Adiabatic Limits and Algebraic Interacting Nets

The algebraic adiabatic limit removes a compact switching function from the description of observables in a bounded region without requiring a global operator limit at constant coupling. If two switchings agree on a suitable causal neighborhood of the region, causal factorization gives an inner intertwiner between their relative S-matrices there. The resulting compatible local algebras form an interacting net even when infrared effects prevent $g\to1$ globally.

Required background. Bogoliubov maps in perturbative AQFT defines relative S-matrices and interacting observables. Causal factorization supplies the support identity used to compare switchings. Helpful background. Thermal nuclearity, return to equilibrium, and mixing explains analytic input needed for some thermal infinite-volume limits. Clustering, vacuum uniqueness, and the mass gap separates local construction from representation-level infrared control. Adiabatic limits and infrared obstructions treats the curved-spacetime and massless qualifications.

Relative S-matrices near a bounded region

Let $O$ be a relatively compact, causally convex region of a globally hyperbolic spacetime, and let

$$V_g=\int_M g(x)\mathcal L_{\mathrm{int}}(x)\,\mathrm d\mu_g(x), \qquad g\in C_c^\infty(M).$$

For a compactly supported local functional $F$, define the relative S-matrix

$$S_g(F)=S(V_g)^{-1}\star S(V_g+F).$$

All inverses and products are coefficientwise in the formal coupling and $\hbar$; no operator norm or strong limit is hidden in this notation. The local algebra $\mathfrak A_g(O)$ is generated by $S_g(F)$ with $\operatorname{supp}F\subset O$.

Choose a neighborhood $N$ containing the causal closure needed to propagate all such supports. Suppose $g$ and $g'$ coincide on $N$, and put $h=g'-g$. A partition of unity adapted to two Cauchy surfaces around $N$ decomposes the change as

$$h=h_-+h_+,$$

where $\operatorname{supp}h_-$ lies to the causal past of $O$ and $\operatorname{supp}h_+$ lies to its future, up to pieces spacelike to the relevant causal hull. Causal factorization cancels the future piece in the retarded relative S-matrix; the past piece acts by conjugation. Consequently there is a formally invertible $U(g',g)$, independent of $F$ in the local generating family, such that

$$S_{g'}(F)=U(g',g)\star S_g(F)\star U(g',g)^{-1}, \qquad \operatorname{supp}F\subset O.$$

Different decompositions change $U$ only by factors that act trivially on the local family. With coherent choices, the intertwiners obey the cocycle relation

$$U(g'',g')\star U(g',g)=U(g'',g).$$

This is the theorem’s mechanism: a geometric support split, followed by two uses of causal factorization. Brunetti and Fredenhagen construct the local interacting net this way in Brunetti and Fredenhagen 2000, §§7–8, pp. 646–655; the relative-S-matrix and algebraic-adiabatic formulation is stated in Brunetti, Dütsch, and Fredenhagen 2009, §6.3, pp. 1574–1576.

The conclusion is an isomorphism of local formal algebras. It is not literal equality of two chosen representatives, not convergence of a Dyson series, and not existence of a vacuum state. A compatible choice of intertwiners under inclusions $O_1\subset O_2$ yields isotony; locality follows because spacelike-supported relative S-matrices commute by causal factorization. Covariance requires the time-ordered products and the choice-independent construction to be locally covariant.

First application: massive φ⁴ theory

Take four-dimensional massive scalar theory with

$$\mathcal L_{\mathrm{int}}=\frac{\lambda}{4!}\phi^4$$

and two compact switchings $g,g'$ equal to one on a neighborhood of the causal closure of a fixed double cone $O$. For $F_f=\int f\phi^2\,\mathrm d^4x$ with $\operatorname{supp}f\subset O$, expand $S_g(F_f)$ to any fixed total order. Every coefficient contains only finitely many compactly supported distribution pairings. Replacing $g$ by $g'$ changes vertices outside $N$; the decomposition above moves future changes out of the retarded product and absorbs past changes into $U(g',g)$. Thus $F_f$ determines the same abstract local observable in both descriptions.

This is the local construction needed before discussing interacting fields and effective descriptions. The positive mass is helpful for later clustering or weak adiabatic limits, but it is not used to prove the local switching isomorphism itself. Conversely, local switching independence does not prove a global massive vacuum exists for the four-dimensional model.

An independent check uses nested regions. Let $O_1\Subset O_2$ and choose $g_2=1$ near the causal hull of $O_2$. Its restriction already serves for $O_1$, so the generators of $\mathfrak A_{g_2}(O_1)$ are a subalgebra of $\mathfrak A_{g_2}(O_2)$. If a second switching is used for $O_1$, the cocycle intertwiner identifies it with this subalgebra. This verifies isotony without taking any limit.

Adversarial test: the massless global limit

Now set the mass to zero and enlarge a sequence $g_R$ toward the constant function one. Long-range correlations can make individual coefficients grow with $R$, depend on the way the temporal and spatial cutoffs are removed, or fail to define a vacuum expectation value. None of this contradicts the local theorem: for fixed $O$, sufficiently large $g_R$ agree on the required neighborhood and their algebras are already isomorphic.

The failed inference is

$$\text{local cocycle equivalence for every bounded }O \;\not\Longrightarrow\; \lim_{R\to\infty}S(V_{g_R})\text{ exists}.$$

A global vacuum, KMS state, or scattering matrix requires additional infrared estimates in a specified representation. The algebraic adiabatic limit deliberately licenses less and therefore survives more generally.

Exercises

1. Future switching changes. Let $h_+$ be supported entirely later than both $\operatorname{supp}F$ and the remaining interaction change. Use causal factorization to explain why it cancels from $S_g(F)$.

Solution

Both $S(V_g+V_{h_+}+F)$ and $S(V_g+V_{h_+})$ factor with the same later factor $S(V_{h_+})$. In the ratio defining the relative S-matrix that factor cancels. A past factor instead remains on opposite sides and produces conjugation.

2. Check the cocycle. Compose the isomorphism from $g$ to $g'$ with that from $g'$ to $g''$.

Solution

Two conjugations give $\operatorname{Ad}_{U(g'',g')\star U(g',g)}$. Direct comparison of $g$ and $g''$ gives $\operatorname{Ad}_{U(g'',g)}$. Coherent representatives may therefore be chosen with $U(g'',g')\star U(g',g)=U(g'',g)$; central factors act trivially on the local algebra.

References

• Brunetti, Romeo, Michael Dütsch, and Klaus Fredenhagen. “Perturbative Algebraic Quantum Field Theory and the Renormalization Groups.” Advances in Theoretical and Mathematical Physics 13 (2009): 1541–1599. DOI; Open PDF.
• Brunetti, Romeo, and Klaus Fredenhagen. “Microlocal Analysis and Interacting Quantum Field Theories: Renormalization on Physical Backgrounds.” Communications in Mathematical Physics 208 (2000): 623–661. DOI.