Lecture 4. Scattering theory.

David Kazhdan

4.1 Introduction.

This important branch of functional analysis is not so popular as it should be; one of the reasons is that the phenomena studied in scattering theory has no finite-dimensional analog.

The situation is roughly as follows.

Let A_{0} be a self-adjoint operator in a Hilbert space H (the function space). Let λ be a point in the continuous spectrum of A_{0} and let f_{λ}^{0} be the corresponding “eigenfunction” (quotation marks are put to remind that f_{λ}^{0} lies not in the Hilbert space H, but rather in an appropriate larger function space, i.e. f_{λ} is a linear functional on a dense subspace in H.)

We consider a “small” perturbation A = A_{0} + B of our self-adjoint operator A_{0}, where B is a self-adjoint operator such that (\sqrt{1 + A_{0}^{2}})^{−1}B is compact, and we look for a λeigenfunction f_{λ} of the operator A.

Remark. This problem makes sense only if λ lies in the continuous spectrum of A_{0}; in particular it never makes sense in finite dimensions. Indeed, the spectrum of an operator moves as we perturb the operator; so an isolated point of the spectrum in general does not lie in the spectrum of perturbed operator.

We look for an eigenfunction of the form f_{λ} = f_{λ}^{0} + g. We have:

Af_{λ} = λf_{λ} ⇔ (A − λ)g = Bf_{λ}^{0} so formally
g = (A − λ)^{−1}B(f_{λ}^{0}) .

Of course the last equality makes no sense since (A − λ) is not invertible. Nevertheless in some cases one can show existence of the limits g_{±} = lim_{ε→±0}(A − λ ± iε)^{−1}B(f_{λ}^{0}). Then we put f_{±}^{λ} = f_{λ}^{0} + g_{±}.

Let us consider a concrete example of the above situation.

In this example one can actually show existence of the limits lim_{ε→±0}(A − λ ± iε)^{−1}B(f_{λ}^{0}). We will present another way to obtain the two eigenfunctions f_{±}^{λ}. We leave it as an exercise to the reader to check that the two procedures lead to the same answer.

We take H = L^{2}(E), where E is a d-dimensional Euclidean space. Let A_{0} = −Δ + m^{2}, and let B be multiplication by a function b ∈ C_{c}^{∞}(E).

Claim. For f ∈ C_{c}^{∞}(E) we have:

∥Be^{itA_{0}}f∥ ∼ |t|^{−d/2}  for  t → ±∞

Corollary. For any f ∈ C_{c}^{∞}(E) the L^{2}-limits lim_{t→±∞} e^{itA}e^{itA_{0}}f exist provided d > 2.

Proof of Corollary. Denote f^{t} = e^{itA}e^{−itA_{0}}f. Then we have:

∥\frac{∂f^{t}}{∂t} _{t_{0}} ∥ = ∥iAe^{it_{0}A}e^{−it_{0}A_{0}}f − ie^{it_{0}A}A_{0}e^{−it_{0}A_{0}}f∥
= ∥e^{it_{0}A}Be^{−it_{0}A_{0}}f∥ = ∥Be^{−t_{0}A_{0}}f∥ .

So by the previous claim f_{+} = ∫_{0}^{∞}(\frac{∂f^{t}}{∂t} ) dt converges if d > 2. The proof for f_{−} is analogous.

Obviously: ∥f_{±}∥_{L^{2}} = ∥f∥_{L^{2}}, hence there exist well-defined operators W_{±}: H → H, such that W_{±}(f) = f_{±}. Moreover, we see that:

∥(AW_{±} − W_{±}A_{0})(f)∥ = lim_{t→±∞} ∥e^{itA}Be^{−itA_{0}}(f)∥ = 0 ,

hence AW_{±} = W_{±}A_{0}.

If the image of W_{+} is equal to the image of W_{−}, we define the S-matrix by: S = W ∗_{+}W^{−}. Thus S ∈ End(H), and S commutes with A_{0}.

4.2 System of n particles (potential scattering). This is a generalization of the last problem. We take H = L^{2}(E^{n}), A = − ∑_{i=1}^{n} Δ_{i} + ∑ B_{ij}, where B_{ij} is multiplication by a function b(v_{i} − v_{j}), for b ∈ C_{c}^{∞}(E), and Δ_{i} is the Laplace operator along the i^{th} copy of E.

