Max R. Niedermaier

Off shell dynamics of the O(3) NLS model beyond Monte Carlo and perturbation theory

The off-shell dynamics of the O(3) non-linear sigma model is probed in terms of spectral densities and two-point functions by means of the form factor approach. The exact form factors of the spin field, Noether current, EM tensor and the topological charge density are computed up to six particles. The corresponding n ⩽ 6 particle spectral densities are used to compute the two-point functions, and are argued to deviate at most a few per mille from the exact answer in the entire energy range below 103 in units of the mass gap. To cover yet higher energies we propose an extrapolation scheme to arbitrary particle numbers based on a novel scaling hypothesis for the spectral densities. It yields candidate results for the exact two-point functions at all energy scales and allows us to exactly determine the values of two, previously unknown, non-perturbative constants.

Does the XY model have an integrable continuum limit

Abstract The quantum field theory describing the massive O(2) non-linear sigma-model is investigated through two non-perturbative constructions: the form factor bootstrap based on integrability and the lattice formulation as the XY model. The S-matrix, the spin and current two-point functions, as well as the 4-point coupling are computed and critically compared in both constructions. On the bootstrap side a new parafermionic super selection sector is found; in the lattice theory a recent prediction for the (logarithmic) decay of lattice artifacts is probed.

Questionable and unquestionable in the perturbation theory of non-Abelian models ∗

We show, by explicit computation, that bare lattice perturbation theory in the two-dimensional O(n) nonlinear σ models with superinstanton boundary conditions is divergent in the limit of an infinite number of points |�|. This is the analogue of David’s statement that renormalized perturbation theory of these models is infrared divergent in the limit where the physical size of the box tends to infinity. We also give arguments which support the validity of the bare perturbative expansion of short-distance quantities obtained by taking the limit |�| → ∞

A Derivation of the Cyclic Form Factor Equation

Abstract:A derivation of the cyclic form factor equation from quantum field theoretical principles is given; form factors being the matrix elements of a field operator between scattering states. The scattering states are constructed from Haag–Ruelle type interpolating fields with support in a “comoving” Rindler spacetime. The cyclic form factor equation then arises from the KMS property of the modular operators Δ associated with the field algebras of these Rindler wedges. The derivation in particular shows that the equation holds in any massive 1+1 dim. relativistic QFT, regardless of its integrability.

The quantum spectrum of the conserved charges in affine Toda theories

The exact eigenvalues of the infinite set of conserved charges on the multi-particle states in affine Toda theories are determined. This is done by constructing a free field realization of the Zamolodchikov-Faddeev algebra in which the conserved charges are realized as derivative operators. The resulting eigenvalues are renormalization group (RG) invariant, have the correct classical limit and pass checks in first-order perturbation theory. For n = 1 one recovers the (RG invariant form of the) quantum masses of Destri and De Vega.

GLM equations, tau function and scattering data

The direct and the inverse scattering problem for affine Toda/mKdV systems is addressed and is found to develop non-standard features within the framework of the inverse scattering method. A solution scheme based on the tau function formalism is described. The inverse problem is shown to be equivalent to a set of decoupled, scalar Gelfand-Levitan-Marchenko-type equations. The Fredholm-Grothendieck determinants of the latter are shown to define tau-functions in the sense of the Kyoto School. In particular, a simple monodromy formula allows the derivation of trace identities.

An algebraic approach to form factors

An associative ∗-algebra is introduced (containing a TTR-algebra as a subalgebra) that implements the form factor axioms, and hence indirectly the Wightman axioms, in the following sense: Each T-invariant linear functional over the algebra automatically satisfies all the form factor axioms. It is argued that this answers the question (posed in the functional Bethe ansatz) how to select the dynamically correct representations of the TTR-algebra. Applied to the case of integrable QFTs with diagonal factorized scattering theory a universal formula for the eigenvalues of the conserved charges emerges.

VARYING THE UNRUH TEMPERATURE IN INTEGRABLE QUANTUM FIELD THEORIES

A computational scheme is developed to determine the response of a quantum field theory (QFT) with a factorized scattering operator under a variation of the Unruh temperature. To this end a new family of integrable systems is introduced, obtained by deforming such QFTs in a way that preserves the bootstrap S-matrix. The deformation parameter β plays the role of an inverse temperature for the thermal equilibrium states associated with the Rindler wedge, β = 2π being the QFT value. The form factor approach provides an explicit computational scheme for the β ≠ 2π systems, enforcing in particular a modification of the underlying kinematical arena. As examples deformed counterparts of the Ising model and the sinh-Gordon model are considered.

The large N expansion in hyperbolic sigma models

Invariant correlation functions for SO(1,N) hyperbolic sigma models are investigated. The existence of a large N asymptotic expansion is proven on finite lattices of dimension d≥2. The unique saddle point configuration is characterized by a negative gap vanishing at least like 1/V with the volume. Technical difficulties compared to the compact case are bypassed using horospherical coordinates and the matrix-tree theorem.

Trace identities from Tau functions

An approach to trace identities is described based on theτ-function formalism for mKdV and affine Toda systems. The set-up is given for discrete and continuous spectrum solutions by exploiting the relation between Jost solutions, diagonal entries of the monodromy matrix, andτ-functions. Examples for solutions with discrete scattering data are worked out.