Perturbative Algebraic Interactions

An interaction need not be switched on throughout an entire curved spacetime in order to define what an observer can measure in a bounded region. Perturbative algebraic quantum field theory instead uses compactly supported interactions, causal factorization, and relative S-matrices to construct interacting local algebras. The result is local and covariant, but deliberately modest: every quantity is a formal power series, not a convergent construction of a nonperturbative Hilbert-space theory or a global scattering operator.

Required background. Flat-Spacetime Quantum Field Theory supplies time ordering and perturbative interaction vertices; Partial Differential Equations and Green Functions provides the retarded-support reasoning used below.

Helpful background. Causality and Global Structure establishes causal futures, pasts, and globally hyperbolic regions; Quantum Fields in Curved Spacetime supplies the local-algebra and Hadamard-state setting.

The needed PDE bridge is short. For a Green-hyperbolic free operator $P$, retarded response to a compact source is supported in its causal future. A Hadamard two-point function then permits products of the free field whose wave-front sets satisfy the microlocal product criterion. Those two facts replace, respectively, a global interaction history and a preferred vacuum.

Compactly supported interactions and local S-matrices

Let $(M,g_{\mu\nu})$ be smooth and globally hyperbolic, and let $\mathcal A_0(M)$ be the free Hadamard algebra. Choose a local scalar interaction $\mathcal L_I$ and a real switching function $g\in C_0^\infty(M)$; the switching function and metric are distinct objects despite the customary shared letter. The smeared interaction is

$$V_g=\int_M g(x)\mathcal L_I(x)\,d\mathrm{vol}_g(x).$$

We adopt the convention $S(V)=\mathcal T\exp(iV)$, with natural units and the product in $\mathcal A_0(M)$ denoted by $\star$. Renormalized time-ordered products $T_n$ define

$$S(V)=\mathbf 1+\sum_{n\geq1}\frac{i^n}{n!}T_n(V^{\otimes n}).$$

This equation lives in $\mathcal A_0(M)[[\lambda]]$ when $V$ carries a formal coupling $\lambda$. It does not assert that the series converges for any nonzero numerical $\lambda$. Compact support makes every coefficient local and infrared-safe enough for the construction; ultraviolet singularities are handled by the declared renormalized $T_n$. This is the local S-matrix of Brunetti–Fredenhagen 2000, §§3.1–3.2, not an asymptotic scattering matrix.

Relative S-matrices and the Bogoliubov field

To probe the interacting theory with another compactly supported functional $F$, form the relative S-matrix

$$S_V(F)=S(V)^{-1}_{\star}\star S(V+F).$$

The interacting version of $F$ is the Bogoliubov derivative

$$R_V(F)=\left.\frac{1}{i}\frac{d}{dt}S_V(tF)\right|_{t=0}.$$

The order of the inverse and the sign $1/i$ follow from $S(V)=\mathcal T e^{iV}$. Changing the sign convention in the exponential requires changing both; mixing conventions reverses the first retarded correction.

Expanding independently to first order in $V$ gives

$$\begin{aligned} S(V)^{-1}_{\star}&=\mathbf 1-iV+O(V^2),\\ S(V+tF)&=\mathbf 1+iV+itF-t\,T_2(V,F)+O(V^2,t^2), \end{aligned}$$

and hence

$$R_V(F)=F+i\bigl[T_2(V,F)-V\star F\bigr]+O(V^2).$$

If the support of $V$ lies entirely later than the support of $F$, time ordering gives $T_2(V,F)=V\star F$, so the correction vanishes. Spacelike-separated vertices also cancel, using local commutativity. Together these facts identify the retarded support: only interaction vertices in the causal past of the probe can contribute.

Causal factorization does the infrared work

The central identity is

$$S(A+B+C)=S(A+B)\star S(B)^{-1}_{\star}\star S(B+C)$$

whenever $\operatorname{supp}A$ is later than $\operatorname{supp}C$. It says that once the middle interaction $B$ is held fixed, future and past changes factor in causal order. The construction and its use in local interacting algebras are given in Brunetti–Fredenhagen 2000, §8, especially Proposition 8.1.

