Interacting Fields on Curved Backgrounds

Interacting quantum field theory on a prescribed curved spacetime reuses the perturbative language of flat-space QFT but changes the global questions one is entitled to ask. Feynman graphs, gauge symmetry, and ordinary ultraviolet power counting transfer; a global S-matrix, preferred vacuum, or constant-coupling momentum-space treatment may not. The essential curved extensions are local covariant deformation, curvature counterterms, Hadamard states, and causal in-in evolution.

Required background. Flat-Spacetime Quantum Field Theory supplies perturbative amplitudes, gauge structure, and standard renormalization; Partial Differential Equations and Green Functions provides causal propagators and initial-value support.

Helpful background. Causality and Global Structure establishes the globally hyperbolic regions used by local algebras; Quantum Fields in Curved Spacetime supplies Hadamard singular structure and the distinction between algebra, state, and representation.

The bounded PDE bridge is the retarded solution of a compact source problem. It tells us how an interaction in the causal past changes a local observable. The bounded state bridge is that Hadamard states share the ultraviolet singularity needed for local renormalization, even though their smooth remainders and physical expectation values differ.

A flat-space derivation worth importing

Consider a scalar theory on Minkowski spacetime with

$$\mathcal L=-\frac12\partial_\mu\phi\,\partial^\mu\phi -\frac12m^2\phi^2-\frac{\lambda}{4!}\phi^4.$$

The interaction Hamiltonian density is $\mathcal H_I=\lambda\phi_I^4/4!$. Dyson expansion gives

$$S=\mathbf1-i\int d^4x\,\mathcal H_I(x)+O(\lambda^2).$$

In a four-point matrix element, contracting the four fields at the vertex with four external legs produces $4!$ assignments, cancelling the normalization in $\mathcal H_I$. The connected tree vertex is therefore

$$\mathcal V_4=-i\lambda.$$

This sign follows jointly from the minus sign in the action interaction, the positive interaction Hamiltonian, and the $e^{-iH_It}$ evolution convention. The combinatorics, loop expansion, gauge-fixing machinery, BRST identities, scattering kinematics, and ordinary flat renormalization belong to Flat-Spacetime Quantum Field Theory. They should be learned once rather than rebuilt for each geometry.

The derivation also shows what does not transfer automatically. It used translation invariance to label external states by four-momentum and assumed asymptotic particle states. A generic expanding or black-hole spacetime has neither global symmetry. The local vertex remains meaningful, but a global amplitude with four on-shell external plane waves may not.

Local deformation replaces the global S-matrix

On a smooth globally hyperbolic $(M,g)$, choose a compact switching function $f$ and write

$$V_f=-\int_M d\mathrm{vol}_g\,f(x)\frac{\lambda}{4!}\phi^4(x).$$

Renormalized time-ordered products define $S(V_f)$ as a formal series. The relative S-matrix

$$S_{V_f}(F)=S(V_f)^{-1}_{\star}\star S(V_f+F)$$

generates observables in the interacting theory. Causal factorization proves that changing $f$ away from the causal neighborhood of $\operatorname{supp}F$ changes only the representative, with local algebras related by intertwiners. The detailed construction is given in Brunetti–Fredenhagen 2000, §§3 and 8.

This is a formal local S-matrix, not the flat asymptotic scattering matrix. The hypothesis includes compact switching, a free Hadamard algebra, a declared renormalized time ordering, and formal power series. If an asymptotically stationary geometry later permits an in/out construction, that is extra structure to be proved, not part of the local theorem.

Curvature enlarges operator mixing

Flat-space power counting remains necessary but is not sufficient. A scalar operator of the appropriate dimension can mix with local curvature expressions. For example, renormalization of scalar interactions generally allows

$$R\phi^2,\qquad R^2\mathbf1,\qquad R_{\mu\nu}R^{\mu\nu}\mathbf1,$$

subject to dimension, covariance, symmetry, and equivalence relations. All vanish or become invisible in sufficiently special flat backgrounds, so flat scattering data alone cannot fix their finite coefficients.

This gives a discriminating counterexample to the claim that flat renormalization conditions determine the curved theory. Take two admissible prescriptions differing by $cR\phi^2$. Their Minkowski correlation functions agree because $R=0$, yet their predictions differ on a spacetime with nonzero scalar curvature. Local covariance permits the difference; a curved-space renormalization condition is needed to fix $c$.

