Quantum Chromodynamics

One research area that makes use of matrix functions is the field of lattice quantum chromodynamics (QCD). QCD computations in chemistry and physics currently occupy large amounts of supercomputer time, with a single simulation taking of the order of days, and computing requirements in the subject are estimated to be of the order of 10 teraflops by 2003. Underlying many of these computations are matrix function evaluations. Much recent work has focused on QCD simulations with the overlap-Dirac operator, which has the form

D = 1/2 ( 1+ gamma5 epsilon (Homega) ),

where gamma5 = diag(plusminus 1), Homega is the Hermitian Wilson-Dirac operator, and epsilon(A) = A(A*A)-1/2 is the unitary polar factor of A (referred to by physicists as the ``step function''). Calculations of quark propagators require the computation of matrix-vector products involving D, so the key computational requirement is to form epsilon (Homega)b efficiently, for many b. Since Homega is large and sparse, this must be done without computing epsilon (Homega) explicitly. Mathematically, the problem is to compute x = f(A)b without explicitly forming f(A), with f(A) = A(A*A)-1/2 and A Hermitian. As physicists refine their models we can expect other f to arise.


  • HEP-LAT(High Energy Physics LATtice gauge theory) preprint archive server.
  • QCD Matrices
  • Stanford Linear Accelerator Center (SLAC), Stanford Public Information REtrieval System (SPIRES). U.K. mirror
  • UKQCD Collaboration
  • The Worldwide Web Virtual Library: High-Energy Physics