Euclidean Tunneling Saddles and Boundary Conditions

Tunneling is invisible to any finite expansion about one classical minimum, yet it can dominate an exponentially small transition amplitude. After analytic continuation, the relevant contribution is organized by a finite-action solution of the Euclidean equations with boundary conditions fixed by the observable. The boundary conditions are decisive: a heteroclinic instanton, a periodic saddle, and a false-vacuum bounce solve related differential equations but compute different quantities.

Required background. Wick rotation and analytic continuation supplies the continuation of amplitudes and their boundary values. Saddles, control parameters, and loop counting supplies the stationary-phase expansion and its evidence limits.

Helpful background. Contour deformation, pinches, and causal prescriptions explains why a Euclidean integral does not by itself specify every Lorentzian observable.

Finite-action Euclidean trajectories

For one degree of freedom with mass $m$, the Euclidean transition kernel is

$$\langle q_f|e^{-HT/\hbar}|q_i\rangle = \int_{q(-T/2)=q_i}^{q(T/2)=q_f}\mathcal Dq\, \exp\!\left[-\frac{1}{\hbar}S_E[q]\right], \qquad S_E=\int_{-T/2}^{T/2}d\tau \left[\frac{m}{2}\dot q^{\,2}+V(q)\right].$$

The stationary equation,

$$m\ddot q=V'(q),$$

is Newton’s equation in the inverted potential $-V$. Multiplying by $\dot q$ gives the conserved Euclidean energy

$$\mathcal E_E=\frac{m}{2}\dot q^{\,2}-V(q).$$

Suppose $q_-$ and $q_+$ are degenerate minima and the vacuum energy has been subtracted so that $V(q_\pm)=0$. A finite-action trajectory on the infinite line must obey

$$q(\tau)\longrightarrow q_\pm,\qquad \dot q(\tau)\longrightarrow0 \quad (\tau\longrightarrow\pm\infty),$$

and hence $\mathcal E_E=0$. Its first-order equation and action are

$$\dot q=\pm\sqrt{\frac{2V(q)}{m}}, \qquad S_I=\int_{q_-}^{q_+}dq\,\sqrt{2mV(q)}.$$

This is a heteroclinic instanton: it joins distinct asymptotic vacua and contributes to an off-diagonal transition amplitude. Translation invariance produces the zero mode $\dot q_I$. The instanton’s exponential weight is $e^{-S_I/\hbar}$; the prefactor comes from the collective-coordinate Jacobian and the determinant over nonzero fluctuations. This separation is derived systematically on Instanton Measures, Zero Modes, and Determinants.

At finite Euclidean period $\beta$, the trace $\operatorname{Tr}e^{-\beta H}$ instead imposes $q(\tau+\beta)=q(\tau)$. Its stationary points may be periodic instanton–anti-instanton configurations or other periodic saddles. They are not single heteroclinic trajectories on $\mathbb R$, and their Euclidean energy need not vanish.

Instanton–bounce boundary and mode comparison

The following comparison keeps the observable, endpoint data, and fluctuation spectrum together. “Negative mode” means a normalizable negative eigenfunction of the gauge-fixed quadratic operator; gauge redundancies are removed before that count.

Boundary data, actions, modes, determinants, and outputs for four Euclidean saddles

Saddle	Boundary conditions	Physical question	Classical action	Moduli	Zero and negative modes	Fluctuation factor	Amplitude or rate	Control failure
Quantum-mechanical instanton	τ ∈ ℝ; q(−∞) = q₋ and q(+∞) = q₊ for degenerate minima	How do two perturbative wells mix?	Sᵢ = ∫ from q₋ to q₊ of dq √(2mV); 4/(3g) in the normalized quartic well	Center τ₀	One translation zero mode; no negative mode for the elementary heteroclinic path	[det′ Mᵢ / det M₀]−1/2 with the translation Jacobian outside the determinant	Off-diagonal amplitude; a dilute alternating sum gives real tunnel splitting ΔE	Higher loops, close instanton–anti-instanton interactions, or loss of diluteness
Four-dimensional gauge instanton	Euclidean ℝ⁴; finite action makes the connection pure gauge at the sphere at infinity	What contributes in a declared topological sector or correlator?	8π²|Q|/g²; the BPST instanton has Q = +1	Position x₀, size ρ, and global gauge orientation modulo its stabilizer	4N bosonic zero modes for SU(N), representation-dependent fermion zero modes, and no physical negative mode for a self-dual minimum in fixed Q	Ghost determinant divided by the square root of the primed vector determinant; collective and fermion zero modes treated separately	Sector weight or a correlation function only after every fermion zero mode is saturated	The ρ integral can reach ρΛ ≈ 1; an ensemble can also cease to be dilute
Fractional caloron constituent	ℝ³ × S¹ with declared circle holonomy and global gauge data; constituents recombine to periodic caloron boundary data	How is a unit caloron resolved in a compactified weak-coupling regime?	8π²νᵢ/g² with holonomy gap νᵢ; constituent charges sum to one	Position and phase-type coordinates, with the detailed count fixed by the compactified theory	Collective zero modes and fermion localization depend on holonomy and circle boundary conditions; no bounce-type negative mode for the BPS constituent saddle	Compactified nonzero-mode determinant plus any correlated-event quasi-zero-mode integral	Constituent amplitude or effective operator inside the declared circle theory	Loss of stabilized holonomy, abelianization, scale separation, or diluteness; not an isolated fractional instanton on ℝ⁴
False-vacuum bounce	Euclidean ℝᵈ; field approaches the same metastable vacuum in every asymptotic direction and turns near the escape configuration	At what rate does a metastable state decay?	B = Sᴱ[bounce] − Sᴱ[false vacuum]	d translations for an O(d)-symmetric bounce, plus any genuine internal moduli	d translation zero modes and exactly one physical negative mode for the leading decay bounce	Primed determinant ratio for nonzero modes; the negative direction is fixed by a contour prescription, not deleted	Imaginary part and decay rate Γ/V proportional to e−B/ℏ, not tunnel splitting	No metastable state, extra negative modes, overlapping bounces, or unaccounted thermal or gravitational effects

