Fields and local operators

Moreover, we note that, in the sense of quadratic forms, (z^†mzⁿ(f))^∗ = z^†nz^m(f^∗), where f^∗(θ,η) = f(η_n, . . . , η₁, θ_m, . . . , θ₁), and where one finds kf^∗kn×m = kfkm×n. Another application of (2.91) then gives

kQ_ke^−ω(H/µ)z^†mzⁿ(f)Q_kk =kQ_kz^†nz^m(f^∗)e^−ω(H/µ)Q_kk ≤2 k!

(k−m)!kfk^ω_m×n, (2.93) and thus (2.82) is proven.

Remark: The equality kf^∗kn×m =kfkm×n follows from a short computation:

kf^∗k_n×m = sup

kgk₂≤1 khk2≤1

R f^∗(θ,η)g(η)h(θ)d^mηdⁿθ kgk₂khk₂

= sup

kgk₂≤1 khk2≤1

R f(η_m, . . . , η₁, θ_n, . . . , θ₁)g(η)h(θ)d^mηdⁿθ kgk₂khk₂

= sup

kgk2≤1 khk2≤1

R f(θ,η)g(θ_m, . . . , θ₁)h(η_n, . . . , η₁)d^mθdⁿη kgk₂khk₂

= sup

kg⁰k2≤1 kh⁰k₂≤1

R f(θ,η)g⁰(θ)h⁰(η)dⁿηd^mθ kg⁰k₂kh⁰k₂

=kfkm×n. (2.94) where in the fourth equality we called g⁰(θ) :=g(θ_m, . . . , θ₁) (similar definition forh⁰) and we used that kg(θ_m, . . . , θ₁)k₂ =kgk₂ (same forh).

2.8 Fields and local operators

Following [Lec06], and analogous to free field theory, we can define a quantum field φ as, f ∈ S(R²),

φ(f) :=z^†(f⁺) +z(f⁻). (2.95) As shown in [Lec06, Proposition 4.2.2], this field has similar mathematical properties to the free scalar field: It is defined on H^f, and essentially selfadjoint for real-valued f. Moreover, φ has the Reeh-Schlieder property, transforms covariantly under the representation U(x, λ) ofP₊, and it solves the Klein-Gordon equation.

However, φ is strictly local only if S = 1. For generic S, the field is only localized in an infinitely extended wedge – rather than at a space-time point – in the following sense. Let us introduce the “reflected” Zamolodchikov operators, ψ ∈ H₁,

z(ψ)⁰ :=J z(ψ)J, z^†(ψ)⁰ :=J z^†(ψ)J, (2.96) and define another field φ⁰ as,f ∈ S(R²),

φ⁰(f) := J φ(f^j)J, f^j(x) :=f(−x). (2.97) It has been shown in [Lec06, Proposition 4.2.6] that the two fields φ, φ⁰ are relatively wedge-local: For real-valued test functions f, g with suppf ⊂ W⁰ and suppg ⊂ W,

one finds that [e^iφ(f)−, e^iφ0(g)−] = 0. Hence, we can understand φ(x) and φ⁰(y) as being localized in the shifted left wedge W_x⁰ and in the shifted right wedgeWy, respectively.

This result is obtained by computing the commutation relations ofz, z^† with z⁰, z^†0 [Lec06, Lemma 4.2.5]: Letg ∈ H₁. The following holds in the sense of operator-valued distributions on H^f:

[z(g)⁰, z^†(θ)] =B^g,θ, [z^†(g)⁰, z(θ)] = −(B^¯g,θ)^∗ (2.98) [z(g)⁰, z(θ)] = 0, [z^†(g)⁰, z^†(θ)] = 0, (2.99) whereB^g,θ =⊕^∞_n=0B_n^g,θ andB_n^g,θacts on then-particle Hilbert space as a multiplication operator:

B^g,θ_n (θ₁, . . . , θ_n) =g(θ)

n

Y

j=1

S(θ−θ_j). (2.100)

