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Essentials of Lattice Gauge Theories (LGT)

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Strings to Strings

Part of the book series: Lecture Notes in Physics ((LNP,volume 1018))

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Abstract

This chapter gives an introduction to Lattice Gauge Theories (LGT). After an introduction into the history of the subject beginning with Wegner’s \(Z_2\) LGT, the three pillars of Euclidean LGT’s, namely, (i) the path-integral formulation of QFT’s, (ii) Euclideanization and (iii) discretization, are very carefully discussed. The exact mapping between QFT’s in D space-time dimensions and classical equilibrium statistical mechanics in D spatial dimensions is explained along with the central role this connection plays in the numerical simulations of LGT’s. The necessity of taking the continuum limit of LGT’s resulting in the concept of the statistical continuum limit is carefully explained and an explicit analytical example is constructed in \(D=1\) to illustrate its nuances. The importance of phase transitions with diverging correlation functions is also thoroughly discussed. The chapter then elaborates the issues involved in discretizing gauge theories, both abelian and non-abelian. A description of Wilson’s LGT is presented. After discussing gauge-invariant objects in continuum gauge theories, the corresponding invariants on the lattice like the Wegner-Wilson loops and the Polyakov Loops (Lines) are constructed and their uses explained. The Euclidean path-integrals for LGT’s with plaquette actions are constructed with the help of the group theoretic Haar measures.

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  1. Institute of Mathematical Sciences (Retired), Chennai, India

    N. D. Hari Dass

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Hari Dass, N.D. (2023). Essentials of Lattice Gauge Theories (LGT). In: Strings to Strings. Lecture Notes in Physics, vol 1018. Springer, Cham. https://doi.org/10.1007/978-3-031-35358-1_20

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