Local Expansions and Global Vacuum Structure

A perturbative expansion around one stationary configuration determines local fluctuation data—masses, vertices, and loop corrections—but not the number of quantum vacua, the tunneling matrix elements between distant configurations, or the branch that minimizes the exact energy. Those global questions require boundary conditions, the full configuration space, and a declared order of limits. A count of classical minima is therefore useful input, not a theorem about the quantum ground state.

Required background. Vacua, states, and representations distinguishes a lowest-energy state from a classical field value and explains why inequivalent infinite-volume representations can occur.

Helpful background. Symmetry realization and order parameters supplies physical diagnostics, while clustering and vacuum assumptions supplies the condition that separates a pure phase from a mixture.

Shared comparison. The sector–theta–branch map places local branches inside the global Fourier and volume-limit structure, while the claim-and-control comparison states how strongly each conclusion is supported.

Local curvature does not determine the ground state

For a scalar coordinate $q$ and a stationary point $q_i$, write

$$V(q_i+\eta)=V(q_i)+\frac12 V''(q_i)\eta^2 +\frac1{3!}V'''(q_i)\eta^3+\cdots.$$

The Hessian $V''(q_i)$ fixes the harmonic frequency or, in field theory, the quadratic fluctuation operator. Higher derivatives generate a local perturbative series. None of these coefficients says whether another minimum exists far away, whether a finite-action interpolation connects the two regions, or which state has the lowest exact energy.

This distinction survives in the effective action. Its stationary points are candidates for homogeneous quantum states only after convexity, boundary conditions, gauge invariance, volume limits, and stability have been addressed. The exact Legendre construction belongs to the 1PI effective action and mean-field equations, while order parameters and broken phases belong to symmetry realization and order parameters. Here the concern is what a single local patch fails to determine.

Exact and branch-restricted effective potentials

The word “effective potential” is used for two objects that agree only with qualifications. At regulated finite volume, couple a real order parameter $\Phi$ to a uniform Euclidean source $J$ and define

$$w_{V_s}(J)=\frac1{V_s}\log Z_{V_s}(J), \qquad \phi(J)=\frac{\partial w_{V_s}}{\partial J}.$$

For a nonnegative measure,

$$\frac{\partial^2w_{V_s}}{\partial J^2} =\frac1{V_s} \left\langle \left(\int_{V_s}\!\Phi-\left\langle\int_{V_s}\!\Phi\right\rangle\right)^2 \right\rangle \geq0.$$

The Legendre–Fenchel transform $U_{V_s}(\phi)=\sup_J[J\phi-w_{V_s}(J)]$ is therefore convex. Where ordinary derivatives exist, $U''=1/w''\geq0$. In the thermodynamic limit, coexistence can produce a flat segment between pure-phase values rather than a literal barrier in this exact convex function.

A loop-expanded or saddle-restricted potential answers a different question: it follows one local branch and can display minima, barriers, and metastability before the global minimization and infinite-volume limit are performed. Such a potential is extremely useful, but its local extrema must not be counted as exact vacua. Weinberg explains how source selection and the infinite-volume limit recover broken pure phases in Weinberg 1996, § 19.1, pp. 163–167.

Symmetric and tilted double wells

This section temporarily restores $\hbar$ to display the semiclassical expansion parameter; the remaining conventions are the site-wide natural-unit conventions.

Take

$$H=\frac{p^2}{2m}+\lambda(q^2-a^2)^2-\varepsilon q, \qquad \lambda>0.$$

For a high barrier, localized states $\lvert L\rangle$ and $\lvert R\rangle$ give a two-state effective Hamiltonian

$$H_{\mathrm{eff}}= \begin{pmatrix} E_L & -t\\ -t & E_R \end{pmatrix}, \qquad t=K e^{-S_0/\hbar}\left[1+O\!\left(\frac{\hbar}{S_0}\right)\right].$$

Its eigenvalues are

$$E_\pm=\frac{E_L+E_R}{2} \pm\sqrt{\left(\frac{E_L-E_R}{2}\right)^2+t^2}.$$

At $\varepsilon=0$, local perturbation theory about either minimum gives the same $E_L=E_R$, while the exact low states are even and odd combinations split by $2t$. At nonzero tilt, the local energy difference is approximately $E_R-E_L\simeq-2\varepsilon a$, and the avoided crossing has minimum gap $2t$. This elementary matrix cleanly separates local information $(E_L,E_R)$ from global information $t$. The underlying instanton transition amplitude is derived in Coleman 1985, ch. 7, § 2.2, pp. 270–277 and Mariño 2015, ch. 1, §§ 1.8–1.9, pp. 38–53.