Let π be a partition of {1, …, n} in a union of k disjoint sets π_{1}, …, π_{k} (“clusters”).

Consider the decomposition E^{π_{ℓ}} ≃ E×\overline{E}_{π_{ℓ}}, where \overline{E}_{π_{ℓ}} = \{(v_{i_{1}}, …, v_{i_{|π_{ℓ}|}}) ∈ E^{π_{ℓ}}| ∑ v_{i} = 0\}, and the isomorphism is given by (v_{i_{1}}, …, v_{i_{|π_{ℓ}|}}) → (w_{ℓ} = \frac{1}{|π_{ℓ}|} ∑ v_{i}; (v_{i_{1}} − w_{ℓ}, v_{i_{2}} − w_{ℓ}, … v_{i_{|π_{ℓ}|}} − w_{ℓ})).

We have obvious decompositions E = Π_{ℓ=1}^{k} E^{π_{ℓ}}, and the corresponding decomposition of the function space:

H = L^{2}(E^{n}) = ⊗_{ℓ=1}^{k} L^{2}(E^{π_{ℓ}}) = ⊗_{ℓ=1}^{k} (L^{2}(E) ⊗ L^{2}(\overline{E}_{π_{ℓ}}))

The operator A_{0} obviously decomposes as: A_{0} = ∑_{ℓ=1}^{k} A_{ℓ}, where A_{ℓ} acts along E^{π_{ℓ}}; and furthermore A_{ℓ} = A′_{ℓ} + A′′_{ℓ}, where A′_{ℓ} is minus the Laplace operator acting along E, and A′′_{ℓ} acts along \overline{E}_{π_{ℓ}}. Let H_{ℓ} = ⊕H_{ℓ}^{λ} ⊂ L^{2}(\overline{E}_{π_{ℓ}}) be the sum of eigenspaces of A′′_{ℓ} where λ runs over the discrete spectrum of A′′_{ℓ}. Put H_{0}(π) = ⊗_{ℓ=1}^{k}(L^{2}(E) ⊗ H_{ℓ}) ⊂ L^{2}(E^{n}) = H. Let J_{π}: H_{0}(π) ↪ H denote the natural isometric inclusion; thus J_{π}(H_{0}(π)) is an A_{0}-invariant subspace in H. We denote J = ⊕J_{π}: ⊕_{π} H_{0}(π) → H.

Theorem 1. a) There exist strong limits W_{±} = lim_{t→±∞} e^{itA}Je^{−itA_{0}}.

b) W_{±} is an isometric imbedding and we have: A ∘ W_{±} = W_{±} ∘ A_{0}.

c) W_{±} is an isomorphism.

We omit the proof. Just remark that the proof of a) and b) is routine, while c) is a difficult theorem, the full proof of which was obtained only recently.

4.3 Haag-Ruelle theory. We now apply the method of scattering theory to QFT.

Claims and methods of scattering theory discussed below apply only to Minkowski, not to the Euclidean picture.

We will describe the following construction.

Let (U, ℋ ⊃ 𝒟 ∋ Ω_{i}) be a scalar QFT. Suppose that:

a) ℋ^{V} = ℂΩ (uniqueness of vacuum).

b) There exists a P -invariant Hilbert subspace H ⊂ ℋ with a P -equivariant isomor-

phism H ≃ L^{2}(𝒪m^{+}), m > 0.

c) Mass spectrum of ℋ/H ⊕ ℂΩ does not contain 0 and m.

d) The span of ⟨ϕ(f)Ω⟩, f ∈ 𝒮 is not contained in H^{⊥}. Let (U_{0}, ℋ_{0} ⊃ 𝒟_{0}, ϕ_{0}) be the free field theory of mass m. Then one can construct two P -equivariant isometric imbeddings W_{±}: ℋ^{0} ↪ ℋ. Remark. We cannot hope that W_{±} is an isomorphism, because □ _{ℋ} can have discrete spectrum outside of {0, m^{2}} while for free theory it does not. An ideal theorem would assert that a free theory constructed on discrete part of the spectral decomposition of □ on ℋ maps isomorphically to ℋ (compare with potential scattering described in the previous section). Unfortunately there is no known way to formulate a precise theorem of this sort. Remark. The construction can be carried out under weaker assumptions. Namely one can omit condition d), and replace condition c) by requirement that the theory has positive mass, and m^{2} is an isolated point of the discrete spectrum of □.

