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Nonperturbative Questions, Vacua, and Sector Dynamics

Nonperturbative vacuum physics begins where a local expansion is no longer the whole question. A candidate minimum can be mixed by tunneling; a path integral can decompose into topological sectors; a theta parameter can Fourier-transform those sectors; and the infinite-volume energy can develop branches constrained, but not uniquely determined, by symmetry and anomaly. This chapter supplies the distinctions, normalizations, and order-of-limits tests needed to move among those statements without conflating them.

Enter this chapter

Section titled “Enter this chapter”

Begin with What Does Nonperturbative Mean? to identify what a chosen perturbative expansion leaves unresolved.

Helpful background. Review Vacua, States, and Representations and Theta Terms, Periodicity, and Vacuum Sectors for vacuum-state language and topological-sector phases.

Topics

Section titled “Topics”
  1. What Does Nonperturbative Mean? — Defines nonperturbative relative to a specified expansion.
  2. Local Expansions and Global Vacuum Structure — Separates local fluctuation data from global vacuum structure.
  3. Topological Sectors, Boundary Data, and Global Form — Determines sectors from spacetime, boundaries, and the gauge group.
  4. Theta Parameters, Theta States, and Sector Sums — Relates theta phases to sector sums and states.
  5. Theta Dependence, CP, and Model-Dependent Branches — Distinguishes periodic vacuum energy from individual branches.
  6. Topological Susceptibility and Vacuum Response — Extracts vacuum response from topological-charge fluctuations.
  7. Tunneling, Superselection, and the Infinite-Volume Limit — Shows why volume and symmetry-breaking limits can fail to commute.
  8. Anomaly and Generalized-Symmetry Constraints on Infrared Phases — Restricts infrared possibilities without selecting a unique dynamical phase.

How the ideas fit together

Section titled “How the ideas fit together”

A symmetric double well makes the distinction between local and global data concrete. Expanding around either minimum gives the same perturbative energies, while tunneling mixes the localized states and splits the lowest pair. In a field theory, the mixing can vanish in the infinite-volume limit, allowing distinct pure vacua. Thus finite-volume uniqueness does not settle spontaneous symmetry breaking, and classical minima cannot simply be counted as quantum vacua. Theta-sector sums add global information of a different kind and require specified boundary and charge data. Coleman 1985, ch. 7, § 2.2.

Where to go next

Section titled “Where to go next”

Continue to Semiclassical Expansions and Integration Cycles to identify the large dimensionless action and the admissible saddles.

References

Section titled “References”
  • Coleman, Sidney. Aspects of Symmetry: Selected Erice Lectures. Cambridge University Press, 1985, ch. 7, § 2.2, pp. 270–277. DOI.