Quantum Field Theory Course

Total: 440+ pages covering classical field theory through the Standard Model

Part III, Chapter 2

Path Integrals in Quantum Mechanics

Semiclassical approximation, instantons, and tunneling

Page 3 of 3

2.7 Semiclassical (WKB) Approximation from the Path Integral

The semiclassical limit arises naturally from the path integral by expanding around the classical solution. For a general potential V(x), we write $x(t) = x_{\text{cl}}(t) + y(t)$and expand the action to second order in fluctuations:

$$S[x_{\text{cl}} + y] = S_{\text{cl}} + \underbrace{\frac{\delta S}{\delta x}\bigg|_{\text{cl}} \cdot y}_{=\,0} + \frac{1}{2}\int dt\,dt'\; y(t)\,\frac{\delta^2 S}{\delta x(t)\delta x(t')}\bigg|_{\text{cl}} y(t') + O(y^3)$$

The linear term vanishes by the equations of motion. The second-order term defines the fluctuation operator:

$$\hat{O} = -m\frac{d^2}{dt^2} - V''(x_{\text{cl}}(t))$$

The Gaussian integral over fluctuations gives:

$$\boxed{K(x_b,t_b;x_a,t_a) \approx \sum_{\text{cl. paths}} \frac{1}{\sqrt{\det(\hat{O}/2\pi i\hbar)}}\; e^{iS_{\text{cl}}/\hbar}}$$

The sum runs over all classical trajectories connecting the endpoints. This is the WKB approximation derived directly from the path integral, valid when $S_{\text{cl}} \gg \hbar$.

💡Geometrical Meaning

The prefactor $(\det\hat{O})^{-1/2}$ measures how "focused" the bundle of nearby paths is around the classical trajectory. A large determinant means paths spread out quickly (weak focusing), giving a small amplitude. This is the path-integral version of the Van Vleck determinant in classical mechanics.

Multiple Classical Paths and Caustics

When multiple classical paths connect the same endpoints (e.g., the harmonic oscillator at times $T = n\pi/\omega$), the semiclassical propagator is a sum over all of them. Each path picks up an additional phase $e^{-i\mu\pi/2}$ where $\mu$ is the Morse index (the number of negative eigenvalues of $\hat{O}$, counting conjugate points):

$$K_{\text{sc}} = \sum_{\gamma}\frac{e^{iS_\gamma/\hbar - i\mu_\gamma \pi/2}}{\sqrt{|\det(\hat{O}_\gamma/2\pi i\hbar)|}}$$

2.8 Instantons in Quantum Mechanics

One of the most powerful applications of the Euclidean path integral is computing tunneling amplitudes between classically disconnected states.

The Double-Well Potential

Consider a symmetric double-well potential:

$$V(x) = \frac{\lambda}{4}(x^2 - a^2)^2$$

This has two degenerate minima at $x = \pm a$. Classically, a particle sitting at$x = -a$ can never reach $x = +a$ if its energy is below the barrier. Quantum mechanically, it can tunnel.

In the Euclidean path integral, the tunneling amplitude is dominated by solutions of the Euclidean equations of motion:

$$m\frac{d^2 x}{d\tau^2} = V'(x)$$

Note the crucial sign flip compared to Minkowski time. The Euclidean equation of motion describes a particle moving in the inverted potential $-V(x)$, where the barrier becomes a valley!

💡Inverted Potential Trick

In real time, the particle cannot classically cross the barrier. But in Euclidean (imaginary) time, the potential is flipped: $V \to -V$. The barrier becomes a hill that the particle can roll over. This "Euclidean classical path" is the instanton.

The Instanton Solution

The instanton interpolates from $x = -a$ at $\tau = -\infty$ to $x = +a$ at$\tau = +\infty$. Using the first integral of the Euclidean equation of motion ($\frac{m}{2}\dot{x}^2 - V(x) = 0$ for zero energy):

$$\boxed{x_{\text{inst}}(\tau) = a\tanh\left[\frac{\omega(\tau - \tau_0)}{2}\right], \qquad \omega = a\sqrt{2\lambda/m}}$$

where $\tau_0$ is the "center" of the instanton (a collective coordinate reflecting time-translation symmetry). The instanton is localized in Euclidean time with width$\sim 1/\omega$, hence the name: it happens in an "instant."