However the assumption that 0 is an isolated point of Spec(U _{V}) is essential. Application of scattering to a massless QFT confronts much more serious difficulties than in the massive case. Remark. The construction does not work for small d. Consider the case d = 1. Then QFT reduces to quantum mechanics. The spectrum of Hamiltonian of the free theory Spec H_{0} = mℤ_{+} is totally discrete, thus scattering theory is not applicable. One can prove a slightly weaker statement in the case d = 2. Our argument will work for d ≥ 4.

We now turn to construction of imbeddings W_{±}. We try to imitate the construction of the previous section (potential scattering).

Let Λ ⊂ V^{+} be a G-invariant neighborhood of 𝒪m^{+}, such that Λ ∩ Spec U _{H^{⊥}} = ∅. For any function f on V such that supp ℱ(f) ⊂ Λ and t ∈ ℝ let f^{t} ∈ 𝒮(V) be the function such that:

ℱ(f^{t})(p_{0}, p⃗) = e^{it(p_{0}−\sqrt{m^{2}−p^{2}} )}ℱ(f)(p_{0}, p⃗) .

Note that f^{t} = f provided supp(ℱ(f)) ⊂ 𝒪m^{+}.

In the rest of the lecture we prove the following Theorem 2. a) If d ≥ 4 then for any f_{1}, …, f_{n} ∈ 𝒮(V) such that supp(ℱ(f_{i})) ⊂ Λ the limits lim_{t→ ± ∞} ϕ(f_{1}^{t}) … ϕ(f_{n}^{t})Ω exist. b) There exists a unique isometric imbedding W_{±}: ℋ_{0} ↪ ℋ, such that for any

f_{1}, …, f_{n} ∈ 𝒮(V) with supp(ℱ(f)) ⊂ λ we have:

W_{±} (ϕ_{0}(f_{1}) … ϕ_{0}(f_{n})Ω) = lim_{t→ ± ∞} ϕ(f_{1}^{t}) … ϕ(f_{n}^{t})Ω

The imbeddings W_{±} are P -equivariant.

Remark. For a free field theory the vector ϕ_{0}(f_{1}^{t}) … ϕ_{0}(f_{n}^{t})Ω does not depend on t. Indeed, the field operator ϕ_{0}(f) for a free field theory depends on ϕ(f) _{𝒪_{m}^{+}} only, and ℱ(f^{t}) _{𝒪_{m}^{+}} does not depend on t.

Remark. Let L be the space of functions satisfying the conditions of Lemma 1. Fourier transform identifies L with the space of Schwartz functions on 𝒪m^{+}. Let χ be a Schwartz function of one variable such that ℱ(χ) is supported in a small neighborhood of 0. For f ∈ L let us put f_{χ}(x) = χ(x_{0}) · f(x) where x = (x_{0}, x⃗). Then f_{χ} ∈ 𝒮(V), and the Fourier transform of f_{χ} is concentrated in a neighborhood of 𝒪m^{+} being the convolution of the function ℱ(f) concentrated on 𝒪_{m}^{+} and a function concentrated in a small neighborhood of 0 on the time-axis.

This way we can fix an initial imbedding j of (a dense subspace of) ℋ_{0} to ℋ such that j(ϕ_{0}(f_{1}) ⋯ ϕ_{0}(f_{k})Ω) = ϕ(f_{1χ}) ⋯ ϕ(f_{kχ})Ω for any functions f_{1}, .., f_{k} ∈ L. Then one checks directly that:

ϕ(f_{1χ}^{t}) ⋯ ϕ(f_{kχ}^{t})Ω = U ((t,0 )) j(U_{0} ((−t,0 )) ϕ_{0}(f_{1}) ⋯ ϕ_{0}(f_{k})Ω).

Thus the situation is quite analogous to the potential scattering considered in the previous section.

Proof of the Theorem. We start with two auxiliary estimates needed in the proof. The first is a standard fact from analysis.

