False Vacuum Decay in Quantum Mechanics

A false vacuum in quantum mechanics is a long-lived state localized near a local minimum that can escape through a barrier. The word “vacuum” is inherited from field theory. In ordinary one-particle quantum mechanics, the relevant object is not the minimum of the potential itself and not an exact normalizable energy eigenstate. It is a metastable wavepacket, a quasibound level, or an outgoing resonance.

The central observable is a decay rate. With the convention

$$\mathcal E_F = E_R - \frac{i\hbar\gamma}{2},$$

the resonance-dominated survival probability behaves as

$$P_F(t) \simeq e^{-\gamma t}, \qquad \tau_F = \frac{1}{\gamma}.$$

Here $\gamma$ has units of inverse time, $\tau_F$ is the lifetime, and $\Gamma_E=\hbar\gamma$ is the corresponding energy width.

Required background. Bounce solutions derives the Euclidean saddle, barrier penetration supplies the transmission exponent, and unitary evolution gives the spectral survival amplitude. Euclidean path integrals explains imaginary-time kernels.

A quasibound level leaks through the forbidden interval between $x_b$ and $x_c$. The resulting exponential law is an intermediate-time approximation: survival begins quadratically for a state with finite energy variance and develops a threshold tail when the spectral assumptions below apply. The displayed time regimes are schematic and are not drawn on a common quantitative scale.

The Metastable Well

Let $V(x)$ have a local minimum at $x_F$:

$$V'(x_F)=0, \qquad V''(x_F)\gt0.$$

A barrier separates this false minimum from an escape region. Near $x_F$,

$$V(x) \simeq V_F + \frac{m\omega_F^2}{2} (x-x_F)^2,$$

so the well supports local oscillator-like levels. For the lowest one,

$$E_F \simeq V_F + \frac{\hbar\omega_F}{2},$$

before tunneling and anharmonic corrections are included.

The local level is not generally an eigenvalue of the full self-adjoint Hamiltonian. If the potential opens into a continuum, probability can escape and an outgoing resonance replaces the putative bound state. If the entire potential is confining, the exact spectrum remains real and discrete; leakage from one well is then followed by interference and recurrence rather than irreversible decay.

This distinction is foundational:

• Open escape geometry: outgoing boundary conditions define a resonance and an approximate decay rate.
• Closed finite geometry: exact evolution is quasiperiodic; a rate can describe only a coarse-grained or intermediate-time loss from a chosen region.
• Open-system geometry: environmental degrees of freedom may make exponential relaxation accurate over a wider interval, but the rate then depends on the system–environment model.

Calling a local minimum a false vacuum therefore abbreviates a state-preparation and boundary-condition problem.

Preparing the false-well state

One practical preparation is a normalized packet $\lvert F\rangle$ concentrated inside the well. Another is to confine the system temporarily, prepare a local eigenstate, and then remove the auxiliary wall. Preparation affects the occupied quasibound levels and early transients. The same resonance exponent is obtained only when both preparations predominantly populate the same narrow level; appreciable high-energy weight can escape much faster.

Choose a dividing point $x_d$ beyond the barrier and define a false-well projector, schematically,

$$\Pi_F = \Theta(x_d-\hat x).$$

The nonescape probability is

$$P_{\mathrm{ne}}(t) = \langle\psi(t)\vert \Pi_F \vert\psi(t)\rangle.$$

The survival probability of the initially prepared state is

$$P_{\mathrm{surv}}(t) = \left| \langle F\vert e^{-iHt/\hbar} \vert F\rangle \right|^2.$$

These observables are not identical. A packet can deform or dephase within the well, reducing its overlap with $\lvert F\rangle$ without having escaped. For an isolated narrow resonance after preparation transients have died away, both quantities often exhibit the same dominant exponential rate.

