Interacting pAQFT on Curved Spacetimes

Perturbative algebraic QFT (pAQFT) constructs interacting local observables on a fixed globally hyperbolic spacetime as formal power series in the coupling and $\hbar$. Causal factorization, microlocal function spaces, and locally covariant extensions of time-ordered products give interacting local algebras without selecting a global interacting vacuum. The construction is mathematically controlled order by order; it does not establish convergence of the series or a nonperturbative continuum theory.

Required background. Formal-power-series construction and state scope supplies the perturbative status. Time-ordered products and the renormalized stress tensor supplies extension to coincident points. Algebraic free fields on curved spacetimes supplies the free algebra.

Helpful background. Perturbative agreement and background independence compares free splittings. Adiabatic limits and interacting nets separates local from global limits. Why local covariant renormalization is needed and the curved-spacetime $\phi^4$ benchmark develop the physical model.

Microcausal functionals and the free product

Let $\mathcal E(M)=C^\infty(M)$ be the off-shell configuration space of a real scalar field. A smooth functional $F:\mathcal E(M)\to\mathbb C$ is microcausal when its functional derivatives are compactly supported distributions whose wavefront sets avoid configurations in which all covectors lie in the same closed future or past cone. This condition makes contractions with a Hadamard two-point function well defined.

Choose a Hadamard bidistribution $H_+$ with antisymmetric part $iE/2$. The free product is the formal series

$$F\star_HG =\sum_{n=0}^\infty\frac{\hbar^n}{n!} \left\langle F^{(n)},H_+^{\otimes n}G^{(n)}\right\rangle.$$

Changing $H_+$ by a smooth symmetric bisolution gives an isomorphic algebra, so this auxiliary choice is not a vacuum selection. Local polynomial functionals have functional derivatives supported on coincident points and require renormalized time ordering. Epstein–Glaser extension increases distributions from configuration space with diagonals removed to the diagonals while preserving causal factorization, covariance, scaling, and the microlocal spectrum condition. Brunetti and Fredenhagen establish this curved-background extension method in Brunetti and Fredenhagen 2000, §§ 3–6, pp. 640–660.

Local S-matrices and interacting fields

For a compactly supported interaction

$$V_g(\phi)=\frac{\lambda}{4!}\int_Mg(x)\phi(x)^4\,\mathrm d\mathrm{vol}_g, \qquad g\in C_0^\infty(M),$$

define the formal local S-matrix

$$S(V)=\sum_{n=0}^\infty\frac{1}{n!} \left(\frac{i}{\hbar}\right)^n\mathrm T_n(V^{\otimes n}).$$

Its central law is causal factorization: if the support of $A$ is not in the causal past of the support of $C$, then

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

The relative S-matrix $S_V(F)=S(V)^{-1}\star S(V+F)$ generates the interacting algebra, and differentiation in $F$ gives the Bogoliubov interacting field. In retarded-product notation,

$$R_V(F)=F+\lambda R_1(V_g;F) +\frac{\lambda^2}{2}R_2(V_g,V_g;F)+O(\lambda^3).$$

Each coefficient has retarded causal support: changing the interaction outside the causal past of the observable cannot affect it. If two cutoff functions equal one on a causally closed neighborhood of a region $O$, causal factorization produces an isomorphism between the corresponding interacting algebras on $O$. This is the algebraic adiabatic limit. It removes dependence on the cutoff locally, not by taking $g\to1$ over the whole spacetime.

Hollands and Wald prove existence of local covariant time-ordered products satisfying the required axioms and classify their finite freedom in Hollands and Wald 2002, Theorems 5.1–5.2, pp. 328–343. Covariance under a causality-preserving isometric embedding follows when the free propagators, Hadamard parametrices, interaction density, and renormalization maps form natural assignments. An arbitrary spacetime-dependent choice of finite constants would break this step.

A second-order curved-spacetime construction

The application on local S-matrices and causal factorization takes compactly supported $\phi^4$ on a globally hyperbolic background. Through second order, the two retarded products above define the interacting local algebra. Starting from a free Hadamard state $\omega$, composition with the perturbative Møller map gives a formal state

$$\omega_V(F)=\omega(R_V(F))$$

through the same order. Renormalized contractions preserve the microlocal spectrum condition order by order, so the singular part remains of Hadamard type in the perturbative sense. This is not an ordinary convergent positive functional at a fixed nonzero $\lambda$; positivity is interpreted in the formal deformation framework unless a separate state-construction theorem is supplied.

Two independent checks are useful. First, the first retarded correction vanishes when the interaction support lies entirely to the future of $F$. Second, under a causally convex embedding $\psi:M\to N$, pushing forward $F$ and $g$ before constructing $R_1$ and $R_2$ must give the same coefficients as constructing them on $M$ and then applying the algebra map. A mismatch localizes a violation of propagator covariance or renormalization naturality.

Failure boundary: the missing global limit

Set $g=1$ on an infinite-volume spacetime and announce a global interacting vacuum. The local proof does not license that step: $V_g$ is no longer compactly supported, the formal S-matrix need not exist as an element of the chosen algebra, infrared contributions need not converge, and no GNS representation or ground-state spectral condition has been constructed. Even if every bounded region has an interacting algebra, their realization in one global representation is an additional problem.

The strongest surviving conclusion is an interacting net of local formal algebras, locally independent of cutoff choices that agree near the region, with formal states or specially constructed thermal states where separate hypotheses apply. Neither the local adiabatic argument nor a second-order calculation proves a global vacuum, convergence in $\lambda$, or nonperturbative existence.

Exercises

Use causal factorization to explain why changing the interaction in the strict future of an observable does not change its retarded interacting field.

Solution

Write the changed interaction as $V+C$, with $\operatorname{supp}C$ later than $\operatorname{supp}F$. Causal factorization separates the factor $S(C)$ on the side that cancels between $S(V+C)^{-1}$ and $S(V+C+F)$. Therefore $S_{V+C}(F)$ is identified with $S_V(F)$ in the local algebra. Differentiating in $F$ gives the same retarded field.

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; Open PDF.
• Fredenhagen, Klaus, and Kasia Rejzner. “Batalin–Vilkovisky Formalism in Perturbative Algebraic Quantum Field Theory.” Communications in Mathematical Physics 317 (2013): 697–725. DOI; Open PDF.
• 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; Open PDF.