Conclusion

(Ω|A_n(f_n,T)^∗. . . A₁(f_1,T)^∗Ab₁(fb_1,T). . .Ab_n(fb_n,T)Ω)

= Xn

k=1

(Ω|A_n(f_n,T)^∗. . . A₂(f_2,T)^∗Ab₁(fb_1,T). . .[A₁(f_1,T)^∗,Ab_k(fb_k,T)]. . .Ab_n(fb_n,T)Ω)

= Xn

k=1

Xⁿ

l=k+1

(Ω|A_n(f_n,T)^∗. . . A₂(f_2,T)^∗Ab₁(fb_1,T). . . . . . [[A₁(f_1,T)^∗,Ab_k(fb_k,T)],Ab_l(fb_l,T)]. . .Ab_n(fb_n,T)Ω)

+ (Ω|A_n(f_n,T)^∗. . . A₂(f_2,T)^∗Ab₁(fb_1,T). . .k . . .ˇ Ab_n(fb_n,T)A₁(f_1,T)^∗Ab_k(fb_k,T)Ω) .(A.3.18) The terms containing double commutators vanish in the limit by Lemma A.2.2 (b) and Lemma A.2.1 (a). The remaining terms factorize by the preceding proposition and by Lemma A.2.1 (b) and (c):

Tlim→∞(Ω|A_n(f_n,T)^∗. . . A₂(f_2,T)^∗Ab₁(fb_1,T). . .ˇk . . .Ab_n(fb_n,T)A₁(f_1,T)^∗Ab_k(fb_k,T)Ω)

= lim

T→∞(Ω|A_n(f_n,T)^∗. . . A₂(f_2,T)^∗Ab₁(fb_1,T). . .ˇk . . .Ab_n(fb_n,T)|Ω)·

·(Ω|A₁(f₁)^∗P_[m]Ab_k(fb_k)Ω). (A.3.19) This quantity factorizes into two-point functions by the induction hypothesis.

It is also evident from the proof that the scalar product of two asymptotic states involving different numbers of operators is zero. Now the Fock structure of the asymptotic states and the construction of the wave operators follows by standard density arguments: By Condition A^′, there exists a regular operator A ∈ A(O) s.t. P_[m]AΩ 6= 0. Since the representation U acts irreducibly in H^[m], the vectors {U(x,Λ)P_[m]AΩ|(x,Λ) ∈ P₊^↑} span a dense set in this subspace. Making use of the fact that a Poincar´e transformation of a regular operator is regular, we obtain that the set of vectors

{P_[m]A(f)Ω|A-regular,f˜∈C₀^∞(R³) vanishes sufficiently fast at 0} (A.3.20) is total in H^[m]. In view of Theorem A.2.3 the wave operator W⁺ can be defined on a dense set in Γ(H^[m]), extending by linearity the following relation

W⁺ a^∗(P_[m]A1(f1)Ω). . . a^∗(P_[m]An(fn)Ω)Ω

= s - lim

T→∞A1(f1,T). . . An(fn,T)Ω, (A.3.21) where A and f satisfy the conditions stated in (A.3.20). Due to Theorem A.3.4, W⁺ preserves norms, thus it extends to an isometry from Γ(H^[m]) to H. The wave operator W⁻ is constructed analogously, making use of the incoming states, and the scattering matrix S: Γ(H^[m])→Γ(H^[m]) is given by S= (W⁺)^∗W⁻.

A.4 Conclusion

We have constructed a scattering theory of massive particles without the lower and upper mass gap assumptions. The Lorentz covariance of the construction can be verified by application of standard arguments [Ar]. Including fermions would cause no additional difficulty, as the fermionic creation operators are bounded uniformly in time [Bu75].

The only remaining restriction is the regularity assumption A^′. We note that it was used only to establish the existence of the scattering states - the construction of the Fock structure was independent of this property. Moreover, we would like to point out that it does not seem possible to derive it from general postulates. In fact, let us consider the generalized free field φwith the commutator fixed by the measureσ:

[φ(x), φ(y)] = Z

dσ(λ)∆_λ(x−y), (A.4.1)

where ∆_λ is the commutator function of the free field of mass √

λ. Suppose that the measure σ contains a discrete massm and in its neighborhood is defined by the function F(λ) = 1/ln|λ−m²|. Then it is easy to find polynomials in the fields smeared with Schwartz class functions which violate the bound from Condition A^′. However, the exis-tence of the scattering states, constructed with the help of such polynomials, can easily be verified using properties of generalized free fields. These observations indicate that Condi-tionA^′ is only of technical nature. To relax it one should probably look for a construction of asymptotic states which avoids Cook’s method - perhaps similarly to the scattering theory of massless particles [Bu75, Bu77].

