arXiv:2610.38717
A Primer on Instanton Physics - A Short Course
Pervez Hoodbhoy
Abstract
These six lectures, delivered at the University of New Brunswick 6-21 October 2026, are an introduction to instantons in quantum mechanics and quantum field theory for senior undergraduate and beginning graduate students. The first four lectures deal solely with particle mechanics, building the necessary background for the last two that cover applications in semi-classical field theory.
AI Summary
This six-lecture primer addresses how instanton methods describe quantum tunneling and nonperturbative effects in particle mechanics and quantum field theory. Intended for senior undergraduates and beginning graduate students, it presents established calculations rather than new research results. The pedagogical emphasis is on evaluating fluctuation determinants through the Gel'fand–Yaglom theorem instead of detailed reflectionless-potential methods. The first four lectures develop path integrals, imaginary-time classical trajectories, Gaussian fluctuations, collective coordinates for zero modes, and dilute instanton gases. For the symmetric double well, they derive the kink trajectory, its action $S_1=2m\omega a^2/3$, and the primed determinant ratio $\det'\mathcal O_c/\det\mathcal O_v=1/(12\omega^2)$. Summing alternating instantons and anti-instantons yields the leading semiclassical splitting $\Delta E=2\hbar\omega\sqrt{6S_1/(\pi\hbar)}e^{-S_1/\hbar}$. Metastable decay is treated using bounce solutions, a negative fluctuation mode, analytic continuation, and a cubic-potential example. The final lectures extend the discussion to scalar-field vacuum decay, gravitational effects, and Yang–Mills topology. They reproduce the four-dimensional thin-wall bubble radius $R_c=3S_1/\epsilon$ and action $B=27\pi^2S_1^4/(2\epsilon^3)$, where $S_1$ denotes wall tension and $\epsilon$ the vacuum-energy difference, and outline winding sectors, the axial anomaly, and the $\theta$ vacuum. The stated approximation limits are central: dilute-gas calculations require $S_1\gg\hbar$ and separations much larger than the instanton width; overlapping configurations require interactions or other nonperturbative methods. The thin-wall treatment assumes a small vacuum-energy difference.
Error found: Part I uses t=iτ to derive decaying Euclidean evolution, although its stated real-time propagator requires t=−iτ. Part II's Gel'fand–Yaglom proof incorrectly asserts that eigenvalue differences vanish and invokes Liouville's theorem without establishing its hypotheses. For the cubic potential and bounce in Part IV, direct integration gives S_b=6m³ω⁵/(5λ²), twice the printed action, changing the leading tunneling exponential. Part IV's false-vacuum decay formulas omit required normalization factors, miss the negative-mode half contribution, and inconsistently treat Γ as an energy width and an inverse-time rate. Part VI conflates the Yang–Mills kinetic density tr(F∧*F) with the topological density tr(F∧F), without the self-duality condition needed to identify them.
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