Expansion of Θ E into Rank-One Mappings

In this section we will decompose the map ΘE : A(O(r)) → B(H), given by ΘE(A) = P_EAP_E, as follows

Θ_E(A) =

∧

Θ_E,1(A) +

∨

Θ_E,1(A) + Θ_E,2(A), A∈A(O). (B.5.1) Here Θ^∧_E,1 is a finite rank map, the part Θ^∨_E,1(A) collects the terms involving high deriva-tives of the field and Θ_E,2(A) contains the contributions to A coming from high Wick powers. In order to construct such a decomposition, we evaluate the map Θ_E on a Weyl operator W(f), f ∈L˚r, given by definition (B.2.10). We obtain the following expansion valid in the sense of quadratic forms on D_F ×D_F:

Θ_E(W(f)) = e^−12kfk22P_E :e^i√2φ+(F+)e^{i√2φ−(F−)}:P_E

= e^−12kfk22 X

k⁺,k⁻∈N₀

(i√

2)^k++k−

k⁺!k⁻! P_E :φ₊(F⁺)^k+φ₋(F⁻)^k−:P_E,(B.5.2) where ˜f^±=ω^∓12F˜^±. Now we introduce natural numbers K⁺,K⁻ and the set of indices S = {(k⁺, k⁻) ∈ N²₀|k⁺ ≤K⁺ and k⁻ ≤ K⁻}. We decompose the above sum into two parts

Θ_E(W(f)) = Θ_E,1(W(f)) + Θ_E,2(W(f)), Θ_E,1(W(f)) = e^−12kfk22 X

k⁺,k⁻∈S

(i√

2)^k++k−

k⁺!k⁻! P_E :φ₊(F⁺)^k+φ₋(F⁻)^k−:P_E, (B.5.3) Θ_E,2(W(f)) = e^−12kfk22 X

k⁺,k⁻∈S^′

(i√

2)^k++k−

k⁺!k⁻! P_E :φ₊(F⁺)^k+φ₋(F⁻)^k−:P_E,(B.5.4) where S^′ is the complement of S and Θ_E,1, Θ_E,2 are understood as linear maps from the

∗-algebra ˚A(O(r)) of finite linear combinations of the Weyl operators, given by

˚A(O(r)) = Span{W(f)|f ∈L˚r}, (B.5.5) to quadratic forms onD_F×D_F. In the sequel we will show that Θ_E,1 and Θ_E,2have their ranges inB(H). Moreover, we will extend their domain, by continuity, to the whole local algebra A(O(r)). For this purpose we will use the expansions of the functions f^± ∈ L˚^±r

introduced in Section B.4.

B.5. Expansion of Θ_E into Rank-One Mappings 89

B.5.1 Expansion of Θ_E,1

First, recalling that ˜F_E^± :=χEF˜^± and exploiting the Wick ordering, we obtain

PE :φ+(F⁺)^k+φ₋(F⁻)^k−:PE =PE :φ+(F_E⁺)^k+φ₋(F_E⁻)^k−:PE. (B.5.6) With the help of the multinomial formula (B.1.4) and expansion (B.4.3) we get the fol-lowing identity

where α^± are multiindices and we introduced the short-hand notation (∂φ_±)^α± =

Y∞ j=1

(∂^κjφ_±)^α±(j), (B.5.8) which refers to the numbering of the s-indices κ, introduced in formula (B.4.3). Iden-tity (B.5.7) relies on the fact that the vectors b^±_κ,r and f^± are real in configuration space and exploits the relation Substituting expansion (B.5.7) to (B.5.3), we obtain

Θ_E,1(W(f)) = X Next, we choose some finite subsetMof the set of all pairs of multiindices and decompose the map Θ_E,1 as follows

The above expressions can be restated in terms of suitable normal functionals on B(H).

We denote by ¯σ_α^(r)+,α⁻ the functionals from Proposition B.3.1, corresponding to the families of vectors{b^±_κj,r}j∈N. Next, we define the normal functionals τ^∧(r)_α+,α⁻ on B(H) given by

τ∧^(r)_α+,α⁻ = (i√

2)^{|α+|+|α−|}

α⁺!α⁻! σ¯^(r)_α+,α−. (B.5.14) Making use of formula (B.5.12), we can write

∧

Θ_E,1(A) = X

(α⁺,α⁻)∈M (|α⁺|,|α⁻|)∈S

τ∧^(r)_α+,α⁻(A)P_E : (∂φ₊)^α+(∂φ₋)^α−:P_E, (B.5.15)

where A ∈ ˚A(O(r)) is any finite linear combination of Weyl operators. The Wick monomials : (∂φ₊)^α+(∂φ₋)^α− : belong to the field content of the theory (see defini-tion (2.3.10) and reladefini-tion (B.2.28)), hence they are elements of T_∞^∗. It follows that kP_E : (∂φ₊)^α+(∂φ₋)^α− : P_Ek < ∞, thus Θ^∧_E,1 is a finite rank map from ˚A(O(r)) to B(H). Since the functionals ^∧τ^(r)_α+,α⁻ are normal, Θ^∧_E,1 extends to a finite rank map from A(O(r)) toB(H).

