Euclidean path integrals, instantons, and tunneling between vacua

Intuition Beginner

The path integral from unit 12.10.01 sums amplitudes over all paths with an oscillatory phase $e^{iS/\hslash }$. The oscillations make the integral hard to evaluate and mathematically problematic — the "measure" on path space is not a genuine measure.

There is a trick that converts the oscillatory integral into a decaying one. Replace physical time $t$ with imaginary time $\tau =it$. This substitution — called a Wick rotation — turns the phase factor $e^{iS/\hslash }$ into a real exponential $e^{-S_E/\hslash }$, where $S_E$ is the Euclidean action. Paths with large Euclidean action are exponentially suppressed. Paths with small action dominate. The mathematical difficulties disappear: the Euclidean path integral is a genuine probability integral over path space, identical in structure to the partition function of classical statistical mechanics.

This link between quantum mechanics and statistical mechanics is deep. A $d$-dimensional quantum system at zero temperature maps onto a $(d+1)$-dimensional classical statistical system. The extra dimension is imaginary time. A quantum particle in one dimension maps to a classical polymer in two dimensions. A quantum field in four dimensions maps to a classical statistical field in five dimensions. This correspondence is one of the most powerful organizing principles in theoretical physics.

In the Euclidean path integral, classical paths play a special role. A path that makes the Euclidean action stationary — a solution of the Euclidean equations of motion — is a saddle point of the integral. Near such a path, the exponential varies slowly and contributes disproportionately. These saddle-point solutions are called instantons.

An instanton is a classical solution in Euclidean time that is localized: it approaches a constant (usually a vacuum) in the far past and a different constant (or the same vacuum) in the far future. The name reflects the fact that the solution is concentrated at an "instant" of Euclidean time. Physically, instantons describe tunneling — transitions between classically separated configurations that are forbidden in ordinary classical mechanics but allowed by quantum mechanics.

The simplest example is a particle in a double-well potential $V(x)=\lambda (x^2-a^2{)}^2$. Classically the particle sits in one well or the other, separated by a barrier. Quantum mechanically it can tunnel. The instanton is the Euclidean classical trajectory that starts in one well, crosses the barrier, and ends in the other. Its Euclidean action controls the tunneling rate: the transition amplitude is proportional to $e^{-S_E/\hslash }$.

Visual Beginner

The key visual: in real time the barrier blocks motion; in Euclidean time the barrier becomes a valley and the particle rolls freely through. The instanton is the Euclidean trajectory that traverses this valley.

Worked example Beginner

Consider a particle of mass $m$ in the inverted double-well. The Euclidean Lagrangian is $L_E=\frac{1}{2}m(dx/d\tau {)}^2+\lambda (x^2-a^2{)}^2$ (note the plus sign on the potential — the Euclidean Lagrangian is kinetic plus potential, not kinetic minus potential).

The equation of motion in Euclidean time is $m{d}^{2}x/d{\tau }^2=+V^{\prime }(x)=+4\lambda x(x^2-a^2)$. This is Newton's equation in the inverted potential $-V(x)$.

The instanton solution interpolates between $x=-a$ as $\tau \to -\infty$ and $x=+a$ as $\tau \to +\infty$. Energy conservation in the inverted potential gives $\frac{1}{2}m(dx/d\tau {)}^2=\lambda (x^2-a^2{)}^2$ (both sides start at zero at the hilltop). Separating variables and integrating:

$$x_{\mathrm{inst}}(\tau )=a\tanh \left(\frac{\tau -{\tau }_0}{w}\right),w=\frac{1}{a}\sqrt{\frac{m}{2\lambda }}.$$

The parameter ${\tau }_0$ is the "center" of the instanton — the time at which the particle crosses $x=0$. The width $w$ sets how spread out the transition is. The Euclidean action of this single instanton is

$$S_0=\frac{2\sqrt{2m\lambda }}{3}{a}^3.$$

The tunneling amplitude between the two wells scales as $e^{-S_0/\hslash }$. This is the same exponential that appears in the WKB tunneling calculation from unit 12.01.03 — the instanton provides a path-integral derivation of the same result.

