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.

Helpful background. Saddles and the semiclassical expansion supplies the elementary Gaussian construction; asymptotic scales, remainders, and uniformity supplies the language for controlling a nonconvergent expansion.

From a saddle to an observable

For a regulated Euclidean integral with weight $e^{-S_E/g}$, the contribution of a saddle family $\sigma$ has the schematic form

$$\mathcal Z_\sigma[\mathcal O] \sim n_\sigma\, e^{-S_{\sigma,\rm ren}/g} e^{i\alpha_{\Gamma,\sigma}} \int_{\mathcal M_\sigma}\mathrm d\mu_\sigma\, \left( \frac{\det{}'M_{\sigma,\rm ren}} {\det M_{\rm ref,ren}} \right)^{-1/2} \mathcal I_\sigma[\mathcal O]\, \bigl[1+O(g)\bigr].$$

Every factor answers a different question:

• $n_\sigma$ says whether the original integration cycle contains the saddle’s thimble;
• $S_{\sigma,\rm ren}/g$ supplies the leading exponential suppression;
• $\alpha_{\Gamma,\sigma}$ is the contour phase, including the treatment of negative directions;
• $\mathrm d\mu_\sigma$ replaces normalizable zero modes by physical moduli;
• $\det{}'M_\sigma$ integrates the remaining Gaussian fluctuations;
• $\mathcal I_\sigma[\mathcal O]$ carries the insertion and its projected propagators;
• $O(g)$ denotes loop corrections only after all soft directions have been removed from ordinary perturbation theory.

The formula is schematic because a false-vacuum rate, a level splitting, and a correlation function use different boundary conditions and normalizations. The pages in this chapter make each specialization explicit.

The Euclidean convention is $e^{-S_E}$ before an explicit small parameter is factored out. Lorentzian formulas inherit the site-wide $(+---)$ metric convention. All control inequalities are stated in dimensionless variables; zero, gauge, negative, and parametrically soft directions are kept distinct.

A route through the chapter

1. Saddles, control parameters, and loop counting identifies the dimensionless large-action limit, derives the loop power $g^{L-1}$, and explains why boundary data and intersection numbers—not real action alone—select saddles. Its quartic-double-well instanton fixes the chapter’s normalization.

2. Fluctuation operators and determinant ratios derives the one-loop ratio from the second variation, states the Gel’fand–Yaglom hypotheses, and evaluates the reduced Pöschl–Teller determinant.

3. Zero modes, collective coordinates, and moduli measures replaces a normalizable symmetry zero mode by its invariant moduli measure and separates physical moduli from gauge redundancy.

4. Negative modes and instability indices relates the Morse index to steepest Gaussian directions and distinguishes a stable tunneling instanton from a one-negative-mode false-vacuum bounce.

5. Complex saddles, Lefschetz thimbles, and integration cycles defines downward and dual cycles, obtains their intersection numbers, and checks a quartic integral against an exact Bessel representation.

6. Multi-saddle sums and dilute ensembles derives ordered-center factors, exponentiation, and the double-well transfer matrix, then states where correlated-event interactions invalidate the ideal gas.

7. Stokes jumps, saddle dominance, and contour dependence distinguishes phase alignment from equal magnitude and uses a tilted double well to show how an exponentially small avoided crossing smooths a limiting saddle cusp.

8. Renormalized saddle contributions and validity tests combines actions, determinants, measures, counterterms, and insertions; it turns the double-well splitting into a quantitative comparison with direct diagonalization and an explicit error budget.

The comparison of canonical saddle calculations provides a compact cross-check of actions, mode counts, determinant prescriptions, contours, renormalization, and failure regimes across several standard examples.

Reading paths by physical question

Stable tunneling and level splitting. Read pages 1–3, then pages 6 and 8. This path derives the instanton action, determinant, translation measure, dilute sum, and spectral test.

Metastable decay. Read pages 1–4 and 8. The negative-mode contour and false-vacuum boundary prescription are indispensable; a real instanton formula cannot simply be reinterpreted as a rate.

Complex saddles and parameter continuation. Read pages 1, 4, 5, and 7. This path separates local complex critical points from their global intersection numbers and keeps Stokes jumps distinct from dominance changes.

QFT prefactors. Read pages 1–4 and 8 together with the linked heat-kernel and renormalization material. The determinant divergence, collective measure, ghost or gauge quotient, and counterterms must be treated in the same regulator and scheme.