In order to simplify expression (B.5.13) defining the map Θ^∨_E,1, we note that for any two families of functions{F_j⁺}j∈N,{F_j⁻}j∈N fromS(R^s)Rand for any pair of multiindices (α⁺, α⁻) there holds the identity

2^12|α+|2^12|α−|:φ₊(F⁺)^α+φ₋(F⁻)^α− :

= X

µ^±,ν^± µ⁺+ν⁺=α⁺ µ⁻+ν⁻=α⁻

α⁺!α⁻!

µ⁺!ν⁺!µ⁻!ν⁻!a^∗( ˜f⁺)^µ+a^∗(if˜⁻)^µ−a( ˜f⁺)^ν+a(if˜⁻)^ν−, (B.5.16)

where ˜f^±=ω^∓12F˜^± and the equality holds in the sense of quadratic forms onD_F ×D_F. Consequently, we obtain from (B.5.13)

∨

Θ_E,1(W(f))

= X

µ,ν µ+ν∈M^′ (|µ⁺|+|ν⁺|,|µ⁻|+|ν⁻|)∈S

i^{|µ+|+|ν+|+2|µ−|}

µ!ν! e^−12kfk22hb_κ,r|fi^µ+νP_Ea^∗(h_κ,E)^µa(h_κ,E)^νP_E. (B.5.17)

We introduce the normal functionalsτ^∨(r)_µ,ν and the quadratic formsS^∨_µ,ν onD_F×D_F given by

τ∨^(r)_µ,ν := i^{|µ+|+|ν+|+2|µ−|}

µ!ν! σ¯^(r)_µ+ν, (B.5.18)

∨

S_µ,ν := P_Ea^∗(h_κ,E)^µa(h_κ,E)^νP_E. (B.5.19)

B.5. Expansion of Θ_E into Rank-One Mappings 91

In order to study its convergence properties we collect several auxiliary results. From Proposition B.3.1 we obtain the following lemma.

Lemma B.5.1. For anyl≥0, the functionalsτ^∨(r)_µ,ν, defined by (B.5.18), satisfy the bound kR^−lτ^∨(r)_µ,νR^−lk ≤(2c_l)^|µ|+|ν|p

(|µ|+|ν|)!kb_κ,rk^µ+ν_2,l , (B.5.21) where c_l= (12 + 2m²)^l/2, and m is the mass of the theory.

Next, to study the formsS^∨_µ,ν, we recall the so-called energy bounds [BP90]:

Lemma B.5.2. For any h₁, . . . , h_n ∈ L²(R^s, d^sp) in the domain of ω¹² there holds the bound

ka(ω¹²h₁). . . a(ω¹²h_n)P_Ek ≤Eⁿ²kh₁k2. . .kh_nk2. (B.5.22) Making use of the above result and Lemma B.4.4, we obtain that the forms

∨

Sµ,ν, defined by (B.5.19), are actually elements of B(H) and satisfy the bound

kS^∨_µ,νk ≤E^|µ|+|ν|2 kω⁻¹²˜h_κ,Ek^µ+ν2 . (B.5.23) Now we are ready to prove the convergence of expansion (B.5.20).

Proposition B.5.3. [Bos00] LetS ={(k⁺, k⁻)∈N²₀|k⁺≤K⁺ and k⁻≤K⁻} for some

Proof. From estimates (B.5.21), (B.5.23) we obtain X

The sum on the r.h.s. can be estimated as follows X

Making use of the fact that the multinomial coefficients are larger or equal to one and of

and similarly for the sum w.r.t. µ⁻. The last expression is finite due to relation (B.4.17).

This proves the uniform convergence of expansion (B.5.17). It follows that the mapΘ^∨_E,1, which was defined on the norm dense subalgebra ˚A(O(r)), can be extended to the full local algebra A(O(r)).

From definition (B.5.19) we obtain that in massive scalar free field theory of mass m > 0 there holds and Lemma 3.5.1 give the known fact [BP90, Bos00] that Condition N_♯ holds in massive scalar free field theory.

Proposition B.5.4. In massive scalar free field theory there holds the identity Θ_E(A) =X However, the methods of the present subsection do not suffice to verify Condition N_♯ in massless free field theory. This goal is accomplished in the next subsection.

B.5.2 Expansion of Θ_E,2

Our last task is to complete the construction of the map Θ_E,2. In Subsection B.4.2 we introduced the p-nuclear positive operator T which has a J-invariant orthonormal basis of eigenvectors {e_i}^∞1 . We expand the functionsf^±∈L˚^±r in this basis

f^±= X∞ j=1

he_j|f^±ie_j (B.5.30)

and make use of the multinomial formula (B.1.4), obtaining the following equality valid on D_F

a^(∗)(f^±)^m± = X

µ^±,|µ^±|=m^±

m^±!

