Scattering Theory: LSZ and Haag–Ruelle¶
Job: construct the S-matrix in axiomatic QFT from the interpolating fields — the Lehmann–Symanzik–Zimmermann (LSZ) reduction formula and the rigorous Haag–Ruelle asymptotic construction — with the obstructions (infraparticle problem, massless / charged scattering) and the analyticity of amplitudes (Bros–Epstein–Glaser, Jost–Lehmann–Dyson).
Core¶
The scattering problem in axiomatic QFT¶
Given a Wightman / Haag–Kastler theory with a mass gap — the joint spectrum of translations is \(\{0\}\cup[m^2,\infty)\) with the one-particle threshold at mass \(m>0\) (the isolated mass hyperboloid \(p^2=m^2\)) — construct the S-matrix: a unitary map \(S:\mathcal{H}_{\mathrm{in}}\to\mathcal{H}_{\mathrm{out}}\) from the asymptotic "in" Fock space (multi-particle states at \(t\to-\infty\)) to the "out" Fock space (\(t\to+\infty\)).
The obstruction. In an interacting theory, the Heisenberg field \(\varphi\) is not the asymptotic creation/annihilation operator: the field-strength renormalization \(Z=\langle 0|\varphi(0)|p\rangle\ne 1\) (the overlap of the field with the one-particle state), and the interacting vacuum / one-particle states live in the same Hilbert space but the field does not create asymptotic particles directly. The LSZ / Haag–Ruelle theories construct the asymptotic fields from the interpolating field.
LSZ reduction (Lehmann–Symanzik–Zimmermann)¶
For a scalar field \(\varphi\) with the one-particle state \(|p\rangle\) (\(p^2=m^2\)) satisfying \(\langle 0|\varphi(0)|p\rangle=Z^{1/2}\) (the field-strength renormalization, \(0<Z\le1\)), the asymptotic fields \(\varphi_{\mathrm{in/out}}\) are the free fields of mass \(m\) defined by the weak limit $$ \varphi(x)\xrightarrow{t\to\mp\infty}Z^{1/2}\,\varphi_{\mathrm{in/out}}(x) $$ (the LSZ asymptotic limit, in the weak / distributional sense on smeared matrix elements).
Theorem (LSZ reduction formula). The S-matrix elements are $$ \langle p_1',\dots,p_m';\mathrm{out}|p_1,\dots,p_n;\mathrm{in}\rangle=\prod_{i=1}^{m}\prod_{j=1}^{n}\frac{i}{Z^{1/2}}\int d^4x\,d^4y\,e^{ip_i'x-ip_jy}(\Box_x+m^2)(\Box_y+m^2)\langle 0|T\varphi(x)\varphi(y)\cdots|0\rangle, $$ i.e. each external leg is amputated by the Klein–Gordon operator \((\Box+m^2)\) and divided by \(Z^{1/2}\); the Fourier transform of the time-ordered correlator (the Green's function) gives the S-matrix element.
LSZ theorem. If the Wightman theory has (i) a mass gap, (ii) a discrete one-particle pole at \(p^2=m^2\) with residue \(Z>0\) (the Källén–Lehmann spectral representation has \(\int\rho(\mu^2)\frac{d\mu^2}{\mu^2-p^2}\) with a \(\delta(\mu^2-m^2)\) term of weight \(Z\)), then the LSZ reduction gives a well-defined unitary S-matrix.
Haag–Ruelle scattering theory¶
The rigorous construction of the asymptotic states without the LSZ weak-limit assumption (which is hard to verify directly). Let \(\varphi\) be a local field with the one-particle form factor \(\langle 0|\varphi(f)|p\rangle\ne0\) for some test function \(f\) (the field creates a non-zero overlap with the one-particle state).
The Haag–Ruelle construction. For a test function \(f\) with \(\mathrm{supp}\,\hat f\subset\{p:p^2>m^2,p^0>0\}\) (Fourier support inside the mass shell), define the almost-local operator $$ A(t)=\int d^dx\,f_t(x)\varphi(x),\qquad f_t(x)=e^{iHt}f(x)e^{-iHt}\text{ (the boosted wave packet)}, $$ with \(f_t(x)=\frac{1}{(2\pi)^{d-1}}\int\tilde f(k)e^{-i(\omega_k t-k\cdot x)}\,dk\) concentrating on the mass shell \(\omega_k=\sqrt{|k|^2+m^2}\) as \(|t|\to\infty\).
Theorem (Haag–Ruelle). The weak limit $$ \lim_{t\to\mp\infty}A(t)\Omega=|f\rangle_{\mathrm{in/out}} $$ exists, defining the asymptotic one-particle state \(|f\rangle_{\mathrm{in/out}}\in\mathcal{H}_{\mathrm{in/out}}\cong\mathcal{H}_{\mathrm{1-particle}}\). The multi-particle states are $$ |f_1,\dots,f_n\rangle_{\mathrm{in}}=\lim_{t\to-\infty}A_1(t)\cdots A_n(t)\Omega\in\mathcal{H}_{\mathrm{in}}, $$ and the S-matrix is \(S:\mathcal{H}_{\mathrm{in}}\to\mathcal{H}_{\mathrm{out}}\), \(|f_1,\dots,f_n\rangle_{\mathrm{in}}\mapsto|f_1,\dots,f_n\rangle_{\mathrm{out}}\) (the limit exists by the Haag–Ruelle estimates).
