Tunneling, Superselection, and the Infinite-Volume Limit

Tunneling usually turns classically degenerate minima into a unique ground state at finite volume. Quantum field theory adds a singular limit: the transition amplitude can fall exponentially with spatial volume, so distinct pure vacua survive as superselection sectors when the infinite-volume limit is taken first. The conclusion depends on locality, boundary conditions, exact conserved charges, and the order in which volume and external sources are removed.

Required background. Local expansions and global vacuum structure distinguishes a local minimum from a quantum vacuum.

Helpful background. Finite-volume, thermodynamic limits, and pure phases supplies the phase-selection language, and clustering and vacuum assumptions explains the factorization criterion used below.

Shared comparison. The sector–theta–branch map separates finite-volume response, the branch envelope, and the conditional superselection limit, while the sector and periodicity comparison keeps the defining global data explicit.

This page temporarily restores $\hbar$ in tunneling exponents and time scales to display the semiclassical limit; the remaining conventions are the site-wide natural-unit conventions.

Finite-volume mixing

Begin with two normalized wave packets $|L\rangle$ and $|R\rangle$ localized near symmetry-related minima. In their span, a reflection-symmetric Hamiltonian has the form

$$H_{\mathrm{eff}} = \begin{pmatrix} E_0 & -t\\ -t^* & E_0 \end{pmatrix}.$$

After a phase choice makes $t>0$, the eigenstates are

$$|+\rangle=\frac{|L\rangle+|R\rangle}{\sqrt2}, \qquad |-\rangle=\frac{|L\rangle-|R\rangle}{\sqrt2}, \qquad \Delta E=E_--E_+=2t.$$

In the quantum-mechanical double well, a one-instanton saddle gives $t\propto e^{-S_0/\hbar}$ with a fluctuation prefactor. The exact finite-volume eigenstates transform irreducibly under the reflection symmetry even though $|L\rangle$ and $|R\rangle$ are the natural long-lived states. Coleman develops this relation between instantons and level splitting in Coleman 1985, ch. 7, § 2.2, pp. 270–277.

For a local field theory in a spatial region of volume $V_s$, a transition between macroscopically different homogeneous phases requires a Euclidean configuration extended across the system. When its action has the asymptotic form

$$S_{\mathrm{mix}}(V_s)=cV_s+o(V_s), \qquad c>0,$$

the level splitting behaves as

$$\Delta E(V_s) \sim A(V_s)e^{-cV_s/\hbar}.$$

The power-law prefactor $A$ cannot compete with the exponential. The scaling must be derived for the relevant transition: defects, interfaces, long-range forces, or gapless modes can change the exponent or prefactor, and an exact conserved charge can make the matrix element vanish already at finite volume.

A domain wall across a finite scalar box

The finite-volume scalar example makes the volume law calculable. Put a broken-$\mathbb Z_2$ scalar on a periodic spatial three-torus of volume $V_s=L^3$ with

$$U(\phi)=\frac\lambda4(\phi^2-v^2)^2, \qquad m_\phi=\sqrt{2\lambda}\,v.$$

Restrict first to a spatially uniform Euclidean history with boundary values $\phi(-\infty)=-v$ and $\phi(+\infty)=+v$. The first-order equation $\dot\phi=\sqrt{2U(\phi)}$ has the kink

$$\phi_{\mathrm{kink}}(\tau) =v\tanh\!\left[\frac{m_\phi}{2}(\tau-\tau_0)\right].$$

Its wall is perpendicular to Euclidean time and spans the entire spatial box. Therefore

$$\begin{aligned} S_{\mathrm{mix}} &=V_s\int\mathrm d\tau \left[\frac12\dot\phi^2+U(\phi)\right] =V_s\int_{-v}^{v}\mathrm d\phi\,\sqrt{2U(\phi)}\\ &=\sigma_\tau V_s, \qquad \sigma_\tau=\frac{2\sqrt{2\lambda}}{3}v^3, \end{aligned}$$

and the associated two-state splitting has the conditional form

$$\Delta E(V_s)\sim A(V_s) \exp\!\left(-\frac{\sigma_\tau V_s}{\hbar}\right).$$

This is the promised QFT realization of the volume exponent, not a universal assertion about every box. It assumes a gapped discrete-symmetry regime and boundary conditions for which this temporal wall is the dominant interpolation. An inhomogeneous nucleation path, a spatial interface, a boundary defect, or a gapless mode can change the leading action and must be compared explicitly.

