Nonperturbative Questions, Vacua, and Sector Dynamics

Nonperturbative vacuum physics begins where a local expansion is no longer the whole question. A candidate minimum can be mixed by tunneling; a path integral can decompose into topological sectors; a theta parameter can Fourier-transform those sectors; and the infinite-volume energy can develop branches constrained, but not uniquely determined, by symmetry and anomaly. This chapter supplies the distinctions, normalizations, and order-of-limits tests needed to move among those statements without conflating them.

Helpful background. Vacua, states, and representations supplies the state-space language, and theta terms, periodicity, and vacuum sectors supplies the general topological-response construction.

Enter the chapter

Start from the question you can actually pose about an observable.

If your question is…	Read…	Result you should be able to use
What is the calculation nonperturbative relative to, and why should I trust it?	What Does Nonperturbative Mean?	State the expansion, observable, regime, control test, error, and falsifier
Can an expansion around one minimum determine the vacuum?	Local Expansions and Global Vacuum Structure	Separate local saddle data from the spectrum and global vacuum choice
What fixes the topological charge lattice?	Topological Sectors, Boundary Data, and Global Form	Derive sectors and periodicity from boundary, bundle, and global-form data
Is theta a coupling, a state label, or a Fourier variable?	Theta Parameters, Theta States, and Sector Sums	Translate consistently among the three roles
How can periodic physics have nonperiodic branches?	Theta Dependence, CP, and Model-Dependent Branches	Distinguish branch relabeling, the lower envelope, CP, and metastability
What do theta derivatives measure?	Topological Susceptibility and Vacuum Response	Derive charge cumulants and handle contact terms and volume limits
Why can finite-volume tunneling coexist with broken phases?	Tunneling, Superselection, and the Infinite-Volume Limit	Compare finite-size level splitting with infinite-volume pure sectors
What does an anomaly determine about the infrared?	Anomaly and Generalized-Symmetry Constraints on Infrared Phases	Exclude impossible phases while keeping all matching alternatives

The shortest conceptual route is

$$\begin{gathered} \text{theory and boundary data} \longrightarrow \text{charge sectors} \longrightarrow Z(\theta) \longrightarrow \text{response and branches},\\ \text{local minima} \longrightarrow \text{tunneling at finite volume} \longrightarrow \text{pure vacua at infinite volume},\\ \text{symmetry and backgrounds} \longrightarrow \text{anomaly class} \longrightarrow \text{allowed infrared realizations}. \end{gathered}$$

The arrows do not all reverse. For example, a susceptibility does not reconstruct the global theta dependence, and an anomaly-compatible phase is not thereby shown to occur.

Shared sector conventions

For an allowed charge set $\Lambda_Q$, the chapter uses the Euclidean Fourier sum

$$Z_{V_4}(\theta) =\sum_{Q\in\Lambda_Q}e^{i\theta Q}Z_{V_4,Q}, \qquad \mathcal E(\theta) =-\lim_{V_4\to\infty}\frac1{V_4} \log\frac{Z_{V_4}(\theta)}{Z_{V_4}(0)}.$$

The normalization of $Q$ and the set $\Lambda_Q$ must be declared before a theta period is inferred. If $\Lambda_Q=\mathbb Z$, the sum is $2\pi$ periodic. Fractional bundles, a different global gauge group, defects, or non-spin manifolds can alter the statement or make a $2\pi$ shift exchange theories with different discrete data.

At finite volume,

$$\frac{\partial^n}{\partial\theta^n} \log Z_{V_4}(\theta) =i^n\langle Q^n\rangle_{\theta,c}.$$

Only after the regulator and contact prescription are fixed should this identity be converted into an integrated correlator of local topological densities. Only after the volume limit is controlled should a nonanalytic vacuum-energy branch or superselection claim be made.

One double-well thread, four lessons

The symmetric potential

$$V(q)=\lambda(q^2-a^2)^2$$

provides the chapter’s simplest recurring example.

1. Expanding around $q=a$ gives a local perturbative series but cannot, at any finite order, produce the exponentially small splitting proportional to $e^{-S_0/\hbar}$.
2. The two minima are not two exact finite-volume ground states. Tunneling produces symmetric and antisymmetric eigenstates.
3. In a field theory, the tunneling action can grow with spatial volume, making the splitting vanish and leaving distinct pure phases after the thermodynamic limit.
4. This example teaches the branch and volume-limit logic, but it has no four-dimensional gauge-bundle data and no one-form-symmetry anomaly. Those conclusions require their own hypotheses.

