Instantons, Fermion Zero Modes, and Tunneling

Instanton calculus begins with a finite-action Euclidean saddle and ends, when it succeeds, with a contribution to a specified amplitude or correlation function. Between those points lie boundary conditions, topology, gauge fixing, collective coordinates, determinants, fermion saturation, renormalization, and integration-range tests. This chapter develops that complete chain and marks the places where a correct saddle does not imply a controlled physical conclusion.

Enter this chapter

Begin with Euclidean Tunneling Saddles and Boundary Conditions to choose Euclidean boundary conditions appropriate to a tunneling observable.

Helpful background. Review Wick Rotation and Analytic Continuation and Theta Terms, Periodicity, and Vacuum Sectors for Euclidean continuation and topological phases.

Topics

1. Euclidean Tunneling Saddles and Boundary Conditions — Selects finite-action paths for a stated transition.
2. Quantum-Mechanical Instantons and Tunnel Splitting — Derives exponentially small level splitting.
3. Gauge Instantons, Topological Charge, and Moduli — Constructs gauge instantons and their collective parameters.
4. Instanton Measures, Zero Modes, and Determinants — Combines collective measures with normalized fluctuation determinants.
5. Dilute Instanton Ensembles and Theta Dependence — Derives theta dependence under dilute-gas assumptions.
6. Fermion Zero Modes, Index Data, and Selection Rules — Uses Grassmann integration to determine allowed insertions.
7. The Instanton Size Modulus and Infrared Limitations — Tests whether the size integral enters an uncontrolled infrared region.
8. Fractional Events, Calorons, and Monopole Constituents — Resolves compactified instantons into holonomy-dependent constituents.

How the ideas fit together

Topology can fix the classical instanton action, but an observable still requires integration over its collective parameters. In an asymptotically free theory, small instantons may be weakly coupled while the large-size portion of that integral is not. Fermion zero modes impose another restriction: without insertions or masses that saturate them, the proposed contribution vanishes. A correct classical solution therefore establishes neither a nonzero amplitude nor a controlled dilute gas. ‘t Hooft 1976, pp. 3432–3450.

Where to go next

Continue to Metastability and Vacuum Decay to construct a saddle that returns to the false vacuum at Euclidean infinity.

References

• ‘t Hooft, Gerard. “Computation of the Quantum Effects Due to a Four-Dimensional Pseudoparticle.” Physical Review D 14 (1976): 3432–3450; erratum 18 (1978): 2199. DOI.