Continuous symmetry breaking supplies an important near miss. Its finite-volume low states can form a quantum-rotor or “tower of states” spectrum with moment of inertia $I_{\mathrm{eff}}\propto V_s$ and gaps $\Delta E\propto1/V_s$, rather than an exponentially split doublet. The infinite-volume conclusion—many pure orientations—can be similar, but the approach to the limit and the appropriate effective degrees of freedom are different. The exponential formula above therefore applies only after a barrier-crossing saddle with $S_{\mathrm{mix}}\propto V_s$ has been established.

The thermodynamic limit creates superselection

The mixing time associated with the two lowest levels is

$$\tau_{\mathrm{mix}}\sim\frac{\hbar}{\Delta E(V_s)}.$$

It diverges exponentially in the regime above. More fundamentally, matrix elements of local observables between different pure vacua vanish as $V_s\to\infty$. The resulting Hilbert-space representations are disjoint: no finite-support physical operation turns one vacuum into the other.

The observation time is another limit. At fixed $V_s$, measurements with $T_{\mathrm{obs}}\gg\tau_{\mathrm{mix}}$ resolve the exact symmetry eigenstates. A localized phase persists operationally only when the thermodynamic limit is taken first, or along a sequence with $T_{\mathrm{obs}}/\tau_{\mathrm{mix}}\to0$. A large but finite lifetime is evidence for suppressed mixing, not exact superselection.

Weinberg gives a locality-based argument that equal-time local-operator matrices can be simultaneously diagonalized on the vacuum subspace and that the diagonal vacua, rather than their generic superpositions, satisfy cluster decomposition Weinberg 1996, § 19.1, pp. 163–167. In this sense superselection is not the assertion that a finite box has several exact ground states. It is a property of the infinite system and its algebra of local observables.

Suppose a $\mathbb Z_2$-odd order parameter $\Phi$ has $\langle L|\Phi|L\rangle=+v$ and $\langle R|\Phi|R\rangle=-v$. The finite-volume symmetric ground state has $\langle+\!|\Phi|+\rangle=0$. Nevertheless, at separations large compared with the correlation length,

$$\langle+\!|\Phi(x)\Phi(0)|+\rangle\longrightarrow v^2,$$

while $\langle+\!|\Phi|+\rangle^2=0$. The symmetric combination therefore fails clustering in the infinite-volume limit. Either pure phase clusters:

$$\langle L|\Phi(x)\Phi(0)|L\rangle \longrightarrow \langle L|\Phi|L\rangle^2=v^2.$$

Clustering is what distinguishes an extremal vacuum from a nonextremal combination of phases. Along the thermodynamic sequence the finite-volume coherent state loses locally measurable phase coherence; its limiting state on the local observable algebra behaves as a mixture. The two pure phases then belong to disjoint representations, so an ordinary normalizable superposition vector spanning both sectors is no longer available inside either representation.

A source selects the phase

Add a uniform symmetry-breaking source $h$ through

$$H_h=H-h\int_{V_s}\mathrm d^d x\,\Phi(x).$$

The energy bias between the two localized states is approximately $2h vV_s$. For every fixed nonzero $h$, this extensive bias eventually dominates the exponentially small tunneling matrix element. The order parameter is therefore defined by the ordered limits

$$\phi_\pm =\lim_{h\to0^\pm}\lim_{V_s\to\infty} \langle\Phi\rangle_{V_s,h} =\pm v.$$

Reversing the limits restores the symmetric finite-volume answer:

$$\lim_{V_s\to\infty}\lim_{h\to0} \langle\Phi\rangle_{V_s,h}=0.$$

This noncommutativity is the operational signature of spontaneous breaking. Boundary conditions can play the same selecting role as the infinitesimal source.

Exact and emergent separation

Three mechanisms should not be conflated.