µ^±!he|fi^µ±a^(∗)(L^±re)^µ±. (B.5.31) Using relation (B.5.16), we obtain from definition (B.5.4) the following expansion, under-stood in the sense of quadratic forms on D_F ×D_F

B.5. Expansion of Θ_E into Rank-One Mappings 93 From the second part of Proposition B.3.1 there follows the existence of normal functionals τ_µ,ν on B(H) which have the property

τ_µ,ν(W(f)) = i^{|µ+|+|ν+|+2|µ−|}

µ!ν! e^−12kfk22he|fi^µ+ν (B.5.33) and satisfy the bound

kτ_µ,νk ≤ 4^|µ|+|ν|

(µ!ν!)¹²

(µ+ν)!

µ!ν! ¹₂

≤ 2^{52(|µ|+|ν|)}

(µ!ν!)¹² , (B.5.34) where we used properties of multinomial coefficients. Finally, we define the quadratic forms on D_F ×D_F given by

Sµ,ν =P_Ea^∗(Lre)^µa(Lre)^νP_E. (B.5.35) We note that a(L^±re)^νP_E = a(Q_EL^±re)^νP_E and ω⁻¹²Q_EL^±re_i = T_E,±e_i, where T_E,± are bounded operators by Lemma B.4.5. Thus we obtain from the energy bounds (B.5.22) and definition (B.4.28) of the operator T

kS_µ,νk ≤E^|µ|+|ν|2 kω⁻¹²Q_ELrek^µ2kω⁻¹²Q_ELrek^ν2 ≤E^|µ|+|ν|2 t^µt^ν, (B.5.36) where{t_j}^∞1 are eigenvalues ofT. We have arrived at the following expansion, still in the sense of quadratic forms on D_F ×D_F

Θ_E,2(A) = X

µ,ν

(|µ⁺|+|ν⁺|,|µ⁻|+|ν⁻|)∈S^′

τ_µ,ν(A)S_µ,ν, (B.5.37)

valid for any A ∈˚A(O(r)) i.e. for any finite, linear combination of Weyl operators. Our task is to establish the convergence of this sum in the norm topology of B(H) and extend this map by continuity to allA∈A(O(r)). It suffices to consider the caseS ={0,0}when there holds

Θ_E(A) =ω₀(A)P_E+ Θ_E,2(A), A∈A(O). (B.5.38) The following proposition verifies the known fact that Condition N_♯ holds in scalar free field theory.

Proposition B.5.5. [BP90, Bos00] In massive and massless scalar free field theory there holds the identity

Θ_E(A) =X

µ,ν

τµ,ν(A)Sµ,ν, A∈A(O(r)), (B.5.39) in the sense of norm convergence in B(H). Moreover, for any 0< p ≤1 there holds the bound

X

µ,ν

kτ_µ,νk^pkS_µ,νk^p<∞. (B.5.40)

Proof. We note the following estimate X

µ,ν

kτ_µ,νk^pkS_µ,νk^p ≤ X

µ,ν

(2⁵E)^{12p(|µ|+|ν|)} (µ!)^12p(ν!)^12p t^pµt^pν

≤ X

m^±,n^±∈N₀

(2^5/2E^1/2kTkp)^{p(m++m−+n++n−)}

(m⁺!m⁻!n⁺!n⁻!)^12p , (B.5.41) where in the first step we used (B.5.34) and (B.5.36). In the second step we made use of the fact that the multinomial coefficients are larger or equal to one, of the identity

X

µ^±,|µ^±|=m^±

m^±!

µ^±!(t^p)^µ± = (kTk^pp)^m± (B.5.42) and its counterpart for the sums w.r.t. ν^±.

Thus the map Θ_E,2, given by (B.5.37), has its range in B(H) and extends by continuity to the whole local algebra A(O(r)). We denote the resulting map by the same symbol.

Appendix C

Verification of Condition L ⁽²⁾ in Scalar Free Field Theory

The goal of this appendix is to verify that ConditionL⁽²⁾, introduced in Section 2.2, holds in scalar free field theory. In the massive case this fact follows from ConditionL⁽¹⁾, verified in Appendix D, and Theorem 2.3.1. Thus our main interest in the present appendix is in massless scalar free field theory, although some results will be stated for general m≥0 to facilitate their application in other contexts. Our aim is to prove Theorems 2.2.5, 2.2.6 and 2.2.7 which are at the basis of our discussion in Subsection 2.2.3.

The proofs of these three statements are given in Section C.1. They rely on the auxiliary Theorem C.1.1, stated below, whose proof is the subject of the later part of this appendix. In Section C.2 we define the functionals τ_k,k∈ {1,2,3}, on ˆAand verify that they have the properties required in the statement of Theorem C.1.1. We also show that the mapR⁽²⁾, defined in Theorem C.1.1, can be expressed in terms of the maps

∨

ΘE,1 and Θ_E,2, introduced in Appendix B. In Section C.3 we show that the range ofR⁽²⁾consists of square-integrable operators. The argument is divided into three parts: In Subsection C.3.1 we prove a variant of Theorem 1.6.1 which is applicable to the present problem. In Subsections C.3.2 and C.3.3 we apply this result to prove the square-integrability of the ranges of the maps

∨

ΘE,1 and ΘE,2, respectively.

