Decay Property

Z

R^s+1

d^s+1x g(x)αx(L) (2.33) is an operator in(A_L,T^u_q), satisfying the estimates

q_∆ αg(L)

6kgk1q_∆(L) (2.34)

for any bounded Borel set∆. The energy-momentum transfer ofαg(L)is contained in supp ˜g, the support of the Fourier transform ˜g of g.

Proof. By translation invariance of the norm k . k as well as of the seminorms q_∆ (cf. Lemma2.15) the (measurable) integrand on the right-hand side of (2.33) is major-ized by the functions x7→ |g(x)|kLk and x7→ |g(x)|q_∆(L)for any bounded Borel set

∆. These are Lebesgue-integrable and therefore αg(L) exists as a unique element of (AL,T^u_q), satisfying the claimed estimates (2.34).

Next, we consider an arbitrary function h∈L¹ R^s+1,d^s+1x

. By Fubini’s Theorem [26, II.16.3] and translation invariance of Lebesgue measure

Z

R^s+1

d^s+1y h(y)αy αg(L)

= Z

R^s+1

d^s+1y h(y)αy

Z

R^s+1

d^s+1x g(x)αx(L)

= Z

R^s+1

d^s+1y Z

R^s+1

d^s+1x h(y)g(x)αx+y(L)

= Z

R^s+1

d^s+1x Z

R^s+1

d^s+1y h(y)g(x−y)

αx(L), where the term in brackets on the right-hand side of the last equation is the convo-lution product h∗g of h and g. Its Fourier transform hg∗g is given by hg∗g(p) = (2π)^(s+1)/2˜h(p)g(˜ p) (cf. [39, Theorem VI.(21.41)]), so that this function vanishes if

˜h and ˜g have disjoint supports. Therefore supp ˜h∩supp ˜g=/0entails Z

R^s+1

d^s+1y h(y)αy αg(L)

=0, and this shows that the Fourier transform of y7→αy αg(L)

has support in supp ˜g, which henceforth contains the energy-momentum transfer ofαg(L).

2.3.4 Decay Property

Eventually we are able to establish a property of rapid decay with respect to the semi-norms q_∆for commutators of elements ofLwhich are almost local.

Lemma 2.19. Let L₁and L₂belong toL₀and let A₁,A₂∈Abe almost local. Then for any bounded Borel subset∆ofR^s+1

R^s3x7→q_∆ αx(A₁L₁),A₂L₂ decreases with|x| →∞faster than any power of|x|⁻¹.

2.3 Characteristics of the Spectral Seminorms 27

Proof. First we consider the special case of two elements La and Lb in L₀ having energy-momentum transfer in compact and convex subsetsΓaandΓbof{V₊, respect-ively, with the additional property thatΓa,b= (Γ. a+Γb)−ΓaandΓb,a= (Γ. a+Γb)−Γb

lie in the complement of V+, too. According to the Lemmas2.14and2.12 q_∆ αx(L_a),L_b2 and we are left with the task to investigate for large|x|the behaviour of the functions q_∆ αx(L_a)^∗αx(L_a),Lb

and q_∆ Lb∗αx(L_a),L_b

. Since the arguments of both terms belong to L₀, having energy-momentum transfer in the compact and convex subsets Γa,bandΓb,aof{V₊, we can apply (2.6) of Proposition2.6in connection with (2.17a) to get the estimate (for the second term)

|x|^2kq_∆ L_b^∗

constitute the large radius part of approximating nets for the almost local operators Lb∗αx(La),Lb for any l∈Nwith suitable C_l >0. Now, as suggested by the remark following Defin-ition 2.1, there exist approximating nets

L(a,b; x)r∈A(Or): r >0 , x∈R^s, with that, according to (2.2c), the integrand of (2.36) is bounded by

|x|^2k

28 Localizing Operators and Spectral Seminorms

Evaluation of the integrals on the right-hand side yields (for l>s+2) polynomials of degree s in|x|, so that, due to the decay properties of x7→

αx(L_a),L_b

, there exists a uniform bound

|x|^kq_∆ L_b^∗αx(L_a),L_b2

6M, x∈R^s. (2.40)

The same reasoning applies to the term q_∆ αx(L_a)^∗

αx(L_a),L_b

, thus establishing the asserted rapid decrease for the mapping x7→q_∆ αx(L_a),L_b

, according to relation (2.35).

