An ode to instantons

This paper presents a formalism for semiclassical time evolution in quantum mechanics that utilizes various complex and real saddle points to accurately reproduce known results and analyze metastable state decay, with the ultimate goal of developing methods to compute decay rates in time-dependent quantum field theories.

The Gist

Imagine you are trying to predict the future of a tiny particle, like an electron, moving through a landscape of hills and valleys. In the quantum world, this particle doesn't just follow one path like a car on a road. Instead, it explores every possible path simultaneously, a concept known as the "path integral."

This paper is a guidebook for calculating what happens when we try to predict this particle's journey, specifically when it has to do something impossible in the classical world: tunnel through a wall.

Here is the story of the paper, broken down into simple concepts and analogies.

1. The Problem: The "Impossible" Tunnel

In the real world, if you roll a ball toward a hill that is too high, it rolls back. In the quantum world, particles can sometimes "ghost" through the hill and appear on the other side. This is called quantum tunneling.

Physicists have been trying to calculate exactly how fast this happens, especially when the environment is changing (like a hill that is shaking or moving). The old methods work great for static situations (a still hill), but they get confused when things are moving or when the particle is in a "metastable" state (a ball sitting in a small dip that could roll out, but hasn't yet).

The authors say: "Let's go back to basics. Let's solve this in the simple world of one-dimensional quantum mechanics first, so we can figure out how to solve the messy, complex problems in the universe later."

2. The Tool: The "Time Travel" Map

To solve this, the authors use a mathematical trick called the Path Integral. Imagine you are trying to find the best route from home to work.

• The Classical Way: You look for the single fastest road.
• The Quantum Way: You imagine taking every road at once, including driving through your neighbor's house, flying over the city, or driving backward. Most of these paths cancel each other out, but a few special paths (called Saddle Points) dominate the result.

The paper's big innovation is how they handle these "special paths" when the particle has to tunnel.

The Two Methods of "Time Travel"

The authors compare two ways to find these special paths:

1. The "Direct" Method (Real Time, Complex Paths):

   • Analogy: Imagine you are walking on a real sidewalk (real time), but your feet are made of ghostly, shifting colors (complex numbers). You stay on the ground, but your steps are weird and mathematical.
   • Pros: It feels more "real" because time moves forward normally.
   • Cons: The paths are hard to calculate because they are "ghostly."

2. The "Indirect" Method (Complex Time, Real Paths):

   • Analogy: Imagine you are walking on a normal path, but you decide to take a "time machine" detour. You step into a tunnel where time flows sideways (imaginary time). In this tunnel, the hill you were trying to climb turns into a valley, and you can roll right through it! Then you step back out into real time on the other side.
   • Pros: The path looks normal and easy to calculate, even though the "time" you spent was weird.
   • Cons: You have to be careful to step back into real time correctly at the end.

The authors show that the "Indirect" method (taking the time machine detour) is often the best way to understand tunneling.

3. The Big Discovery: The "Bounce" and the "Crowd"

The most famous part of this paper deals with a metastable state.

• The Metaphor: Imagine a ball sitting in a small dip on the side of a mountain. It's stable for now, but if it gets a little push, it can roll down the mountain forever. This is a "metastable state."
• The Old View: Physicists used to think the ball would sit there forever until it suddenly "tunneled" out in a single, magical instant called a "bounce."
• The New View: The authors show that the ball doesn't just do one bounce. It's more like a crowd of possibilities.
  • The ball can try to roll out, get stuck, roll back, try again, roll back, and try again.
  • Each time it tries, it leaves a "ghostly footprint" (a mathematical contribution).
  • At first, these footprints are tiny. But as time goes on, there are so many different ways the ball can try and fail that they add up to a massive probability.
  • The Result: This "crowd" of attempts creates a smooth, exponential decay. The ball doesn't just vanish; it slowly leaks out because there are millions of ways for it to try to escape.

4. The "Resonant" Super-Highway

The paper also looks at a scenario with two hills (a double barrier).

• The Analogy: Imagine a particle trying to get through two walls. Usually, it's very hard.
• The Magic: If the particle's energy matches a specific "tune" (like a singer hitting the right note to shatter a glass), the two walls act like a super-highway.
• The Mechanism: The particle bounces back and forth between the two walls millions of times. Each bounce adds a little bit of "constructive interference" (like waves in a pool adding up to a giant wave).
• The Result: Instead of a tiny chance of getting through, the particle gets through with 100% certainty. It's as if the universe conspired to open a door just for that specific energy.

5. Why This Matters

The authors are essentially saying: "We have a new, robust way to calculate these quantum jumps."