Instead of working with unbounded (closed) operators, we can also work with associated von Neumann algebras: We define the “wedge algebra” as:

M={e^iφ(f)−|f ∈ S_R(R²),suppf ⊂ W⁰}⁰. (2.101) Remark: we can restrict this definition to smaller sets f ∈ D_R(W⁰), or even to f ∈ D^ω

R(W⁰). This does not change Msince D^ω

R(W⁰) is dense inD_R(W⁰) in theD-topology (see [Bj¨o65]) andD_R(W⁰) is dense inS_R(R²) with respect to test functions with support inW⁰. Moreover, the set of operatorse^iφ(f) with f in these restricted domains is dense in M because the map f 7→ e^iφ(f) is continuous in the strong operator topology (see for example [RS75]).

We can extend this definition to define algebras associated with any wedge in R²: As shown in [Lec06, Proposition 4.4.1], the triple (M, U(x),H) satisfies the defining properties of a standard right wedge algebra in the sense of [Lec06, Definition 2.1.1]

and the associated mapW 7→ A(W) (where here W is a generic wedge) is a local net of von Neumann algebras with the properties in [Lec06, Proposition 4.4.1].

We can extend this definition to bounded regions by taking intersections of wedge algebras. Namely, the local algebra of a double cone O_x,y = W_x ∩ W_y⁰, x, y ∈ R², y−x∈ W, is defined as

A(O_x,y) := A(W_x)∩ A(W_y)⁰. (2.102) It has been shown in [Lec06] that O 7→ A(O), where

A(O) := [

O_x,y⊂O

A(O_x,y)⁰⁰

, (2.103)

is a covariant, local net of von Neumann algebras fulfilling the standard axioms of a local quantum field theory. Here it is not a priori clear that the algebras A(O) are nontrivial, i.e., that they contain any operator except for multiples of the identity.

However, Lechner proved [Lec08] that at least for regionsO of a certain minimum size, the vacuum vector Ω is indeed cyclic for A(O), of which it follows that the algebras are type III₁ factors [BL04].

Chapter 3

The Araki expansion

We consider quantum field theory on 1+1 dimensional Minkowski space. We know that in the case of a real scalar free field, any operator Aon Fock space can be decomposed as

A=

∞

X

m,n=0

Z d^mθ dⁿη

m!n! f_m,n(θ,η)a^†(θ₁)· · ·a^†(θ_m)a(η₁)· · ·a(η_n), (3.1) where θ_j, η_j are rapidities and where the (generalized) functions f_m,n can be written down explicitly in terms of a string of nested commutators:

f_m,n(θ,η) =

Ω,[a(θ₁),[. . . a(θ_m),[. . .[A, a^†(η_n)]. . . a^†(η₁)]. . .]Ω

. (3.2) Araki has shown in [Ara63] (in a different notation) that every bounded operators A has such decomposition.

In the following section we aim to establish an analogue of the series expansion (3.1) in terms of the deformed creators and annihilatorsz, z^† in our models with factorizing scattering matrix. Moreover, we aim to establish this expansion for arbitrary bounded operators, and more generally for unbounded operators and quadratic forms. This is an important ingredient for a characterization theorem for local operators which we will formulate in Sec. 5.

3.1 Contractions

In this section we will introduce some of our notation, similar to [Lec08] but with conventions slightly more convenient for our purposes.

We consider forA∈ Q^ω the matrix element

hz^†(θ₁)· · ·z^†(θ_m)Ω, Az^†(η_n)· · ·z^†(η₁)Ωi=:h`(θ), Ar(η)i, (3.3) A contraction C is a triple C = (m, n,{(l₁, r₁), . . . ,(l_k, r_k)}), where m, n ∈ N0, 1 ≤ l_j ≤ m and m + 1 ≤ r_j ≤ m+n, and both the l_j and the r_j are pairwise different among each other. We denote C_m,n the set of all contractions for fixed m and n, and write |C|:=k for the length of a contraction (in other words, the number of elements of the set in the third entry in the definition of C).