Haag–Ruelle estimates. The existence of the limit follows from the clustering (the mass gap controls the decay of correlations at large spacelike separation): the overlap \(\langle A_i(t)A_j(t)\Omega,\,A_k(t')A_l(t')\Omega\rangle\) factorizes as \(t,t'\to-\infty\) with the error \(O(e^{-m|t-t'|})\) from the gap, so the multi-particle state asymptotically factorizes into a product of one-particle states (the in/out Fock structure).
Asymptotic completeness. The in/out states span \(\mathcal{H}\): \(\mathcal{H}=\mathcal{H}_{\mathrm{in}}=\mathcal{H}_{\mathrm{out}}\) (the scattering is elastic, the S-matrix is unitary). Proven for massive theories with a mass gap; open for massless / charged (the infraparticle problem).
Asymptotic completeness and obstructions¶
| Regime | Status |
|---|---|
| Massive with mass gap | Haag–Ruelle asymptotic completeness proven; S-matrix unitary |
| Massive charged (with mass gap) | proven (DHR sectors; Buchholz–Fredenhagen) |
| Massless neutral (e.g. massless \(\lambda\varphi^4\)) | partial; the IR divergences |
| Massless charged (QED) | fails — the infraparticle problem (Buchholz) |
| QCD (confinement) | open (no asymptotic quarks / gluons) |
The infraparticle problem (Buchholz). In QED, the electron is an infraparticle: the spectrum at \(p^2=m^2\) is a continuous threshold (not a discrete eigenvalue), due to the soft-photon cloud (the massless photons dress the electron, smearing the mass shell). The LSZ / Haag–Ruelle asymptotic completeness fails — there is no discrete one-particle pole, the field-strength renormalization \(Z=0\) (the overlap \(\langle 0|\varphi(0)|p\rangle=0\) for an infraparticle), and the S-matrix is not unitary without the inclusive / Bloch–Nordsieck construction (sum over soft photons to cancel the IR divergences).
Faddeev–Kulish. The asymptotic states for charged particles are constructed as dressed states (the electron + a coherent cloud of soft photons), and the S-matrix between dressed states is unitary; the exclusive S-matrix (fixed photon number) is not. The Buchholz scattering theory for massless particles uses the long-time limit of the current algebra (the "scattering of massless particles" via the asymptotic electromagnetic current).
The analytic S-matrix: crossing and dispersion relations¶
The crossing symmetry: the S-matrix element with a crossed incoming ↔ outgoing leg (e.g. \(\langle p_1',p_2';\mathrm{out}|p_1,p_2;\mathrm{in}\rangle\) with \(p_1'\to-p_1'\) crossing to an incoming anti-particle) is the analytic continuation of the time-ordered correlator across the physical region. This follows from the LSZ reduction + the edge-of-the-wedge theorem (the Jost–Lehmann–Dyson representation): the Wightman functions' analyticity domains glue across the mass shell, giving the crossing relation.
Bros–Epstein–Glaser analyticity. The scattering amplitude \(T(s,t)\) (Mandelstam variables) is the boundary value of a holomorphic function on a domain (the Martin ellipse in \(t\) at fixed \(s\), the Lehmann ellipse), derived from the Wightman axioms (locality + spectrum). The Froissart bound \(|T(s,0)|\le C\,s\log^2 s\) (the total cross-section grows at most logarithmically) follows from the analyticity + the unitarity.
| Analytic property | Wightman input |
|---|---|
| Crossing symmetry | LSZ + edge-of-the-wedge (Jost–Lehmann–Dyson) |
| Lehmann–Martin ellipse | locality + spectrum |
| Froissart bound | analyticity + unitarity |
| Dispersion relations (subtracted) | analyticity in \(t\) + polynomial boundedness |
See it¶
Haag–Ruelle construction of the two-particle in-state. For two local fields \(A_1(t),A_2(t)\) with the one-particle form factors \(\langle 0|\varphi(f_i)|p\rangle\ne0\), the two-particle state $$ |f_1,f_2\rangle_{\mathrm{in}}=\lim_{t\to-\infty}A_1(t)A_2(t)\Omega $$ is shown to exist by the Haag–Ruelle estimates: the norm \(\|A_1(t)A_2(t)\Omega-|f_1\rangle|f_2\rangle\|\to0\) as \(t\to-\infty\) because the overlap of the wave packets at large time separation factorizes (the clustering / mass gap controls the connected part: \(\langle\Omega,A_1(t)^*A_1(t)A_2(t)^*A_2(t)\Omega\rangle=\langle A_1(t)^*A_1(t)\rangle\langle A_2(t)^*A_2(t)\rangle+O(e^{-m|t|})\) from the spectral representation + locality). The inner product $$ \langle f_1',f_2';\mathrm{out}|f_1,f_2;\mathrm{in}\rangle=\langle f_1',f_2'|S|f_1,f_2\rangle $$ is the two-body S-matrix element. For the free field (\(Z=1\), no interaction), the S-matrix is the identity (the in/out Fock spaces coincide: \(\varphi_{\mathrm{in}}=\varphi_{\mathrm{out}}\)). For the interacting \(\varphi^4_2\) constructive theory (with the mass gap, Glimm–Jaffe), the S-matrix is the non-trivial two-body amplitude computed via the LSZ amputated time-ordered four-point function \(\langle 0|T\varphi\varphi\varphi\varphi|0\rangle\) — the LSZ formula gives the interacting two-body amplitude (the \(2\to2\) scattering, with the phase shift determined by the interacting spectral density).