• For Cosmology: This helps us understand how the early universe might have changed phases (like water turning to ice) or how black holes might form.
• For the Future: By mastering these calculations in simple quantum mechanics, they hope to apply the same logic to Quantum Field Theory (the physics of the whole universe), solving long-standing mysteries about how the universe decays or evolves over time.

Summary in One Sentence

This paper teaches us how to calculate the odds of a quantum particle escaping a trap by realizing that it doesn't just make one magical jump, but rather explores a vast, complex landscape of "time-traveling" paths, where the sheer number of failed attempts eventually adds up to a successful escape.

Technical Summary

Here is a detailed technical summary of the paper "An ode to instantons" by Janssen, Karlsson, Riccardi, and Varrone.

1. Problem Statement and Motivation

The paper addresses longstanding open problems in the semiclassical limit of quantum mechanics and quantum field theory (QFT), specifically regarding time-dependent decay rates and tunneling phenomena in non-trivial time-dependent backgrounds.

• The Core Issue: Standard semiclassical methods (like the WKB approximation or Euclidean instanton calculus) often rely on rotating to imaginary time and assuming the system starts in a "would-be" ground state (false vacuum). However, many physical scenarios—such as an oscillating scalar field in the early universe or inflationary models with time-dependent potentials—involve excited states and real-time evolution.
• Specific Open Questions:
  1. What is the time-dependent decay probability $\Gamma(t)$ for a scalar field with an oscillating initial profile?
  2. How does tunneling occur in multi-field inflation models where the potential changes dynamically?
• The Gap: Existing calculations for these problems yield conflicting results (e.g., additive vs. multiplicative corrections to the decay rate) because the analytic continuation to imaginary time is ambiguous in time-dependent settings. The authors aim to resolve this by returning to first principles in one-dimensional quantum mechanics (QM) to develop a robust real-time path integral formalism before applying it to QFT.

2. Methodology: Semiclassical Time Evolution from the Path Integral

The authors develop a formalism to compute the time-evolved wave function $\psi(x, t)$ from an initial semiclassical state $\psi(x_0, 0) = g(x_0)e^{-f(x_0)/\hbar}$ using the path integral:
$$\psi(x, t) = \int dx_0 \int_{y(0)=x_0}^{y(t)=x} [dy(t')] e^{\frac{i}{\hbar}S[y]} \psi(x_0, 0)$$

Key Technical Steps:

1. Complex Saddle Points: The authors identify that for semiclassical evolution, the relevant saddle points (classical trajectories) may be complex-valued paths in real time, or real-valued paths in complex time.
2. Steepest Descent Expansion: They expand the action $S$ and the initial state exponent $f$ around a saddle point $(\bar{z}_0, \bar{z})$.
   • The trajectory $\bar{z}(t')$ satisfies the classical equation of motion: $m\ddot{\bar{z}} = -V'(\bar{z})$.
   • The initial condition is determined by the gradient of the initial state: $m\dot{\bar{z}}(0) = i f'(\bar{z}_0)$.
3. One-Loop Prefactor: The authors derive a compact formula for the one-loop prefactor (the determinant of fluctuations). They define a fluctuation operator $F = -m\partial_{t'}^2 - V''(\bar{z})$ and solve for a function $\gamma(t)$ satisfying $F\gamma = 0$ with specific boundary conditions. The wave function is approximated as:
   $$\psi(x, t) \approx \frac{N_P}{\sqrt{\gamma(t)}} g(\bar{z}_0) \exp\left( \frac{i}{\hbar}S[\bar{z}] - \frac{f(\bar{z}_0)}{\hbar} \right)$$
   where $N_P$ is a normalization constant (discussed as an open question).
4. Two Approaches:
   • Direct Method: Keep time $t$ real, but allow the trajectory $\bar{z}(t)$ to be complex.
   • Indirect Method: Keep the trajectory $\bar{z}$ real-valued but complexify the time variable $t \to u$. The result is then analytically continued back to real time. This is particularly useful for tunneling problems.

3. Key Contributions and Results

The paper applies this formalism to five classic problems in 1D QM, demonstrating how it reproduces known WKB results and provides new insights into the structure of saddle points.

A. Harmonic Oscillator (Coherent States)

• Result: The formalism correctly reproduces the time evolution of coherent states.
• Insight: Even in simple quadratic potentials, the saddle point trajectories are complex ellipses in the complex plane, validating the necessity of complex paths even for real-time evolution.