Control boundaries

A semiclassical answer should state which of the following can invalidate it:

$$\frac{\operatorname{Re}\Delta S}{g}\not\gg1, \qquad \frac{\lambda_{\rm soft}}{\lambda_{\rm gap}}\not\ll1, \qquad \kappa\xi\not\ll1, \qquad g_{\rm ren}(1/\rho)\not\ll1.$$

These tests respectively diagnose unsuppressed neighboring saddles, loss of Gaussian mode separation, overlap of localized events, and strong running coupling at saddle size $\rho$. Additional boundaries include a Stokes connection, a saddle coalescence, an infrared-divergent moduli integral, an unstable regulator limit, or uncanceled renormalization-scale dependence.

The relevant remedy depends on the failure:

Failure	Required reorganization
Isolated zero mode	Collective coordinate and primed determinant
Parametrically soft nonzero mode	Explicit non-Gaussian integral or uniform approximation
One physical negative mode	Contour continuation tied to the boundary prescription
Stokes connection	Simultaneous update of thimble basis and intersection numbers
Overlapping instantons	Connected-cluster or correlated-event expansion
Ultraviolet determinant divergence	Local counterterms in a fixed renormalization scheme
Strong-coupling or infrared moduli endpoint	New effective description, regulator, or non-semiclassical method

No single “large action” statement substitutes for this classification.

Review the chapter

By the end of the chapter, you should be able to:

• expose the parameter multiplying a dimensionless action and derive loop counting;
• decide which critical points contribute to a specified cycle and boundary problem;
• compute a determinant ratio with its domain, zero-mode projection, and dimensions;
• derive a moduli measure from zero-mode norms and quotient gauge redundancy;
• interpret negative modes without discarding their contour phases;
• sum a dilute ensemble while enforcing sector and endpoint constraints;
• distinguish Stokes jumps, equal-magnitude curves, and saddle coalescence;
• extract a renormalized observable with an honest error budget.

Exercises

1. Assemble the leading one-instanton fugacity for the chapter’s dimensionless double well from its action, determinant ratio, and translation norm.

Solution

The ingredients are

$$\mathcal S_I=\frac43, \qquad \frac{\det{}'M_I}{\det M_0}=\frac1{48}, \qquad G_{\tau_0\tau_0}=\frac43.$$

Therefore, per unit Euclidean time,

$$\kappa =\left(\frac{4/3}{2\pi g}\right)^{1/2} \sqrt{48}\,e^{-4/(3g)} =4\sqrt{\frac{2}{\pi g}}\,e^{-4/(3g)}.$$

2. A saddle has one exact translation zero mode, one eigenvalue $\lambda_{\rm soft}\sim g$, and otherwise order-one positive spectrum. Which directions belong in the determinant?

Solution

The exact zero mode is removed and replaced by the translation collective coordinate. The soft mode cannot be treated uniformly by the ordinary Gaussian determinant because its quadratic term is comparable to interaction corrections; retain it as an explicit non-Gaussian coordinate or construct a uniform approximation. Only the remaining gapped spectrum belongs in the ordinary determinant.

3. Two contributing saddles satisfy $\operatorname{Im}(\Delta S/g)=0$ and $\operatorname{Re}(\Delta S/g)=7$. What additional information is needed before changing the saddle sum?

Solution

Phase alignment alone is insufficient. One must determine whether an oriented connecting flow exists and, if it does, its integer incidence. The thimble basis and intersection coefficients must then be transformed together. The real part only says that the second saddle is suppressed by about $e^{-7}$ before prefactors; the desired accuracy decides whether it may be omitted.

4. Explain why a numerical agreement in the instanton exponent does not validate a decay rate.

Solution

The exponent tests only the classical action. A decay rate also requires exactly the relevant negative-mode structure, the false-vacuum contour prescription, translation Jacobians, the renormalized determinant prefactor, and control of multi-bounce and finite-volume corrections. A stable instanton can have the same exponential scale in a related potential while producing a real level splitting rather than a rate.

References

• Callan, Curtis G., Jr., and Sidney Coleman. “Fate of the False Vacuum. II. First Quantum Corrections.” Physical Review D 16 (1977): 1762–1768. DOI.
• Coleman, Sidney. Aspects of Symmetry: Selected Erice Lectures. Cambridge University Press, 1985, ch. 7, pp. 265–350. DOI.
• Dunne, Gerald V. “Functional Determinants in Quantum Field Theory.” Journal of Physics A: Mathematical and Theoretical 41 (2008): 304006. DOI.
• Mariño, Marcos. Instantons and Large N: An Introduction to Non-Perturbative Methods in Quantum Field Theory. Cambridge University Press, 2015, ch. 1, pp. 3–61. DOI.
• Witten, Edward. “Analytic Continuation of Chern–Simons Theory.” In Chern–Simons Gauge Theory: 20 Years After, AMS/IP Studies in Advanced Mathematics 50 (2011): 347–446. arXiv:1001.2933.