Lemma 1 (locality for free fields). Let f ∈ C^{∞}(V) be a solution of the Klein-Gordon equation, i.e. supp ℱ(f) ⊂ 𝒪m^{+}. Assume that f(0, x⃗) and \frac{∂f}{∂x_{0}} (x_{0}, x⃗) _{x_{0}=0} are Schwartz functions on ℝ^{d−1}. Then for some constant C we have:

a) sup_{x⃗∈ℝ^{d−1}} |f(x_{0}, x⃗)| ≤ C (1 + |x_{0}|)^{−(d−1)/2}

b) ∫_{ℝ^{d−1}} |f(x_{0}, x⃗)|dx⃗ ≤ C(1 + |x_{0}|)^{(d−1)/2}

We omit the proof; it is an exercise in stationary phase method (see lecture 1, §1.6, Claim 1b) for a similar argument).

The second ingredient is the following estimate on truncated Wightman functions (see lecture 3, §3.5 for the definition).

Proposition 1. There exist a differential operator with constant coefficients D, a C^{0} function of tempered growth F, and a polynomial function P on V_{n} such that 𝒲_{n}^{tr} = D(F) and the following estimate holds.

Let [1..n] = I ∪ J be a partition of [1..n] into union of 2 disjoint subsets; let a_{1}, …, a_{n} ∈ ℝ^{d−1} ⊂ V be such that a_{i} ≠ a_{j} for i ∈ I, j ∈ J. Then for all t_{1}, ..., t_{n} ∈ ℝ ⊂ V we have:

|F (t_{1} + a_{1}, …, t_{n} + a_{n})| < |P (t_{1} + a_{1}, …, t_{n} + a_{n})|e^{−m min_{i∈I,j∈J} (\sqrt{−(a_{i}−a_{j})^{2}}−|t_{i}−t_{j}|)}.

Remark. The particular case n = 2 of the Proposition is implied by lecture 1, §1.6, Claim 1b). Also part a) of the same Claim guarantees that at least 𝒲_{n}^{tr}(v_{1}, .., v_{i}, v_{i+1} +λa, .., v_{n} + λa) → 0 for any v_{1}, …, v_{n} provided a^{2} < 0, which obviously follows from the Proposition.

The statement we will actually need is the following immediate

Corollary. Let i_{t} denote the embedding of (ℝ^{d−1})^{n} to V_{n}, ∏ v⃗_{i} → ∏(t, v⃗_{i}), and let π : (ℝ^{d−1})^{n} → (ℝ^{d−1})^{n−1} = (ℝ^{d−1})^{n}/Δ be the projection to quotient modulo diagonal. For a Schwartz function h on V_{n} the restriction i_{t}^{∗}(h ∗ 𝒲_{n}^{tr}) is a smooth function invariant under diagonal translations, hence i_{t}^{∗}(h ∗ 𝒲_{n}^{tr}) = π^{∗}(F_{t}^{h}) for some function F_{h}^{t} on (ℝ^{d−1})^{n−1}. Then F_{h}^{t} ∈ L^{1} ((ℝ^{d−1})^{n−1}) for any t, and |F_{t}^{h}|_{L^{1}} is bounded by a constant independent of t.

Proof of the Proposition. We start with an auxiliary result. Recall from the first lecture that the Fourier transform of Wightman functions has the property: ℱ(𝒲_{n}(p_{1}, …, p_{n})) = 0 if p_{1} + ⋯ + p_{k} ∉ \overline{V}_{+} for some k. One can easily show:

Lemma 2. Let V_{+}(m) be {p ∈ V_{+}|p^{2} > m}. Then we have ℱ(𝒲_{n}^{tr}(p_{1}, …, p_{n})) = 0 provided p_{1} + ⋯ + p_{k} ∉ \overline{V_{+}(m)} for some k where m is the mass of the theory. Also ℱ(𝒲_{n}^{tr}(p_{1}, …, p_{n})) is concentrated on the hyperplane ∑_{i=1}^{n} p_{i} = 0.

We are ready to prove the Proposition.