In the general case of almost local elements A₁,A₂∈Aand L₁,L₂∈L₀one has, by Lemma2.12

q_∆

αx(A₁L₁),A₂L₂ 6kA1k

αx(L1),A2

q_∆(L2) +kA1kkA2kq_∆

αx(L1),L2

+

αx(A1),A2

kL2kq_∆(L1) +kA2k

αx(A1),L2

q_∆(L1), and rapid decay is an immediate consequence of almost locality for all terms but the second one on the right-hand side of this inequality. Using suitable decompositions of L₁ and L₂ in terms of elements ofL₀ complying pairwise with the special properties exploited in the previous paragraph, the remaining problem of decrease of the mapping x7→q_∆ αx(L1),L2

reduces to the case that has already been solved above, thus completing the proof.

Chapter 3

Particle Weights as Asymptotic Plane Waves

Having analysed in great detail the nets of seminorms q_∆ and p_∆, indexed by the bounded Borel sets ∆⊆R^s+1, on L andC, respectively, we now turn to the invest-igation of the topological dual spaces:

Definition 3.1. (a) The linear functionals onCwhich are continuous with respect to the seminorm p_∆constitute a vector spaceC_∆^∗, which is a normed space via

kςk∆ .

=sup

|ς(C)|: C∈C,p_∆(C)61 , ς∈C_∆^∗.

(b) The topological duals of the locally convex spaces (L₀,T_q), (L,T_q) and(C,T_p) are denotedL₀^∗,L^∗andC^∗, respectively.

Remark. Due to the net property (Proposition 2.9) of the family of seminorms p_∆, a linear functional belongs to the topological dual C^∗ of (C,T_p) if and only if it is continuous with respect to one specific seminorm p_∆⁰, ∆⁰ a bounded Borel subset of R^s+1[42, Proposition 1.2.8]. Hence

C^∗=[

C_∆^∗:∆⊆R^s+1a bounded Borel set . (3.1) By continuous linear extension [44, Chapter One, § 5, 4.(4)], the functionals from C^∗ are moreover in one-to-one correspondence with the elements of the topological dual C^∗ of the complete locally convex space (C,Tp). By the same argument, they are furthermore embedded in the topological dualA_C^∗ of(A_C,T^u_p). We shall make use of these properties without special mention.

3.1 General Properties

Before proceeding to extract certain elements fromC^∗to be interpreted, on the grounds of their specific properties, as representing asymptotic mixtures of particle-like quant-ities, we are first going to collect a number of important properties common to all functionals from the topological dual of C whose proof does not depend on special assumptions. First of all, continuity as established in Proposition2.16directly carries over to functionals inC^∗.

30 Particle Weights as Asymptotic Plane Waves

Lemma 3.2. Continuous linear functionalsς∈C^∗have the following properties.

(i) The mappingP₊^↑ 3(Λ,x)7→ς L₁^∗α_(Λ,x)(L₂)

is continuous for arbitrary but fixed L₁,L₂∈L.

(ii) The mappingP₊^↑ 3(Λ,x)7→ς α_(Λ,x)(C)

is continuous for given C∈C.

Proof. Due to the assumed continuity ofς, the assertions follow from Proposition2.16 in connection with Corollary2.13.

Every positive functional ςon the^∗-algebraC=L^∗Ldefines a non-negative ses-quilinear form onLthrough

h.|.iς:L×L→C (L₁,L₂)7→ hL₁|L₂iς .

=ς(L1∗L₂), (3.2a) and thus induces a seminorm q_ςonLvia

q_ς:L→R₊ L7→q_ς(L) .

