John R. Hiller, Associate Fellow

Nonperturbative Analysis of Field Theories Quantized on the Light Cone

  These researchers continued the development of nonperturbative techniques for the analysis of quantum field theories in the context of Yukawa theory and supersymmetric Yang-Mills theories. Light-cone coordinates, for which ct + z plays the role of time, are used to allow separation of internal and external motions in relativistic systems. Successful techniques will eventually be applied to bound-state problems for the strong interaction, in order to understand the structure of protons, neutrons, and other hadrons.

  The field-theoretic bound-state problem reduces to a set of coupled integral equations for wavefunctions (of momentum) that describe states with a fixed number of particles. The complete eigenstate is an expansion in a set of states with different numbers of particles. The wavefunctions are the coefficients of this expansion and specify the probability distributions for quantum numbers such as spin and momentum. The eigenvalue of the coupled system is the square of the mass of the eigenstate.

  The bound-state problem is not automatically well defined and is usually plagued by infinities. Two approaches were studied to handle this. One was the use of Pauli-Villars regularization, which is accomplished by adding heavy particles to the basis and adding interaction terms to the Hamiltonian. The norm of the basis state is arranged to produce cancellations of the divergent contributions from the original interactions. Thus the masses of the Pauli-Villars particles act as regulators of the theory. Bare parameters are obtained as functions of these masses by imposing physical renormalization conditions. With the fit to physical conditions established, the regulators can be removed in an infinite-mass limit.

  The other approach taken by the researchers was to consider supersymmetric theories in which infinities typically do cancel automatically. Because the symmetries involved do not appear in nature, they must be broken in some way that retains the beneficial aspects. A theory with broken supersymmetry is analogous to a Pauli-Villars regulated theory, with heavy (unobserved) particles providing the regulation.

  The numerical treatment of the bound-state integral equations is based on discretization of the momentum integrals and Lanczos diagonalization of the resulting matrix problem. The matrices are very sparse, which makes the Lanczos algorithm an ideal choice. In the case of Pauli-Villars regularization, the matrix problem has an indefinite metric, with the negatively-normed particles providing the needed cancellations. An efficient variant of the bi-orthogonal Lanczos algorithm was specifically tailored to this situation.

  The development of nonperturbative techniques for the analysis of quantum field theories continued in the context of models useful for the understanding of Pauli-Villars regularization, scattering processes, and zero modes. Yukawa theory was investigated in a two-fermion truncation, and a single-electron trancation of quantum electrodynamics was explored.

  Renormalization conditions include constraints on scattering processes. Some effort was made towards collaborating in an exploration of supersymmetric theories with essentially the same numerical techniques. These are of interest because a supersymmetric form of Quantum Chromodynamics may turn out to be the best avenue of attack on the strong interactions.