For the one-dimensional instanton connecting $-a$ to $a$,

$$S_0 =\int_{-a}^{a}\mathrm dq\,\sqrt{2mV(q)} =\frac{4}{3}a^3\sqrt{2m\lambda}.$$

The exponential is a precise beyond-all-orders contribution relative to the local $\hbar$ expansion. The prefactor, multi-instanton corrections, and error estimate are further calculations, not consequences of the label “nonperturbative.” Coleman gives the classic semiclassical treatment in Coleman 1985, ch. 7, § 2.2, pp. 270–277, and Mariño develops the same quantum-mechanical setting in Mariño 2015, ch. 1, §§ 1.8–1.9, pp. 38–53.

Four checks before accepting a vacuum claim

Name the comparison. State the expansion variable or formulation relative to which the effect is nonperturbative.

Specify the theory globally. Include the gauge-group global form, allowed bundles and defects, boundary conditions, and spacetime structure relevant to the charge lattice.

Order the limits. Keep regulator removal, Euclidean-time projection, thermodynamic limits, source removal, large-$N$ limits, and mass limits in an explicit sequence.

Separate constraints from dynamics. Periodicity is a property of the complete theory; an anomaly excludes infrared responses; neither statement alone selects an energy branch or phase.

These checks are deliberately observable-centered. A method can be exact under its hypotheses and still answer the wrong observable or the wrong limiting problem.

Review the chapter

Why does the function $e^{-A/g}$ with $A>0$ escape every power series about $g=0^+$?

Solution

For every fixed $N$,

$$\lim_{g\to0^+}\frac{e^{-A/g}}{g^N}=0.$$

Repeated differentiation produces only powers of $1/g$ multiplying the same exponential, so every right derivative of its continuous extension at zero vanishes. Its Taylor series is therefore zero even though the function is nonzero for $g>0$.

If $Q\in\mathbb Z$, prove the $2\pi$ periodicity of the sector sum. Identify the premise that fails for a fixed theory with half-integral sectors.

Solution

For integer $Q$,

$$e^{i(\theta+2\pi)Q}=e^{i\theta Q}e^{2\pi iQ}=e^{i\theta Q},$$

term by term, hence $Z(\theta+2\pi)=Z(\theta)$. If half-integral charges are included, $e^{2\pi iQ}$ can equal $-1$, so the termwise argument fails. One must include the discrete theta data and derive whether the shift changes a parameter of the same theory or exchanges distinct theories.

Two models have the same susceptibility $\chi$ at theta zero:

$$\mathcal E_{\mathrm{quad}}(\theta) =\min_k\frac{\chi}{2}(\theta+2\pi k)^2, \qquad \mathcal E_{\mathrm{gas}}(\theta) =\chi(1-\cos\theta).$$

Give one observable distinction.

Solution

The quadratic envelope has a cusp at $\theta=\pi$, whereas the cosine is smooth there. Locally, their fourth derivatives also differ: $\mathcal E_{\mathrm{quad}}^{(4)}(0)=0$ but $\mathcal E_{\mathrm{gas}}^{(4)}(0)=-\chi$. Thus equal curvature at the origin does not determine the higher cumulants or global branch structure.

Why can a unique symmetric ground state in every finite box coexist with spontaneous symmetry breaking?

Solution

The splitting between symmetric and antisymmetric states can vanish as $e^{-cV_s/\hbar}$. An arbitrarily weak extensive source then dominates the splitting when $V_s\to\infty$. Taking the volume limit before sending the source to zero selects one clustering pure phase; reversing the limits retains the symmetric finite-volume expectation value.

A nonzero anomaly excludes a unique trivial symmetric gapped vacuum. Why does it not prove spontaneous symmetry breaking?

Solution

Gapless degrees of freedom, topological order, symmetry extension, or a genuine higher-dimensional bulk boundary can also reproduce the anomaly. Spontaneous breaking is established only after additional dynamics selects it and its vacua or walls match the same background response.

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.