Let us denote: 𝒲′(v_{1}, …, v_{n}) = 𝒲^{tr}(v_{i_{1}}, …, v_{i_{k}}; v_{j_{1}}, v_{j_{2}}, …, v_{j_{n−k}}) and 𝒲′′(v_{1}, …, v_{n}) = 𝒲^{tr}(v_{j_{1}}, …, v_{j_{n−k}}; v_{i_{1}}, …, v_{i_{k}}) where i_{1} < i_{2} < … < i_{k} are the elements of I, and j_{1} < … < j_{n−k} are the elements of J. From the Lemma it follows that

ℱ(𝒲′)(p_{1}, …, p_{n}) = 0 if p_{i_{1}} + ⋯ + p_{i_{k}} ∉ \overline{V_{+}(m)};
ℱ(𝒲′′)(p_{1}, …, p_{n}) = 0 if − (p_{i_{1}} + ⋯ + p_{i_{k}}) ∉ \overline{V_{+}(m)}

Let χ̃ be a smooth function of one variable, such that χ̃(x) = 1 for x > m, χ̃(x) = −1 for x < −m, and its Fourier transform has asymptotics φ̃ = ℱ(χ̃(t)) ∼ e^{−m|t|} as t → ±∞. [It is a standard fact from calculus that such a function exists].

Let us put χ(p_{1}, …, p_{n}) = χ̃(p_{i_{1}}^{(0)}+ ⋯ +p_{i_{k}}^{(0)}), where p_{i} = (p_{i}^{(0)}, p⃗_{i}) for p_{i}^{(0)} ∈ ℝ, p⃗_{i} ∈ ℝ^{d−1} is the usual decomposition. Then we see that χℱ(𝒲′) = ℱ(𝒲′), χℱ(𝒲′′) = −ℱ(𝒲′′). Let φ be Fourier transform of χ; so φ is concentrated on a 1-dimensional subspace. Then we have φ ∗ 𝒲′ = 𝒲′, φ ∗ 𝒲′′ = −𝒲′′, where ∗ stands for convolution.

Since 𝒲_{n} is a tempered distribution it can be represented as 𝒲_{n} = D(F) where F is a continuous function of polynomial growth and D is a differential operator with constant coefficients. We can assume that D is invariant under the action of symmetric group Σ_{n+1}. Then the functions F′ = F (v_{i_{1}}, .., v_{i_{k}}; v_{j_{1}}, .., v_{j_{n−k}}) and F′′ = F (v_{j_{1}}, .., v_{j_{n−k}}; v_{i_{1}}, .., v_{i_{k}}) satisfy: D(F′) = 𝒲′ and D(F′′) = 𝒲′′. Then obviously: φ ∗ F′ = F′ and φ ∗ F′′ = −F′′

The space-locality property implies that: F′(v_{1}, …, v_{n}) = F′′(v_{1}, …, v_{n}) = F (v_{1}, …, v_{n}) provided (v_{i} − v_{j})^{2} < 0 for i ∈ I, j ∈ J.

So for |(a_{i} − a_{j})^{2}| > (t_{i} − t_{j})^{2} we have:

F (t_{1} + a_{1}, …, t_{n} + a_{n}) = \frac{1}{2}(F′(t_{1} + a_{1}, …, t_{n} + a_{n}) + F′′(t_{1} + a_{1}, …, t_{n} + a_{n}))

= \frac{1}{2}φ ∗ (F′ − F′′) = \frac{1}{2} ∫_{−∞}^{∞}(F′ − F′′)((t_{1} + a_{1})^{t}, (t_{2} + a_{2})^{t}, …, (t_{n} + a_{n})^{t})φ̃(t)dt

where we used the notation: (t_{i} + a_{i})^{t} = t_{i} + a_{i} − t if i ∈ I, and (t_{j} + a_{i})^{t} = t_{j} + a_{j} for j ∈ J. The expression in the integral vanishes if t < min(\sqrt{−(a_{i} − a_{j})^{2}} − |t_{i} − t_{j}|). So since F is of polynomial growth we get the desired upper bound for the integral.

The crucial step of the proof is the following

Lemma 3.

|𝒲_{n}^{tr}(g_{1}^{t}, …, g_{n}^{t})| ≤ C(1 + |t|)^{(1−d)(n−2)/2}

for some C and all t, where g_{j}^{t} = f_{j}^{t} or g_{j}^{t} = \frac{∂f_{j}^{t}}{∂t} or complex conjugate for each j.