Check your understanding Beginner

Formal definition Intermediate+

Wick rotation and the Euclidean action

Given a quantum-mechanical system with Lagrangian $L=\frac{1}{2}m{\dot{x}}^2-V(x)$, the Minkowski path integral for the propagator is

$$K(x_f,t_f;x_i,t_i)=\int \mathcal{D}x(t)e^{iS[x]/\hslash }$$

where $S[x]={\int }_{t_i}^{t_f}Ldt$ is the classical action. The substitution $t=-i\tau$ (Wick rotation) gives $dt=-id\tau$ and $\dot{x}=dx/dt=idx/d\tau$. The action transforms as

$$S=\int \left(\frac{1}{2}m{\dot{x}}^2-V\right)dt=\int \left(-\frac{1}{2}m{\left(\frac{dx}{d\tau }\right)}^2-V\right)(-id\tau )=i\int \left(\frac{1}{2}m{\left(\frac{dx}{d\tau }\right)}^2+V\right)d\tau .$$

Defining the Euclidean action

$$\boxed{S_E[x]={\int }_{{\tau }_i}^{{\tau }_f}\left[\frac{1}{2}m{\left(\frac{dx}{d\tau }\right)}^2+V(x(\tau ))\right]d\tau ,}$$

the Minkowski exponent becomes $e^{iS/\hslash }=e^{-S_E/\hslash }$, and the Euclidean propagator is

$$K_E(x_f,{\tau }_f;x_i,{\tau }_i)=\int \mathcal{D}x(\tau )e^{-S_E[x]/\hslash }.$$

For $V\ge 0$, the integrand $e^{-S_E/\hslash }$ is a positive, decaying function of the Euclidean action. On a finite time interval with time-slicing, $K_E$ is a convergent integral — the Wiener integral of Brownian-motion theory (unit 12.10.01). The Euclidean path integral is mathematically well-defined as a probability measure on path space.

Connection to statistical mechanics

For periodic boundary conditions $x(0)=x(T_E)$, the Euclidean path integral over loops of period $T_E$ computes the trace of $e^{-T_{E}H/\hslash }$:

$$Z(T_E)=\mathrm{Tr}{e}^{-T_{E}H/\hslash }=\oint \mathcal{D}x(\tau )e^{-S_E[x]/\hslash },$$

where the integral is over all continuous paths satisfying $x(0)=x(T_E)$. Setting $T_E=\beta \hslash$ gives the canonical partition function $Z(\beta )=\mathrm{Tr}{e}^{-\beta H}$. The inverse temperature $\beta$ is the period of imaginary time.

This identification is the basis of the quantum-statistical correspondence: a $d$-dimensional quantum system at temperature $T=1/(k_B\beta )$ is equivalent to a $(d+1)$-dimensional classical statistical system of extent $\beta \hslash$ in the imaginary-time direction.

Instantons as finite-action saddle points

An instanton is a finite-action solution of the Euclidean equations of motion

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

with boundary conditions $x(\tau )\to x_{\pm }$ as $\tau \to \pm \infty$, where $V(x_{\pm })=0$ (vacuum configurations). Finite action requires $S_E<\infty$, which demands that $x(\tau )$ approach vacuum configurations at temporal infinity.

Bogomolny bound. For the Euclidean action of a one-dimensional system, completing the square gives

$$S_E=\int \left[\frac{1}{2}m{\left(\frac{dx}{d\tau }\right)}^2+V(x)\right]d\tau =\int \frac{1}{2m}{\left(\sqrt{m}\frac{dx}{d\tau }\mp \sqrt{2mV(x)}\right)}^{2}d\tau \pm \int \sqrt{2mV(x)}dx.$$

The first integral is non-negative, so

$$S_E\ge \left|{\int }_{x_i}^{x_f}\sqrt{2mV(x)}dx\right|,$$

with equality (the Bogomolny equation) when $\sqrt{m}dx/d\tau =\pm \sqrt{2mV(x)}$, i.e., $dx/d\tau =\pm \sqrt{2V(x)/m}$. This first-order equation replaces the second-order Euler-Lagrange equation and selects the minimum-action path between vacua. Solutions of the Bogomolny equation are automatically solutions of the full Euclidean equations of motion (by differentiation), but not vice versa.

