Gandalf Lechner

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An object of central importance in the first part of the thesis is a so-called standard right wedge algebra, defined as follows.

Definition: A standard right wedge algebra is a triple $({\cal M},U,{\cal H})$ consisting of a Hilbert space $\cal H$ with ${\rm dim}{\cal H}>1$, a representation $U$ of the translation group $({\mathbb R}^2,+)$ on $\cal H$, and a von Neumann algebra ${\cal M}\subset {\cal B}({\cal H})$ such that the following conditions are satisfied.

• $U$ is strongly continuous and unitary. The joint spectrum of the generators $P_0, P_1$ of $U({\mathbb R}^2)$ is contained in the forward light cone.
  There is an up to a phase unique unit vector $\Omega\in\cal H$ which is invariant under the action of $U$.
• $\Omega$ is cyclic and separating for $\cal M$.
• For each $x\in \overline{W_R}=\{y\in{\mathbb R}^2\,:\,y_1\geq |y_0|\}$, the adjoint action of $U(x)$ induces endomorphisms on $\cal M$.

Starting from a standard right wedge algebra, a net ${\cal O}\mapsto {\cal A}({\cal O})$ of local observable algebras on ${\mathbb R}^2$ can be defined as follows [cf. Borchers 92]:
$${\cal A}(W_R) := {\cal M},\qquad {\cal A}(W_L) := {\cal M'},$$ $${\cal A}(W_{R/L}+x) := U(x){\cal A}(W_{R/L})U(x)^{-1},$$ $${\cal A}(W_1 \cap W_2) := {\cal A}(W_1) \cap {\cal A}(W_2)$$
(Here $W_1, W_2$ are arbitrary wedges in ${\mathbb R}^2$) The main result of the first part of my thesis is contained in the following theorem:

Theorem: Consider a standard right wedge algebra $({\cal M},U,{\cal H})$ with the property that the maps

$$\Xi(x):{\cal M}\longrightarrow{\cal H},\qquad \Xi(x)A := \Delta^{1/4}U(x)A\Omega$$

are nuclear for $x\in W_R$ (modular nuclearity condition, [Buchholz, d'Antoni, Longo 90].)
Then

• The map ${\cal O}\rightarrow{\cal A}({\cal O})$ is a local net of von Neumann algebras which transforms covariantly under a representation of the proper Poincare group.
• The net $\cal A$ has the split property for inclusions of wedges.
• The wedge algebras and double cone algebras of $\cal A$ are isomorphic to the unique hyperfinite type $III_1$ factor.
• Haag duality for double cones holds.
• Strong additivity holds (i.e. the double cone algebras generate the wedge algebras.)
• The vacuum vector $\Omega$ is cyclic and separating for all double cone algebras.

Some statements of this theorem have been known before and are due to Borchers, Buchholz, d'Antoni, Doplicher, Longo, and Müger, see the thesis for references.

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The main input in the construction of model theories with factorizing S-matrices is a so-called scattering function to parametrize an S-matrix of a single species of neutral, scalar, massive particles.

Definition: A scattering function $S_2$ is a bounded analytic function on the strip $\{\zeta\,:\,0\leq{\rm Im}\zeta\leq\pi\}$ which satisfies $$\overline{S_2(\theta)}=S_2(-\theta)=S_2(\theta)^{-1}=S_2(\theta+i\pi),\qquad \theta\in\mathbb R.$$

Starting from such a scattering function, a wedge-localized quantum field $\phi$ is defined (for details, see the main text), which can be shown to generate a standard right wedge algebra $\cal M$. The construction of $\phi$ is due to Schroer [Schroer 97], and uses ideas of the form factor program (Zamolodchikov's algebra.)

To prove the existence of the model theories associated to a scattering function $S_2$, it is necessary to verify the modular nuclearity condition in these models. For a large class ${\cal S}_0^-$ of scattering functions (defined in the main text), the following theorem holds:

Theorem: Consider a model theory with scattering function $S_2\in{\cal S}_0^-$.

• The modular nuclearity condition for wedge algebras holds. In particular, the model exists as a local, covariant net of observable algebras which satisfies the Reeh-Schlieder property.
• All $n$-particle scattering states and the S-matrix of the model can be computed. It is shown that the construction solves the inverse scattering problem for the underlying scattering function, and that the theory is asymptotically complete.
• A proof for formulae for asymptotic collision states, which is usually employed in the form factor program, is proven.

The family ${\cal S}_0^-$ of scattering functions contains in particular the well-known scattering functions of the Sinh-Gordon and scaling Ising model.

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