Appendix B

Scalar Free Field Theory and its Phase Space Structure

In this appendix we collect some known results on the phase space structure of scalar free field theory in a slightly modified form, suitable for our purposes. They provide a basis for the proofs that the new ConditionsL⁽²⁾,L⁽¹⁾,N_♮ andC_♭, introduced in the main body of this work, hold in the model of scalar, non-interacting particles. These arguments are given in the subsequent appendices. For ConditionsL⁽¹⁾andC_♭only the massive case will be considered.

Verification of phase space conditions in models is an integral part of phase space analysis in QFT as it proves consistency of the introduced criteria with the basic postulates of quantum field theory. This issue was treated already in the seminal paper of Haag and Swieca [HS65], who verified their compactness condition in massive scalar free field theory. Serious technical improvements, including the reduction of the problem to a single-particle question, appeared in the work of Buchholz and Wichmann [BWi86], who noted the importance of nuclearity. The massless case was included for the first time in [BJa87].

Conditions C_♯ and N_♯, stated in Sections 3.1 and 3.5, respectively, were first verified in [BP90]. A particularly flexible and explicit formulation of the subject was given by Bostelmann in [Bos00] and our presentation relies primarily on this work.

The goal is to construct an expansion of the map Θ_E : A(O(r)) → B(H), given by Θ_E(A) =P_EAP_E, into rank-one mappings. More precisely, we are looking for functionals τ_i ∈A(O(r))^∗ and operatorsS_i∈B(H),i∈N, s.t.

Θ_E(A) =X

i

τ_i(A)S_i, A∈A(O(r)), (B.0.1) X

i

kτ_ik^pkS_ik^p<∞, 0< p≤1. (B.0.2) Thus, in view of Lemma 3.5.1, we corroborate the well known fact that ConditionN_♯holds in scalar free field theory [BP90]. Moreover, we establish properties of the functionals {τ_i}^∞1 and {S_i}^∞1 which will be needed to verify the new conditions introduced in this Thesis.

This appendix is organized as follows: In Section B.1 we explain our multiindex no-tation. Section B.2 introduces scalar free field theory, its even part and its sub-theory generated by the derivatives of the field. In Section B.3 we construct the functionals

75

{τ_i}^∞1 and estimate their norms. Apart from the material familiar from Section 7.2.B of [Bos00], we establish certain energy bounds on these functionals which we need to verify Condition L⁽¹⁾. In Section B.4 two expansions of the single-particle wavefunctions are developed, following Sections 7.2.2 and 7.2.3 of [Bos00]. They give rise to an expansion of the map Θ_E into rank-one mappings, which we introduce in Section B.5. It is a variant of the ’mixed expansion’ from Section 7.2.6 of [Bos00]. In order to avoid the unduly compli-cated infrared structure and phase space properties of low dimensional massless models, in this appendix we adopt the following:

Standing assumption: Unless stated otherwise, all statements concerning scalar free field theory hold either for m >0 and s≥1 or m= 0 ands≥3.

B.1 Multiindex Notation

A multiindex is a sequence µ={µ(i)}^∞1 of elements from N₀ s.t. only a finite number of components is different from zero. Addition of multiindices is performed component-wise.

The length of a multiindex is given by

|µ|=X

i

µ(i). (B.1.1)

The factorial of a multiindex is defined as µ! = Q

iµ(i)!. Given a sequence a = {a_i}^∞1 , valued in any set with multiplication, its multiindex power is defined as

a^µ=Y

i

a^µ(i)_i . (B.1.2)

It is convenient to extend the above conventions to pairs of multiindices µ = (µ⁺, µ⁻):

The length of such 2-multiindex is given by |µ|=|µ⁺|+|µ⁻| and the factorial is defined as µ! = µ⁺!µ⁻!. Given any sequence of pairs b ={b⁺_i , b⁻_i }^∞1 , we define the 2-multiindex power of bas follows

b^µ= (b⁺)^µ+(b⁻)^µ−. (B.1.3) Finally, an n-index κ is a multiindex s.t. κ(i) = 0 for i > n. All the above conventions extend naturally to n-indices.

We recall that for any sequence of complex numbers{tj}^∞1 andk∈Nthere holds the multinomial formula

Xn

j=1

t_jk

= X

µ,|µ|=k

|µ|

µ!t^µ, (B.1.4)

where the sum on the r.h.s. extends over all n-indices of length k. Assuming that the sequence {t_j}^∞1 is absolutely summable, we can take the limit n→ ∞, obtaining on the r.h.s. the sum over all multiindices µof length k.