Worldline Localization
Changha Choi, Leon A. Takhtajan
TL;DR
This work reveals a localization mechanism for worldline path integrals arising from hidden fermionic BRST/BV symmetries, distinct from physical supersymmetry. By treating the worldline trajectory and the proper-time modulus as coupled yet separately localizable sectors, the authors derive exact semiclassical results for two paradigmatic problems: the harmonic oscillator’s thermal partition function and the scalar QED one-loop effective action in a constant field. The harmonic oscillator case yields an alternative Jacobi–Poisson localization with a further modulus localization via modular invariance, while the scalar QED case provides a controlled, fully localized computation of both the real Euler–Heisenberg integrand and the imaginary Schwinger pair-production contribution, anchored by worldline instantons. The results offer a fresh localization perspective on semiclassical exactness and the AAM observation, clarifying how moduli spaces of worldline instantons drive vacuum decay in strong fields and how zero-dimensional SUSY manifests in the localization of the modulus.
Abstract
We show that two elementary worldline path integrals-the thermal partition function of the harmonic oscillator and the one-loop effective action of scalar QED in a constant field strength-exhibit a natural form of supersymmetric localization. The mechanism relies on hidden fermionic symmetries of the worldline BRST formulation, rather than on standard BRST structure or physical supersymmetry. These symmetries localize the target-space trajectory. For the harmonic oscillator this yields an alternative localization derivation of the Jacobi-Poisson formula. Moreover, after the trajectory is localized, the remaining proper-time integral exhibits an emergent zero-dimensional supersymmetry generated by modular invariance, allowing the modulus T itself to be localized. For scalar QED the same structure provides a controlled computation of both the real and imaginary parts of the Euler-Heisenberg effective action. In particular, the imaginary part arises from a moduli space of circular worldline instantons, offering a localization perspective on the semiclassical exactness of the Schwinger pair-production result observed by Affleck-Alvarez-Manton.