Using this notion of a contraction, we can consider (“contracted”) matrix elements h`_C(θ), Ar_C(η)i, where

`_C(θ) :=z^†(θ₁)· · ·z\^†(θ_l1)· · ·z\^†(θ_l|C|)· · ·z^†(θ_m)Ω, (3.4) r_C(η) :=z^†(η_n)· · ·z^†\(η_r1−m)· · ·z^†(η\_r|C|−m)· · ·z^†(η₁)Ω, (3.5)

and where the hats indicate that the marked elements have been omitted in the se-quence.

We note that`C(·) is anH-valued distribution onD(R<sup>m−|C|</sup>), namely when smear-ing each z<sup>†</sup> with test functions in D(R), `_C(f) is a vector in H. Actually, its values are in H^ω,f; indeed, given any function f smooth and of compact support, `_C(f) :=

R d^mθ f(θ)`<sub>C</sub>(θ) is a vector of finite particle number, and has the norm ke<sup>ω(H/µ)</sup>`_C(f)k ≤p

(m− |C|)!kfk^ω₂. (3.6) This inequality is a generalization of (2.48) first part, in the case ` = 0. Namely, we can follow the same computation as in the proof of (2.48), but setting ` = 0 and consideringz^†m(f) instead of z^†(f). By explicit computation:

ke^ω(H/µ)`_C(f)k² = Z

d^m−|C|θdˆ ^m0−|C|θˆ⁰f(ˆθ⁰)f(ˆθ)he^ω(H/µ)z^†(θ⁰₁)· · ·

· · ·z\^†(θ⁰_l

1)· · ·z\^†(θ_l⁰

|C|)· · ·z^†(θ_m⁰ )Ω, e^ω(H/µ)z^†(θ₁)· · ·z\^†(θ_l1)· · ·z\^†(θ_l|C|)· · ·z^†(θ_m)Ωi

≤(m− |C|)!

Z

d^m−|C|θ|f(ˆˆ θ)|²e^{2ω(Pmj=1coshθbj)}

= (m− |C|)!(kfk^ω₂)². (3.7) where in the second inequality we used theS-symmetry of f.

This holds similarly for r_C(·). Therefore, for fixed A ∈ Q^ω, the matrix element h`_C(θ), Ar_C(η)i is a well-defined distribution onD(R^m+n−2|C|)⁰.

We associate with a contraction C ∈ C_m,n the following quantities:

δ_C(θ,η) :=

|C|

Y

j=1

δ(θ_lj −η_rj−m), (3.8)

S_C(θ,η) :=

|C|

Y

j=1 rj−1

Y

mj=lj+1

S_m^(m)

j,lj · Y

ri<rj

li<lj

S_l^(m)

j,ri, (3.9)

where we used the notation

S_a,b(θ) :=S(θ_a−θ_b), S_a,b^(m) :=

S_b,a a≤m < b, b≤m < a

S_a,b otherwise (3.10)

We will often not write down explicitly the argumentsθ,η where they are clear from the context. We will see the use of the above expressions later in the present thesis.

It will become also very useful the fact that we can express the factors S_C in terms of the expressionsS^σ associated with permutations σ, as the following Lemma shows.

Lemma 3.1. There holds

δ_CS_C(θ,η) = δ_CS^σ(θ)S^ρ(η), (3.11) where

σ=

1 . . . m

1 . . . ˆl . . . m l₁ . . . l|C|

, ρ=

m+ 1 . . . m+n

r|C| . . . r1 m+ 1 . . . r . . . mˆ +n

.

(3.12)

3.1. CONTRACTIONS

Remarks: ˆl indicates that we leave out thel_j from the sequence; ˆranalogously. The permutations σ, ρ are not unique since one can permute the pairs of the contraction.

However, the right hand side of (3.11) is independent of this choice since the extra S-factors associated with different permutations σ, ρ of the same pairs would cancel each other due to the delta distributions.