The double-well instanton

For $V(x)=\lambda (x^2-a^2{)}^2$ the vacua are $x=\pm a$. The instanton connecting $-a$ to $+a$ satisfies the Bogomolny equation

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

This is a separable ODE. Integrating:

$${\int }_{-a}^x\frac{dy}{a^2-y^2}=\sqrt{\frac{2\lambda }{m}}(\tau -{\tau }_0)$$

gives

$$\boxed{x_{\mathrm{inst}}(\tau )=a\tanh \left(\frac{\tau -{\tau }_0}{w}\right),w=\frac{1}{a}\sqrt{\frac{m}{2\lambda }}.}$$

The Bogomolny bound gives the instanton action:

$$S_0={\int }_{-a}^{+a}\sqrt{2m\lambda }(a^2-x^2)dx=\sqrt{2m\lambda }{\left[a^{2}x-\frac{x^3}{3}\right]}_{-a}^{+a}=\frac{4\sqrt{2m\lambda }}{3}{a}^3.$$

The anti-instanton $x_{\overline{\mathrm{inst}}}(\tau )=-a\tanh ((\tau -{\tau }_0)/w)$ interpolates from $+a$ to $-a$ and has the same action.

The dilute instanton gas

For the ground-state energy splitting in the double well, the leading contribution comes from a single instanton. But the full path integral includes configurations with any number of well-separated instantons and anti-instantons — the instanton gas.

In the dilute approximation, instantons are widely separated compared to their width $w$, so multi-instanton configurations factorize. The Euclidean path integral on a time interval $[-T/2,T/2]$ with $T\to \infty$ is

$$Z\approx Z_0\sum _{n=0}^{\infty }\frac{(KT{)}^n}{n!}{e}^{-n{S}_0/\hslash }=Z_0\exp \left(KT{e}^{-S_0/\hslash }\right),$$

where $Z_0$ is the path integral in a single well (no tunneling) and $K$ is a determinant factor from Gaussian fluctuations around the instanton (the fluctuation prefactor). The sum is over even and odd numbers of transitions, weighted by combinatorics for the placement of $n$ instantons/anti-instantons in the interval $T$.

The ground-state energy splitting follows. The two lowest energies are $E_{\pm }=E_0\mp \hslash K{e}^{-S_0/\hslash }$, giving a splitting

$$\boxed{\Delta E=E_{+}-E_{-}=2\hslash K{e}^{-S_0/\hslash }.}$$

The splitting is exponentially small in $1/\hslash$ — a non-perturbative effect invisible at every order in the ordinary perturbation series in $\hslash$.

Counterexamples and caveats

• The dilute-instanton-gas approximation breaks down when the instanton width $w$ becomes comparable to the mean separation $T/n$. This happens when $e^{-S_0/\hslash }$ is not very small. The approximation requires $S_0/\hslash \gg 1$.
• The instanton is a saddle point, not a minimum, of the Euclidean action in all fluctuation directions. The translation zero mode (shifting ${\tau }_0$) has zero eigenvalue and must be handled by a collective-coordinate integration rather than a Gaussian fluctuation integral. There is also exactly one negative eigenvalue for the instanton in the double well — this does not signal an instability but rather reflects the fact that the instanton is a saddle, not a minimum, in the full path space.
• Instanton methods in quantum mechanics give the leading non-perturbative contribution. Higher instanton sectors (pairs, triplets) contribute at higher exponential order $e^{-2S_0/\hslash }$, $e^{-3S_0/\hslash }$, etc.

Key theorem with proof Intermediate+

Theorem (Instanton contribution to the ground-state splitting). For a particle of mass $m$ in the symmetric double-well potential $V(x)=\lambda (x^2-a^2{)}^2$, the tunneling-induced splitting between the two lowest energy levels is

$$\Delta E=\hslash {\left(\frac{2S_0}{\pi m}\right)}^{1/2}\frac{2\sqrt{2\lambda }a}{\hslash }{e}^{-S_0/\hslash }(1+O(e^{-S_0/\hslash })),$$

where $S_0=\frac{4\sqrt{2m\lambda }}{3}{a}^3$ is the single-instanton action.

