H.J. de Vega

Light-cone lattice approach to fermionic theories in 2D: the massive Thirring model

Abstract Two-dimensional fermionic field theories are defined on a diagonal lattice obtained by discretizing Minkowski space-time in light-cone coordinates. This approach leads to local second-quantized equations of motion on the lattice. The continuum limit is carefully performed, yielding the massive Thirring model whenever fermions without internal structure are considered. The exact eigenstates and eigenvalues constructed in this lattice formalism confirm the known Bethe ansatz equations of the massive Thirring model. The light-cone lattice approach brings a class of integrable fermion models within the general algebraic scheme of the quantum inverse scattering method.

A new approach to string quantization in curved spacetimes

We develop a general scheme for solving the equations of motion and constraints of strings in curved spacetimes, both classically and quantum mechanically. We treat the spacetime geometry exactly and the string excitations small as compared with the energy scales of the metric. [The perturbation (dimensionless) parameter is g=πα′Rc where α′=lPl is the Planck length and Rc the typical curvature radius of the geometry.] This formalism is particularly well suited to properly consider strings in the context of quantum black holes and cosmology. As an illustration we apply it to de Sitter spacetime and find the mass spectrum and vertex operator. The lower mass states are the same as in flat space up to corrections of order g2 whereas heavy states deviate significantly from the linear Regge trajectories. We find a maximum (very large) value of order 1g2 for the quantum number N and spin J of particles. The critical dimension for bosonic strings is found to be 25 in de Sitter spacetime.

Conformal invariance and integrable theories

Abstract Integrable theories on the lattice having zero energy gap exhibit conformal invariance for long distances. It is shown here how to extract its conformal properties (central charge c and scaling dimensions) from the Bethe ansatz equations. The methods here exposed are applied to the six-vertex model and to the critical Potts model.

Light Cone Lattices and the Exact Solution of Chiral Fermion and

A rich set of integrable two-dimensional quantum field theories are obtained from integrable lattice vertex models with q states per bound (q>or=2) in the scaling limit by a generalisation of the light-cone lattice approach. Chiral fermion models with any simple Lie group of symmetry arise in this way (for finite q) as well as bosonic models like the principal chiral model (for q= infinity ). The Hamiltonian, momentum and colour-conserved currents are constructed on the lattice and the bare equations of motion are derived. The renormalised mass spectrum is given explicitly for the set of models considered here. All these integrable vertex models yield conformal invariant theories if one takes the scalding limit in an appropriate different way. It is argued that the values one obtains for the central charges are the same as those provided by the Sugawara construction (in the continuum) for all simple Lie algebras.

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Abstract We quantize a closed bosonic string in a light-cone gauge in Rindler (uniformly accelerated) space-time and apply it to the Schwarzschild-Kruskal manifold. Inertial and accelerated particle states of the string associated to positive frequency modes with respect to the inertial and Rindler times respectively, are defined. There is a stretching effect of the string due to the presence of an event horizon. We explicitly solve the dynamical constraints leaving as physical degrees of freedom only those transverse to the acceleration. Different mass formulae are introduced depending on whether the centre of mass of the string has uniform speed or uniform acceleration. The expectation value of the Rindler (Schwarzschild) number-mode operator in the string around state (tachyon) results equal to a thermal spectrum at the Hawking-Unruh temperature T s = α /2 π (∼ M Pl ( M Pl / M ) 1/( D −3 ), where M is the black hole mass). We find T 0 = M ′/2 π where M ′ is the accelerated ground state string mass and T 0 the temperature T s in dimensionless frequency units. Correlation functions of string coordinates and vertex operators and their Fourier transforms in accelerated time (string response functions) are computed and their thermal properties analyzed.

Models

Abstract The light-cone lattice approach to two-dimensional quantum field theories is generalized to a large class of vertex models with any number of possible states per link and any simple Lie group of symmetry. Starting from a given lattice model, different scaling limits are defined leading to conformal field theories or to massive integrable quantum field theories, for which the lattice hamiltonian, momentum and currents are constructed. For a large set of models, the complete mass spectrum is also exhibited. Our approach applies equally well to chiral fermionic theories (like the chiral Gross-Neveu) and to bosonic models like the principal chiral model.

String Quantization in Accelerated Frames and Black Holes

The method presented permits us to systematically compute physical magnitudes for large but finite size in theories solvable by the nested Bethe ansatz. The value of the central charge c and the surface tension are explicitly calculable in this way. For the fundamental vertex and spin models associated to simply laced Lie algebras, c turns out to be equal to the rank of the algebra.

Integrable quantum field theories and conformal field theories from lattice models in the light-cone approach

It is shown that local gauge transformations preserve the integrability of one-dimensional quantum Heisenberg chains. Abelian U(1) gauge transformations associated to z-rotations appear in the XXZ model which is worked out in detail. The exact energy spectrum derived by the Bethe ansatz turns out to be gauge-invariant whereas the eigenvectors are explicitly gauge-dependent. Isotropic XXX chains exhibit SU(2) ⊗ Z2 gauge invariance properties and anisotropic XYZ chains possess discrete Z2 ⊗ Z2 gauge invariance.

Finite-size corrections for nested Bethe ansatz models and conformal invariance

New solutions of self-dual Yang-Mills (SDYM) equations are constructed in Minkowski space-time for the gauge groupSL(2, ℂ). After proposing a Lorentz covariant formulation of Yangs equations, a set of Ansätze for exact non-linear multiplane wave solutions are proposed. The gauge fields are rational functions ofex·ki(Ki2=0, 1≦i≦N) for these Ansätze. At least, three families of multisoliton type solutions are derived explicitly. Their asymptotic behaviour shows that non-linear waves scatter non-trivially in Minkowski SDYM.