Flux, Survival, and Resonance

The three most useful descriptions emphasize different data:

Description	Defining quantity	What it measures
Nonescape	$\langle\Pi_F\rangle_t$	Probability remaining in a chosen well region
Survival	$\lvert\langle F\vert e^{-iHt/\hbar}\vert F\rangle\rvert^2$	Persistence of the specific prepared state
Resonance	Pole $\mathcal E_F=E_R-i\hbar\gamma/2$	Energy, width, and intermediate-time decay scale

For one escape boundary, the continuity equation gives

$$\frac{dP_{\mathrm{ne}}}{dt} = - j(x_d,t),$$

where

$$j(x,t) = \frac{\hbar}{m} \operatorname{Im} \left[ \psi^*(x,t) \frac{\partial\psi(x,t)}{\partial x} \right].$$

An instantaneous loss rate can be defined whenever $P_{\mathrm{ne}}(t)\gt0$:

$$\gamma_{\mathrm{inst}}(t) = - \frac{d}{dt} \ln P_{\mathrm{ne}}(t) = \frac{j(x_d,t)}{P_{\mathrm{ne}}(t)}.$$

It is approximately constant only in the exponential window. Outside that window the flux can reverse, so this instantaneous quantity need not remain positive. This flux definition makes clear that a decay rate is an observable statement about probability leaving a region, not merely the exponential of a Euclidean action.

Outgoing resonances

A one-dimensional resonance solves the stationary Schrödinger equation with an outgoing-wave condition in the escape channel. Schematically,

$$\psi_{\mathrm{res}}(x) \sim e^{+ikx} \qquad (x\to+\infty),$$

for an asymptotically constant potential, with no incoming component. Long-range or unbounded escape potentials require their corresponding outgoing asymptotics. The resulting complex energy

$$\mathcal E_F = E_R - \frac{i\Gamma_E}{2}$$

is a pole of the analytically continued resolvent or scattering matrix, usually on a nonphysical sheet. Its formal time dependence is

$$e^{-i\mathcal E_F t/\hbar} = e^{-iE_Rt/\hbar} e^{-\Gamma_Et/(2\hbar)}.$$

Taking the modulus squared gives

$$P_F(t) \propto e^{-\Gamma_Et/\hbar} = e^{-\gamma t}, \qquad \gamma = \frac{\Gamma_E}{\hbar}.$$

The outgoing resonance function is generally not square-integrable. It is a compact representation of the pole contribution to physical wavepacket evolution, not an ordinary Hilbert-space eigenvector.

Why Decay Is Only Approximately Exponential

Define the survival amplitude

$$\mathcal A_F(t) = \langle F\vert e^{-iHt/\hbar} \vert F\rangle.$$

By the spectral theorem,

$$\mathcal A_F(t) = \int_{E_{\min}}^\infty \rho_F(E) e^{-iEt/\hbar} \,dE,$$

where

$$\rho_F(E) = \langle F\vert \delta(E-H) \vert F\rangle.$$

Normalization requires

$$\int_{E_{\min}}^\infty \rho_F(E)\,dE = 1.$$

A narrow resonance produces an approximately Breit–Wigner peak,

$$\begin{aligned} \rho_F(E) &\simeq \rho_{\mathrm{pole}}(E) + \rho_{\mathrm{bg}}(E), \\ \rho_{\mathrm{pole}}(E) &= \frac{Z\Gamma_E}{2\pi D(E)}, \\ D(E) &= (E-E_R)^2 + (\Gamma_E/2)^2. \end{aligned}$$

Fourier transforming the isolated pole contribution gives the exponential law. The background, the physical threshold $E_{\min}$, and the details of state preparation generate deviations.

Short times: the quadratic law

Assuming the required energy moments through fourth order are finite, expanding the unitary evolution gives

$$\mathcal A_F(t) = 1 - \frac{i\langle H\rangle_F}{\hbar}t - \frac{\langle H^2\rangle_F}{2\hbar^2}t^2 + O(t^3).$$

Consequently,

$$P_{\mathrm{surv}}(t) = 1 - \frac{(\Delta H)_F^2}{\hbar^2}t^2 + O(t^4).$$

Thus the initial slope vanishes:

$$\left. \frac{dP_{\mathrm{surv}}}{dt} \right|_{t=0} = 0.$$

The scale

$$\tau_Z = \frac{\hbar}{(\Delta H)_F}$$

is often called the Zeno time. An exact exponential would have a nonzero negative slope at $t=0$, so it cannot describe sufficiently short unitary evolution of a state with finite energy variance.

