1,983 research outputs found
Massive-Conformal Dictionary
The finite-volume spectrum of an integrable massive perturbation of a
rational conformal field theory interpolates between massive multi-particle
states in infinite volume (IR limit) and conformal states, which are approached
at zero volume (UV limit). Each state is labeled in the IR by a set of `Bethe
Ansatz quantum numbers', while in the UV limit it is characterized primarily by
the conformal dimensions of the conformal field creating it. We present
explicit conjectures for the UV conformal dimensions corresponding to any IR
state in the -perturbed minimal models and . The
conjectures, which are based on a combinatorial interpretation of the
Rogers-Ramanujan-Schur identities, are consistent with numerical results
obtained previously for low-lying energy levels.Comment: 18/11 pages in harvmac, Tel-Aviv preprint TAUP 2109/9
Spectral methods in time for hyperbolic equations
A pseudospectral numerical scheme for solving linear, periodic, hyperbolic problems is described. It has infinite accuracy both in time and in space. The high accuracy in time is achieved without increasing the computational work and memory space which is needed for a regular, one step explicit scheme. The algorithm is shown to be optimal in the sense that among all the explicit algorithms of a certain class it requires the least amount of work to achieve a certain given resolution. The class of algorithms referred to consists of all explicit schemes which may be represented as a polynomial in the spatial operator
High degree interpolation polynomial in Newton form
Polynomial interpolation is an essential subject in numerical analysis. Dealing with a real interval, it is well known that even if f(x) is an analytic function, interpolating at equally spaced points can diverge. On the other hand, interpolating at the zeroes of the corresponding Chebyshev polynomial will converge. Using the Newton formula, this result of convergence is true only on the theoretical level. It is shown that the algorithm which computes the divided differences is numerically stable only if: (1) the interpolating points are arranged in a different order, and (2) the size of the interval is 4
Polynomial approximation of functions of matrices and its application to the solution of a general system of linear equations
During the process of solving a mathematical model numerically, there is often a need to operate on a vector v by an operator which can be expressed as f(A) while A is NxN matrix (ex: exp(A), sin(A), A sup -1). Except for very simple matrices, it is impractical to construct the matrix f(A) explicitly. Usually an approximation to it is used. In the present research, an algorithm is developed which uses a polynomial approximation to f(A). It is reduced to a problem of approximating f(z) by a polynomial in z while z belongs to the domain D in the complex plane which includes all the eigenvalues of A. This problem of approximation is approached by interpolating the function f(z) in a certain set of points which is known to have some maximal properties. The approximation thus achieved is almost best. Implementing the algorithm to some practical problem is described. Since a solution to a linear system Ax = b is x= A sup -1 b, an iterative solution to it can be regarded as a polynomial approximation to f(A) = A sup -1. Implementing the algorithm in this case is also described
The eigenvalues of the pseudospectral Fourier approximation to the operator sin (2x) d/dx
It is shown that the eigenvalues Z sub i of the pseudospectral Fourier approximation to the operator sin(2x) curly d/curly dx satisfy (R sub e) (Z sub i) = + or - 1 or (R sub e)(Z sub I) = 0. Whereas this does not prove stability for the Fourier method, applied to the hyperbolic equation U sub t = sin (2x)(U sub x) - pi x pi; it indicates that the growth in time of the numerical solution is essentially the same as that of the solution to the differential equation
Sine-Gordon =/= Massive Thirring, and Related Heresies
By viewing the Sine-Gordon and massive Thirring models as perturbed conformal
field theories one sees that they are different (the difference being
observable, for instance, in finite-volume energy levels). The UV limit of the
former (SGM) is a gaussian model, that of the latter (MTM) a so-called {\it
fermionic} gaussian model, the compactification radius of the boson underlying
both theories depending on the SG/MT coupling. (These two families of conformal
field theories are related by a ``twist''.) Corresponding SG and MT models
contain a subset of fields with identical correlation functions, but each model
also has fields the other one does not, e.g. the fermion fields of MTM are not
contained in SGM, and the {\it bosonic} soliton fields of SGM are not in MTM.
Our results imply, in particular, that the SGM at the so-called ``free-Dirac
point'' is actually a theory of two interacting bosons with
diagonal S-matrix , and that for arbitrary couplings the overall sign of
the accepted SG S-matrix in the soliton sector should be reversed. More
generally, we draw attention to the existence of new classes of quantum field
theories, analogs of the (perturbed) fermionic gaussian models, whose partition
functions are invariant only under a subgroup of the modular group. One such
class comprises ``fermionic versions'' of the Virasoro minimal models.Comment: 50 pages (harvmac unreduced), CLNS-92/1149, ITP-SB-92-3
- …