Monte Carlo Renormalization Group: A Review

Abstract

The logic and the methods of Monte Carlo Renormalization Group (MCRG) are reviewed. A status report of results for 4-dimensional lattice gauge theories derived using MCRG is presented. Existing methods for calculating the improved action are reviewed and evaluated. The Gupta-Cordery improved MCRG method is described and compared with the standard one.The development of Monte Carlo Renormalization group method (MCRG) was essentially complete in 1979 with the work of Wilson1, Swendsen2 and Shenker and Tobochnik3. Prior to this Ma4 and Kadanoff5 had provided key ingredients. The method is therefore relatively new, furthermore its application to field theories has been carried out only since 1982. In this short period there has been considerable activity and I shall review the methodology and summarize the status with emphasis on 4-dimensional gauge theories. There already exists extensive literature on MCRG and I direct the reader to it1,3,6,7 for details and for a wider exposure. Similarly, the reviews8,9 are a good starting point for background on Lattice Gauge Theories and on spin systems. The topics I shall cover are 1) Introduction to MCRG and its methodology. 2) Renormalization Group Transformations fo d = 4 lattice gauge theories. 3) U(1) Lattice Gauge theory. 4) β-function and Scaling for SU(3) Lattice Gauge Theory. 5) Improved Actions and Methods to calculate them. 6) Improved Monte Carlo Renormalization Group. 7) Effective Field Theories. The main results in QCD from MCRG are the determination of the β-function and the consequent prediction for the value of the coupling at which asymptotic scaling sets in and second an estimate of the improved gauge action10. These results are not spectacular in the sense of confirming that QCD is the correct theory of strong interactions, however they have led to a deeper understanding of the lattice theory and provided a quantitative estimate of the approach to the continuum limit. I shall attempt to show that his method is as yet in its infancy and should be used to tackle a number of problems.KeywordsRenormalization GroupWilson LoopIsing ModelTruncation ErrorEffective Field TheoryThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.