Intermediate times: pole dominance

After preparation transients and before threshold effects dominate, a narrow isolated pole can control the Fourier integral:

$$\mathcal A_F(t) \simeq Z_{\mathrm{pole}} e^{-iE_Rt/\hbar} e^{-\gamma t/2},$$

and therefore

$$P_{\mathrm{surv}}(t) \simeq \lvert Z_{\mathrm{pole}}\rvert^2 e^{-\gamma t}.$$

This is the regime computed by standard WKB and bounce methods. A useful lifetime requires

$$\gamma \ll \omega_F,$$

so intrawell motion and preparation transients occur on a much shorter timescale than escape.

Long times: the threshold tail

For a lower-bounded Hamiltonian, the spectral density cannot be an exact Lorentzian over the entire real axis. Assume the prepared state has no bound-state component, its continuum spectral density is sufficiently smooth away from its lowest threshold, and other singularities give faster-decaying contributions. Suppose near that threshold

$$\rho_F(E) \sim C(E-E_{\min})^\alpha, \qquad \alpha\gt-1.$$

Then the endpoint of the spectral integral gives

$$\mathcal A_F(t) \sim C\, \Gamma(\alpha+1) e^{-iE_{\min}t/\hbar} \times e^{-i\pi(\alpha+1)/2} \left( \frac{\hbar}{t} \right)^{\alpha+1}.$$

so

$$P_{\mathrm{surv}}(t) \propto t^{-2(\alpha+1)}.$$

The crossover usually occurs only after many lifetimes because the pole contribution must first become extremely small. Nevertheless, the tail is conceptually essential: exact exponential decay for all $t\geq0$ is incompatible with a lower-bounded self-adjoint Hamiltonian.

WKB Escape Rate

Consider a quasibound level of energy $E$ with three turning points

$$x_a \lt x_b \lt x_c.$$

The interval $x_a\lt x\lt x_b$ is the classically allowed false well, while $x_b\lt x\lt x_c$ is the forbidden barrier. Define

$$p(x;E) = \sqrt{2m[E-V(x)]}$$

in the well and

$$\kappa(x;E) = \sqrt{2m[V(x)-E]}$$

under the barrier. The one-way forbidden action is

$$W(E) = \int_{x_b}^{x_c} \kappa(x;E)\,dx.$$

To leading exponential accuracy, the transmission probability per encounter with the barrier is

$$\mathcal T(E) \sim e^{-2W(E)/\hbar}.$$

The classical period of motion inside the false well is

$$T_{\mathrm{cl}}(E) = 2 \int_{x_a}^{x_b} \frac{m\,dx}{p(x;E)}.$$

For one escape barrier encountered once per period, the Gamow estimate is

$$\gamma_{\mathrm{WKB}}(E) \simeq \frac{1}{T_{\mathrm{cl}}(E)} e^{-2W(E)/\hbar}.$$

Near a harmonic false minimum,

$$T_{\mathrm{cl}} \simeq \frac{2\pi}{\omega_F},$$

and hence

$$\gamma_{\mathrm{WKB}} \simeq \frac{\omega_F}{2\pi} e^{-2W(E_F)/\hbar}.$$

The interpretation is “attempt frequency times escape probability per attempt.” It applies to a narrow quasibound resonance with weak transmission and an outgoing channel that does not return appreciable amplitude during the decay window. Environmental dephasing is not required: coherent Schrödinger evolution with outgoing boundary conditions gives the same leading resonance width.

What the simple prefactor omits

The factor $1/T_{\mathrm{cl}}$ is not a universal one-loop prefactor. Quantitative work may require:

• connection formulas at all turning points;
• the correct outgoing normalization and channel multiplicity;
• the energy shift of the quasibound level;
• barrier-top corrections when $W/\hbar$ is not large;
• interference from several escape paths;
• multidimensional stability determinants;
• environmental recrossing or dissipation.

The robust statement is

$$\ln\left( \frac{\gamma}{\omega_F} \right) = - \frac{2W}{\hbar} + O\!\left( \ln\frac{W}{\hbar} \right)$$

in a controlled high-barrier family. Prefactor accuracy requires more information than the leading exponent.

