Following Thijssen Ch. 13 we discuss very briefly the physics of lattice gauge theories and aspects of computational algorithms on lattices.

1. Algorithms for lattice field theories
   • QMC
     • Metropolis
     • Heat Bath
     • Acceleration
       Closely related to Poison problem
       Different algorithms can change power by which methods scale
       Gauss-Seidel, SOR, ... (Alder's work Ref. 269)
   • Molecular Dynamics
     • How to use classical MD on a quantum lattice problem?
     • Just like Car-Parrinello! (Car started in high energy physics.)
   • Critical slowing down
     • If one make only local updates, the algorithm should scale as time scale tau ~ (length)^z
     • Correlation length diverges near transition Always problem in lattice field theory if one wants scale to approach the continuum
     • Cluster algorithms
       • Swendsen-Wang cluster update algorithm
         Improves z from 2.125 (2 + 1/8) to 0.35 in 2d Ising model
         Still satisfies detailed balance
       • Wolfe single cluster variation
2. QED on a lattice
   • QED Action (13.7.1)
     Gauge invariance (but must fix gauge in integrals over fields so integrals converge)
   • Coupling to Fermions
     Action (13.115) leads to Dirac Eq. 13.112)
     Grassman variables for non-commuting fields
   • Action on a lattice - plaquets
   • Wilson loops - like Feynman diagrams
   • Leads to "extra" solution - confined phase - not correct
3. QCD
   • QCD Action (13.174) analogous to QED (13.115)
     Except the the commuting phase factors of the U(1) symmetry of QED is replaced by the non-commuting SU(2) or SU(3) gauge theories for weak or strong interactions
     Quarks, gluons, ...
     Key point: Gluons interact unlike photons
   • Simulations must sample over matrices (like in the nuclear physics case described by Pandharipande)