Quantum field theories are used to describe interactions between fundamental particles. In determining the properties of the bound states that these particles can form, the use of light-cone coordinates, with t+z/c playing the role of time, can be advantageous. The state of the system can then be expanded in a basis of momentum eigenstates, with wave functions as the coefficients in the expansion. The wave functions satisfy a coupled system of integral equations that almost always require numerical techniques for their solution. Within the integrals there are infinities that must be regulated in some way in order to properly define the given theory.
In this project, two methods are considered: Pauli–Villars regularization, which requires the introduction of unphysical massive particles, and supersymmetry. These methods have been applied to various field theories, in particular Yukawa theory, quantum electrodynamics (QED), super Yang-Mills (SYM) theory, and phi4 theory, and continue to be explored, with the ultimate goal of applying them to quantum chromodynamics (QCD), the theory of the strong interactions that determine the properties of mesons and baryons. The work on Pauli-Villars regularization has included development of a mechanism to extend this technique to non-Abelian gauge theories, such as QCD. For QED, this approach has already yielded a direct check on the gauge independence of a nonperturbative calculation of the electron's anomalous magnetic moment. Other recent progress has been in the development and application of the light-front coupled-cluster (LFCC) method, which applies the mathematics of the nonrelativistic coupled-cluster method to the light-front Hamiltonian problem in a way that avoids truncations in particle number. The LFCC method allows calculations to avoid difficulties with uncanceled divergences in the integral equations. A study of the convergence of the LFCC method in the context of quenched scalar Yukawa theory was recently completed. In phi4 theory, basis function methods, utilizing newly developed symmetric multivariate polynomials, have been applied to the calculation of massive eigenstates and estimation of the critical coupling; work on improved estimates and better understanding of the massless vacuum state has made significant progress, particularly with respect to a method for including and regulating vacuum transitions in light-front quantization. This new method has already reproduced some key results previously unobtainable in light-front coordinates. An extension of earlier work on the Casimir effect in light-front coordinates has been published as part of a broader review of the topic. The connections between the LFCC method and possible field-theoretic forms of density functional theory (DFT) are being explored. Recent calculations of the effective potential between static sources in both scalar and ordinary Yukawa theory yield exact eigenstates within a quenched approximation and will provide a proving ground for a DFT-like method; it is also the starting point for work on effective potentials in unquenched Yukawa theory, to include the effects of pair production and annihilation, and then to consider QED and QCD where the exchanged bosons are vectors rather than scalars.