Proof of the Lemma. Let us work in Fourier coordinates; let pr be the projection pr : V_{n} → (ℝ^{d−1})^{n}.

Of course 𝒲_{n}^{tr}(g_{1}^{t}, …, g_{n}^{t}) = ∫ ℱ(𝒲_{n}^{tr})(−p)ℱ(g_{1}^{t} × ⋯ × g_{n}^{t}) = ∫ pr_{∗}(ℱ(W_{n}^{tr})(−p) · ℱ(g_{1}^{t} × ⋯ × g_{n}^{t})). Let us look more closely at the last integral.

For this purpose it is convenient to write ℱ(f_{i}) (respectively ℱ(\overline{f}_{i})) in the form ψ̃_{i} · η̃_{i} where ψ̃_{i} is a Schwartz function with support in Λ (respectively in −Λ), and η̃_{i} is the pull-back of a Schwartz function under the projection on ℝ^{d−1}.

Let η_{i} be given by ℱ(η_{i}) = δ_{𝒪_{m}^{+}} · η̃_{i} (respectively ℱ(η_{i}) = δ_{𝒪_{m}^{−}} · η̃_{i}).

Using the projection formula we can rewrite:

pr_{∗} (ℱ(𝒲_{n})(−p) · ℱ(∏ g_{i}^{t})) = pr_{∗}[T_{t} (ℱ(𝒲_{n})(−p) · ∏ h̃_{i})] · pr_{∗}[T_{−t} (ℱ(∏ η_{i}))]

Here T_{t}(f)(p) = exp(it · p_{0})f(p) and h̃_{i} = ψ̃_{i} if g_{i}^{t} = f_{i}^{t} or complex conjugate, and h̃_{i} = ψ̃_{i} · ±i(p_{0} − \sqrt{m^{2} − p^{2}}) if g_{i}^{t} = \frac{∂f_{i}^{t}}{∂t} or complex conjugate.

Let h denote ℱ(∏ h̃_{i}). Then the integral we need to estimate equals to

∫_{(ℝ^{d−1})^{n}} pr_{∗}[T_{t} (ℱ(𝒲_{n})(−p) · ℱ (∏ h̃_{i}))] · pr_{∗}[T_{t} (\overline{ℱ(∏ η_{i}}))] = ∫_{(ℝ^{d−1})^{n}} i_{t}^{∗}(F ∗ h) · i_{−t}^{∗} (∏ η_{i})

By Corollary to the Proposition i_{t}^{∗}(𝒲_{n}∗h) = π^{∗}(F_{t}^{h}), for some function F_{t}^{h} ∈ L^{1}((ℝ^{d−1})^{n−1}), and |F_{h}^{t}|_{L^{1}} < C where C does not depend on t.

Thus we have:

∫_{(ℝ^{d−1})^{n}} |pr^{∗}i_{t}^{∗}(𝒲_{n} ∗ h) · pr_{1}i_{−t}^{∗}(η_{1})| ≤ C · ∫_{ℝ^{d−1}} |j_{t}^{∗}(η_{1})| ≤ C · (1 + |t|)^{(d−1)/2}

by Lemma 1b), where j_{t} is the imbedding ℝ^{d−1} ↪ V, v⃗ → (t, v⃗)), and pr_{i} : (ℝ^{d−1})^{n} → ℝ^{d−1} is the projection to the i-th component. Hence

\begin{matrix}∫_{(ℝ^{d−1})^{n}} i_{t}^{∗}(𝒲_{n} ∗ h)· & i_{−t}^{∗} ( ∏ η_{i}) ≤ C(1 + |t|)^{(d−1)/2} \\ · max_{(ℝ^{d−1})^{n}} ∏_{k=2}^{n} pr_{k}^{∗}j_{−t}^{∗}(η_{j}) ≤ C(1 + |t|)^{−(n−2)(d−1)/2} \end{matrix}

where the last inequality follows from Lemma 1a). Lemma 3 is proved.

