Instantons in Quantum Mechanics

Here an instanton is a nonconstant finite-action Euclidean classical trajectory connecting two distinct degenerate minima. A return trajectory describing metastable escape is instead called a bounce. Instantons give semiclassical estimates of tunneling effects that are exponentially small and invisible to ordinary perturbation theory around a single classical minimum. An instanton is a saddle of an imaginary-time amplitude calculation, not a particle’s literal real-time route through the barrier.

Required background. Euclidean time and action derives the continuation and variational equation; double-well tunneling explains the spectral doublet.

The Euclidean action for one coordinate is

$$S_E[x] = \int d\tau \left[ \frac{m}{2} \left(\frac{dx}{d\tau}\right)^2 + V(x) \right].$$

Euclidean path integrals are weighted schematically by

$$e^{-S_E[x]/\hbar},$$

so finite-action saddle points can dominate exponentially small effects.

Double-Well Splitting evaluates the quartic trajectory and $S_0=4/3$ in a convention designed for direct comparison with numerical spectral gaps. The reusable saddle construction remains here.

Euclidean Equation of Motion

Stationarity of $S_E$ gives

$$m\frac{d^2x}{d\tau^2} = \frac{dV}{dx}.$$

This differs in sign from the real-time Newton equation. Equivalently, Euclidean motion in $V(x)$ resembles real-time motion in the inverted potential $-V(x)$.

Assume $V\geq0$ with isolated nondegenerate minima, after subtracting their common minimum value. A finite-action saddle connecting these minima approaches them with vanishing velocity as $\tau\to\pm\infty$. Without the subtraction, a nonzero vacuum energy produces an infinite background action.

Boundary Conditions

For a degenerate double well with minima at $x=-a$ and $x=a$, an instanton satisfies

$$x(\tau)\to -a \quad \text{as} \quad \tau\to-\infty,$$

and

$$x(\tau)\to a \quad \text{as} \quad \tau\to+\infty.$$

The anti-instanton has the opposite orientation. In a two-well system, repeated crossings alternate instantons and anti-instantons. Widely separated events form approximate saddles in a dilute expansion; their simple superposition is not an exact finite-separation solution.

First Integral

For a double well whose minima have $V=0$, the Euclidean energy

$$\frac{m}{2} \left(\frac{dx}{d\tau}\right)^2 - V(x)$$

is conserved. The finite-action boundary conditions set this conserved value to zero, so

$$\frac{m}{2} \left(\frac{dx}{d\tau}\right)^2 = V(x).$$

Therefore the instanton action can be written as

$$S_0 = \int_{x_-}^{x_+} dx\, \sqrt{2mV(x)}.$$

This is the zero-energy forbidden-region action. For the lowest quantum doublet, WKB instead uses an energy of order $\hbar\omega/2$. The leading $1/\hbar$ exponential agrees, but the energy dependence of the action must be matched carefully to compare prefactors.

Quartic Double-Well Instanton

For

$$V(x)=\lambda(x^2-a^2)^2, \qquad \lambda>0,\quad a>0,\quad m>0,$$

the minima are at $x=\pm a$. The small-oscillation frequency near a minimum is

$$\omega = \sqrt{\frac{8\lambda a^2}{m}}.$$

For the increasing path between the minima, the zero-energy first integral becomes

$$\frac{dx}{d\tau} =\sqrt{\frac{2\lambda}{m}}(a^2-x^2), \qquad -a<x<a.$$

The instanton solution can be written

$$x_{\mathrm{inst}}(\tau) = a \tanh \left[ \frac{\omega}{2} (\tau-\tau_0) \right].$$

The parameter $\tau_0$ is the instanton center. Translating the instanton in Euclidean time costs no action, so $\tau_0$ is a collective coordinate.

The action is

$$S_0 = \int_{-a}^{a} dx\, \sqrt{2m\lambda}(a^2-x^2) = \frac{4}{3} a^3 \sqrt{2m\lambda}.$$

Tunneling Splitting

Suppose a symmetric double well has an isolated lowest doublet of normalized exact eigenstates: an even state $|0\rangle$ with energy $E_0$ and an odd state $|1\rangle$ with energy $E_1>E_0$. Define the orthonormal combinations

$$|L\rangle=\frac{|0\rangle+|1\rangle}{\sqrt2}, \qquad |R\rangle=\frac{|0\rangle-|1\rangle}{\sqrt2}.$$

For a high, broad barrier, these states are localized near opposite minima, with the relative phase chosen to assign the labels $L$ and $R$. Their Euclidean transfer amplitude, for $T>0$, is exactly

$$\begin{aligned} \mathcal K_{RL}(T) &=\langle R|e^{-HT/\hbar}|L\rangle\\ &=\frac12\left(e^{-E_0T/\hbar}-e^{-E_1T/\hbar}\right). \end{aligned}$$

This identity follows from the specified exact doublet states. For arbitrary localized trial packets, retaining only the two lowest levels is an approximation and their overlaps set the spectral weights; suppressing higher states alone does not make those weights equal to $1/2$. The Euclidean kernel ratio explains the corresponding distinction for position-space kernels.

The off-diagonal amplitude therefore contains the splitting $\Delta E=E_1-E_0$. In the semiclassical regime $S_0/\hbar\gg1$, instanton calculus gives

$$\Delta E \sim A e^{-S_0/\hbar}.$$

The exponent $S_0$ comes from the classical Euclidean saddle. The prefactor $A$ has units of energy and comes from quadratic fluctuations around the saddle, normalization of the zero mode, and multi-instanton combinatorics. The factor $e^{-S_0/\hbar}$ is an amplitude-level saddle weight, not a tunneling probability or a normalized transfer amplitude by itself. The boundary conditions and observable determine which vacuum, instanton, anti-instanton, and repeated-event sectors must be assembled.