The transfer to QFT is structural rather than literal. A field has infinitely many degrees of freedom, interpolating configurations may be domain walls or other extended objects, and the action that suppresses mixing can grow with spatial volume. The double well shows why local series can miss a splitting; it does not prove that every pair of field-theory extrema defines two vacua.

A finite-volume scalar QFT

To make the transfer concrete, place a $3+1$-dimensional real scalar field on a spatial three-torus of volume $V_s=L^3$ and use the Euclidean action

$$S_E[\phi] =\int\mathrm d\tau\,\mathrm d^3x \left[ \frac12(\partial\phi)^2 +\frac\lambda4(\phi^2-v^2)^2 -h\phi \right], \qquad \lambda>0.$$

At $h=0$, the two homogeneous classical minima have $\phi=\pm v$ and the same local fluctuation mass $m_\phi^2=2\lambda v^2$. Retaining the spatially homogeneous mode $\phi(\mathbf x,\tau)=q(\tau)$ gives

$$S_E[q] =V_s\int\mathrm d\tau \left[ \frac12\dot q^2 +\frac\lambda4(q^2-v^2)^2 -hq \right].$$

The homogeneous trajectory from $-v$ to $+v$ has action at zero source

$$S_{\mathrm{hom}} =V_s\int_{-v}^{v}\!\mathrm dq\, \sqrt{\frac\lambda2}(v^2-q^2) =\frac{2\sqrt{2\lambda}}{3}v^3V_s.$$

The dimensions close in four spacetime dimensions: $v^3V_s$ is dimensionless. When the homogeneous mode is a reliable finite-volume truncation, it produces a mixing element $t(V_s)\sim A(V_s)e^{-S_{\mathrm{hom}}/\hbar}$. The two low states are then described by

$$H_{\mathrm{eff}}(h) =\begin{pmatrix} \mathcal E_0V_s-hvV_s & -t(V_s)\\ -t(V_s) & \mathcal E_0V_s+hvV_s \end{pmatrix}.$$

At fixed $V_s$ and $h=0$, the eigenstates are symmetry-even and symmetry-odd and are split by $2t(V_s)$. At fixed $h\neq0$, the extensive bias $2hvV_s$ eventually dominates the mixing as the box grows. Thus

$$\lim_{h\to0^\pm}\lim_{V_s\to\infty} \langle\phi\rangle_{V_s,h}=\pm v, \qquad \lim_{V_s\to\infty}\lim_{h\to0} \langle\phi\rangle_{V_s,h}=0$$

within this two-state description. The calculation exhibits the missing global inputs—mixing and limit order—that the identical local masses do not contain. It also states its failure boundary: in a large box an inhomogeneous interface can beat the homogeneous path, so the dominant action and prefactor must be recalculated rather than inferred from $S_{\mathrm{hom}}$. The general saddle machinery is developed in Semiclassical Expansions and Integration Cycles.

Branches and the lower envelope

Suppose a parameter $\alpha$ admits locally smooth candidate energies $\mathcal E_k(\alpha)$. The physical vacuum energy is the lowest admissible value,

$$\mathcal E(\alpha)=\min_k \mathcal E_k(\alpha).$$

Each branch can be analytic even when the lower envelope is not. If two branches cross at $\alpha_c$ with unequal slopes, then $\mathcal E$ has a cusp and the conjugate response $\partial\mathcal E/\partial\alpha$ jumps. A local series on one branch cannot predict the crossing unless it also contains information about the competing branch and its domain of existence.

The same separation appears in large-$N$ models of theta dependence, where individual functions of $(\theta+2\pi k)/N$ are exchanged by $\theta\mapsto\theta+2\pi$ while their minimum is periodic. Witten’s construction is an important controlled organizing example, not a universal formula for finite-$N$ gauge theories Witten 1998, §§ 1–2.