Check¶
Why does the LSZ formula fail for the infraparticle (QED electron), and what replaces it?
The LSZ formula requires a discrete one-particle pole \(Z>0\) in the Källén–Lehmann spectral representation: \(\langle 0|\varphi\varphi|0\rangle=\int\rho(\mu^2)\frac{d\mu^2}{\mu^2-p^2}\) with \(\rho(\mu^2)=Z\delta(\mu^2-m^2)+\rho_{\mathrm{cont}}(\mu^2)\). For the infraparticle, the soft-photon cloud smears the mass shell: \(Z=0\) (the discrete pole is replaced by a continuous threshold \(\rho_{\mathrm{cont}}\) starting at \(m^2\)), so the amputation \((\Box+m^2)/Z^{1/2}\) is ill-defined (\(0/0\)-type). The replacement is the Faddeev–Kulish dressed state (electron + soft-photon coherent cloud) with a well-defined inclusive S-matrix (sum over soft photons, the Bloch–Nordsieck cancellation), or the Buchholz massless scattering theory via the asymptotic current.
Show that for the free field, the Haag–Ruelle asymptotic limit gives \(S=\mathbf{1}\).
For the free field \(\varphi_{\mathrm{free}}\) of mass \(m\), \(Z=1\) and \(\varphi\) is already the free field: \(\varphi(x)=\varphi_{\mathrm{in}}(x)=\varphi_{\mathrm{out}}(x)\) (the field decomposes into creation/annihilation parts \(a^\dagger(k)+a(k)\) acting on Fock space). The Haag–Ruelle limit \(A(t)\Omega\to|f\rangle_{\mathrm{in}}\) gives the same one-particle state as \(A(t)\Omega\to|f\rangle_{\mathrm{out}}\) (since the field doesn't interact, the late-time wave packet is the same free state). Hence \(\mathcal{H}_{\mathrm{in}}=\mathcal{H}_{\mathrm{out}}=\mathcal{H}\) and \(S=\mathbf{1}\) (the in/out Fock spaces coincide, no scattering).
Pitfalls¶
- The LSZ limit assumes the mass gap + a discrete one-particle pole (\(Z>0\)); the infraparticle has \(Z=0\) and LSZ fails — do not apply LSZ to QED without the Bloch–Nordsieck / Faddeev–Kulish dressing.
- Haag–Ruelle asymptotic completeness is proven for massive but OPEN for massless / charged — the infraparticle problem is the obstruction; do not state "asymptotic completeness holds" for QED.
- The Faddeev–Kulish construction handles the inclusive / soft-photon infrared (sum over soft photons, dressed states), not the exclusive S-matrix (fixed photon number, which has IR divergences and is not unitary).
- Haag's theorem: there is no interaction-picture unitary intertwining free and interacting Wightman fields with the same vacuum — the interacting theory lives in a different (unitarily inequivalent) representation of the CCR. The LSZ / Haag–Ruelle constructions work in the Heisenberg picture (no interaction picture), avoiding Haag's theorem.
- Crossing symmetry is a consequence of LSZ + edge-of-the-wedge, not an independent axiom; it requires the Wightman analyticity (spectrum + locality) and the LSZ pole structure.
- The Froissart bound requires the full Wightman analyticity + unitarity; it fails in theories without a mass gap or with non-local interactions.
Refs¶
- Lehmann–Symanzik–Zimmermann, "Zur Formulierung quantisierter Felder" (Nuovo Cimento 1955)
- Haag, "Quantum field theories with composite particles and the asymptotic condition" (Phys. Rev. 1958); Ruelle, "On the asymptotic condition in quantum field theory" (Helv. Phys. Acta 1962)
- Buchholz, "Gauss' law and the infraparticle problem" (Phys. Lett. B 1986); Faddeev–Kulish, "Asymptotic states and operator products" (TMP 1970)
- Bros–Epstein–Glaser, "On the analyticity properties of the scattering amplitude in QFT" (Nuovo Cimento 1964–1967); Jost–Lehmann–Dyson representation (1957)
- Reed–Simon, Methods of Modern Mathematical Physics III: Scattering Theory; Derezinski–Jaksic, "Asymptotic completeness of long-range QFT"
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