Bounce Estimate

The infinite-time bounce approaches the false minimum and returns to it; its subtracted action uses the minimum energy $V_F$. At an energy $E>V_F$, the Euclidean round trip instead joins the two finite-energy barrier turning points in finite time. Its fixed-energy abbreviated action, rather than its untransformed Euclidean action, is

$$\begin{aligned} B(E) &= 2 \int_{x_b}^{x_c} \sqrt{2m[V(x)-E]} \,dx \\ &= 2W(E). \end{aligned}$$

Thus

$$e^{-B(E)/\hbar} = e^{-2W(E)/\hbar},$$

matching the WKB probability suppression. For a round trip of duration $T(E)$, this action is $B(E)=S_E(T)-ET$; it is not obtained by imposing the false-minimum boundary condition at a different energy. In the limit $E\to V_F$, $T\to\infty$ and the background-subtracted infinite-time bounce action is recovered. The factor of two comes from the outward and return branches.

At leading semiclassical order,

$$\gamma_{\mathrm{bounce}} \simeq A e^{-B/\hbar},$$

where $A$ has units of inverse time. The saddle action controls the exponential; the translation zero mode, the one negative mode, the determinant ratio, and normalization of the false-well state determine how the saddle contribution becomes a physical rate.

Equivalently, the metastable energy acquires

$$\operatorname{Im}\mathcal E_F = - \frac{\hbar\gamma}{2}.$$

In the standard Euclidean treatment, the negative fluctuation direction is handled by a contour prescription and supplies the imaginary contribution. It should not be discarded as a failed Gaussian integral. At the same time, an imaginary part does not arise from an ordinary finite-dimensional Hermitian eigenvalue problem without analytic continuation or outgoing boundary conditions. The physical observable and contour prescription are part of the calculation.

Bounce Solutions derives $B$, the boundary conditions, and the mode count. Fluctuation Determinants Preview explains why $A$ is a normalized determinant ratio rather than an arbitrary attempt frequency.

Three Routes to the Same Lifetime

Route	Primary input	Output	Strongest check
Real-time flux	Prepared packet and dividing surface	$P_{\mathrm{ne}}(t)$ and $\gamma_{\mathrm{inst}}(t)$	Plateau in the instantaneous rate
Resonance pole	Outgoing boundary condition	$E_R$ and $\Gamma_E=\hbar\gamma$	Stability under numerical exterior treatment
WKB or bounce	Barrier action and semiclassical fluctuations	Asymptotic exponent and prefactor	Agreement of $\ln\gamma$ across parameter scans

These methods need not agree pointwise outside their common regime. A time-dependent packet contains preparation transients and background continuum contributions. A resonance pole isolates one analytic structure. A bounce expansion is asymptotic in a semiclassical parameter. Agreement should be demanded after the same potential, energy convention, channel count, and normalization are imposed.

Numerical extraction

Several complementary calculations are useful:

1. Outgoing resonance: use complex scaling, an absorbing exterior, or a direct outgoing boundary condition to find $\mathcal E_F$.
2. Real-time propagation: initialize a packet in the false well and monitor $P_{\mathrm{ne}}(t)$ and the outgoing flux.
3. WKB quadrature: evaluate $W(E_R)$ and $T_{\mathrm{cl}}(E_R)$ using the same quasibound energy.
4. Euclidean saddle: solve the bounce problem and compare $B$ with $2W$.
5. Parameter scan: vary mass, barrier height, or a small coupling and compare the logarithmic slope of the exact and semiclassical rates.

A fitted exponential should be accompanied by a time window. Fitting across the initial quadratic region, a reflection from a numerical boundary, or the onset of a recurrence produces a number that is not a resonance lifetime.

