Ultraviolet cascade in the thermalization of the classical phi^4 theory in 3+1 dimensions

We investigate the dynamics of thermalization and the approach to equilibrium in the classical phi^4 theory in 3+1 spacetime dimensions. The non-equilibrium dynamics is studied by numerically solving the equations of motion in a light- cone-like discretization of the model for a broad range of initial conditions and energy densities.A smooth cascade of energy towards the ultraviolet is found to be the basic mechanism of thermalization.After an initial transient stage,at a time scale of several hundreds inverse masses,the squared of the field gradient becomes larger than the nonlinear term and a stage of universal cascade emerges. As the cascade progresses, the modes with higher wavenumbers exhibit weaker and weaker nonlinearities well described by the Hartree approximation while the infrared modes retain strong selfinteractions. Two timescales for equilibration appears.For k^2>(t) we observe an effective thermalization with a time scale in the thousands of inverse masses and the Hartree approximation holds. By effective thermalization we mean that the observable acquires the equilibrium functional form with an effective time dependent temperature Teff, which slowly decreases with time. Infrared modes with k^2 (t) equilibrate only by time scales in the millions of inverse masses. Infrared modes with k^2 (t) equilibrate only by time scales in the millions.Virialization and the equation of state start to set much earlier than effective thermalization.The applicability of these results in quantum field theory for large occupation numbers and small coupling is analyzed.Comment: 47 pages, 31 figures. Presentation improved, 4 new figure

Yang--Baxter symmetry in integrable models: new light from the Bethe Ansatz solution

We show how any integrable 2D QFT enjoys the existence of infinitely many non--abelian {\it conserved} charges satisfying a Yang--Baxter symmetry algebra. These charges are generated by quantum monodromy operators and provide a representation of $q-$deformed affine Lie algebras. We review and generalize the work of de Vega, Eichenherr and Maillet on the bootstrap construction of the quantum monodromy operators to the sine--Gordon (or massive Thirring) model, where such operators do not possess a classical analogue. Within the light--cone approach to the mT model, we explicitly compute the eigenvalues of the six--vertex alternating transfer matrix \tau(\l) on a generic physical state, through algebraic Bethe ansatz. In the thermodynamic limit \tau(\l) turns out to be a two--valued periodic function. One determination generates the local abelian charges, including energy and momentum, while the other yields the abelian subalgebra of the (non--local) YB algebra. In particular, the bootstrap results coincide with the ratio between the two determinations of the lattice transfer matrix.Comment: 30 page

String dynamics in cosmological and black hole backgrounds: The null string expansion

We study the classical dynamics of a bosonic string in the $D$--dimensional flat Friedmann--Robertson--Walker and Schwarzschild backgrounds. We make a perturbative development in the string coordinates around a {\it null} string configuration; the background geometry is taken into account exactly. In the cosmological case we uncouple and solve the first order fluctuations; the string time evolution with the conformal gauge world-sheet $\tau$--coordinate is given by $X^0(\sigma, \tau)=q(\sigma)\tau^{1\over1+2\beta}+c^2B^0(\sigma, \tau)+\cdots$, $B^0(\sigma,\tau)=\sum_k b_k(\sigma)\tau^k$ where $b_k(\sigma)$ are given by Eqs.\ (3.15), and $\beta$ is the exponent of the conformal factor in the Friedmann--Robertson--Walker metric, i.e. $R\sim\eta^\beta$. The string proper size, at first order in the fluctuations, grows like the conformal factor $R(\eta)$ and the string energy--momentum tensor corresponds to that of a null fluid. For a string in the black hole background, we study the planar case, but keep the dimensionality of the spacetime $D$ generic. In the null string expansion, the radial, azimuthal, and time coordinates $(r,\phi,t)$ are $r=\sum_n A^1_{n}(\sigma)(-\tau)^{2n/(D+1)}~,$ $\phi=\sum_n A^3_{n}(\sigma)(-\tau)^{(D-5+2n)/(D+1)}~,$ and $t=\sum_n A^0_{n} (\sigma)(-\tau)^{1+2n(D-3)/(D+1)}~.$ The first terms of the series represent a {\it generic} approach to the Schwarzschild singularity at $r=0$. First and higher order string perturbations contribute with higher powers of $\tau$. The integrated string energy-momentum tensor corresponds to that of a null fluid in $D-1$ dimensions. As the string approaches the $r=0$ singularity its proper size grows indefinitely like $\sim(-\tau)^{-(D-3)/(D+1)}$. We end the paper giving three particular exact string solutions inside the black hole.Comment: 17 pages, REVTEX, no figure