C.1 Proofs of Theorems 2.2.5, 2.2.6 and 2.2.7 based on The-orem C.1.1

Our discussion in this section is based on the following theorem, whose proof is given in Sections C.2 and C.3.

Theorem C.1.1. In massless scalar free field theory in s ≥ 3 dimensional space there exist linear functionals τ₁, τ₂, τ₃ on A, invariant under translations in space, s.t. for anyˆ A∈Aˆ the quantity

R⁽²⁾(A) :=A−ω₀(A)I−τ₁(A)φ₊−τ₂(A) :φ²₊:−τ₃(A) :φ³₊:, (C.1.1) defined as a quadratic form on the domain DB×DB of vectors of bounded energy, satisfies kR⁽²⁾(A)kE,2 < ∞ for any E ≥ 0. Moreover, ω₀, τ₁, τ₂, τ₃ form a linearly independent

95

family. In addition, there hold the following assertions:

Aˆ^(e)⊂kerτ₁∩kerτ₃ and τ₂|Aˆ^(e)6= 0, (C.1.2) Aˆ^(d)⊂kerτ₁∩kerτ₂∩kerτ₃. (C.1.3) It is an immediate consequence of this theorem that the theory (A^(d), α,H), generated by the derivatives of the massless scalar free field, satisfies Condition L⁽²⁾ and has trivial point-continuous subspace. In fact:

Proof of Theorem 2.2.7: The statement follows directly from relations (C.1.3) and (C.1.1).

Let us now consider the full massless scalar free field theory (A, α,H). Theorem C.1.1 reduces the analysis of the point-continuous subspace in this model to the study of the three pointlike-localized fields: φ₊, :φ²₊:, :φ³₊:. We will show below that the concepts of square-integrability and of the infrared order of an operator, defined for observables from ˆA_c by relations (2.2.17) and (2.2.22), respectively, can be extended to the fields in question. Moreover, there holds the following proposition, whose proof is given in the later part of this section.

Proposition C.1.2. In massless scalar free field theory the following statements hold true:

(a) If s≥3, then ord(φ₊) = 2.

(b) If s= 3, then ord(:φ²₊:) = 1.

If s= 4, then ord(:φ²₊:) = 0.

If s≥5, then k:φ²:kE,2<∞ for anyE ≥0.

(c) If s= 3, then ord(:φ³₊:) = 0.

If s≥3, then k:φ³:kE,2<∞ for anyE ≥0.

Part (b) also holds if :φ²₊: is replaced with :φ²₊:∈Φ^(e)_FH.

We note that vanishing of the infrared order does not imply that a given operator is square-integrable. There remains an open question if :φ²₊: for s= 4 and :φ³₊: for s= 3 have the property of square-integrability. Its resolution would allow one to determine exactly the dimensions of the point-continuous subspaces in Theorem 2.2.5 (a) and (b), and Theorem 2.2.6 (b).

After this preparation we estimate the dimension of the point-continuous subspace in (full) massless scalar free field theory and compute the infrared orders of its elements. As expected, the infrared structure improves with increasing dimension of space, in the sense that the dimension of the point-continuous subspace decreases. However, this subspace remains non-trivial for any s≥3.

Proof of Theorem 2.2.5: We consider only the cases= 3 as the remaining cases can be proven analogously. Since we do not know whether :φ³₊:, whose infrared order is zero by Proposition C.1.2, is also square-integrable, we have to consider both possibilities. First, we show that if k :φ³₊:kE,2 = ∞ for some E ≥ 0, then for any A ∈ Aˆ_c there hold the following statements:

(i) τ₁(A)6= 0⇔ord(A) = 2.

(ii) (τ₁(A) = 0 and τ₂(A)6= 0)⇔ord(A) = 1.

C.1. Proofs of Theorems 2.2.5, 2.2.6 and 2.2.7 based on Theorem C.1.1 97 (iii) (τ₁(A) = 0, τ₂(A) = 0 and τ₃(A)6= 0)⇔(ord(A) = 0 and ∃E≥0 s.t. kAkE,2=∞).

(iv) (τ₁(A) = 0, τ₂(A) = 0 and τ₃(A) = 0)⇔ ∀E≥0 kAkE,2 <∞.

To justify these claims, suppose that τ₀(A) =· · ·= τ_l−1(A) = 0 for some l∈ {1,2,3,4}. (τ₀ := ω₀ is understood here). Then relation (C.1.1) gives the following bounds for any ϕ∈ TE,1

± Z

d^sp|~p|^β|ϕ(A(~e p))|²¹₂

− |τ_l(A)| Z

d^sp|p~|^β|ϕ(:φ]^l₊:(~p))|²¹₂

≤ X3 k=l+1

|τ_k(A)| sup

ϕ^′∈TE,1

Z

d^sp|~p|^β|ϕ^′(]

:φ^k₊:(~p))|²¹₂

+E^β2kR⁽²⁾(A)kE,2. (C.1.4) We note that all the terms in this expression are finite for sufficiently large β by esti-mate (2.2.15) and Proposition C.1.2. We will now study their behavior with decreasingβ.