The one-negative-mode statement is what converts the bounce saddle into an imaginary part. For a single coordinate, a bounce leaves and returns to the false vacuum, so $\dot q_B$ has a node. Sturm–Liouville ordering then puts a lower, nodeless eigenfunction below the translation zero mode: the negative mode. A heteroclinic instanton has monotone $q_I$, so $\dot q_I$ is nodeless and is the lowest mode; there is no lower negative eigenvalue. The semiclassical decay interpretation is derived in Coleman 1977, pp. 2929–2936; the field-theory prefactor additionally requires the contour deformation and normalization developed by Callan and Coleman 1977, pp. 1762–1768.

What analytic continuation does—and does not do

Euclidean time changes the oscillatory factor $e^{iS/\hbar}$ into a decaying weight only after the state, contour, and continuation have been specified. It makes a classically forbidden path accessible as a real saddle of $S_E$, but it does not license the following replacements:

• A finite-action solution is not automatically a tunneling saddle for the observable under study.
• A real Euclidean saddle need not lie on the integration cycle obtained from the Lorentzian problem.
• A bounce contribution is not a real energy correction; its negative mode encodes metastability.
• A periodic thermal saddle is not an infinite-time vacuum instanton.

The semiclassical hierarchy also has two logically separate requirements: $S_\star/\hbar\gg1$ suppresses higher loops near one saddle, while a small event density times the interaction volume controls a dilute multi-event sum. Either condition can fail independently.

Worked example: the quartic heteroclinic path

Take

$$V(q)=\lambda(q^2-a^2)^2,\qquad \lambda>0.$$

The zero-energy equation integrates to

$$q_I(\tau)=a\tanh\!\left[\frac{\omega}{2}(\tau-\tau_0)\right], \qquad \omega^2=\frac{8\lambda a^2}{m}.$$

Indeed,

$$\dot q_I = \frac{a\omega}{2}\operatorname{sech}^2\!\left[\frac{\omega}{2}(\tau-\tau_0)\right] = \sqrt{\frac{2\lambda}{m}}\left(a^2-q_I^2\right).$$

The first-order action formula gives

$$S_I = \sqrt{2m\lambda}\int_{-a}^{a}dq\,(a^2-q^2) = \frac{4}{3}a^3\sqrt{2m\lambda}.$$

The center $\tau_0$ is arbitrary, so $\partial_{\tau_0}q_I=-\dot q_I$ is a normalizable zero mode. The determinant and dilute sum that turn this saddle into an actual spectral splitting are carried out on Quantum-Mechanical Instantons and Tunnel Splitting. A detailed one-dimensional derivation with this normalization appears in Mariño 2015, §§ 1.8–1.9, pp. 38–53.

Common pitfalls

Confusing the inverted-potential picture with a change of physics. It is a mnemonic for the Euclidean Euler–Lagrange equation. The observable still comes from a specified continuation of the original quantum theory.

Calling every finite-action Euclidean solution a bounce. “Bounce” is reserved for a saddle that returns to a metastable vacuum and has the negative mode appropriate to decay. A vacuum-changing heteroclinic solution is an instanton.

Dropping endpoint data. The same differential equation supports different saddles under transition, trace, and false-vacuum boundary conditions. The endpoints determine the physical interpretation.

Exercises

1. Show that any finite-action trajectory approaching degenerate minima on $\mathbb R$ has $\mathcal E_E=0$, and derive the line-integral expression for $S_I$.

Solution

At either end, finite action requires $V(q)\to0$ and $\dot q\to0$, hence the conserved quantity $\mathcal E_E=m\dot q^2/2-V$ is zero everywhere. Thus $m\dot q^2/2=V$. Along the increasing path,

$$S_I=\int d\tau\,[m\dot q^2/2+V] =\int d\tau\,2V =\int_{q_-}^{q_+}dq\,\frac{2V}{\dot q} =\int_{q_-}^{q_+}dq\,\sqrt{2mV}.$$

2. Why does the nodal structure of the translation mode distinguish an elementary instanton from a bounce in one-dimensional quantum mechanics?

Solution

The quadratic operator is a one-dimensional Schrödinger operator, and its eigenfunctions are ordered by their number of nodes. For a monotone heteroclinic instanton, $\dot q_I$ never changes sign, so the zero mode is nodeless and is the lowest eigenfunction; there is no negative mode. A bounce reverses direction, so $\dot q_B$ has a node. A nodeless eigenfunction therefore lies below it and has a negative eigenvalue.

References

• Callan, Curtis G., Jr., and Sidney Coleman. “Fate of the False Vacuum. II. First Quantum Corrections.” Physical Review D 16 (1977): 1762–1768. DOI.
• Coleman, Sidney. “Fate of the False Vacuum: Semiclassical Theory.” Physical Review D 15 (1977): 2929–2936; erratum 16 (1977): 1248. DOI.
• Mariño, Marcos. Instantons and Large N: An Introduction to Non-Perturbative Methods in Quantum Field Theory. Cambridge: Cambridge University Press, 2015. DOI.