We are now ready to prove the Theorem. Denote v^{t} = ϕ(f_{1}^{t}) … ϕ(f_{n}^{t})Ω. To show that the limit lim_{t→∞} v^{t} exists it suffices to check that the integral ∫_{0}^{∞}|\frac{∂v^{t}}{∂t} | dt converges. We have:

\begin{matrix}(\frac{∂v^{t}}{∂t} )^{2} = (∑_{i} & ϕ(f_{1}^{t}) … ϕ(ḟ_{i}^{t}) … ϕ(f_{n})Ω, ∑_{j} ϕ(f_{1}^{t}) … ϕ(ḟ_{j}^{t}) … ϕ(f_{n})Ω) \\ = ∑_{i,j} 𝒲_{2n} (\overline{f}_{n}^{t}, \overline{f}_{n−1}^{t}, …, ¯ ḟti, …, \overline{f}_{1}^{t}, f_{1}^{t}, ḟ_{j}^{t}, …, f_{n}) \end{matrix}

Let us decompose 𝒲_{2n} as sum of products of truncated Wightman functions, and consider the corresponding decomposition of each summand in the last sum.

Each factor containing 𝒲_{1}^{tr} obviously vanishes. By the previous lemma each factor containing 𝒲_{i}^{tr} is bounded by const·(1+|t|)^{(i−2)(1−d)/2}; in particular each factor is bounded.

Let us look at a factor of the form 𝒲_{k}^{tr}(g_{1}^{t}, ..., g_{k}^{t}) where at least one of g_{i}^{t} is a time derivative of f_{ℓ}^{t} (or complex conjugate). If k = 2 then such a factor vanishes. Indeed, if say g_{1}^{t} = \overline{f_{i}^{t}} and g_{2}^{t} = ḟ_{j}^{t} then 𝒲_{2}^{tr}(g_{1}^{t}, g_{2}^{t}) = ⟨ϕ(f_{i}^{t})Ω, ϕ(ḟ_{j}^{t})Ω⟩ = 0 since ϕ(f_{j}^{t})Ω does not depend on t. (Here we use that supp ℱ(f_{j}) ∩ supp U _{V} = 𝒪m^{+}.) The other cases are absolutely parallel.

We now see that each non-zero summand contains a factor of the form 𝒲_{k}^{tr}(g_{1}^{t}, …, g_{k}^{t}) with k ≥ 3. Furthermore, it either contains at least 2 factors 𝒲_{k}^{tr} with k ≥ 3 or at least one factor 𝒲_{k} with k ≥ 4, because the term has the form 𝒲_{k_{1}}^{tr} … 𝒲_{k_{ℓ}}^{tr} with k_{1} + ⋯ + k_{ℓ} = 2n being even, and each k_{i} ≥ 2. In any case it is bounded by const · (1 + |t|)^{(1−d)/2·∑(k_{i}−2)} ≤ const · (1 + |t|)^{1−d}. Thus |v̇^{t}| ≤ const · (1 + |t|)^{(1−d)/2} and ∫_{−∞}^{∞} v̇^{t} dt converges absolutely if d ≥ 4.

Proof of b). Let us check that if v^{t} = ϕ(f_{1}^{t}) … ϕ(f_{k}^{t})Ω, w^{t} = ϕ(g_{1}^{t}) … ϕ(g_{ℓ}^{t})Ω for f_{i}, g_{j} as above then lim_{t→ ± ∞}(v^{t}, w^{t}) = (ϕ_{0}(f_{1}) … ϕ_{0}(f_{k})Ω, ϕ_{0}(g_{1}) … ϕ_{0}(g_{ℓ})Ω). Let us again decompose (v^{t}, w^{t}) = 𝒲_{ℓ+k}(\overline{f}_{k}^{t}, \overline{f}_{k−1}^{t}, …, \overline{f}_{1}^{t}, g_{1}^{t}, …, g_{ℓ}^{t}) into sum of products of truncated Wightman functions. From the above arguments we see that every term containing 𝒲_{1}^{tr} vanishes, and every term containing 𝒲_{k}^{tr} for k ≥ 3 tends to 0 as t → ± ∞. Besides, 𝒲_{2}^{tr}(\overline{f}, g) = (ϕ(f)Ω, ϕ(g)Ω) = (ϕ_{0}(f)Ω, ϕ_{0}(g)Ω). Thus we have

lim_{t→ ± ∞}(ϕ(f_{1}^{t}) … ϕ(f_{k}^{t})Ω, ϕ(g_{1}^{t}) … ϕ(g_{ℓ}^{t})Ω) = lim_{t→ ± ∞}(ϕ_{0}(f_{1}^{t}) … ϕ_{0}(f_{k}^{t})Ω,