By Proposition C.1.2, ord(:φ^l₊:)> ord(:φ^k₊:), for k > l in the above formula. Thus, by consideringβ in a small neighborhood of ord(:φ^l₊:) and taking supremum w.r.t. ϕ∈ TE,1, we easily obtain that ord(A) = ord(:φ^l₊:) if and only if τ_l(A)6= 0. Hence, there holds (i) and (ii). In part (iii) we set β= 0 and make use of our assumption that k:φ³₊:kE,2=∞ for some E≥0. In (iv) the implication (⇒) follows trivially from the square-integrability of R⁽²⁾(A). The opposite implication is a consequence of (i), (ii) and (iii).

Thus we have verified that Ord( ˆA_c) ={0,1,2} and the subspace ˆA_ac, consisting of the square-integrable observables, can be expressed as follows

Aˆ_ac= kerω₀∩kerτ₁∩kerτ₂∩kerτ₃, (C.1.5) Now computation of the dimension of the point-continuous subspace is a simple exercise in linear algebra: Since the above functionals are linearly independent (by Theorem C.1.1), we can findA1, A2, A3 ∈Aˆ_cs.t. τi(Aj) =δi,j. For anyA∈Aˆ_cwe obtain the decomposition

A= A−A₁τ₁(A)−A₂τ₂(A)−A₃τ₃(A)

+A₁τ₁(A) +A₂τ₂(A) +A₃τ₃(A), (C.1.6) where the term in bracket belongs to ˆA_ac due to (C.1.5). Choosing the point-continuous subspace as ˆA_pc = Span{A₁, A₂, A₃} and noting that {A_j}³1 are linearly independent, we obtain that dim ˆA_pc = 3.

Assuming that for any E ≥ 0 there holdsk :φ³₊: kE,2 <∞, we can incorporate the term τ3(·) :φ³₊: to R⁽²⁾(·) in formula (C.1.1). Thus, proceeding analogously as in the previous case, we verify the following facts for any A∈Aˆ_c:

(i^′) τ₁(A)6= 0⇔ord(A) = 2.

(ii^′) (τ₁(A) = 0 and τ₂(A)6= 0)⇔ord(A) = 1.

(iii^′) (τ₁(A) = 0 and τ₂(A) = 0)⇔(∀E≥0 kAkE,2 ≤ ∞).

Again, it follows that Ord( ˆA_c) ={0,1,2}. However, the absolutely continuous subspace is now given by

Aˆ_ac = kerω₀∩kerτ₁∩kerτ₂. (C.1.7) Hence the point-continuous subspace is two-dimensional in this case. In view of rela-tions (C.1.5) and (C.1.7) the theory satisfies ConditionL⁽²⁾.

The last example which we consider is the even part of massless scalar free field theory.

Making use of Theorem C.1.1, we will show that the theory (A^(e), α,H^(e)), introduced in Section B.2, satisfies Condition L⁽²⁾ and we will analyze the resulting point-continuous subspace. To this end, we define the functionalsω₀ =ω₀◦π⁻_(e)¹,τ₂ =τ₂◦π⁻_(e)¹ and set for any A∈A^(e)

R⁽²⁾_(e)(A) :=A−ω₀(A)I−τ₂(A):φ²₊: (C.1.8) as a quadratic form on states of bounded energy in H^(e). Then, due to relation (B.2.18) and the fact that ϕ(:φ²₊:) =ι_(e)(ϕ)(:φ²₊:) for anyϕ∈ T_E^(e), there holds

ϕ R⁽²⁾_(e)(A)

=ι_(e)(ϕ) R⁽²⁾(π⁻¹_(e)(A))

, A∈A^(e), ϕ∈ T_E^(e). (C.1.9) Finally, making use of relation (B.2.20), we obtain kR_(e)⁽²⁾(A)kE,2 ≤ kR⁽²⁾(π⁻¹_(e)(A))kE,2 <

∞, where the last bound follows from Theorem C.1.1. With the help of Proposition C.1.2 we obtain the description of the point-continuous subspace in the even part of massless scalar free field theory. Here the only possible obstruction to the square-integrability of observables form ˆA^(e)_c is the presence of the term :φ²₊: in relation (C.1.8).

Proof of Theorem 2.2.6: Exploiting relation (C.1.8) and the subsequent discussion, and proceeding as in the proof of Theorem 2.2.5 above, we obtain the result.

The remaining part of this section is devoted to the proof of Proposition C.1.2 which was the main technical input of the above discussion. We note that any field φ ∈ Φ_FH belongs to T_∞^∗ and therefore ϕ(φ(~x)) is a bounded, continuous function for any ϕ∈ TE. We are interested in the regularity properties of its Fourier transform ϕ(φ(~ep)) which is a tempered distribution. To begin with, we prove a simple, technical lemma.

