Thijssen Ch. 13 discusses gauge theories and computational algorithms on lattices. Many of the examples an discussion are closely related to the Rev Mod Phys Paper of Kogut. WE will go into more detail on the "Ising model in a transverse field" (see Kogut, work by Fradkin, and thesis of Roomany, UIUC, 1980) that illustrates physical conclusions on phase transitions.

1. Field theories in 4 dimensional space time
   • Euclidean vs Minkowski metric
     • "Systems live in Minkowski space" with real time
     • But in imaginary time tau, space-time is Euclidean
     • Two uses of studies in Euclidean space:
       Imaginary time is real temperature - i.e. the mapping to statistical mechanics which has real consequences
       Trick to study Hamiltonian even when one is not interested in thermal averages
   • Correspondence of Stat. Mech and Field Theory:

     Correspondence (following Kogut p. 668

     Stat Mech	Field Theory
     Free Energy Density	Vacuum Energy Density
     Correlation Function	Propagator
     Inverse Correlation length	Mass Gap

2. The 1d Ising model in a transverse field
   • Maps onto model for quantum double-well tunneling model for ferroelectrics
   • Hamiltonian for interacting quantum Ising spins
   • Perturbation theory analysis - shows the key points of ground state energy and correlation functions
   • Phase transition at critical coupling constant
   • Duality of Ising model shows analytically where transition is expected
   • A gap vanishes at this point (gap between ground state and lowest excited state.)
3. Solution by Lanczos methods
   • Solution from Roomany for finite systems
   • Finite size scaling and the phase transition at infinite size
4. References
   • J. Kogut, Rev. Mod. Phys. 51, 651 (1979).
   • H. Roomany, H. W. Wyld, and L. Holloway, Phys. Rev. D21, 1557 (1980), and thesis, U of I, 1980.