Hollands and Wald prove that local covariant time-ordered products can be constructed by a scaling expansion and that the remaining freedom is local and covariant Hollands–Wald 2002, §§2–4, especially §4.1, Theorem 4.1. Their result does not turn curvature counterterms into arbitrary functions: the coefficients must arise naturally from the background fields.

States and contours are part of the physical question

An algebra does not choose a state. On a stationary background, symmetry and regularity may distinguish a ground or KMS state; on a generic background there need be no preferred vacuum. Interacting expectation values therefore require an initial state or density operator in addition to the local algebra. The state must have suitable ultraviolet behavior, normally expressed through a Hadamard condition or its interacting analogue.

Likewise, the contour depends on the observable. In-out methods generate transition amplitudes and Feynman boundary conditions. An expectation value evolved from an initial density matrix requires a forward and backward Schwinger–Keldysh contour. Equal branch sources enforce normalization, and the physical effective equation has retarded support. Importing an in-out self-energy into an initial-value mean-field equation can produce complex or acausal terms even when the underlying flat diagram was evaluated correctly.

Thus “use the flat Feynman rule” is only the first line of a curved calculation. One must also specify the state, local renormalization prescription, observable, contour, background, and causal domain.

Prescribed geometry versus backreaction

This volume treats $g_{\mu\nu}$ as prescribed. For constant physical couplings, it may compute a renormalized matter expectation value such as $\langle T_{\mu\nu}\rangle$ and verify

$$\nabla^\mu\langle T_{\mu\nu}\rangle=0$$

when the matter equations and renormalization conditions imply it. A spacetime-dependent external coupling can exchange energy with the field. For the classical action interaction $-\lambda(x)\phi^4/4!$, the on-shell diffeomorphism identity instead reads

$$\nabla^\mu T_{\mu\nu} =-\frac{\phi^4}{4!}\nabla_\nu\lambda.$$

The quantum identity uses the corresponding renormalized composite operator and counterterms. Thus one checks conservation in a constant-coupling region, or the source-dependent balance law where the interaction is being switched. Neither check solves for a geometry responding to the field.

The self-consistent question instead has the schematic form

$$G_{\mu\nu}+\Lambda g_{\mu\nu} +\text{higher-curvature terms} =8\pi G\,\langle T_{\mu\nu}\rangle_{\mathrm{ren}}.$$

Existence, stability, order reduction, fluctuations, and evaporation then depend on the gravitational equation as well as the state. Those questions pass to Semiclassical Gravity. Treating a fixed-background matter calculation as a solved backreaction problem would silently change the dynamical system.

A practical division of questions

Use Flat-Spacetime Quantum Field Theory for generic perturbative graph combinatorics, scattering and LSZ reduction, gauge and BRST foundations, ordinary flat-space renormalization, effective field theory, and reusable many-body methods. Retain here the deformation of local algebras on globally hyperbolic backgrounds, local covariant time ordering, curvature-dependent counterterms, Hadamard state dependence, and causal closed-time-path expectation values. Pass a dynamically responding metric to Semiclassical Gravity.

The bridge remains self-contained because the crucial formulas and hypotheses appear here. For example, a flat one-loop calculation can determine the short-distance coefficient of a four-point counterterm, but it cannot choose an initial state in an expanding geometry or fix a finite $R\phi^2$ coupling. Geometry, state, causal support, and the role of the metric must accompany the imported calculation before it yields a physical prediction.

Solved diagnostic: route three calculations

Classify the following tasks and justify each choice: deriving the tree-level four-scalar vertex; determining whether $R\phi^2$ is required by local covariant renormalization; and deciding whether Hawking flux shrinks a black hole self-consistently.

Solution

The tree vertex follows from flat-spacetime perturbation theory: Dyson expansion and Wick contractions give $-i\lambda$ without requiring curvature. The $R\phi^2$ question is intrinsically curved because the operator vanishes in Minkowski space and is selected by local covariance and power counting. Self-consistent shrinking changes the metric in response to the quantum stress tensor, so it requires the coupled equations of semiclassical gravity. The three calculations differ in whether curvature enters and whether the metric is prescribed or determined dynamically, not merely in which diagrams appear.

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.