ϕ_{0}(g_{1}^{t}) … ϕ_{0}(g_{ℓ}^{t})Ω) = (ϕ_{0}(f_{1}) … ϕ_{0}(f_{k})Ω, ϕ_{0}(g_{1}) … ϕ_{0}(g_{ℓ})Ω)

We can now finish the proof. For free field theory the space 𝒟 is generated by the vectors ϕ_{0}(f_{1}) … ϕ_{0}(f_{n})Ω for f_{i} as above, because ϕ_{0}(f)x depends only on ℱ(f) _{𝒪_{m}^{+}} for f ∈ 𝒮, x ∈ ℋ.

For x ∈ ℋ let us choose functions f_{i} such that x = ∑ ϕ_{0}(f_{i_{1}}) … ϕ_{0}(f_{i_{s}})Ω, and consider the limit lim_{t→ ± ∞} ∑ ϕ(f_{i_{1}}^{t}) … ϕ(f_{i_{s}}^{t})Ω. We proved already that for any y ∈ ℋ, and functions g_{j} such that y = ∑ ϕ_{0}(g_{j_{1}}) … ϕ_{0}(g_{j_{r}})Ω we have

( lim_{t→ ± ∞} ∑ ϕ(f_{i_{1}}^{t}) … ϕ(f_{i_{s}}^{t})Ω, lim_{t→ ± ∞} ∑ ϕ(g_{j_{1}}^{t}) … ϕ(g_{j_{r}}^{t})Ω) = (x, y).

Hence the limits do not depend on the choice of f_{i}, g_{j}, and give correctly defined isometric imbedding. The Theorem is proved.

4.4 Scattering matrix.

Proposition 2. We have: Im(W_{+}) = Im(W_{−}).

Thus we can define the S-matrix S = W ∗_{+}W_{−} ∈ End(ℋ_{0}). It is interesting that S can be described in terms of the Wightman functions of the theory.

The precise formulation is as follows. Let f̃ denote the partial Fourier transform of f in space variables. Vector v⃗ ∈ ℝ^{d−1} is called a velocity of f if we have v⃗ = p⃗/\sqrt{p^{2} + m^{2}} for p⃗ such that (p_{0}, p⃗) ∈ supp(f̃) for some real number p_{0}.

Theorem. (Lehmann-Symanzik-Zimmerman)

a) The residue ℱ(𝒲_{n}^{T}) _{(𝒪_{m}^{+})^{n}} is a well-defined distribution on V_{n} supported on (𝒪m^{+})^{n}, where 𝒲_{n}^{T} is the time-ordered Wightman function (see lecture 2, §2.4 for the definition).

b) Assume f_{i} ∈ 𝒮 and the set of velocities of f_{i} and f_{j} does not intersect for i ≠ j. Then:

\begin{aligned}∑_{I⊂(1,...,n)}(−1)^{|I|} & ⟨W_{−} (∏_{i∈I} ϕ_{0}(f_{i})Ω) , W_{+} (∏_{j∉I} ϕ_{0}(f_{j})Ω)⟩ = \\ & = ⟨Res(ℱ(𝒲_{n}^{T})), ℱ(f_{1}) × … × ℱ(f_{n})⟩ . \end{aligned}

It is an elementary combinatorial problem to deduce an expression for matrix coefficients of the S-matrix: ⟨W_{−} ∏ ϕ_{0}(f_{1})Ω, W_{+} ∏ ϕ_{0}(f_{j})Ω⟩ in terms of ℱ(𝒲^{T}) from the last formula.

References

M. Reed, B. Simon,spaceMethods of Modern Mathematical Physics, vol. 4. Analysis of operators; §XI.16,spaceAcademic Press,space1978.

J. Glimm, A. Jaffe,spaceQuantum Physics. (A functional integral point of view); §§13.5– 14.2,spaceSpringer-Verlag,space1987.