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Global Symmetry, Local Symmetry, and the Lattice

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An Introduction to the Confinement Problem

Part of the book series: Lecture Notes in Physics ((LNP,volume 972))

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Abstract

Introducing the basics: gauge invariance and the lattice formulation, the Monte Carlo method, and the impossibility of breaking a local gauge symmetry spontaneously. The possible phases of a gauge theory. Wilson loops, and the concept of magnetic disorder.

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Notes

  1. 1.

    The construction was first introduced by Wegner [2].

  2. 2.

    This leading-order (in 1∕m 2) result is obtained by neglecting the one-link term in S matter everywhere except along the timelike links from t = 0 to t = T, at x = 0 and x = R. On these links, expand \(\exp [\phi ^\dagger U_0 \phi + h.c.] \approx 1 + \phi ^\dagger U_0 \phi + h.c.\). Integration over the scalar field then yields the result (2.32).

  3. 3.

    One approach, based on the non-abelian Stokes Law, derives an area law for a large Wilson loop from an assumed finite range behavior of field strength correlators, which means that field strengths are uncorrelated, i.e. disordered, at sufficiently large separations. This “field correlator” approach to magnetic disorder has been pursued by Simonov and co-workers [10].

  4. 4.

    The Wilson loop calculation is a little easier in two dimensions with free boundary conditions. Periodic boundary conditions introduce a correction which is irrelevant for N p large.

References

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Greensite, J. (2020). Global Symmetry, Local Symmetry, and the Lattice. In: An Introduction to the Confinement Problem. Lecture Notes in Physics, vol 972. Springer, Cham. https://doi.org/10.1007/978-3-030-51563-8_2

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