Abstract
A formally exact discrete multi-resolution representation of quantum field theory on a light front is presented. The formulation uses an orthonormal basis of compactly supported wavelets to expand the fields restricted to a light front. The representation has a number of useful properties. First, light front preserving Poincar\'e transformations can be computed by transforming the arguments of the basis functions. The discrete field operators, which are defined by integrating the product of the field and the basis functions over the light front, represent localized degrees of freedom on the light-front hyperplane. These discrete fields are irreducible and the vacuum is formally trivial. The light-front Hamiltonian and all of the Poincar\'e generators are linear combinations of normal ordered products of the discrete field operators with analytically computable constant coefficients. The representation is discrete and has natural resolution and volume truncations like lattice formulations. Because it is formally exact it is possible to systematically compute corrections for eliminated degrees of freedom.
Highlights
A discrete multiresolution representation of quantum field theory on a light front is presented
The purpose of this work is to investigate a representation of quantum field theory that has some of the advantages of both approaches, this initial work is limited to canonical field theory rather than gauge theories
In 1939, Wigner [1] showed that the independence of quantum observables in different inertial reference frames related by Lorentz transformations and space-time translations requires the existence of a dynamical unitary representation of Poincaregroup on the Hilbert space of the quantum theory
Summary
A discrete multiresolution representation of quantum field theory on a light front is presented. In [35], flow equation methods are used to block diagonalize the Hilbert space of a truncated free-field theory by resolution, constructing an effective Hamiltonian that involves only coarse-scale degrees of freedom, but includes the dynamics of the eliminated degrees of freedom This calculation provided some insight into the complementary roles played by volume and resolution truncations. Variational methods could be employed for low-lying composite states Another method that takes advantage of the discrete nature of the wavelet representation is to use the light-front Heisenberg equations to generate an expansion of the field as a linear combination of products of fields restricted to the light front. A detailed study of the scaling properties could help to formulate efficient approximations to the solution of the light-front Heisenberg field equations by eliminating irrelevant degrees of freedom Another potential use of the wavelet representation would be in quantum computing.
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