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Euclidean quantum fields obtained as solutions of stochastic partial pseudo differential equations driven by a Poisson white noise have paths given by locally integrable functions. This makes it possible to define a class of ultra-violet finite local interactions for these models (in any space-time dimension). The corresponding interacting Euclidean quantum fields can be identified with systems of classical "charged" particles in the grand canonical ensemble with an interaction given by a nonlinear energy density of the "static field" generated by the particles' charges via a "generalized Poisson equation". A new definition of some well-known systems of statistical mechanics is given by formulating the related field theoretic local interactions. The infinite volume limit of such systems is discussed for models with trigonometric interactions using a representation of such models as Widom–Rowlinson models associated with (formal) Potts models at imaginary temperature. The infinite volume correlation functional of such Potts models can be constructed by a cluster expansion. This leads to the construction of extremal Gibbs measures with trigonometric interactions in the low-density high-temperature (LD-HT) regime. For Poissonian models with certain trigonometric interactions an extension of the well-known relation between the (massive) sine-Gordon model and the Yukawa particle gas connecting characteristic and correlation functionals is given and used to derive infinite volume measures for interacting Poisson quantum field models through an alternative route. The continuum limit of the particle systems under consideration is also investigated and the formal analogy with the scaling limit of renormalization group theory is pointed out. In some simple cases the question of (non-) triviality of the continuum limits is clarified.

In the bootstrap approach to integrable quantum field theories in the $(1+1)$-dimensional Minkowski space, one conjectures the two-particle S-matrix and tries to study local observables. We find a family of two-particle S-matrices parametrized by two positive numbers, which are separated from the free field or any other known S-matrix. We propose candidates for observables in wedge-shaped regions and prove their commutativity in the weak sense.

The sine-Gordon model is conjectured to be equivalent to the Thirring model, and its breather–breather S-matrix components (where the first breather corresponds to the scalar field of the sine-Gordon model) are closed under fusion. Yet, the residues of the poles in this breather–breather S-matrix have wrong signs and cannot be considered as a separate model. Our S-matrices differ from the breather–breather S-matrix in the sine-Gordon model by CDD factors which adjust the signs, so that this sector alone satisfies reasonable assumptions.

The formalism of space–time dependent Lagrangians developed in Ref. 1 is applied to the sine-Gordon and massive Thirring models. It is shown that the well-known equivalence of these models (in the context of weak–strong duality) can be understood in this approach from the same considerations as described in Ref. 1 for electromagnetic duality. A further new result is that all these can naturally be linked to the fact that the holographic principle has analogues at length scales much larger than quantum gravity. There is also the possibility of noncommuting coordinates residing on the boundaries.

The functional renormalization group treatment is presented for the two-dimensional sine-Gordon model including a bilocal term in the potential, which contributes to the flow at the tree level. It is shown that the flow of the bilocal term can substitute the evolution of the wave function renormalization constant, since it can recover the Kosterlitz–Thouless type phase transition. The flows can also reveal the connection between the sine-Gordon and the noninteracting Thirring models at a special value of the wave number parameter.

We study the structure of superselection sectors of an arbitrary perturbation of a conformal field theory. We describe how a restriction of the q-deformed sl(2) affine Lie algebra symmetry of the sine-Gordon theory can be used to derive the S-matrices of the Φ^(1,3) perturbations of the minimal unitary series. This analysis provides an identification of fields which create the massive kink spectrum. We investigate the ultraviolet limit of the restricted sine-Gordon model, and explain the relation between the restriction and the Fock space cohomology of minimal models. We also comment on the structure of degenerate vacuum states. Deformed Serre relations are proven for arbitrary affine Toda theories, and it is shown in certain cases how relations of the Serre type become fractional spin supersymmetry relations upon restriction.

We use the two-dimensional Fermion-Boson mapping to perform a field theory analysis of the effective Lagrangian model for incommensurate charge-density waves (ICDW) in one-dimensional systems. We consider an approach in which both the phase of the complex phonon field and the electron field are dynamical degrees of freedom contributing to the quantum dynamics and symmetry-related features of the ICDW phenomenon. We obtain the bosonized and fermionized versions of the effective electron–phonon Lagrangian. The phase of the phonon field and the phase of the bosonized chiral density of the electron field condense as a soliton order parameter, carrying neither the charge nor the chirality of the electron–phonon system, leading to a periodic sine-Gordon potential. The phonon field is fermionized in terms of a chiral fermionic condensate and the effective model is mapped into the chiral Gross–Neveu (GN) model with two Fermi field species. The linked electron–phonon symmetry of the ICDW system is mapped into the chiral symmetry of the GN model. Within the functional integral formulation, we obtain for the vacuum expectation value of the phonon field < ϕ > = 0 and < ϕ ϕ* > ≠ 0, due to the charge selection rule associated with the chiral electron–phonon symmetry. We show that the two-point correlation function of the phonon field satisfies the cluster decomposition property, as required by the chiral symmetry of the underlying GN model. The quantum description of the ICDW corresponds to charge transport through the lattice, due to the propagation of a "Goldstone mode" carrying the effective charge of the electron–phonon system, is accomplished by an electron–lattice energy redistribution. This accounts for a dynamical Peierls's energy gap generation.

Neutral excitations present in the repulsive regime (1/2 < β²/8π < 1) of the sine–Gordon/massive–Thirring model and its study of the massless limit by the thermodynamic Bethe ansatz is revisited. At β²/8π = 1-1/(p+1) the solitons become infinitely heavy, forcing truncation to the neutral excitations alone. The central charge in this limit is calculated to be c = 1-6/p(p+1); the mass and S-matrices of the truncated theories are identified as those of the minimal conformal theory M_p perturbed by the ϕ_{(1, 3)} operator.

We apply the fermionic description of CFT developed in our previous work to the computation of the one-point functions of the descendent fields in the sine-Gordon model.

For the integrable 6 vertex model, the expectation values of local operators are known to be given by complicated multiple integrals. We show that there exists a basis of (quasi) local operators for which the expectation values simplify drastically. Such a basis is constructed out of a simple ‘tail’ operator (analogous to the disorder field in the Ising model) by acting with integrals of motion and a newly introduced set of fermions. The expectation values for their generating functions are given by determinants with explicit entries. This fermionic structure is present at a generic coupling, away from the usual ‘free fermion point’.

Taking the continuum limit to CFT and the sine-Gordon model, we formulate conjectural explicit formulas for the one-point functions of all descendant fields in both cases, generalizing the remarkable formulas due to Lukyanov, Zamolodchikov and others. We argue also that at the level of form factors our fermions coincide with yet another fermions which have been introduced some time ago by Babelon, Bernard and Smirnov.

This talk is based on a series of joint works with H. Boos, T. Miwa, F. Smirnov and T. Takeyama.

Note from Publisher: This article contains the abstract only.

Neutral excitations present in the repulsive regime (1/2 < β²/8π < 1) of the sine– Gordon/massive–Thirring model and its study of the massless limit by the thermodynamic Bethe ansatz is revisited. At β²/8π = 1 - 1/(p + 1) the solitons become infinitely heavy, forcing truncation to the neutral excitations alone. The central charge in this limit is calculated to be c = 1 - 6/p(p + 1); the mass and S-matrices of the truncated theories are identified as those of the minimal conformal theory M_p perturbed by the ϕ_(1,3) operator.