1. Finite-volume tunneling. Candidate vacua mix, producing a small but nonzero splitting.
2. Thermodynamic superselection. Local mixing vanishes only after $V_s\to\infty$; pure phases become distinct representations.
3. Exact sector separation. A conserved charge, gauge constraint, or imposed boundary datum forbids mixing even before the volume limit.

Topological sector labels in a Euclidean path integral are also not automatically Hilbert-space superselection charges. The full path integral may deliberately sum over them, and instantons can mediate transitions between semiclassical gauge-field configurations. One must identify the Lorentzian observable algebra and allowed finite-action histories before using the word “superselection.”

Common pitfalls

Calling each classical minimum a vacuum. The finite-volume energy eigenstates are determined only after tunneling is included.

Taking the limits silently. Sending the source to zero before the volume to infinity gives a different state from the standard symmetry-breaking prescription.

Using a macroscopic lifetime as an exact conservation law. Exponentially slow mixing is not zero mixing. Exact separation requires an independent symmetry, constraint, or limiting construction.

Exercises

Diagonalize

$$H_{\mathrm{eff}}= \begin{pmatrix} E_0-hvV_s & -t\\ -t & E_0+hvV_s \end{pmatrix}$$

and determine the crossover source at which the ground state becomes localized.

Solution

The eigenvalues are

$$E_\pm=E_0\pm\sqrt{t^2+(hvV_s)^2}.$$

The ground-state expectation of the two-state order parameter $\Phi=v\,\mathrm{diag}(1,-1)$ is

$$\langle\Phi\rangle =v\,\frac{hvV_s}{\sqrt{t^2+(hvV_s)^2}}.$$

Localization sets in when $|h|vV_s\gg t$, so the crossover scale is $|h_*|\sim t/(vV_s)$. If $t\sim e^{-cV_s/\hbar}$, then $h_*$ vanishes exponentially fast.

Show explicitly why the symmetric combination of two clustering vacua with order parameters $\pm v$ fails cluster decomposition when cross matrix elements of local operators vanish.

Solution

For $|+\rangle=(|L\rangle+|R\rangle)/\sqrt2$, vanishing cross terms give

$$\langle+\!|\Phi|+\rangle =\tfrac12(v-v)=0.$$

At large separation, clustering inside each pure phase gives

$$\langle+\!|\Phi(x)\Phi(0)|+\rangle \longrightarrow\tfrac12(v^2+v^2)=v^2.$$

Since $v^2\neq0=\langle\Phi\rangle^2$, the symmetric state is not an extremal clustering vacuum.

A broken continuous symmetry has a finite-volume zero mode described by a rotor with $H=\mathbf L^2/(2I_{\mathrm{eff}})$ and $I_{\mathrm{eff}}=\rho V_s$. Determine the first gap and compare its volume dependence with the discrete two-vacuum splitting.

Solution

For an $O(3)$ rotor, $\mathbf L^2$ has eigenvalues $\ell(\ell+1)$, so the gap from $\ell=0$ to $\ell=1$ is

$$\Delta E_{\mathrm{rotor}} =\frac{1}{I_{\mathrm{eff}}} =\frac{1}{\rho V_s}.$$

It closes algebraically. A discrete broken phase with a verified interface-crossing saddle instead has $\Delta E\sim A(V_s)e^{-cV_s/\hbar}$. Both gaps vanish, but fitting one form with the other would give the wrong finite-size extrapolation and the wrong mechanism.

Continue with the correct finite-size problem

Use Finite Volume as a Controlled Deformation to design the boundary conditions and volume sequence that distinguish exponential mixing, rotor gaps, and exact charge sectors. Use Semiclassical Expansions and Integration Cycles when the transition action and prefactor themselves must be calculated.

References

• Coleman, Sidney. Aspects of Symmetry: Selected Erice Lectures. Cambridge University Press, 1985, ch. 7, § 2.2, pp. 270–277. DOI.
• Weinberg, Steven. The Quantum Theory of Fields. Volume II: Modern Applications. Cambridge University Press, 1996, § 19.1, pp. 163–167. Chapter DOI.