Toda lattice field theories, discrete W algebras, Toda lattice hierarchies and quantum groups

In analogy with the Liouville case we study the $sl_3$ Toda theory on the lattice and define the relevant quadratic algebra and out of it we recover the discrete $W_3$ algebra. We define an integrable system with respect to the latter and establish the relation with the Toda lattice hierarchy. We compute the the relevant continuum limits. Finally we find the quantum version of the quadratic algebra.Comment: 12 pages, LaTe

Long-Time Asymptotics for Solutions of the NLS Equation with a Delta Potential and Even Initial Data

We consider the one-dimensional focusing nonlinear Schr\"odinger equation (NLS) with a delta potential and even initial data. The problem is equivalent to the solution of the initial/boundary problem for NLS on a half-line with Robin boundary conditions at the origin. We follow the method of Bikbaev and Tarasov which utilizes a B\"acklund transformation to extend the solution on the half-line to a solution of the NLS equation on the whole line. We study the asymptotic stability of the stationary 1-soliton solution of the equation under perturbation by applying the nonlinear steepest-descent method for Riemann-Hilbert problems introduced by Deift and Zhou. Our work strengthens, and extends, earlier work on the problem by Holmer and Zworski

The Hamiltonian formulation of General Relativity: myths and reality

A conventional wisdom often perpetuated in the literature states that: (i) a 3+1 decomposition of space-time into space and time is synonymous with the canonical treatment and this decomposition is essential for any Hamiltonian formulation of General Relativity (GR); (ii) the canonical treatment unavoidably breaks the symmetry between space and time in GR and the resulting algebra of constraints is not the algebra of four-dimensional diffeomorphism; (iii) according to some authors this algebra allows one to derive only spatial diffeomorphism or, according to others, a specific field-dependent and non-covariant four-dimensional diffeomorphism; (iv) the analyses of Dirac [Proc. Roy. Soc. A 246 (1958) 333] and of ADM [Arnowitt, Deser and Misner, in "Gravitation: An Introduction to Current Research" (1962) 227] of the canonical structure of GR are equivalent. We provide some general reasons why these statements should be questioned. Points (i-iii) have been shown to be incorrect in [Kiriushcheva et al., Phys. Lett. A 372 (2008) 5101] and now we thoroughly re-examine all steps of the Dirac Hamiltonian formulation of GR. We show that points (i-iii) above cannot be attributed to the Dirac Hamiltonian formulation of GR. We also demonstrate that ADM and Dirac formulations are related by a transformation of phase-space variables from the metric $g_{\mu\nu}$ to lapse and shift functions and the three-metric $g_{km}$, which is not canonical. This proves that point (iv) is incorrect. Points (i-iii) are mere consequences of using a non-canonical change of variables and are not an intrinsic property of either the Hamilton-Dirac approach to constrained systems or Einstein's theory itself.Comment: References are added and updated, Introduction is extended, Subsection 3.5 is added, 83 pages; corresponds to the published versio

The Hamiltonian of Einstein affine-metric formulation of General Relativity

It is shown that the Hamiltonian of the Einstein affine-metric (first order) formulation of General Relativity (GR) leads to a constraint structure that allows the restoration of its unique gauge invariance, four-diffeomorphism, without the need of any field dependent redefinition of gauge parameters as is the case for the second order formulation. In the second order formulation of ADM gravity the need for such a redefinition is the result of the non-canonical change of variables [arXiv: 0809.0097]. For the first order formulation, the necessity of such a redefinition "to correspond to diffeomorphism invariance" (reported by Ghalati [arXiv: 0901.3344]) is just an artifact of using the Henneaux-Teitelboim-Zanelli ansatz [Nucl. Phys. B 332 (1990) 169], which is sensitive to the choice of linear combination of tertiary constraints. This ansatz cannot be used as an algorithm for finding a gauge invariance, which is a unique property of a physical system, and it should not be affected by different choices of linear combinations of non-primary first class constraints. The algorithm of Castellani [Ann. Phys. 143 (1982) 357] is free from such a deficiency and it leads directly to four-diffeomorphism invariance for first, as well as for second order Hamiltonian formulations of GR. The distinct role of primary first class constraints, the effect of considering different linear combinations of constraints, the canonical transformations of phase-space variables, and their interplay are discussed in some detail for Hamiltonians of the second and first order formulations of metric GR. The first order formulation of Einstein-Cartan theory, which is the classical background of Loop Quantum Gravity, is also discussed.Comment: 74 page

Elementary Excitations in Dimerized and Frustrated Heisenberg Chains

We present a detailed numerical analysis of the low energy excitation spectrum of a frustrated and dimerized spin $S=1/2$ Heisenberg chain. In particular, we show that in the commensurate spin--Peierls phase the ratio of the singlet and triplet excitation gap is a universal function which depends on the frustration parameter only. We identify the conditions for which a second elementary triplet branch in the excitation spectrum splits from the continuum. We compare our results with predictions from the continuum limit field theory . We discuss the relevance of our data in connection with recent experiments on $CuGeO_{3}$, $NaV_2O_5$, and $(VO)_2P_2O_7$.Comment: Corrections to the text + 1 new figure, will appear in PRB (august 98

Automatic regularization by quantization in reducible representations of CCR: Point-form quantum optics with classical sources

Electromagnetic fields are quantized in manifestly covariant way by means of a class of reducible representations of CCR. $A_a(x)$ transforms as a Hermitian four-vector field in Minkowski four-position space (no change of gauge), but in momentum space it splits into spin-1 massless photons (optics) and two massless scalars (similar to dark matter). Unitary dynamics is given by point-form interaction picture, with minimal-coupling Hamiltonian constructed from fields that are free on the null-cone boundary of the Milne universe. SL(2,C) transformations and dynamics are represented unitarily in positive-norm Hilbert space describing $N$ four-dimensional oscillators. Vacuum is a Bose-Einstein condensate of the $N$-oscillator gas. Both the form of $A_a(x)$ and its transformation properties are determined by an analogue of the twistor equation. The same equation guarantees that the subspace of vacuum states is, as a whole, Poincar\'e invariant. The formalism is tested on quantum fields produced by pointlike classical sources. Photon statistics is well defined even for pointlike charges, with UV/IR regularizations occurring automatically as a consequence of the formalism. The probabilities are not Poissonian but of a R\'enyi type with $\alpha=1-1/N$. The average number of photons occurring in Bremsstrahlung splits into two parts: The one due to acceleration, and the one that remains nonzero even if motion is inertial. Classical Maxwell electrodynamics is reconstructed from coherent-state averaged solutions of Heisenberg equations. Static pointlike charges polarize vacuum and produce effective charge densities and fields whose form is sensitive to both the choice of representation of CCR and the corresponding vacuum state.Comment: 2 eps figures; in v2 notation in Eq. (39) and above Eq. (38) is correcte