From Quantum Mechanics to Field Theory

The quantum-mechanical problem is the zero-spatial-dimensional model of false-vacuum decay. The structural dictionary is:

Quantum mechanics	Scalar field theory
Coordinate $x(\tau)$	Field configuration $\phi(\tau,\mathbf x)$
False-well packet or resonance	State localized near a false-vacuum configuration
Barrier in $V(x)$	Barrier in the field configuration-space energy functional
Bounce $x_B(\tau)$	Euclidean bubble $\phi_B(\tau,\mathbf x)$
Rate $\gamma$	Decay rate per spatial volume $\Gamma/V$
One translation zero mode	Spacetime translation zero modes

For a scalar field in $3+1$ dimensions without gravity, the leading zero-temperature result has the form

$$\frac{\Gamma}{V} \simeq \mathcal A e^{-B/\hbar},$$

with

$$B = S_E[\phi_B] - S_E[\phi_F].$$

For a local canonically normalized real scalar with a regular potential admitting a bounce, in flat space at zero temperature and without gravity, the least-action nontrivial bounce can be chosen $O(4)$ symmetric. Writing

$$\rho = \sqrt{\tau^2+\mathbf x^2},$$

its radial equation is

$$\frac{d^2\phi}{d\rho^2} + \frac{3}{\rho} \frac{d\phi}{d\rho} = \frac{dV}{d\phi},$$

with

$$\left. \frac{d\phi}{d\rho} \right|_{\rho=0} = 0, \qquad \phi(\rho\to\infty) = \phi_F.$$

The extra term $3\phi'/\rho$ has no analogue in one-dimensional Euclidean quantum mechanics; it acts like radial friction in the inverted-potential picture. Four spacetime translation zero modes replace the single bounce-center zero mode, and the completed prefactor must have units of rate per volume.

The bridge has limits:

• finite temperature makes Euclidean time periodic with period $\hbar/(k_B T)$; a static $O(3)$ thermal bounce is appropriate in a sufficiently high-temperature regime, while low temperatures can retain time-dependent nearly $O(4)$ bounces and crossover solutions;
• gauge fields and multiple scalar fields complicate the fluctuation problem;
• gravity changes both the saddle equations and the interpretation of decay;
• in finite volume, exact energy eigenstates remain real and the irreversible rate is again an asymptotic, large-volume, or coarse-grained concept.

Bridge to QFT Instantons places this dictionary beside other Euclidean saddles. The field-theory construction itself should be studied in its canonical QFT treatment rather than inferred by replacing $x$ with $\phi$ mechanically.

Exercises

1. A normalized threshold model

Let the prepared state’s spectral density be

$$\rho(E)=\frac{\epsilon^\alpha e^{-\epsilon/\Lambda}} {\Gamma(\alpha+1)\Lambda^{\alpha+1}}, \quad \epsilon=E-E_{\min}\geq0,\quad\alpha>-1.$$

Compute its exact survival probability and compare its short- and long-time limits. Does a threshold alone ensure an exponential window?

Solution

The gamma integral gives

$$\mathcal A(t)=e^{-iE_{\min}t/\hbar} (1+i\Lambda t/\hbar)^{-(\alpha+1)}, \qquad P(t)=[1+(\Lambda t/\hbar)^2]^{-(\alpha+1)}.$$

The energy variance is $(\alpha+1)\Lambda^2$, agreeing with the quadratic short-time term. The long-time probability is proportional to $t^{-2(\alpha+1)}$. This smooth density has no narrow resonance peak and need not have an exponential window: threshold behavior and pole dominance are distinct assumptions.

2. Compare survival and nonescape

Let $\phi(x)$ solve the stationary Schrödinger equation for a real potential with outgoing resonance energy $\mathcal E_F=E_R-i\hbar\gamma/2$. On a finite interval $[a,x_d]$, assume a reflecting left boundary, so $j_\phi(a)=0$. Derive a relation between the right-boundary flux and the integrated density. Does this make $\phi$ a normalized initial state with an exactly exponential survival probability?

Solution

Multiply the stationary equation by $\phi^*$ and subtract its complex conjugate. With $j_\phi=(\hbar/m)\operatorname{Im}(\phi^*\phi')$, the result is

$$\frac{dj_\phi}{dx}=\gamma|\phi(x)|^2, \qquad \frac{j_\phi(x_d)}{\displaystyle\int_a^{x_d}|\phi(x)|^2\,dx}=\gamma.$$

Both numerator and denominator scale with the arbitrary resonance amplitude, so their ratio is normalization independent. The formal solution $e^{-i\mathcal E_Ft/\hbar}\phi(x)$ has density and current proportional to $e^{-\gamma t}$, and its finite-region norm obeys the nonescape loss law.

