Complexified path integrals, exact saddles and supersymmetry

TL;DR

This work finds new exact complex saddles, and shows that without their contribution the semiclassical expansion is in conflict with basic properties such as the positive semidefiniteness of the spectrum, as well as constraints of supersymmetry.

Abstract

In the context of two illustrative examples from supersymmetric quantum mechanics we show that the semi-classical analysis of the path integral requires complexification of the configuration space and action, and the inclusion of complex saddle points, even when the parameters in the action are real. We find new exact complex saddles, and show that without their contribution the semi-classical expansion is in conflict with basic properties such as positive-semidefiniteness of the spectrum, and constraints of supersymmetry. Generic saddles are not only complex, but also possibly multi-valued, and even singular. This is in contrast to instanton solutions, which are real, smooth, and single-valued. The multi-valuedness of the action can be interpreted as a hidden topological angle, quantized in units of $π$ in supersymmetric theories. The general ideas also apply to non-supersymmetric theories.

Figures (5)

• Figure 1: Real and complex solutions in the inverted tilted double well potential. The inverted potential (on the real axis) is shown in black, the real bounce and associated critical and turning points are shown in red, and the pair of complex bions and turning and critical points are blue. The blue points correspond to $z_1^{\rm cr}$ and $z_T, z_T^*$ in \ref{['dw_cb']}. Note that the motion takes place in the real and imaginary parts of the complex potential, as explained in the text.
• Figure 2: Complex bion solution in supersymmetric quantum mechanics with a double well potential. The black and red lines show the real and imaginary part of the solution for $pg=1\cdot 10^{-6}$. The characteristic size of the solution is ${\rm Re}[2t_{0}]\simeq { \frac{1}{2}} \log\frac{16}{pg}$. For larger values of $pg$ the two tunneling event merge.
• Figure 3: Real and complex solutions in the quantum modified inverted Sine-Gordon potential. The inverted potential (on the real axis) is shown in black, the real bounce and associated critical and turning points are shown in red, the pair of complex bions and associated turning as well as critical points are blue, and the real bion is shown in green. In order to smoothen the (singular) complex bion the solution is plotted at $\theta= 0.95 \pi$. The singular limit is shown as the dashed line. Note that the vacuum properties are governed by the real and complex bion solutions.
• Figure 4: Complex solutions in the quantum modified Sine-Gordon potential with $p g=2\cdot 10^{-5}$. $\theta=0$ correspond to the real bounce solution. At $\theta=\pi^{-}$, the real bounce turns into a complex bion. The characteristic size of the solution is ${\rm Re}[2t_{0}] \approx \ln\frac{32}{pg}$.
• Figure 5: Action of the complex saddle solution in the Sine-Gordon potential as a function of $\theta$ for $p g=0.1$. $\theta=0$ corresponds to the real bounce, and $\theta=\pi$ is the complex bion, a multi-valued singular solution.