An ode to instantons

Abstract

We present a formalism for semiclassical time evolution in quantum mechanics, building on a century of work. We identify complex saddle points in real time, real saddle points in complex time, and complex saddle points in complex time that reproduce the known answers in classic problems. For the decay of a metastable state, we find finite time and finite energy analogs of the "bounce" which do not have strict zero or negative modes. The one-loop phase of the wave function and the multiplicity of bounce solutions at late times are discussed. The motivation of this work is to learn how to compute decay rates in quantum field theory in situations with non-trivial time dependence, by first taking a humble step backwards to the fascinating world of quantum mechanics.

Figures (16)

• Figure 1: A semiclassical wave with energy $E_0 < \max_\mathbb{R} V$ incident upon a potential barrier from the left. We are interested in computing the semiclassical wave function $\psi(x,t)$ at locations $x$ to the right of the barrier for times $t$ when the amplitude at $x$ is near its maximum.
• Figure 2: The complexified time contour we use in our solution.
• Figure 3: A semiclassical wave with energy $E_0 > \max_\mathbb{R} V$ incident upon a potential barrier from the left. We want to know what the reflected wave looks like.
• Figure 4: On the right we show solutions to $V(z_c) = E_0$ (colored dots). The green points give exponentially suppressed contributions to $\psi$ while the red points would give exponentially enhanced contributions, but they must be excluded as they violate \ref{['simpler0condition']}. We consider a particular solution $z_c$ (here, the green point closest to the real axis) and compute for it the complex turning point $u_c$ in \ref{['tstarcontour']}; this is the point reached exactly halfway through the complex time contour on the left plot. Following the contour in \ref{['straightlinepath']}, the solution reaches the turning point and then traces its steps back. For this solution $\operatorname{Re} \left( i S \right) = -0.46$; it is the dominant contribution to $\psi$.
• Figure 5: A slight deformation of the $u'$-contour corresponding to a subleading saddle, keeping the imaginary displacement $\Delta \operatorname{Im} u = 2 \operatorname{Im} u_c$ fixed but passing $u_c$ to the right. This causes the semiclassical particle to travel a bit further on the real line before making the detour in the complex plane, so that the turning point $z_c \in \mathbb{C}$ is encircled in the counter-clockwise direction.