Nonperturbative Dynamics

Nonperturbative dynamics studies information that a fixed perturbative expansion cannot supply: tunneling between vacua, stable field configurations, generated mass scales, and collective behavior at strong coupling. Some of these effects are already accessible at weak coupling. The useful question is therefore which method controls the observable in the regime of interest?

This volume connects saddle expansions, topology, large-$N$ limits, functional equations, resurgence, and integrability. A particle in a symmetric double well supplies a first example: expanding around either minimum gives the same local energy series, while tunneling splits the low-lying levels. The task is to calculate that missing contribution and state why the approximation works.

Where to begin

• First study: begin with What Does Nonperturbative Mean?, then Semiclassical Expansions and Integration Cycles. Follow Quantum-Mechanical Instantons and Tunnel Splitting for a complete example with an independent spectral check.
• For strong dynamics: start with Strong-Coupling Laboratories and Dual Variables to see how controlled models generate gaps and screening. Continue to the confinement chapter when your question concerns a static potential, line operator, or gauge-invariant spectrum.
• For an existing calculation or research claim: enter Continuum Functional Equations and Controlled Truncations if your method closes an exact hierarchy, or Evidence, Validation, and Nonperturbative Status to compare methods at fixed observables and limits.

Helpful background. Saddles and the semiclassical expansion provides Gaussian fluctuations; asymptotic scales and remainders explains controlled nonconvergent expansions; and EFT as a controlled expansion clarifies regimes, omitted terms, and breakdown. Use the preparation relevant to your chosen route.

Chapters

1. Nonperturbative Questions, Vacua, and Sector Dynamics — Connect local expansions to global sectors, theta dependence, tunneling, and constraints on possible phases.
2. Semiclassical Expansions and Integration Cycles — Turn a contributing saddle into an observable using fluctuation determinants, collective coordinates, and controlled multi-saddle sums.
3. Solitons, Defects, and Collective Dynamics — Distinguish topological classification, existence, stability, and the slow dynamics of localized excitations.
4. Instantons, Fermion Zero Modes, and Tunneling — Calculate tunneling contributions and selection rules while testing the instanton measure and its integration range.
5. Metastability and Vacuum Decay — Extract decay rates from a false-vacuum boundary problem and assess negative modes and thin-wall control.
6. Strong-Coupling Laboratories and Dual Variables — Use sigma models, Gross–Neveu theory, and the Schwinger model to investigate generated scales and alternative variables.
7. Confinement, Screening, and Mass Gaps in Controlled Regimes — Match distinct confinement and screening diagnostics to the probes, theory, and controlled limit.
8. Large-N Dynamics — Reorganize vector, matrix, gauge, and tensor theories through large-$N$ counting, saddle points, and corrections.
9. Continuum Functional Equations and Controlled Truncations — Solve selected finite systems while tracking renormalization, symmetry identities, solution branches, and truncation dependence.
10. Resurgence and Transseries — Relate large-order behavior to exponentially small sectors and test cancellation of summation ambiguities.
11. Integrable Relativistic QFT — Use special conservation laws in $1+1$ dimensions to connect factorized scattering, finite-volume energies, and form factors.
12. Evidence, Validation, and Nonperturbative Status — Compare analytic, exact, numerical, and rigorous results without confusing agreement with independent control.

How the ideas connect

For the double well, introduce a small dimensionless semiclassical parameter $\epsilon$. A Euclidean path crossing the barrier has an action contribution $A/\epsilon$, with $A>0$, and therefore produces a factor $e^{-A/\epsilon}$. For every fixed nonnegative integer $n$, $e^{-A/\epsilon}/\epsilon^n\to0$ as $\epsilon\to0^+$. Thus no finite power expansion about one minimum can resolve the splitting.

A global boundary condition distinguishes the even and odd wavefunctions. The leading splitting combines the instanton exponential with a prefactor from fluctuations and the translation zero mode, whose coordinate is the center of the tunneling event. A dilute-event expansion is useful when typical events are well separated relative to their duration. Uniform WKB provides another route to the same spectral problem: the global quantization condition generates opposite leading shifts about the common perturbative energy Dunne and Ünsal 2014, §III.C, pp. 19–21, Eqs. (43) and (48), arXiv v2 (PDF).

This example joins vacua, semiclassics, instantons, and resurgence. At higher orders, summing the perturbative series may require choosing a prescription; a consistent physical energy must combine the corresponding sectors so that the ambiguity cancels. The double-well construction is a model-specific result. In field theory, additional effects such as renormalons can obstruct a direct transfer of the quantum-mechanical instanton analysis Mariño 2014, §§3.1–3.2, p. 37, arXiv v2 (PDF).

Changing the physical question changes the calculation. A stable double well has real stationary energy levels and a tunnel splitting. A metastable state instead motivates a survival probability and a decay rate, with a different saddle and negative-mode treatment. A static soliton is a finite-energy field configuration, whose topology alone does not establish dynamical stability. These distinctions explain why the volume gives solitons, instantons, and vacuum decay separate chapters.

Beyond semiclassics, control can come from another source. Large-$N$ methods organize corrections after specifying which couplings and observables are held fixed. Integrability uses special conservation laws and scattering consistency in selected models. Functional equations start from exact relations, but a finite closure introduces assumptions that need their own tests. A small numerical residual checks the solution of that closure; it does not bound the discarded part of the hierarchy.

Carry one observable through the comparison. For a claimed gap, specify the operator channel and the excitation energy above the chosen vacuum. For a static potential, specify the external probes and whether dynamical matter can screen them. Match volume, regulator, normalization, and limit order before comparing results. Agreement becomes more informative when the methods do not share the same uncontrolled input.

Where to go next

Lattice and Hamiltonian QFT develops regulated calculations, finite-volume spectra, and continuum extrapolation. Gauge Theories and the Standard Model places strong-dynamics results in specific physical models. Mathematical QFT treats existence, constructive limits, and theorem-level control. For changing evidence and open interpretations, use Research.

References

• Dunne, Gerald V., and Mithat Ünsal. “Uniform WKB, Multi-instantons, and Resurgent Trans-Series.” Physical Review D 89 (2014): 105009. DOI; Open PDF, arXiv:1401.5202v2.
• Mariño, Marcos. “Lectures on Non-Perturbative Effects in Large N Gauge Theories, Matrix Models and Strings.” Fortschritte der Physik 62 (2014): 455–540. DOI; Open PDF, arXiv:1206.6272v2.