Lemma C.1.3. Let 0< β < s,φ∈ TE^∗ and D⊂ H be a domain s.t. P_EH ∩Dis dense in P_EH and |~p|^β2(Ψ₁|φ(~e p)Ψ₂) is square-integrable, uniformly in Ψ₁,Ψ₂ ∈(P_EH ∩D)₁. Then

|~p|^β2ϕ(φ(~e p)) is square-integrable, uniformly inϕ∈ TE,1, and sup

Ψ1,Ψ2∈(PEH∩D)1

Z

d^sp|~p|^β|(Ψ₁|φ(~ep)Ψ₂)|² = sup

ϕ∈TE,1

Z

d^sp|p~|^β|ϕ(φ(~ep))|². (C.1.10) Proof. By the Cauchy-Schwarz inequality, there holds for any g ∈ S(R^s) and Ψ₁,Ψ₂ ∈ (P_EH ∩D)

|(Ψ₁|φ(g)Ψ₂)|

≤ kΨ₁k kΨ₂k sup

Ψ^′1,Ψ^′2∈(P_EH∩D)1

Z

d^sp|~p|^β|(Ψ^′₁|φ(~e p)Ψ^′₂)|²¹₂ Z

d^sp 1

|~p|^β|g(~˜ p)|²¹₂

.(C.1.11) The above bound extends, by continuity, to any Ψ1,Ψ2 ∈PEH and we can proceed as in the proof of Theorem 2.5 from [Bu90]: Let L²(R^s)_(β) be the Hilbert space of (classes of) functions h onR^s for which

khk²2,(β) = Z

d^sp|~p|^−β|h(~p)|² <∞. (C.1.12)

C.1. Proofs of Theorems 2.2.5, 2.2.6 and 2.2.7 based on Theorem C.1.1 99 The opposite inequality is trivial, since the supremum on the r.h.s extends over a smaller set.

Setting in the above lemmaφ=φ₊andD=D_S, we obtain a prescription for computation of the infrared order of the field φ₊:

Proof of Proposition C.1.2 (a): We will establish the bound sup

ϕ∈TE,1

Z

d^sp|~p|²|ϕ(φe₊(~p))|² <∞ (C.1.15) and show that the power of the mollifier |~p|² cannot be reduced. By Lemma C.1.3, it suffices to consider ϕ(·) = (Ψ₁| · Ψ₂), where Ψ₁,Ψ₂ ∈D_S∩P_EH. Making use of the fact that

φe₊(~p) = 1

p2|p~| a^∗(~p) +a(−~p)

, (C.1.16)

in the sense of quadratic forms on D_S×D_S, we obtain the estimate Z

where we used representation (B.2.23) of the Hamiltonian. To show that the bound for φ₊ is optimal, we construct a suitable sequence of functionals: We choose a positive function h(~p) ∈ C₀^∞(R^s) s.t. supph(~p) ⊂ {~p ∈ R^s| |~p| ≤ E}, R

In order to compute the infrared orders of higher Wick powers ofφ+, we need the following lemma.

Lemma C.1.4. Let Ψ₁,Ψ₂ be normalized vectors fromD_S∩P_EH. Then Z

d^sp1. . . d^spn|~p1|². . .|~pn|²|(Ψ1|:φe+(~p1). . .φe+(~pn) : Ψ2)|² ≤cn,s,E, (C.1.20) for some constant c_n,s,E independent of Ψ₁, Ψ₂.

Proof. Due to formula (C.1.16), it is clear that the expression on the l.h.s. of (C.1.20) can be bounded by a linear combination of terms of the form

Z where in the first step we made use of the Cauchy-Schwarz inequality and in the second step of the representation (B.2.23) of the Hamiltonian.

After this preparation we turn to the Wick powers of the field φ+.

Lemma C.1.5. Let n >1, s≥3. Then, for any β ≥0 s.t. β >2−(s−2)(n−1), there Here in the first step we made use of the fact that the Fourier transform of a product is a convolution of the Fourier transforms of the factors. We also used the support properties

C.2. Proof of Theorem C.1.1 (I): Functionals {τ_i}³1 101 of the wavefunctions corresponding to Ψ₁, Ψ₂. In the second step we applied the Cauchy-Schwarz inequality and the bound |~p|^β ≤n^β(|p~−~q₁|^β +|~q₁−~q₂|^β· · ·+|~q_n−1|^β) valid for any β ≥ 0. To show that expression (C.1.24) is bounded in the cases considered in the lemma, we make use of the Young inequality [RS2] which implies that forf_i ∈L^ri(R^s, d^sp), easily checks that these parameters satisfy the conditions specified above and, moreover, (2−β)r₁< s, 2r_i< s fori∈ {2, . . . , n}.

Proof of Proposition C.1.2 (b) and (c): Upper bounds from part (b) and part (c) follow from Lemma C.1.5. It remains to prove the lower bound in part (b) for s= 3. Let H_nbe defined as in the proof of Proposition C.1.2 (a) above and let us consider the family of functionals from TE,1 given by

ϕ⁽²⁾_n (·) = (H_n^⊗(n−2)| · H_n^⊗n). (C.1.26) :φ²₊: ∈Φ^(e)_FH the lower bound is established analogously, restricting attention to even n.