An outgoing Gamow function is generally not square-integrable over the complete escape geometry. A finite-region norm does not turn it into a normalized Hilbert-space state, and truncating it changes its evolution. A physically prepared packet can follow this pole contribution during the exponential window, but its survival overlap also contains preparation and continuum-background effects. Thus the exact stationary flux identity fixes the rate of the resonance contribution without proving exponential survival at every time. See Andreassen et al. (2017), Section 2.2, for the connection between outgoing poles and regional probability loss.

3. Explain why a closed two-level model does not decay

Let a normalized initial state have support on two exact eigenstates:

$$\lvert F\rangle = c_1\lvert E_1\rangle + c_2\lvert E_2\rangle.$$

Compute its survival probability and explain why no constant positive decay rate exists.

Solution

The survival amplitude is

$$\mathcal A_F(t) = \lvert c_1\rvert^2 e^{-iE_1t/\hbar} + \lvert c_2\rvert^2 e^{-iE_2t/\hbar}.$$

Therefore

$$P_{\mathrm{surv}}(t) = \lvert c_1\rvert^4 + \lvert c_2\rvert^4 + 2 \lvert c_1c_2\rvert^2 \cos\left( \frac{E_2-E_1}{\hbar}t \right).$$

The result oscillates and recurs. It cannot equal $e^{-\gamma t}$ with constant $\gamma\gt0$. Irreversible decay requires a continuum, an appropriate large-system limit, environmental coarse-graining, or an explicitly limited time window.

References

• G. Álvarez, “Coupling-Constant Behavior of the Resonances of the Cubic Anharmonic Oscillator”, Physical Review A 37, 4079–4083 (1988). Resonance asymptotics for a standard metastable quantum-mechanical model.
• A. Andreassen, D. Farhi, W. Frost, and M. D. Schwartz, “Direct Approach to Quantum Tunneling”, Physical Review Letters 117, 231601 (2016). Physical decay-rate definition from real-time evolution and its relation to Euclidean methods.
• A. Andreassen, D. Farhi, W. Frost, and M. D. Schwartz, “Precision Decay Rate Calculations in Quantum Field Theory”, Physical Review D 95, 085011 (2017). Careful normalization and higher-order structure of semiclassical decay rates.
• C. G. Callan Jr. and S. Coleman, “Fate of the False Vacuum. II. First Quantum Corrections”, Physical Review D 16, 1762–1768 (1977). Fluctuation prefactor and decay-rate normalization.
• S. Coleman, “Fate of the False Vacuum: Semiclassical Theory”, Physical Review D 15, 2929–2936 (1977), with erratum in 16, 1248. Canonical field-theory bounce and false-vacuum construction.
• S. Coleman, “The Uses of Instantons”, in Aspects of Symmetry, Cambridge University Press, 1985, pp. 265–350. Pedagogical quantum-mechanical bounce and dilute-gas treatment.
• L. Fonda, G. C. Ghirardi, and A. Rimini, “Decay Theory of Unstable Quantum Systems”, Reports on Progress in Physics 41, 587–631 (1978). Comprehensive account of survival amplitudes, resonances, and nonexponential regimes.
• G. Gamow, “Zur Quantentheorie des Atomkernes”, Zeitschrift für Physik 51, 204–212 (1928). Foundational barrier-penetration treatment of nuclear decay.
• L. A. Khalfin, “Contribution to the Decay Theory of a Quasi-Stationary State”, Soviet Physics JETP 6, 1053–1063 (1958). Classic proof of nonexponential long-time behavior for lower-bounded spectra.
• J. S. Langer, “Statistical Theory of the Decay of Metastable States”, Annals of Physics 54, 258–275 (1969). Foundational metastability, analytic-continuation, and rate framework.
• J. Zinn-Justin, Path Integrals in Quantum Mechanics, Oxford University Press, 2005. Systematic treatment of Euclidean saddles, metastable potentials, and semiclassical expansions.