continuity to $\mathcal H_1\otimes_s\mathcal H_1$ , including states with overlapping packet supports. Dybalski and Gérard 2012, Proposition 6.6 and its proof, PDF gives this extension. It extends a state map; it does not retroactively prove convergence of an arbitrary overlapping-velocity product of the original approximants.

Two outgoing particles in P(φ)₂

Choose a weak-coupling massive $P(\phi)_2$ vacuum model in the regime with an isolated scalar mass $m$ . This is a model-specific spectral assumption: boundedness below of the interaction polynomial alone is insufficient. The existence of isolated vacuum and positive mass eigenvalues at small coupling is part of Glimm, Jaffe, and Spencer 1974, abstract.

Use bounded local interpolators with nonzero overlap onto two compact shell patches and perform the spectral smearing above. In one spatial dimension choose smooth nonzero packets with velocity supports contained in

\Gamma(f_1)\subset[-0.6,-0.4],\qquad \Gamma(f_2)\subset[0.4,0.6].

The inverse map $p=mv/\sqrt{1-v^2}$ puts their momentum supports inside $[-3m/4,-2m/\sqrt{21}]$ and $[2m/\sqrt{21},3m/4]$ . Select the interpolators so that the filtered vectors are nonzero, and rescale each to $\|\psi_j\|=1$ . Their disjoint momentum supports give $\langle\psi_1,\psi_2\rangle=0$ .

For this example the entire convergence calculation reduces to

\frac{d}{dt}\bigl[B_1(t)B_2(t)\Omega\bigr] =[\dot B_1(t),B_2(t)]\Omega, \qquad \|[\dot B_1(t),B_2(t)]\Omega\| \leq C_N(1+|t|)^{-N}.

Taking any $N>1$ proves a norm limit, and the inner-product formula gives $\|\Psi_{12}^{\rm out}\|=1$ . The state has the total energy–momentum support obtained by adding the two packet supports. At positive times the left-moving and right-moving packets recede from each other; at negative times the same rays describe packets approaching from opposite sides. This constructs the in and out states without solving the interacting Hamiltonian or calculating an amplitude. The incoming and outgoing vectors need not be equal.

Where the proof stops

Overlapping velocities. Without separated supports there is no positive $\delta$ for the locality estimate. The direct Cook proof given here then stops. The two-particle isometry still defines states for overlapping one-particle inputs by continuity, as explained above; that is a different construction from asserting convergence of the unmodified product.

No isolated shell. A mass gap above the vacuum alone does not supply $P_1$ . Without the isolated shell, a fixed spectral smearing need not produce the exact identity $\dot B_t(f)\Omega=0$ . Charged infraparticles require different asymptotic objects; specialized extensions for other spectra need additional estimates.

Completeness and statistics. The finite-product construction gives scattering states in the specified neutral bosonic sector. It does not prove that they exhaust $\mathcal H$ , nor supply charged or fermionic scattering states by changing a sign in the final formula. See asymptotic completeness for the former problem, and the chapter’s hypothesis table and failure map for the distinct conclusions.

Exercises

Momentum and velocity for one mass. Prove that disjoint momentum supports and disjoint velocity supports are equivalent for $\omega_m(\mathbf p)=\sqrt{\mathbf p^2+m^2}$ with one fixed $m>0$ . Does disjoint momentum support suffice for two different masses?

Solution

The map $\mathbf p\mapsto\mathbf v=\mathbf p/\sqrt{\mathbf p^2+m^2}$ is a bijection from $\mathbb R^d$ onto the open unit ball, with inverse

\mathbf p=\frac{m\mathbf v}{\sqrt{1-|\mathbf v|^2}}.

It is injective, so two momentum sets intersect if and only if their velocity images intersect. For compact supports, disjoint images also have a strictly positive separation. Thus neither condition is stronger for the fixed massive dispersion used in this page.

For masses $m$ and $2m$ , momenta $p_1=m/\sqrt3$ and $p_2=2m/\sqrt3$ in one dimension are distinct but both give $v=1/2$ . Small disjoint momentum neighborhoods of those points can have overlapping velocity images. For multiple species it is the actual group velocities that must be separated.

The exponent needed by Cook’s criterion. Suppose the commutator bound were available only as $C(1+|t|)^{-q}$ , while each ordinary factor had the polynomial bound $(1+|t|)^d$ . What condition would this proof require for an $n$ -particle product?