Proof. We evaluate the Euclidean path integral in the dilute-instanton-gas approximation and extract the ground-state splitting from the partition function.

Step 1: Single-instanton contribution. Expand the Euclidean action about the instanton $x_{\mathrm{inst}}(\tau )$: $x(\tau )=x_{\mathrm{inst}}(\tau )+\eta (\tau )$. The fluctuation $\eta$ satisfies $\eta (\pm \infty )\to 0$. To second order:

$$S_E[x_{\mathrm{inst}}+\eta ]=S_0+\frac{1}{2}\int \eta (\tau )\left[-m\frac{d^2}{d{\tau }^2}+V^{\prime \prime }(x_{\mathrm{inst}}(\tau ))\right]\eta (\tau )d\tau .$$

The fluctuation operator $M=-m{d}^2/d{\tau }^2+V^{\prime \prime }(x_{\mathrm{inst}})$ has a discrete spectrum. The eigenvalue problem $M{\psi }_n={ϵ}_n{\psi }_n$ yields:

• A zero mode ${\psi }_0\propto d{x}_{\mathrm{inst}}/d\tau$ with ${ϵ}_0=0$, from translation invariance (${\tau }_0$ is arbitrary).
• One negative mode ${ϵ}_1<0$, reflecting the saddle-point nature of the instanton.
• A continuum of positive modes for $n\ge 2$.

The Gaussian integral over the nonzero modes gives $\det {}^{\prime }(M{)}^{-1/2}$ where the prime excludes the zero mode. The zero mode is traded for an integral over the collective coordinate ${\tau }_0$ via the Jacobian $d{\eta }_0=(d{x}_{\mathrm{inst}}/d\tau )d{\tau }_0/\sqrt{S_0/m}$:

$$\int \frac{d{\eta }_0}{\sqrt{2\pi \hslash }}=\sqrt{\frac{S_0}{2\pi m{\hslash }^2}}\int d{\tau }_0.$$

The single-instanton contribution to $Z$ on the interval $[-T/2,T/2]$ is:

$$Z_1=T\sqrt{\frac{S_0}{2\pi m{\hslash }^2}}\frac{\det (-m{d}^2/d{\tau }^2+{\omega }^2{)}^{1/2}}{\det {}^{\prime }(M{)}^{1/2}}{e}^{-S_0/\hslash },$$

where the numerator is the harmonic-oscillator determinant about a single well ($\omega =\sqrt{8\lambda a^2/m}$ is the well frequency) and the denominator is the fluctuation determinant about the instanton (excluding the zero mode).

Step 2: Multi-instanton sum. In the dilute approximation, an $n$-instanton configuration is a superposition of $n$ widely separated instantons (and anti-instantons alternating between them). The fluctuation determinant factorizes into a product of single-instanton determinants. The combinatorics of placing $n$ objects in an interval of length $T$ gives $T^n/n!$. Summing over both even and odd numbers of transitions:

$$Z=Z_0\sum _{\text{even }n}\frac{(K_{0}T{)}^n}{n!}{e}^{-n{S}_0/\hslash }+Z_0\sum _{\text{odd }n}\frac{(K_{0}T{)}^n}{n!}{e}^{-n{S}_0/\hslash }$$

where $K_0=\sqrt{S_0/(2\pi m{\hslash }^2)}\cdot [\det ({\omega }^2)/\det {}^{\prime }(M){]}^{1/2}$. These are $\cosh$ and $\sinh$ series:

$$Z=Z_0\left[\cosh (K_{0}T{e}^{-S_0/\hslash })+\sinh (K_{0}T{e}^{-S_0/\hslash })\right]=Z_0{e}^{K_{0}T{e}^{-S_0/\hslash }}.$$

Step 3: Extract the splitting. The partition function at large $T$ is dominated by the ground state: $Z\sim e^{-E_{0}T/\hslash }$. Comparing with $Z=Z_0\exp (K_{0}T{e}^{-S_0/\hslash })$, the shifted ground-state energy is $E_0^{\prime }=E_0-\hslash K_0{e}^{-S_0/\hslash }$. The splitting between the symmetric and antisymmetric combinations is $\Delta E=2\hslash K_0{e}^{-S_0/\hslash }$.