Finite volume and the source order

Let $\Phi$ be a physical order parameter and add a uniform source $h$ through $H_h=H-h\int_{V_s}\mathrm d^d x\,\Phi(x)$. A symmetry-invariant finite system often has a unique ground state and hence $\langle\Phi\rangle_{V_s,0}=0$. A broken phase is instead selected by the ordered limit

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

Reversing the limits can give zero:

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

The difference is not a mathematical inconvenience; it is part of the definition of the selected phase. Weinberg develops the state and ordered-source construction in Weinberg 1996, § 19.1, pp. 163–167.

Finite-volume uniqueness is not universal. Exact conserved charges, imposed boundary conditions, gauge constraints, or topological sectors can prevent mixing already at finite volume. One must state the Hilbert space and allowed operators before applying the familiar double-well intuition.

What global data must be supplied?

A vacuum claim should specify at least:

• the theory, geometry, and boundary conditions;
• the candidate states or branches and the physical observable that distinguishes them;
• any exact charge or selection rule that permits or forbids mixing;
• the regulator and the order of source, time, and volume limits;
• a clustering or purity criterion for an infinite-volume phase; and
• the approximation that estimates an interpolation or branch energy.

These data determine whether two extrema are coordinate descriptions of one state, approximately localized finite-volume states, exactly distinct charge sectors, or inequivalent infinite-volume phases.

Common pitfalls

Counting minima as vacua. A classical minimum is a stationary field configuration. Quantum fluctuations can lift it, tunneling can mix it, gauge transformations can identify it, and the exact ground state can lie elsewhere.

Calling every cusp a local instability. A cusp of the lower envelope may arise from a crossing of two individually stable branches. The Hessian within either branch need not vanish at the crossing.

Suppressing the order of limits. “Take the infinite-volume vacuum” is incomplete when a source or boundary condition selects a phase. Reversing the limits can change the state and the observable.

Exercises

Diagonalize $H_{\mathrm{eff}}$ above at $E_R-E_L=\delta$. Show that the probability of finding the lower eigenstate in the left well changes smoothly through $\delta=0$ for $t\neq0$, but becomes discontinuous if $t\to0$ first.

Solution

Write $H_{\mathrm{eff}}-(E_L+E_R)I/2=-(\delta/2)\sigma_3-t\sigma_1$ in the basis where $\sigma_3\lvert L\rangle=\lvert L\rangle$. The lower state has

$$P_L=\frac12\left(1+\frac{\delta}{\sqrt{\delta^2+4t^2}}\right).$$

For fixed $t>0$ this is smooth. If $t\to0$ first, it tends to $1$ for $\delta>0$ and $0$ for $\delta<0$, with a branch switch at the origin. The noncommuting limits model finite-volume mixing versus an infinite-volume phase transition.

Use $w''(J)=V_s^{-1}\operatorname{Var}(\int\Phi)$ to explain what happens to the exact effective potential when the susceptibility grows extensively near phase coexistence.

Solution

Where the Legendre transform is differentiable, $U''(\phi)=1/w''(J)$. If fluctuations between competing phases make $w''$ grow without bound as $V_s\to\infty$, then $U''$ tends to zero over the corresponding range of $\phi$. The limiting exact potential develops a flat coexistence segment. This does not erase the interface tension or the branch-restricted barrier; those are encoded in finite-size and nonuniform configurations rather than in the homogeneous convex envelope alone.

Continue from local data to dynamics

Use Tunneling, Superselection, and the Infinite-Volume Limit to decide when the off-diagonal matrix element vanishes and pure phases become disjoint. Use Semiclassical Expansions and Integration Cycles to calculate that matrix element rather than merely name its exponential form.

References

• Coleman, Sidney. Aspects of Symmetry: Selected Erice Lectures. Cambridge University Press, 1985, ch. 7, § 2.2, pp. 270–277. DOI.
• Mariño, Marcos. Instantons and Large N: An Introduction to Non-Perturbative Methods in Quantum Field Theory. Cambridge University Press, 2015, ch. 1, §§ 1.8–1.9, pp. 38–53. DOI.
• Weinberg, Steven. The Quantum Theory of Fields, Volume II: Modern Applications. Cambridge University Press, 1996, § 19.1, pp. 163–167. Chapter DOI.
• Witten, Edward. “Theta Dependence in the Large N Limit of Four-Dimensional Gauge Theories.” Physical Review Letters 81 (1998): 2862–2865. arXiv. DOI.