B. Under-Barrier Transmission (Tunneling)

• Method: Uses the Indirect Method. The particle travels in real time to the turning point, then evolves in negative imaginary time through the barrier (inverted potential), and returns to real time.
• Result: Reproduces the standard WKB transmission coefficient $e^{-S/\hbar}$.
• Significance: This approach avoids the ambiguity of complexifying the potential $V(z)$ in the complex plane, which can plague the "Direct Method."

C. Over-Barrier Reflection

• Method: Uses complex time contours to reach complex turning points $z_c$ where $V(z_c) = E$.
• Result: Derives the exponentially suppressed reflection coefficient for energies above the barrier, matching the result from [28].
• Insight: The sign of the prefactor is determined by the direction of winding around turning points in the complex time plane (counter-clockwise vs. clockwise).

D. Decay of a Metastable State

This is the central result of the paper.

• Finite-Time/Finite-Energy "Bounces": Unlike the standard Coleman bounce (which is a zero-energy solution in infinite imaginary time), the authors find finite-time, finite-energy analogs.
  • The particle oscillates in the metastable well (real time).
  • Every time it hits the turning point, it can undergo a "bounce" (negative imaginary time evolution) across the barrier.
• Combinatorial Enhancement: While individual bounce solutions are subdominant (exponentially suppressed), the multiplicity of these solutions grows combinatorially with time.
  • Summing over all paths with $\ell$ bounces yields a factor proportional to $(t/\tau)^\ell / \ell!$.
  • This sum exponentiates to produce the correct exponential decay law: $\psi \sim e^{-iEt/\hbar} e^{-\Gamma t/2}$.
• Zero Modes and Negative Modes:
  • The authors argue that in their real-time formalism, the fluctuation operator does not have zero modes (because the energy is finite, not zero) and no negative modes (in the strict Hermitian sense).
  • The "negative mode" physics is implicitly captured by the combinatorial counting of bounce solutions and the phase factors acquired when winding around turning points.
  • The factor of $1/2$ usually associated with the bounce (Coleman's factor) arises here as a normalization factor for the path integral measure of subleading saddles, determined by probability conservation.

E. Resonant Transmission

• Setup: Tunneling through a double-barrier system.
• Result: By summing over paths with multiple bounces in the intermediate well and through the barriers, the authors derive the Breit-Wigner resonance profile.
• Significance: They show that unit transmission probability ($|T|^2=1$) at resonance arises from the constructive interference of infinitely many geometric bounce paths, captured naturally by the path integral summation.

4. Comparison of Methods (Section 3.4.4)

The authors explicitly compare four methods for a specific tunneling potential:

1. WKB: Analytic, standard.
2. Indirect (Complex Time) Path Integral: Analytic, matches WKB.
3. Numerical Solution: Discretized Schrödinger equation.
4. Direct (Real Time) Path Integral: Uses complex saddle points in real time.

Findings:

• The Direct Method reveals a complex landscape of saddle points. At early times, the "trivial" saddle dominates. As time evolves, this saddle ceases to exist (hits a pole), and dominance exchanges with other complex saddles.
• The Indirect Method is more robust for tunneling as it avoids the sensitivity to the analytic properties of the potential in the complex plane.
• Both path integral methods successfully reconstruct the numerical solution in the limit $\hbar \to 0$.

5. Significance and Open Questions

• Significance:
  • Provides a rigorous real-time path integral framework for calculating decay rates and tunneling without relying on the ad-hoc rotation to imaginary time.
  • Resolves the "bounce" paradox by showing that finite-time bounces and their combinatorial multiplicity naturally generate exponential decay, removing the need for artificial zero-mode integration in this context.
  • Offers a potential roadmap for solving the time-dependent decay problems in Quantum Field Theory (e.g., vacuum decay in oscillating backgrounds) that have plagued cosmology.
• Open Questions:
  • Normalization ($N_P$): The authors leave the normalization constant for subleading saddles as an open question. They speculate it is 1 for all saddles (Direct method view) or $1/2^\ell$for$\ell$-bounce solutions (Indirect method view).
  • Selection Criterion: A general, practical criterion for determining which complex saddle points contribute to the path integral (based on steepest descent contours) remains unsolved for general theories.

Conclusion

The paper successfully demonstrates that by carefully analyzing complex saddle points in real-time path integrals, one can reproduce all standard semiclassical results (WKB, instantons, resonances) while gaining a deeper understanding of the time-dependent dynamics of tunneling. It suggests that the "bounce" is not a singular Euclidean object but a manifestation of a sum over real-time paths with imaginary excursions, a perspective that may be crucial for tackling non-equilibrium QFT problems.