=hL|Li^1/2ς . (3.2b) Denoting byN_ςthe null space of q_ς, one can construct the quotientL_ς .

=L/N_ς, which is a normed space through the definition

k.kς:L/N_ς→R₊ [L]_ς7→ k[L]_ςkς .

=q_ς(L), (3.2c)

where we used square brackets to designate the cosets inL/N_ς. These concepts can be applied to formulate, parallel to Proposition2.16, differentiability of the Poincaré automorphisms with respect to continuous positive functionals onC.

Lemma 3.3. Letςbe a continuous positive functional on the^∗-algebraC, i. e.ς∈C^∗₊. Then the restriction of the canonical homomorphism

Qς:L→L/N_ς L7→Qς(L) .

= [L]_ς

to the subspaceL₀isXL0-differentiable in the sense of DefinitionA.16, where XL0 =

ΞL0: L₀∈L₀

is the family of infinitely often differentiable mappings defined in Proposition2.16.

Proof. Due to the assumed continuity of the functionalς, there exists a bounded Borel set∆such that, according to (3.2) in connection with Definition3.1and Lemma2.14, for any L∈Lthere holds the inequality

k[L]_ςk²_ς =q_ς(L)²=ς(L^∗L)6kςk_∆p_∆(L^∗L) =kςk_∆q_∆(L)². Therefore the linear operator

QςL₀:(L₀,T_q)→(L/N_ς,k.kς)

turns out to be continuous, so that the assertion follows by an application of Corol-laryA.15from the result of Proposition2.16, stating that the mappings

ΞL0:P₊^↑ →(L0,T_q) (Λ,x)7→ΞL0(Λ,x) .

=α_(Λ,x)(L0) are differentiable for any L0∈L₀(cf. the remark of that place).

3.1 General Properties 31

The next lemmas are concerned with integrability properties of functionalsς∈C^∗, parallel to those established in Subsection2.3.3. The first one, Lemma3.4, is an im-mediate consequence of Lemmas2.17and2.18, whereas the second one, Lemma3.5, prepares the proof of a kind of Cluster Property for positive functionals inC^∗, formu-lated in the subsequent Proposition3.6.

Lemma 3.4. Letς∈C^∗, L1,L2∈Land C∈C.

(i) Let F∈L¹ P₊^↑,dµ(Λ,x)

have compact supportS, then ς L₁^∗αF(L₂)

= Z

dµ(Λ,x)F(Λ,x)ς L₁^∗α_(Λ,x)(L₂)

, (3.3a)

ς αF(C)

=

Z dµ(Λ,x)F(Λ,x)ς α(Λ,x)(C)

, (3.3b)

and there hold the estimates

ς L1∗αF(L₂)

6kFk1kςk∆q_∆(L₁) sup

(Λ,x)∈Sq_∆ α_(Λ,x)(L₂)

, (3.4a)

ς αF(C)

6kFk1kςk∆ sup

(Λ,x)∈Sp_∆ α_(Λ,x)(C)

(3.4b) for any∆such thatς∈C_∆^∗.

(ii) For any function g∈L¹ R^s+1,d^s+1x ς L1∗αg(L₂)

= Z

R^s+1

d^s+1x g(x)ς L1∗αx(L₂)

, (3.5)

and a bound is given by

ς L1∗αg(L₂)

6kgk1kςk_∆q_∆(L₁)q_∆(L₂) (3.6) for any∆satisfyingς∈C_∆^∗.

Proof. Lemmas2.17and2.18state that αF(L₂) =

Z dµ(Λ,x)F(Λ,x)α(Λ,x)(L₂), αF(C) =

Z dµ(Λ,x)F(Λ,x)α_(Λ,x)(C), αg(L2) =

Z

R^s+1

d^s+1x g(x)αx(L2)

exist in the complete locally convex spaces(A_L,T^u_q)and(A_C,T^u_p), respectively. Now, the functionalς, which lies in A_C^∗ according to the remark following Definition 3.1, is linear and continuous with respect toαF(C)∈A_Cand, by Corollary2.13, also with respect to bothαF(L₂),αg(L₂)∈A_L. Therefore it commutes with the locally convex integrals [26, Proposition II.5.7 adapted to integrals in locally convex spaces], which proves the assertion. The annexed estimates are a further simple application of the results contained in Lemmas2.17and2.18.

