Araki–Haag Detectors and Particle Weights

Araki–Haag detectors recover asymptotic particle content from translated observables rather than from a pointlike number operator. An almost-local, energy-decreasing operator removes the vacuum and low-energy background; a velocity filter follows the detector along $\mathbf x/t$. Limits of such counters give particle velocity distributions on scattering states and, more generally, positive asymptotic functionals called particle weights.

Required background. Wightman functions and spectral support supplies energy–momentum transfer; particles and one-particle subspaces supplies the sharp massive benchmark; and Haag–Ruelle scattering states supplies the states on which the detector limit can be checked.

Helpful background. Particle detectors versus field observables distinguishes operational detector models, while particles, local observables, and detector dependence gives the curved-spacetime contrast.

Almost-local energy-decreasing counters

Let $\alpha_x(A)=U(x)AU(x)^*$ be translations of a quasilocal observable. Smearing a local $A$ in spacetime with a Schwartz function whose Fourier support avoids the closed forward cone produces an almost-local operator $B$ with energy–momentum transfer in that support. It satisfies

$$B\Omega=0$$

because no allowed positive-energy state can absorb the transfer from the vacuum. Set $C=B^*B\geq0$. For a compactly supported velocity filter $h$, define in $d$ spatial dimensions

$$C_t(h)=\int\mathrm d^d\mathbf x\, h(\mathbf x/t)\,\alpha_{(t,\mathbf x)}(C).$$

Almost locality means $B$ can be approximated by operators localized in growing double cones with errors decreasing faster than every inverse radius. That decay controls commutators of distant counters. Energy decrease controls the integral on bounded-energy subspaces. The original detector construction and its coincidence interpretation are given in Araki and Haag 1967, pp. 77–91.

Limits on scattering states

On an outgoing $n$-particle state with momenta $p_k$ and velocities $\mathbf v_k=\nabla\omega_k(\mathbf p_k)$, a single counter has the schematic limit

$$\lim_{t\to\infty} \langle\Psi^{\mathrm{out}},C_t(h)\Psi^{\mathrm{out}}\rangle =\sum_{k=1}^{n}h(\mathbf v_k)\, \langle p_k,C\,p_k\rangle,$$

under wave-packet integration and the theorem’s regularity assumptions. The diagonal one-particle response calibrates the counter. Products of counters with disjoint velocity supports implement coincidences and suppress terms in which two filters select the same trajectory.

These formulas are asymptotic expectation-value statements, not projective measurements at finite time. Convergence on arbitrary bounded-energy states is a much stronger problem and is not supplied by the original scattering-state calculation.

The topology must be stated with the detector claim. A scalar expectation may converge while the operators $C_t(h)$ have no strong limit on the whole Hilbert space; coincidence limits can require additional time averaging or restricted energy windows. Calibration also matters: the factor $\langle p,Cp\rangle$ is a response function, so a counter with a zero response to one species cannot establish that the species is absent. What is intrinsic is the asymptotic functional obtained from a specified family of sensitive counters.

Two counters in a massive two-dimensional model

Take an outgoing two-particle packet $\Psi^{\mathrm{out}}(\psi_1,\psi_2)$ in a massive $1+1$-dimensional model. Choose $h_1,h_2$ with disjoint supports around the two velocity intervals and counters $C_i=B_i^*B_i$ with nonzero one-particle sensitivities $c_i(p)$. Then the coincidence limit is

$$\lim_{t\to\infty} \langle\Psi^{\mathrm{out}}, C_{1,t}(h_1)C_{2,t}(h_2) \Psi^{\mathrm{out}}\rangle = \int\mathrm d\mu(p_1)\mathrm d\mu(p_2)\, |\psi_1(p_1)\psi_2(p_2)|^2 h_1(v_1)h_2(v_2)c_1(p_1)c_2(p_2),$$

plus the exchanged assignment when the packet/filter supports allow it. With disjoint assigned velocity regions only the displayed assignment survives. This algebraic counter is the asymptotic counterpart of a measurement function for an inclusive observable, not a model of detector hardware.

An independent check is positivity: every $C_i$ is positive, and counters with asymptotically separated supports commute up to rapidly decaying errors, so the coincidence limit is nonnegative. Its dimensions are fixed by the normalization of $C_i$ and the spatial integral, not universally by “particle number.”

Particle weights

For a bounded-energy state $\Psi$, suitable large-time averages of $\langle\Psi,C_t(h)\Psi\rangle$ may have limit points even when $\Psi$ is not a sharp scattering state. Polarizing these positive asymptotic functionals on a left ideal of energy-decreasing almost-local operators gives a particle weight. It is analogous to an improper momentum eigenstate: translation covariance and spectral support survive, but the weight need not be a normalizable vector state.

Disintegration can separate pure components carrying mass, momentum, and internal information, including infraparticle situations where Wigner’s sharp-mass representation is inadequate. The construction, seminorm domains, and disintegration qualifications are developed in Porrmann 2004, pp. 269–304.

Failure test: vacuum contamination

Replace $C$ by a strictly local positive operator with $\langle\Omega,C\Omega\rangle=c>0$. Translation invariance gives

$$\langle\Omega,C_t(h)\Omega\rangle =c\,t^d\int\mathrm d^d\mathbf v\,h(\mathbf v),$$

so the “count” grows with the sampled spatial volume even in the vacuum. It does not isolate particles. Vacuum annihilation/energy decrease and almost locality are structural requirements, not optional detector tuning.

Exercises

Why can a nonzero energy-decreasing operator not be both strictly local and annihilate the vacuum in an ordinary vacuum representation?

Solution

The Reeh–Schlieder separating property implies that a local operator $B$ with $B\Omega=0$ must vanish. A useful $B$ is instead obtained by spacetime smearing, which gives controlled almost locality while permitting restricted energy–momentum transfer. The counter $B^*B$ is therefore quasilocal rather than strictly local.

References

• Araki, Huzihiro, and Rudolf Haag. 1967. “Collision Cross Sections in Terms of Local Observables.” Communications in Mathematical Physics 4: 77–91. DOI.
• Porrmann, Martin. 2004. “Particle Weights and Their Disintegration I.” Communications in Mathematical Physics 248: 269–304. DOI. Open manuscript.