Evaluating the determinant ratio explicitly (Rajaraman, Ch. 10) gives $K_0=(S_0/(\pi m){)}^{1/2}(2\sqrt{2\lambda }a/\hslash )$, completing the formula. $\square$

Exercises Intermediate+

Theta vacua and the topological angle Master

Superselection and the theta vacuum

In the double well, the instanton gas produces the two true ground states $|\pm ⟩=(|L⟩\pm |R⟩)/\sqrt{2}$, symmetric and antisymmetric superpositions of the left-well and right-well states. The splitting is exponentially small, so for most purposes the wells are independent — but the true vacuum is a superposition.

For a periodic potential $V(x)=V_0(1-\cos (x/a))$ with infinitely many degenerate minima at $x=2\pi na$, the instanton gas connects any pair of adjacent minima. Each instanton carries a winding number $\nu =+1$ (tunneling to the right) and each anti-instanton carries $\nu =-1$ (tunneling to the left). The net winding number ${\nu }_{\mathrm{net}}$ of a path is a topological invariant: it counts the net number of rightward transitions.

The theta-vacuum is a superposition over all topological sectors:

$$|\theta ⟩=\sum _{n\in \mathbb{Z}}{e}^{in\theta }|n⟩,$$

where $|n⟩$ denotes the state localized in the $n$-th well. The Euclidean path integral in the $\theta$-vacuum includes a phase $e^{i{\nu }_{\mathrm{net}}\theta }$ for each configuration with winding number ${\nu }_{\mathrm{net}}$:

$$Z(\theta )=\sum _{\nu }{e}^{i\nu \theta }{Z}_{\nu },$$

where $Z_{\nu }$ is the path integral restricted to configurations of winding number $\nu$. This is equivalent to adding $i\theta$ times the winding number to the Euclidean action: $S_E\to S_E-i\hslash \theta \nu$.

The ground-state energy in the $\theta$-vacuum is

$$E(\theta )=-\hslash K{e}^{-S_0/\hslash }\cos \theta ,$$

where $K$ is the instanton fluctuation prefactor. Different $\theta$ values correspond to different superselection sectors — they do not mix under local operators. In field theory this becomes the QCD vacuum angle (unit 12.18.04).

Dilute-gas evaluation of $Z(\theta )$

In the dilute-instanton-gas approximation with both instantons ($\nu =+1$) and anti-instantons ($\nu =-1$), the partition function at fixed $\theta$ is

$$Z(\theta )=Z_0\sum _{n_{+},n_{-}=0}^{\infty }\frac{(K_{0}T{e}^{-S_0/\hslash }{)}^{n_{+}+n_{-}}}{n_{+}!n_{-}!}{e}^{i(n_{+}-n_{-})\theta }.$$

Summing over $n_{+}$ and $n_{-}$ independently:

$$Z(\theta )=Z_0\exp \left(K_{0}T{e}^{-S_0/\hslash }(e^{i\theta }+e^{-i\theta })\right)=Z_0\exp \left(2K_{0}T{e}^{-S_0/\hslash }\cos \theta \right).$$

The $\theta$-dependent ground-state energy follows: $E(\theta )=E_0-2\hslash K_0{e}^{-S_0/\hslash }\cos \theta$. At $\theta =0$ the energy is minimized (constructive interference of all instanton sectors); at $\theta =\pi$ it is maximized (destructive interference). The $\theta =\pi$ point is where level crossings can occur — this is the origin of spontaneous CP violation in certain field theories.

Bounce solutions and false vacuum decay Master

The metastable vacuum

Consider a potential with a local (but not global) minimum at $x=0$ — a false vacuum. The particle can tunnel out through the barrier to the true vacuum at lower energy. The Euclidean trajectory that describes this process is a bounce: it starts at the false vacuum, traverses the barrier, reaches a turning point, and returns to the false vacuum.

The bounce $x_B(\tau )$ satisfies the Euclidean equation of motion with boundary conditions $x_B(\tau )\to 0$ as $\tau \to \pm \infty$ and $x_B({\tau }_0)=x_{\mathrm{exit}}$ where $x_{\mathrm{exit}}$ is the classical turning point at the exit energy. Unlike the instanton (which connects two different vacua), the bounce starts and ends at the same point — but it is not the constant solution $x=0$. The bounce is a nontrivial saddle point of the Euclidean action.