Let $O$ be a relatively compact causally convex region and let two switching functions $g$ and $g'$ both equal one on a neighborhood of its closure. Split their difference into $g'-g=a+b$, where $a$ has no intersection with the causal past of $O$ and $b$ has no intersection with its causal future. Causal factorization first removes $a$ from relative S-matrices localized in $O$ and then shows that the remaining change $b$ acts by an inner intertwiner. Consequently,

$$\mathcal A_{g\mathcal L_I}(O)\simeq \mathcal A_{g'\mathcal L_I}(O).$$

The equality sign would be too strong: the two realizations inside the free algebra depend on the cutoff, while their local interacting content is canonically related up to the specified intertwining maps. No limit $g\to1$ on all of $M$ is needed.

From cutoff-dependent representatives to a local net

For each suitable region $O$, choose any $g$ that is one near $O$ and generate an algebra from $S_{V_g}(F)$ with $\operatorname{supp}F\subset O$. The previous argument identifies different choices of $g$. If $O_1\subset O_2$, restriction of these compatible families gives the inclusion map

$$\mathcal A_I(O_1)\longrightarrow\mathcal A_I(O_2).$$

This produces an interacting net order by order. Local covariance further requires that a causality-preserving isometric embedding of backgrounds carry the interaction density, test functions, and renormalized products functorially. The construction concerns the algebra first. A state on that algebra and its GNS representation are additional choices; neither a preferred interacting vacuum nor a global particle interpretation follows from the existence of the net.

One useful special case is an action interaction $\mathcal L_I=-\lambda\phi^4/4!$, corresponding to a stable quartic potential for positive $\lambda$. Since $S(V)=\mathcal T e^{iV}$, its first-order vertex is $-i\lambda$. On a prescribed curved background, admissible local counterterms include curvature couplings as well as flat-space terms. These choices enter the time-ordered products; definite numerical predictions require renormalization conditions in addition to the existence of the formal net.

Hypotheses, scope, and failure tests

The construction assumes global hyperbolicity, compact switching, a free Hadamard algebra, finite-order local functionals, causal factorization, and renormalized time-ordered products satisfying the relevant microlocal conditions. Hollands and Wald establish existence of local covariant time-ordered products for the scalar theory under a precise list of axioms; their scaling expansion controls the extension to the total diagonal Hollands–Wald 2002, §§2–4 and §4.1, Theorem 4.1.

Three failures discriminate the theorem from stronger claims. First, replacing $g$ by the constant function one is an infrared or adiabatic-limit question and may fail, especially for massless fields. Second, abandoning the Hadamard/microlocal condition can make products of distributions undefined before any renormalization choice is considered. Third, dropping causal factorization allows a future change of the interaction to alter an earlier observable. None of these failures is repaired merely by truncating at a higher perturbative order.

The formal perturbative scope is essential. Local S-matrices establish coefficientwise consistency and local covariance; they do not prove convergence, positivity of a resummed state, nonperturbative existence, or unitary equivalence of global Hilbert-space representations. Backreaction is also absent because $g_{\mu\nu}$ is prescribed rather than solved from an interacting expectation value.

Solved diagnostic: a future change of switching

Suppose $g'=g+a$, where $a$ is compactly supported strictly to the future of a causally closed region $O$, and let $F$ be supported in $O$. Show that the interacting observable represented by $S_{V_g}(F)$ is unchanged.

Solution

Set $A=V_a$, $B=V_g$, and $C=F$ in causal factorization. Because $A$ is later than $C$, the identity gives $S(V_a+V_g+F)=S(V_a+V_g)\star S(V_g)^{-1}_{\star}\star S(V_g+F)$. Left multiplication by $S(V_a+V_g)^{-1}_{\star}$ therefore yields $S_{V_{g'}}(F)=S(V_g)^{-1}_{\star}\star S(V_g+F)=S_{V_g}(F)$. Thus a change confined to the probe’s future cancels from the relative S-matrix, rather than merely disappearing at first order. The conclusion follows coefficient by coefficient and uses both compact support and the stated causal ordering. If the added interaction overlapped the causal past of the probe, the factorization could not be applied in this order and a genuine retarded correction would generally remain.

References

• 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.
• Hollands, Stefan, and Robert M. Wald. “Existence of Local Covariant Time Ordered Products of Quantum Fields in Curved Spacetime.” Communications in Mathematical Physics 231 (2002): 309–345. DOI.