Instanton Action

The Euclidean action of the instanton is:

$$S_{\text{inst}} = \int_{-\infty}^{\infty} d\tau\left[\frac{m}{2}\dot{x}_{\text{inst}}^2 + V(x_{\text{inst}})\right] = \int_{-a}^{a} dx\,\sqrt{2mV(x)} = \frac{2\sqrt{2\lambda}\,m\,a^3}{3}$$

2.9 Functional Determinants and the Tunneling Amplitude

To compute the full tunneling amplitude, we need the Gaussian fluctuation determinant around the instanton. Expanding $x(\tau) = x_{\text{inst}}(\tau) + y(\tau)$:

$$S_E[x_{\text{inst}} + y] = S_{\text{inst}} + \frac{1}{2}\int d\tau\; y(\tau)\left[-m\frac{d^2}{d\tau^2} + V''(x_{\text{inst}}(\tau))\right]y(\tau) + \cdots$$

The fluctuation operator has a zero mode $y_0(\tau) \propto \dot{x}_{\text{inst}}(\tau)$, corresponding to translations of the instanton center $\tau_0$. This zero eigenvalue must be treated separately by converting to a collective coordinate integral:

$$\int \mathcal{D}y \to \int_{-\beta\hbar/2}^{\beta\hbar/2} d\tau_0\, \sqrt{\frac{S_{\text{inst}}}{2\pi\hbar}} \times \int' \mathcal{D}y$$

where the prime denotes omission of the zero mode. The factor $\sqrt{S_{\text{inst}}/(2\pi\hbar)}$comes from the Jacobian of the change of variables. The remaining determinant (with zero mode removed) gives a finite number that we call K:

$$\mathcal{K} = \sqrt{\frac{S_{\text{inst}}}{2\pi\hbar}}\left[\frac{\det'(-m\partial_\tau^2 + V''(x_{\text{inst}}))}{\det(-m\partial_\tau^2 + \omega^2)}\right]^{-1/2}$$

Dilute Gas Approximation

For large Euclidean time $\beta\hbar$, the dominant configurations are sequences of widely separated instantons and anti-instantons. Summing over n instanton-anti-instanton pairs in the dilute gas approximation:

$$\langle a | e^{-\beta\hat{H}} | {-a}\rangle = \frac{C}{2}\left[e^{-\beta E_+} - e^{-\beta E_-}\right]$$

where the energy splitting between the symmetric and antisymmetric ground states is:

$$\boxed{\Delta E = E_- - E_+ = 2\hbar\,\mathcal{K}\, e^{-S_{\text{inst}}/\hbar}}$$

💡Non-perturbative Physics

The tunneling splitting $\Delta E \propto e^{-S_{\text{inst}}/\hbar}$ is non-perturbative: it vanishes to all orders in perturbation theory since$e^{-1/g}$ has a zero Taylor expansion around g = 0. Instantons capture physics invisible to Feynman diagrams. This is why path integral methods are indispensable for tunneling, vacuum decay, and confinement in QCD.

2.10 Worked Example: Tunneling in the Double Well

Let us assemble the complete calculation for the quartic double well$V(x) = \lambda(x^2 - a^2)^2/4$ with $m = 1$.

Step 1: Identify the instanton

$x_{\text{inst}}(\tau) = a\tanh[\omega(\tau - \tau_0)/2]$ with $\omega = a\sqrt{2\lambda}$.

Step 2: Compute the instanton action

$S_{\text{inst}} = \int_{-a}^{a} dx\,\sqrt{2V(x)} = \frac{2\sqrt{2\lambda}\,a^3}{3} = \frac{\omega^3}{3\lambda}$

Step 3: Evaluate the fluctuation determinant

The fluctuation operator around the instanton has a Poschl-Teller potential. The ratio of determinants (with zero mode removed) gives:

$$\left[\frac{\det'(-\partial_\tau^2 + V''(x_{\text{inst}}))}{\det(-\partial_\tau^2 + \omega^2)}\right]^{-1/2} = \frac{1}{2}\sqrt{\frac{6}{\pi}}$$

Step 4: Assemble the result

Combining the zero-mode Jacobian with the fluctuation determinant:

$$\mathcal{K} = \sqrt{\frac{S_{\text{inst}}}{2\pi\hbar}} \times \frac{1}{2}\sqrt{\frac{6}{\pi}} = \frac{1}{2}\sqrt{\frac{3S_{\text{inst}}}{\pi^2\hbar}}$$

Step 5: Energy splitting

$$\boxed{\Delta E = \hbar\omega\sqrt{\frac{6S_{\text{inst}}}{\pi\hbar}}\; e^{-S_{\text{inst}}/\hbar}}$$

This non-perturbative result captures the quantum tunneling between the two wells and has been verified numerically to excellent precision.

Key Concepts (This Page)

  • Semiclassical approximation: expand to quadratic order around classical path
  • WKB prefactor is $(\det\hat{O})^{-1/2}$, the Van Vleck determinant
  • Instantons are finite-action solutions of Euclidean equations of motion
  • Double-well instanton: $x(\tau) = a\tanh[\omega(\tau-\tau_0)/2]$
  • Zero modes from symmetries require collective coordinate treatment
  • Tunneling splitting $\Delta E \propto e^{-S_{\text{inst}}/\hbar}$ is non-perturbative
  • Dilute instanton gas gives the complete tunneling amplitude
Prev: Harmonic Oscillator & Euclidean PINext Chapter: Path Integrals in QFT