Proof. Considering the above remark, we can assume that r₁ < . . . < r|C|. From the definition of S^σ,S^ρ in Eq. (2.5) with σ, ρgiven by (3.12), we can read off that

S^σ =

|C|

Y

j=1 m

Y

pj=lj+1

Spj,lj · Y

i<j li<lj

Sli,lj, (3.13)

S^ρ=

|C|

Y

j=1 rj−1

Y

qj=m+1

Srj,qj. (3.14)

Computing the product S^σS^ρ from (3.13) and (3.14), and taking the factor δ_C into account, we find that δ_CS^σS^ρ=δ_CS_C with S_C defined as in (3.9).

We will also need to consider compositions of contractions. Given the contrac-tions C ∈ C_m,n and C⁰ ∈ Cm−|C|,n−|C|, the composed contraction, where the indices are contracted first with C, then with C⁰, is defined as C∪C˙ ⁰ ∈ C_m,n, C∪C˙ ⁰ = (m, n,{(l₁, r₁), . . . ,(l_k, r_k),(l⁰₁, r⁰₁), . . . ,(l⁰_k0, r_k⁰0)}). This definition should be intuitively clear; on the other hand, note that it involves a renumbering of the indices inC⁰ before taking the set union C∪C˙ ⁰; we will often avoid to indicate this renumbering explicitly.

With respect to this composition of contractions, also the factors δ_C and S_C compose in a certain way, as the following lemma shows. Here ˆθ ∈R^m−|C| indicates thatθ has the components θ_l1, . . . , θ_l|C| left out; analogously for ˆη.

Lemma 3.2. Let C ∈ C_m,n and C⁰ ∈ Cm−|C|,n−|C|. There holds

δC(θ,η)δC⁰(ˆθ,η)Sˆ C(θ,η)SC⁰(ˆθ,η) =ˆ δ_C∪C˙ ⁰(θ,η)S_C∪C˙ ⁰(θ,η). (3.15) Proof. From the definition (3.8) it is clear thatδCδC⁰ =δ_C∪C˙ ⁰. Using this and Lemma 3.1, it remains to show that

S^σ(θ)S^σ0(ˆθ) =S^σ00(θ), S^ρ(η)S^ρ0(ˆη) = S^ρ00(η), (3.16) whereσ, ρ,σ⁰, ρ⁰,σ⁰⁰, ρ⁰⁰ are the permutations associated withC,C⁰, andC∪C˙ ⁰, respec-tively, by Eq. (3.11). We note that σ⁰ is given explicitly by

σ⁰ = 1 . . .ˆl . . . m

1 . . .ˆl ˆl⁰. . . m l₁⁰ . . . l⁰_|C0|

!

∈Sm−|C|. (3.17) We can consider σ⁰ as an element of S_m by extending the permutation matrix in the following way:

σ⁰ = 1 . . .ˆl . . . m l₁ . . . l_|C|

1 . . .lˆ⁰ ˆl . . . m l₁⁰ . . . l⁰_|C0| l₁ . . . l|C|

!

. (3.18)

With this, S^σ0(ˆθ) =S^σ0(θ^σ). Using the composition law in Eq. (2.6), one has

S^σ00(θ) = S^σ0(θ^σ)S^σ(θ) =S^σ0(ˆθ)S^σ(θ). (3.19)

whereσ⁰⁰ is given by:

σ⁰⁰ :=σ◦σ⁰ =

1 . . . m

1 . . . ˆl⁰ ˆl . . . m l₁⁰ . . . l_|C⁰ 0|l₁ . . . l_|C|

(3.20) One notices that this permutation is indeed associated with C∪C˙ ⁰ by Eq. (3.11).

We obtain in a similar way the second part of Eq. (3.16), and hence we find the result of this lemma.

Im Dokument A Characterization Theorem for Local Operators in Factorizing Scattering Models (Seite 31-36)