Hamiltonian lattice field theory: Computer calculations using variational methods

Technical Report · Tue Dec 03 04:00:00 EST 1991

DOI:https://doi.org/10.2172/5736347· OSTI ID:5736347

Zako, R L

I develop a variational method for systematic numerical computation of physical quantities -- bound state energies and scattering amplitudes -- in quantum field theory. An infinite-volume, continuum theory is approximated by a theory on a finite spatial lattice, which is amenable to numerical computation. I present an algorithm for computing approximate energy eigenvalues and eigenstates in the lattice theory and for bounding the resulting errors. I also show how to select basis states and choose variational parameters in order to minimize errors. The algorithm is based on the Rayleigh-Ritz principle and Kato's generalizations of Temple's formula. The algorithm could be adapted to systems such as atoms and molecules. I show how to compute Green's functions from energy eigenvalues and eigenstates in the lattice theory, and relate these to physical (renormalized) coupling constants, bound state energies and Green's functions. Thus one can compute approximate physical quantities in a lattice theory that approximates a quantum field theory with specified physical coupling constants. I discuss the errors in both approximations. In principle, the errors can be made arbitrarily small by increasing the size of the lattice, decreasing the lattice spacing and computing sufficiently long. Unfortunately, I do not understand the infinite-volume and continuum limits well enough to quantify errors due to the lattice approximation. Thus the method is currently incomplete. I apply the method to real scalar field theories using a Fock basis of free particle states. All needed quantities can be calculated efficiently with this basis. The generalization to more complicated theories is straightforward. I describe a computer implementation of the method and present numerical results for simple quantum mechanical systems.

• Research Organization: Lawrence Berkeley Lab., CA (United States)
• Sponsoring Organization: DOE; USDOE, Washington, DC (United States)
• DOE Contract Number: AC03-76SF00098
• OSTI ID: 5736347
• Report Number(s): LBL-31589; UCB-PTH--91/69; ON: DE92010479
• Country of Publication: United States
• Language: English

Related Subjects

661100 -- Classical & Quantum Mechanics-- (1992-)
662110* -- General Theory of Particles & Fields-- Theory of Fields & Strings-- (1992-)
71 CLASSICAL AND QUANTUM MECHANICS
GENERAL PHYSICS
72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS
COMPUTER CALCULATIONS
CONVERGENCE
ERRORS
FIELD THEORIES
FUNCTIONS
GREEN FUNCTION
HAMILTONIANS
LATTICE FIELD THEORY
MATHEMATICAL OPERATORS
OPTIMIZATION
QUANTUM FIELD THEORY
QUANTUM OPERATORS
RENORMALIZATION
VARIATIONAL METHODS