Nathan Salwen

The diagonalization of quantum field Hamiltonians

Abstract We introduce a new diagonalization method called quasi-sparse eigenvector diagonalization which finds the most important basis vectors of the low energy eigenstates of a quantum Hamiltonian. It can operate using any basis, either orthogonal or non-orthogonal, and any sparse Hamiltonian, either Hermitian, non-Hermitian, finite-dimensional, or infinite-dimensional. The method is part of a new computational approach which combines both diagonalization and Monte Carlo techniques.

Introduction to stochastic error correction methods

We propose a method for eliminating the truncation error associated with any subspace diagonalization calculation. The new method, called stochastic error correction, uses Monte Carlo sampling to compute the contribution of the remaining basis vectors not included in the initial diagonalization. The method is part of a new approach to computational quantum physics which combines both diagonalization and Monte Carlo techniques.

A Canonical Quantization of the Baker's Map

Abstract We present here a canonical quantization for the bakers map. The method we use is quite different from that used in Balazs and Voros and Saraceno. We first construct a natural “baker covering map” on the plane R 2. We then use as the quantum algebra of observables the subalgebra of operators onL2( R ) generated by {exp(2πix), exp(2πip)}. We construct a unitary propagator such that as ħ→0 the classical dynamics is returned. For Plancks constanth=1/N, we show that the dynamics can be reduced to the dynamics on anN-dimensional Hilbert space, and the unitaryN×Nmatrix propagator is the same as given by Balazs and Voros, except for a small correction of orderh. This correction is shown to preserve the classical symmetryx→1−xandp→1−pin the quantum dynamics for periodic boundary conditions.

Classical limits for quantum maps on the torus

We provide a rigorous canonical quantization for the following toral automorphisms: cat maps, generalized kicked maps, and generalized Harper maps. For each of these systems we construct a unitary propagator and prove the existence of a well-defined classical limit. We also provide an alternative derivation of Hannay and Berry results for the cat map propagator on the plane.

RENORMALIZATION IN SPHERICAL FIELD THEORY

Abstract We derive several results concerning non-perturbative renormalization in the spherical field formalism. Using a small set of local counterterms, we are able to remove all ultraviolet divergences in a manner such that the renormalized theory is finite and translationally invariant. As an explicit example we consider massless φ 4 theory in four dimensions.

EQSE diagonalization of the Hubbard model

Abstract The application of enhanced quasi-sparse eigenvector methods (EQSE) to the Hubbard model is attempted. The ground state energy for the 4 × 4 Hubbard model is calculated with a relatively small set of basis vectors. The results agree to high precision with the exact answer. For the 8 × 8 case, exact answers are not available but a simple first order correction to the quasi-sparse eigenvector (QSE) result is presented.

Modal expansions and nonperturbative quantum field theory in Minkowski space

We introduce a spectral approach to non-perturbative field theory within the periodic field formalism. As an example we calculate the real and imaginary parts of the propagator in 1+1 dimensional phi^4 theory, identifying both one-particle and multi-particle contributions. We discuss the computational limits of existing diagonalization algorithms and suggest new quasi-sparse eigenvector methods to handle very large Fock spaces and higher dimensional field theories.

The massless Thirring model in spherical field theory

Abstract We use the massless Thirring model to demonstrate a new approach to non-perturbative fermion calculations based on the spherical field formalism. The methods we present are free from the problems of fermion doubling and difficulties associated with integrating out massless fermions. Using a non-perturbative regularization, we compute the two-point correlator and find agreement with the known analytic solution.