Real-Time Feynman Path Integral Realization of Instantons

Abstract

In Euclidean path integrals, quantum mechanical tunneling amplitudes are associated with instanton configurations. We explain how tunneling amplitudes are encoded in real-time Feynman path integrals. The essential steps are borrowed from Picard-Lefschetz theory and resurgence theory.

Figures (2)

• Figure 1: (Color Online.) The complexified instanton configuration \ref{['eq:ComplexInstanton']}. Left column:$\alpha = \pi/2$ (Euclidean instanton). Center column:$\alpha = \pi/4$. Right column:$\alpha = \pi/8$. The top row shows the trajectories of \ref{['eq:ComplexInstanton']} for $\tau \in (-\infty, +\infty)$, while the bottom row shows the real and imaginary parts of \ref{['eq:ComplexInstanton']} as a function of $\tau$.
• Figure 2: (Color Online.) Upper-left: trajectory of \ref{['eq:ComplexInstanton']} for $\alpha = 4 \times 10^{-2} \times \frac{\pi}{2}$ in complex configuration space. Upper-right: real and imaginary parts of \ref{['eq:ComplexInstanton']} for $\alpha = 1 \times 10^{-2} \times \frac{\pi}{2}$, with a green $\alpha=\pi/2$ trajectory shown for comparison. Lower-left: illustration of the manner in which the imaginary part of \ref{['eq:ComplexInstanton']} approaches the singular configuration \ref{['eq:SingularInstanton']} for small $\alpha$ (here $\alpha = 1 \times 10^{-2} \times \frac{\pi}{2}$), while the real part approaches to a distribution, a signed Dirac comb. Lower-right: the trajectory for $\alpha = 1 \times 10^{-3} \times \frac{\pi}{2}$, with red dots marking the centers of the wells $x = \pm 1$. In the real-time limit the real-time instanton appears to become a space-filling curve in complexified configuration space.