For many purposes, the exponent is the main physical information. Precise prefactors require more careful analysis than the leading saddle picture.

Tunneling Splittings derives how the dilute instanton sum becomes an even–odd spectral splitting and matches that result to Herring and WKB formulas.

Double-Well Instanton Numerical Check tests the normalized one-loop formula against converged parity gaps while keeping exponent accuracy, prefactor accuracy, and numerical resolution separate.

A metastable well poses a different boundary-value problem. Bounce Solutions develops the Euclidean round trip, its decay exponent, and the additional negative fluctuation mode.

Collective Coordinate Preview

Because the instanton can be centered at any $\tau_0$, the quadratic fluctuation operator around the instanton has a zero mode proportional to

$$\frac{dx_{\mathrm{inst}}}{d\tau}.$$

On the infinite Euclidean line the zero mode is nodeless for a monotone instanton. For the scalar fluctuation operator

$$\mathcal L=-\partial_\tau^2+V''(x_{\mathrm{inst}})/m,$$

the node theorem therefore places this mode at the bottom of the spectrum: there is no negative mode. A bounce has a different spectrum. Fixed endpoints at finite Euclidean time can lift translation invariance, so a small eigenvalue there must be studied as the interval grows.

The zero mode is replaced by integration over $\tau_0$, rather than an ordinary Gaussian integral. The normalization follows from

$$\int m(x'_{\mathrm{inst}})^2d\tau=S_0,$$

and yields $\sqrt{S_0/(2\pi\hbar)}\,d\tau_0$ with the usual mass-normalized mode measure. Fluctuation determinants explains the remaining normalization.

Relation to Perturbation Theory

Perturbation theory around one minimum produces a series in local fluctuations. It does not connect the two classical minima at any finite order. The instanton contribution has the form

$$e^{-S_0/\hbar},$$

which is nonanalytic at $\hbar=0$. This is why tunneling splittings are called nonperturbative.

The same saddle-sector logic extends to field theory, but spatial dependence, gauge constraints, extra zero modes, and renormalization change the calculation. Bridge to QFT Instantons explains those qualifications; quantum-mechanical prefactors cannot be transferred without rebuilding the fluctuation problem.

Exercises

1. Verify the quartic trajectory using both the zero-energy first-order equation and the second-order Euclidean equation. Check its boundary conditions and find its exponential tail scale.

Solution

Writing $u=\omega(\tau-\tau_0)/2$, one has $x'=a\omega\operatorname{sech}^2u/2$ and $x''=-a\omega^2\operatorname{sech}^2u\tanh u/2$. Since $a\omega/2=a^2\sqrt{2\lambda/m}$ and $a^2-x^2=a^2\operatorname{sech}^2u$, the increasing first-order equation holds. Equivalently,

$$\frac{m}{2}(x')^2 =\frac{ma^2\omega^2}{8}\operatorname{sech}^4u =\lambda a^4\operatorname{sech}^4u =V(x).$$

Substitution into $m x''=4\lambda x(x^2-a^2)$ also works because $m\omega^2=8\lambda a^2$. The limits $\tanh u\to\pm1$ give the required endpoints and $x'\to0$. As $\tau\to+\infty$, $a-x\simeq2a e^{-\omega(\tau-\tau_0)}$.

2. A single tunneling event has weight $k\,d\tau_0$, where $k$ has units of inverse time and includes $e^{-S_0/\hbar}$. What controls the dilute approximation? Must $kT$ be small?

Solution

The event width is of order $1/\omega$ and the typical separation is of order $1/k$. Diluteness requires $k/\omega\ll1$. The observation interval may be long enough that $kT$ is large: summing many widely separated events is precisely what produces the split spectral exponentials. Diluteness concerns local overlap, not the total number of events.

3. Starting from the Euclidean equation, show that the connecting instanton has zero Euclidean energy and derive its quartic action. Does reversing its orientation change that action?

Solution

Multiplying $mx''=V'(x)$ by $x'$ gives

$$\frac{d}{d\tau}\left[\frac{m}{2}(x')^2-V(x)\right]=0.$$

The asymptotic conditions $x'\to0$ and $V\to0$ set the conserved Euclidean energy to zero. For the increasing solution, $x'=\sqrt{2V/m}$, so

$$\begin{aligned} S_0 &=\int d\tau\,2V(x) =\int_{-a}^{a}dx\,\sqrt{2mV(x)}\\ &=\sqrt{2m\lambda}\left[a^2x-\frac{x^3}{3}\right]_{-a}^{a} =\frac{4a^3}{3}\sqrt{2m\lambda}. \end{aligned}$$

For the anti-instanton, both $x'$ and the orientation of the spatial integration reverse. The positive Euclidean action is unchanged; equivalently, the action integral uses $|dx|\sqrt{2mV(x)}$ along either orientation.

References

• S. Coleman, “The Uses of Instantons”, in Aspects of Symmetry, Cambridge University Press, 1985, pp. 265–350.
• H. Kleinert, Path Integrals in Quantum Mechanics, Statistics, Polymer Physics, and Financial Markets, 5th ed., World Scientific, 2009.
• R. Rajaraman, Solitons and Instantons, North-Holland, 1982.
• L. S. Schulman, Techniques and Applications of Path Integration, Dover, 2005. Euclidean propagators and their spectral interpretation.
• J. Zinn-Justin, Path Integrals in Quantum Mechanics, Oxford University Press, 2005.