Similarly, the upper bound established in Lemma C.1.5 still holds, since the supremum in relation (C.1.22) extends now over a smaller set of functionals.

C.2 Proof of Theorem C.1.1 (I): Functionals { τ

_i

}

³1

In Section B.5 we introduced the decomposition of the map Θ_E :A(O(r))→B(H), given by Θ_E(A) =P_EAP_E, into three components

Θ_E =Θ^∧_E,1+Θ^∨_E,1+ Θ_E,2, (C.2.1) which are determined by the sets S and M. With the statement of Theorem C.1.1 in mind, we choose K⁺= 3, K⁻= 0 in the definition of the set S and define the set Mas consisting of four 2-multiindices

M={α₀, . . . , α₃} (C.2.2)

which are given by: α⁺_k(j) =kδ_j,1,α⁻_k(j) = 0 for k∈ {0,1,2,3},j∈N. Then, making use of definition (B.5.15) of the map

∧

ΘE,1 and recalling that our numbering of the s-indices {κ_j}^∞1 , introduced after formula (B.4.3), was chosen so that κ₁ = 0, we obtain

∧

Θ_E,1(A) =P_E ω₀(A)I+τ^∧(r)_α1(A)φ+τ^∧(r)_α2(A) :φ²: +τ^∧(r)_α3(A) :φ³:

P_E, (C.2.3) for anyA∈A(O(r)). Now we are in position to construct the functionalsτ_k,k∈ {1,2,3}, appearing in Theorem C.1.1, and verify that they have the required properties.

Proposition C.2.1. In massless scalar free field theory for s≥3 there exist linear func-tionals τ_k, k∈ {1,2,3} on Aˆ which satisfy

τ_k|^A(O(r))=τ^∧(r)_αk, (C.2.4)

τ_k(A(~x)) =τ_k(A) (C.2.5)

for any r > 0, A ∈ Aˆ and ~x ∈ R^s. Moreover, ω₀, τ₁, τ₂, τ₃ form a linearly independent family. In addition there holds:

Aˆ^(e)⊂kerτ₁∩kerτ₃ and τ₂|Aˆ^(e)6= 0, (C.2.6) Aˆ^(d)⊂kerτ₁∩kerτ₂∩kerτ₃. (C.2.7) Proof. Let α_k, k∈ {1,2,3} be the multiindices introduced in (C.2.2) above. By defini-tion (B.5.14), there holds

∧τ^(r)_αk = (i√ 2)^|k|

k! σ¯_α^(r)

k, (C.2.8)

where ¯σ_α^(r)_k are the functionals introduced in Lemma B.3.1, corresponding to the families of vectors{b^±_κj,r}j∈N. Thus for anyf ∈L˚r0 andr ≥r0 we obtain

¯ σ_α^(r)

k(W(f)) =e^−12kfk22hb⁺_0,r|f⁺i^k, (C.2.9) where we made use of our convention that κ₁ = 0. We recall that ˜f^± =ω^∓12F˜^±, where F^±∈D(Or0)R. Moreover, by definition (B.4.1),

˜b^±_0,r(~p) = 1

(2π)^s2ω(~p)^±12χ(^Or)(~p), (C.2.10) where χ(Or) = 1 on Or. We note that forr≥r0 and any ~y∈R^s s.t. Or0+~y⊂ Or there holds

hb⁺_0,r|U(~y)f⁺i= 1

(2π)^s2hχ(Or)|U(~y)F⁺i = 1 (2π)^s2

Z

d^sx χ(Or)(~x)F⁺(~x−~y)

= 1

(2π)^s2 Z

d^sx F⁺(~x), (C.2.11) i.e the functions r → hb⁺_0,r|U(~y)f⁺i are constant for r ≥r₀ and independent of ~y within the above restrictions. Thus for any r, r^′ ≥ r0, A ∈ ˚A(O(r0)) (i.e. A is a finite linear combination of Weyl operators) and ~y as specified above, we obtain

τ∧^(r)_αk(A)−τ^∧(r

′)

α (A) = 0, (C.2.12)

τ∧^(r)_αk(A)−τ^∧(r)_α (A(~y)) = 0. (C.2.13)

C.2. Proof of Theorem C.1.1 (I): Functionals {τ_i}³1 103 Since the functionals above are normal, the identities extend to any Afrom A(O(r₀)). In view of the above relation and the fact that anyA∈Aˆ belongs to A(O(r₀)) for sufficiently large r₀, the following formulas

τ_k(A) = lim

r→∞

∧τ^(r)_αk(A), A∈Aˆ (C.2.14) define linear functionals on ˆAwhich satisfy conditions (C.2.4) and (C.2.5). By this former condition, relation (C.2.8) and formula (C.2.11), we obtain

τ_k(W(f)) =^∧τ^(r_αk⁰⁾(W(f)) = (i√ 2)^|k|

k! e^−12kfk22 1

(2π)^s2 Z

d^sx F⁺(~x) k

, (C.2.15) for f, F⁺ defined as above. The last integral vanishes if F⁺(~x) = Ps

j=1∂_xjF_j⁺(~x) for some F_j⁺ ∈ D(Or0)R. From this fact and the strong continuity of τ^∧(r_αk⁰⁾ there follows statement (C.2.7). Next, we note that, by formula (C.2.15),