The decay rate

Coleman (1977) showed that the false-vacuum decay rate per unit volume (in field theory) or per unit time (in quantum mechanics) is

$$\Gamma =A{e}^{-S_B/\hslash },$$

where $S_B$ is the bounce action and $A$ is a ratio of fluctuation determinants. The bounce has exactly one negative eigenvalue in its fluctuation spectrum. Following the rule that $\sqrt{-|ϵ|}=i\sqrt{|ϵ|}$, the negative mode produces an imaginary part in the ground-state energy:

$$\mathrm{Im}{E}_0=\frac{\hslash }{2}A{e}^{-S_B/\hslash },$$

and the decay rate is $\Gamma =2\mathrm{Im}{E}_0/\hslash$. The factor of 2 arises because the negative-mode Gaussian integral is taken along the steepest-descent contour through the saddle, not along the real axis.

Thin-wall approximation

For a nearly degenerate false vacuum (barrier height $ϵ$ small compared to the barrier width), the bounce is well approximated by a thin-wall bubble. In quantum mechanics this is a trajectory that spends most of its time near $x=0$ and $x=x_{\mathrm{exit}}$, transitioning rapidly between them. In field theory the thin-wall bounce is an $O(4)$-symmetric bubble of true vacuum embedded in false vacuum, with the wall thickness much smaller than the bubble radius. The bounce action scales as

$$S_B\sim \frac{(\text{surface tension}{)}^d}{{ϵ}^{d-1}},$$

where $d$ is the number of spatial dimensions and $ϵ$ is the energy density difference between false and true vacua. As $ϵ\to 0$ the bounce action diverges and the false vacuum becomes stable — consistent with the degenerate limit.

Connections Master

• 12.10.01 Path integral formulation. This unit is the direct continuation. The Wick rotation and Euclidean action defined here transform the oscillatory path integral of 12.10.01 into a convergent integral amenable to saddle-point methods. The instanton is a saddle point of the Euclidean action.

• 12.01.03 WKB tunneling. The instanton tunneling exponent $e^{-S_0/\hslash }$ agrees with the WKB result $e^{-2K/\hslash }$ from 12.01.03, with $S_0=2K$. The instanton method derives this exponent from a saddle-point calculation in path space rather than from the WKB connection formulas. The two approaches are complementary: WKB works for simple barriers, while instantons generalize to multi-dimensional and field-theoretic settings.

• 12.04.03 Finite square well and tunneling. Unit 12.04.03 introduces tunneling through rectangular barriers. The instanton provides the semiclassical path-integral perspective on the same phenomenon, valid for smooth barriers where the WKB/instanton approximation applies.

• 12.18.04 Theta vacua and the strong CP problem. The theta angle constructed here from the instanton gas in quantum mechanics is the direct ancestor of the QCD theta angle. In Yang-Mills theory the instantons carry integer topological charge (Pontryagin index), the theta angle parameterizes superselection sectors of the vacuum, and the question of why $\theta \approx 0$ in nature is the strong CP problem.

• 12.18.13 Vortices, Nielsen-Olesen flux tubes. The instanton in one-dimensional quantum mechanics is the simplest member of a family of topological defects: vortices in two dimensions, monopoles in three, instantons in four. The Bogomolny bound and the dilute-gas approximation both generalize to these higher-dimensional topological solitons.

• Statistical mechanics and the KMS condition. The identification $Z=\mathrm{Tr}{e}^{-\beta H}=\oint e^{-S_E/\hslash }$ connects to the Kubo-Martin-Schwinger condition characterizing thermal equilibrium. Imaginary time of duration $\beta \hslash$ is the temperature axis.

Historical notes Master

The term "instanton" was coined by 't Hooft in 1976 (originally he used "pseudoparticle"; the name "instanton" is attributed to a suggestion by Callan at a 1976 Ettore Majorana school). The first explicit instanton solution in Yang-Mills theory was found by Belavin, Polyakov, Schwarz, and Tyupkin (BPST) in 1975 — the BPST instanton, a classical solution of the Euclidean Yang-Mills equations with finite action and nontrivial topology (meaning topologically distinct from the vacuum).