32 Particle Weights as Asymptotic Plane Waves

Lemma 3.5. Let L⁰ ∈Land let L∈L(Γ) =L∩A(Γ),e Γ⊆R^s+1 compact, i. e. L has energy-momentum transfer inΓ. Ifς∈C^∗+ is a positive functional which belongs to C_∆^∗and∆⁰denotes any bounded Borel set containing∆+Γ, then

Z

R^s

d^sxς L^∗αx(L^0∗L⁰)L

6kςk_∆q_∆(L)²q_∆0(L⁰)². (3.7) Proof. Let K be an arbitrary compact subset ofR^sand note that

Z

K

d^sxαx(L^0∗L⁰)∈A.

Thus, according to the construction ofC, Z

belongs to the algebra of counters and exists furthermore as an integral in the locally convex space(AC,T^u_p). Therefore the functionalς∈A_C^∗ can be interchanged with the integral [26, Proposition II.5.7] to give

Z Application of Lemma2.12then leads to the estimate

06 where we made use of the positivity ofς. The above inequality survives in the limit K%R^sand the convergence of the right-hand side to a finite real number establishes the integrability of the function

R^s3x7→ς L^∗αx(L^0∗L⁰)L

as a consequence of the Monotone Convergence Theorem [26, II.2.7]. In view of (2.17a), one finally arrives at the asserted bound

Z

After these preparations we are in a position to prove the announced Cluster Prop-erty for positive functionals inC^∗.

Proposition 3.6 (Cluster Property). Let L_i and L_i⁰ be elements ofL₀ and let A_i ∈A, i=1,2, be almost local operators, then the function

R^s3x7→ς (L1∗A1L⁰₁)αx(L2∗A2L⁰₂)

for any bounded Borel set∆for whichςbelongs toC_∆^∗, where the constant M_∆depends on∆and the operators involved.

3.1 General Properties 33 where the first term on the right-hand side is evidently integrable overR^s, due to almost locality of the operators encompassed by the commutator. For ς∈C_∆^∗ we have the estimate The second term can be estimated by use of the Cauchy-Schwarz inequality applied to the positive functionalς: Integration of the first term on the right-hand side is possible according to the previous Lemma3.5and gives

Z

R^s

d^sxς L1∗αx(L2∗A2A2∗L2)L1

6kςk∆q_∆(L1)²q_∆1(A2∗L2)², (3.13) where∆1is any bounded Borel set containing the sum of∆and the energy-momentum transfer Γ1 of L₁. Concerning the second term on the right of (3.12), we get, upon commutingαx(L⁰₂^∗)andαx(L⁰₂)to the interior, where again use was made of the positivity ofς. The rapid decay of commutators of almost local operators with respect to the q_∆-seminorm established in Lemma2.19of Subsection2.3.4can be combined with Lemma3.5to show integrability overR^s:

Z

34 Particle Weights as Asymptotic Plane Waves

which holds for any bounded Borel set ∆⁰₁⊇∆+Γ⁰₁, whereΓ⁰₁ denotes the energy-momentum transfer of L⁰₁. By (3.14) and (3.15), the left-hand side of (3.12) turns out to be integrable, and a bound for this integral is proportional tokςk_∆. In connection with (3.11) this establishes the assertion for a suitable constant M_∆that can be deduced from relations (3.11), (3.12), (3.14) and (3.15).

