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Lattice Regulators and Continuum Targets

A lattice replaces continuous spacetime by a regulated system whose relation to a continuum quantum field theory must be demonstrated. Using a free massive scalar as a recurring test, this chapter separates discretization, boundaries, spectral positivity, symmetry restoration, and parameter tuning.

Enter this chapter

Section titled “Enter this chapter”

Begin with Lattice Regulators and Target Continuum Theories to specify the regulated model and the continuum observable it should approximate.

Topics

Section titled “Topics”
  1. Lattice Regulators and Target Continuum Theories — Defines the fields, measure, and target theory.
  2. Lattice Geometry, Boundaries, and Anisotropy — Tracks spacing, volume, boundary conditions, and anisotropy.
  3. Scalar Lattice Actions and Difference Operators — Builds local difference operators with the correct adjoints.
  4. Lattice Momentum, Propagators, and Cutoff Dispersion — Derives periodic momentum kernels and cutoff dispersion.
  5. Exact Symmetries, Broken Spacetime Symmetries, and Restoration — Distinguishes exact lattice symmetry from continuum restoration.
  6. Reflection Positivity and Transfer-Matrix Criteria — Tests whether Euclidean data admit a positive-state interpretation.
  7. Bare Parameters, Tuning Conditions, and Continuum Targets — Tunes renormalized conditions while changing the cutoff.

How the ideas fit together

Section titled “How the ideas fit together”

Taking the lattice spacing to zero changes the ultraviolet regulator, whereas increasing the box size changes the infrared problem. Holding bare parameters fixed need not approach a line of constant physics: renormalized masses and other chosen inputs must be held fixed through tuning. A correct small-momentum dispersion is one check, but positivity and restoration of broken spacetime symmetries pose additional conditions. The free scalar separates these issues before interactions make their numerical control harder. Montvay and Münster 1994, chs. 1–5.

Where to go next

Section titled “Where to go next”

Continue to Lattice Observables and Continuum Inference to read finite-volume energy and overlap information from Euclidean correlators.

References

Section titled “References”
  • Montvay, István, and Gernot Münster. Quantum Fields on a Lattice. Cambridge Monographs on Mathematical Physics. Cambridge University Press, 1994. doi:10.1017/CBO9780511470783.