τ_k(W(f) +W(−f)) =τ^∧(r_αk⁰⁾(W(f) +W(−f)) = (1 + (−1)^k) τ^∧(r_αk⁰⁾(W(f)). (C.2.16) Thus for k ∈ {1,3} the functionals τ^∧(r_αk⁰⁾ are zero on A^(e)(O(r₀)), since they vanish on a strongly dense subspace of this algebra. This implies that ˆA^(e) belongs to kerτ₁∩kerτ₃. Choosing f so that R

d^sx F⁺(~x) 6= 0, we obtain from formulas (C.2.15), (C.2.16) that τ₂((W(f) +W(−f))6= 0, what concludes the proof of statement (C.2.6).

We still have to show that the functionalsω0, τ1, τ2, τ3 are linearly independent. Sup-pose that

c0ω0+c1τ1+c2τ2+c3τ3 = 0 (C.2.17) for some complex numbersc₀, . . . , c₃. Evaluating this expression on the unity operator we obtain from relation (C.2.9) that c0 = 0. Since there exist A ∈ Aˆ^(e) s.t. τ2(A) 6= 0, we get c₂ = 0. It remains to find B ∈ Aˆ s.t. τ₁(B) = 0 and τ₃(B) 6= 0. Given f, F⁺ as introduced above, with the additional condition that R

d^sx F⁺(~x)6= 0, we pick a function h∈S(R) s.t.

Z

e^−12u2kfk22uh(u)du= 0, (C.2.18) Z

e^−12u2kfk22u³h(u)du= 1. (C.2.19) (The existence of such function can be established with the help of the Gram-Schmidt procedure as in the proof of Lemma D.3.1). The weak integral

B :=

Z

W(uf)h(u)du (C.2.20)

defines an element ofA(O(r₀)) by the von Neumann bicommutant theorem. With the help of relations (C.2.8), (C.2.9) and (C.2.11) as well as properties (C.2.18), (C.2.19) of the function h we obtain that τ1(B) = 0 andτ3(B) 6= 0. Sinceτ1 is non-zero, this concludes the proof of linear independence of the functionals.

The functionalsτ_k,k∈ {1,2,3}, constructed in the above proposition, determine the map R⁽²⁾ introduced in the statement of Theorem C.1.1. For anyA∈A(O(r)) we obtain

R⁽²⁾(A) =A−ω₀(A)I −τ₁(A)φ₊−τ₂(A) :φ²₊:−τ₃(A) :φ³₊:, (C.2.21) in the sense of quadratic forms on the domain D_B×D_B of vectors of bounded energy. In view of property (C.2.4), formula (C.2.3) and decomposition (C.2.1) there holds

P_ER⁽²⁾(A)P_E =Θ^∨_E,1(A) + Θ_E,2(A), A∈A(O(r)). (C.2.22) Thus to conclude the proof of Theorem C.1.1, we have to verify that the ranges of the maps Θ^∨_E,1, Θ_E,2 are square-integrable. This is the subject of the next section.

C.3 Proof of Theorem C.1.1 (II): Square-Integrability of R

⁽²⁾

In this section we complete the proof of Theorem C.1.1. After introducing the necessary technical background in Subsection C.3.1, we prove the square-integrability of the ranges of the maps Θ^∨_E,1 and Θ_E,2 in Subsections C.3.2 and C.3.3, respectively.

C.3.1 Key Lemma

The main goal of this subsection is to prove Lemma C.3.1 below, which is inspired by Lemma 2.2 of [Bu90]. To state and prove this result, it is convenient to proceed to the full (non-symmetrized) Fock space Hb on which there act the (prototype) creation operators b^∗(Ψ), Ψ∈L²(R^s, d^sp), given by

b^∗(Ψ)Φ = Ψ⊗Φ, Φ∈Hb (C.3.1)

and their adjoints b(Ψ). Upon restriction to the symmetric Fock space H, the formula a(Ψ) =p

N b(Ψ), whereb Nb is the number operator, gives the standard annihilation oper-ator introduced in Subsection B.2. With these definitions at hand we proceed to the main technical result of this appendix.

Lemma C.3.1. Let E ≥ 0 and h be a Borel function on R^s which is bounded on {~p ∈ R^s|ω(~p)≤E}. We denote the operator of multiplication by h on L²(R^s, d^sp)by the same symbol. Let {g_1,i}^∞1 , {g_2,i}^∞1 be two families of functions from L²(R^s, d^sp) which belong

Lemma C.3.1. Let E ≥ 0 and h be a Borel function on R^s which is bounded on {~p ∈ R^s|ω(~p)≤E}. We denote the operator of multiplication by h on L²(R^s, d^sp)by the same symbol. Let {g_1,i}^∞1 , {g_2,i}^∞1 be two families of functions from L²(R^s, d^sp) which belong

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