Semiclassical Expansions and Integration Cycles
Semiclassical physics is not obtained by solving a classical equation and attaching a determinant. A complete contribution requires a dimensionless control parameter, boundary data and an integration cycle, the correct nonzero-mode determinant, collective-coordinate measures, negative-mode phases, renormalization, and an error estimate. This chapter develops those ingredients in the order needed to turn a saddle into a controlled observable.
Enter this chapter
Section titled “Enter this chapter”Begin with Saddles, Control Parameters, and Loop Counting to identify the large dimensionless action and the admissible saddles.
Helpful background. Review Saddles and the Semiclassical Expansion and Asymptotic Scales, Remainders, Uniformity, and Optimal Truncation for saddle expansion and asymptotic error estimates.
Topics
Section titled “Topics”- Saddles, Control Parameters, and Loop Counting — Defines the parameter controlling fluctuations.
- Fluctuation Operators and Determinant Ratios — Normalizes nonzero-mode Gaussian contributions.
- Zero Modes, Collective Coordinates, and Moduli Measures — Replaces flat Gaussian directions by collective-coordinate integrals.
- Negative Modes and Instability Indices — Relates unstable directions to a specified contour.
- Complex Saddles, Lefschetz Thimbles, and Integration Cycles — Determines which complex saddles belong to the integration cycle.
- Multi-Saddle Sums and Dilute Ensembles — Tests when separated saddle events form a dilute ensemble.
- Stokes Jumps, Saddle Dominance, and Contour Dependence — Separates contour rearrangements from changes of exponential dominance.
- Renormalized Saddle Contributions and Validity Tests — Combines renormalization and approximation checks in an observable.
How the ideas fit together
Section titled “How the ideas fit together”A vanishing Hessian eigenvalue cannot be included in an ordinary Gaussian determinant. When an exact symmetry or modulus generates a saddle family, replace its zero mode by a collective-coordinate integral with the corresponding measure. An isolated degenerate saddle instead requires higher-order fluctuation terms; a zero eigenvalue alone does not establish a family. A negative eigenvalue raises a different question, because the integration contour determines the phase of its contribution. The same saddle machinery can describe real tunnel splitting or false-vacuum decay only after the boundary data and contour are specified. Coleman 1985, ch. 7.
Where to go next
Section titled “Where to go next”Continue to Solitons, Defects, and Collective Dynamics to specify the configuration space before discussing topological protection.
References
Section titled “References”- Coleman, Sidney. Aspects of Symmetry: Selected Erice Lectures. Cambridge University Press, 1985, ch. 7, pp. 265–350. DOI.