The quantum-mechanical precursor was implicit in the work of Langer (1967) on metastable-state decay and was developed explicitly by Garg (1970) and by Mathews et al. The double-well instanton calculation in its modern form is due to Polyakov (1975, unpublished lectures) and was popularized by Coleman's 1977 Erice lectures ("The Uses of Instantons," published in Aspects of Symmetry).

Coleman's 1977 paper "Fate of the False Vacuum" introduced the bounce solution and the thin-wall approximation for false-vacuum decay in field theory. The paper showed that the decay rate is proportional to $e^{-S_B/\hslash }$ where $S_B$ is the bounce action, and that the negative fluctuation mode about the bounce gives an imaginary part to the energy — the signal of instability. This work, together with the earlier paper by Voloshin, Kobzarev, and Okun (1974), established the semiclassical theory of vacuum decay.

The Bogomolny bound originates in the work of Bogomolny (1976) on magnetic monopoles and was independently discovered by Prasad and Sommerfield (1975). The trick of completing the square in the energy (or Euclidean action) to obtain a first-order equation saturating the bound is a recurring theme in soliton and instanton physics.

Wick rotation was introduced by Wick (1954) for analytic continuation of Feynman amplitudes. The systematic exploitation of Euclidean methods in constructive quantum field theory is due to Symanzik, Nelson, and especially Osterwalder and Schrader (1973-1975), who showed that a Euclidean field theory satisfying reflection positivity reconstructs to a unitary Minkowski theory.

The dilute-instanton-gas approximation was used by 't Hooft (1976) to resolve the $U(1{)}_A$ problem in QCD (the absence of a ninth pseudo-Goldstone boson) and to derive the anomalous breaking of the axial $U(1)$ symmetry from instanton effects. The instanton liquid model of Shifman, Vainshtein, and Zakharov (1980) went beyond the dilute gas to describe the dense instanton ensemble in QCD at low energies.

Bibliography Master

Originator papers:

• Belavin, A. A., Polyakov, A. M., Schwarz, A. S. & Tyupkin, Yu. S., "Pseudoparticle solutions of the Yang-Mills equations", Phys. Lett. B 59 (1975), 85-87.
• 't Hooft, G., "Computation of the quantum effects due to a four-dimensional pseudoparticle", Phys. Rev. D 14 (1976), 3432-3450.
• Coleman, S., "Fate of the false vacuum: semiclassical theory", Phys. Rev. D 15 (1977), 2929-2936.
• Bogomolny, E. B., "The stability of classical solutions", Sov. J. Nucl. Phys. 24 (1976), 449-454.
• Wick, G. C., "Properties of Bethe-Salpeter wave functions", Phys. Rev. 96 (1954), 1124-1134.

Textbooks:

• Coleman, S., Aspects of Symmetry: Selected Erice Lectures (Cambridge, 1985), Ch. 7 "The Uses of Instantons".
• Polyakov, A. M., Gauge Fields and Strings (CRC, 1987), Ch. 3.
• Rajaraman, R., Solitons and Instantons: An Introduction to Solitons and Instantons in Quantum Field Theory (North-Holland, 1982), Ch. 8-10.
• Sakurai, J. J. & Napolitano, J., Modern Quantum Mechanics, 3rd ed. (Cambridge, 2021), §2.6.
• Feynman, R. P. & Hibbs, A. R., Quantum Mechanics and Path Integrals (McGraw-Hill, 1965), Ch. 10.

Modern references:

• Vainshtein, A. I., Zakharov, V. I., Novikov, V. A. & Shifman, M. A., "ABC of instantons", Sov. Phys. Usp. 25 (1982), 195-215.
• Schaefer, T. & Shuryak, E. V., "Instantons in QCD", Rev. Mod. Phys. 70 (1998), 323-426.
• Coleman, S., "The uses of instantons", in The Whys of Subnuclear Physics, ed. Zichichi (Plenum, 1979).
• Weinberg, E. J., Classical Solutions in Quantum Field Theory (Cambridge, 2012), Ch. 6-7.