The Cluster Property has been proved above under the fairly general assumption of almost locality of the operators involved. If for given L₁,L₂∈Lthe mapping

R^s3x7→p_∆ L₁^∗αx(L₂) happens to belong to the space L¹ R^s,d^sx

for the bounded Borel set∆, (3.8) is obvi-ously fulfilled in case thatς∈C^∗belongs toC_∆^∗. As an example consider almost local operators L⁰₁,L⁰₂∈Lhaving energy-momentum transfer Γ1andΓ2, respectively, such that(∆+Γ1+Γ2)∩V+= /0. This implies L⁰₁αx(L⁰₂)E(∆) =0 for any x∈R^sand, by Lemmas2.11and2.12, p_∆ L⁰₁^∗αx(L⁰₂^∗)L⁰₁αx(L⁰₂)

=0. An application of Lemma2.12 in connection with translation invariance of q_∆(Lemma2.15) then yields for the coun-ters C_i⁰ .

=L⁰_i^∗L⁰_i, i=1,2, p_∆ C₁^0∗αx(C₂⁰)

=p_∆ L₁^0∗L⁰₁αx(L⁰₂^∗L⁰₂)

=p_∆ L⁰₁^∗

L⁰₁,αx(L⁰₂^∗)αx(L⁰₂) 6

L⁰₁,αx(L⁰₂^∗)

q_∆(L⁰₁)q_∆(L⁰₂), where, due to the assumed almost locality of L⁰₁ and L⁰₂, the right-hand side is seen to belong to L¹ R^s,d^sx

. The integrability of a mapping x7→p_∆ L₁^∗αx(L₂)

, L₁,L₂∈L, has another consequence concerning weakly convergent netsς_ι:ι∈J of functionals fromC^∗, which are contained in bounded subsets ofC_∆^∗with respect to the normk.k∆: a kind of Dominated Convergence Theorem.

Lemma 3.7. Let L1,L2∈Lbe such that x7→p_∆ L1∗αx(L₂)

is integrable and consider the weakly convergent net

ς_ι:ι∈J in the D-ball ofC_∆^∗with limitς.This means that for any C∈C

limι ς_ι(C) =ς(C) and for anyι∈J

|ς_ι(C)|6D·p_∆(C), (3.16a)

|ς(C)|6D·p_∆(C), (3.16b) the latter relation being implied by the former. Then

Z

R^s

d^sxς L₁^∗αx(L₂)

=lim

ι

Z

R^s

d^sxς_ι L₁^∗αx(L₂)

. (3.17)

Proof. As implied by Proposition2.16and Corollary2.13, x7→L₁^∗αx(L₂)is a continu-ous mapping onR^swith respect to the p_∆-topology, hence it is uniformly continuous on any compact set K. This means that toε>0 there existsδ>0 such that x,x⁰∈K and|x−x⁰|<δimply

p_∆ L1∗αx(L₂)−L1∗αx⁰(L₂)

< ε 6D|K|,

3.1 General Properties 35

where |K| denotes the s-dimensional volume of K. Consequently, under the above assumption on x and x⁰, we infer from (3.16) δ-balls around these points cover all of K; moreover, sinceςis the weak limit of the net ς_ι:ι∈J , we can findι0∈J such thatιι0implies and, selecting for x∈K an appropriate x_kin a distance less thanδ, we can put the above results together to get the estimate

ς_ι:ι∈J is indeed uniform convergence on compact subsets ofR^s. Upon integration over K we arrive at

so that toε>0 there exists a compact subset K_εsatisfying

Z

36 Particle Weights as Asymptotic Plane Waves

By arbitrariness ofεthis proves the possibility to interchange integration and the limit with respect toιas asserted in (3.17).

The spectral support of not necessarily positive functionalsς∈C^∗ (considered as distributions) depends, as expressed in the subsequent proposition, on the bounded Borel sets∆for whichς∈C_∆^∗. This property will prove to be of importance when it comes to defining the energy-momentum of particle weights.

Proposition 3.8 (Spectral Property). Let L₁,L₂∈Landς∈C^∗. Then the support of the Fourier transform of the distribution

R^s+13x7→ς L₁^∗αx(L₂)

lies inA_L, according to Lemma2.18, and has energy-momentum transfer in supp ˜g, the support of the Fourier transform of g. If this happens to satisfy supp ˜g⊆{(V+−∆), which, according to the preceding considerations, entails

Z

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