Perturbative Agreement and Background Independence

Perturbative agreement says that an exactly tractable quadratic term may be placed either in the free operator or in the interaction without changing the perturbative algebra, provided the classical and quantum Møller maps exist on the declared support domain and the renormalization prescription satisfies the agreement condition. It is a coherent comparison of two splittings of the same classical action, not a nonperturbative uniqueness theorem and not global background independence.

Required background. The pAQFT Bogoliubov map supplies the quantum Møller map, and the Stückelberg–Petermann group supplies the necessary finite adjustment. Helpful background. Relative Cauchy evolution treats controlled background changes, stress-energy response gives their infinitesimal meaning, and background splitting in curved spacetime develops the physical application.

Two splittings of one quadratic theory

Let

$$P_0=\Box_g+m_0^2+\xi_0R, \qquad P_1=P_0+Q,$$

where $Q(x)=\delta m^2(x)+\delta\xi(x)R(x)$ is a smooth compactly supported multiplication operator. The actions satisfy $S_1=S_0+Q_2$ with

$$Q_2(\varphi)=\frac12\int\varphi Q\varphi\,\mathrm d\mu_g.$$

The retarded classical Møller map pulls $P_1$ solutions to $P_0$ configurations; in one convention it is generated by $r_Q=1-\Delta_1^R Q$, with inverse $1+\Delta_0^R Q$ on compactly supported variations. Retarded support makes it the identity outside the future of $\operatorname{supp}Q$.

The quantum products built from $P_0$ and $P_1$ use different Hadamard/Feynman kernels. Their comparison is a contraction exponential $\beta_Q=\alpha_{d_Q}$ with smooth or appropriately microcausal kernel $d_Q$ given by the difference after pullback. The principle of perturbative agreement requires the quantum Møller map for the quadratic interaction to factor as

$$\mathcal R_{0,Q}=\beta_Q\circ r_Q^*.$$

For an additional local interaction $V$, the two routes—treating $Q+V$ perturbatively over $P_0$, or treating $V$ perturbatively over $P_1$—then intertwine order by order. Drago, Hack, and Pinamonti prove the generalized scalar-field statement for quadratic interactions without derivatives and specify its domain in Drago, Hack, and Pinamonti 2017, §§3–4, pp. 824–851.

The factorization separates two effects that must not be conflated. The pullback $r_Q^*$ changes the classical equation and causal propagator; $\beta_Q$ changes the quantum contraction convention. On regular functionals this can be checked directly. Extension to local functionals requires renormalized time-ordered products, and the finite discrepancy is removed only by imposing perturbative agreement as a normalization condition. Therefore classical equivalence of the splittings is necessary but not by itself sufficient for equality of the quantum prescriptions.

Compact mass and curvature shift

Choose compact $\chi$ and set

$$Q=\chi(x)(\delta m^2+\delta\xi R).$$

Route A quantizes $P_0$ and inserts $Q_2$ through the Bogoliubov map. Route B quantizes $P_1$ exactly and inserts only the remaining interaction. For a regular functional $F$, expansion of the classical pullback gives

$$r_Q^*F=F-\left\langle F^{(1)}, \Delta_1^R Q\varphi\right\rangle+O(Q^2).$$

The contraction map adds the first quantum comparison term

$$\beta_QF=F+\frac{\hbar}{2} \left\langle d_Q,F^{(2)}\right\rangle+O(\hbar^2).$$

Their composition reproduces the first-order Bogoliubov response to $Q_2$. At second and higher orders, nested retarded insertions and repeated contractions match after the time-ordered products are normalized to satisfy perturbative agreement. This explicitly constructs the intertwiner for moving the mass and curvature-coupling perturbation between the two splittings, as required by the curved-spacetime comparison.

An independent check uses a linear observable $F_f(\varphi)=\langle f,\varphi\rangle$. Since $F_f^{(2)}=0$, $\beta_Q$ acts trivially on it, and the entire comparison is the classical retarded Møller map. For a quadratic observable the contraction term is nonzero, exactly where the two normal-ordering prescriptions differ.

There is also a product check. Transport the $P_1$ star product back with $r_Q^*$ and then apply $\beta_Q$. The contraction exponential adds exactly the pulled-back difference of the two-point kernels, so the result equals the $P_0$ star product on regular functionals. This verifies that $\mathcal R_{0,Q}$ is an algebra intertwiner before the local extension problem is introduced.

Coherence and its limit

For successive compact quadratic changes $Q_1,Q_2$, the comparison maps must satisfy a cocycle relation: comparing $P_0$ to $P_2$ directly agrees with comparison through $P_1$. This is not automatic for arbitrary extensions; it is a renormalization condition. Any residual local discrepancy is a Stückelberg–Petermann transformation and must be fixed coherently.

Adversarial test. Change the metric or coefficients without compact support or another hypothesis ensuring retarded Møller maps act on the chosen compact/microcausal domain. Propagator differences may fail to have the proper support or smoothness needed for $r_Q^*$ and $\alpha_{d_Q}$, and the cocycle proof loses its domain. One cannot infer a global isomorphism between arbitrary backgrounds.

Even when the formal intertwiner exists, it compares perturbative algebras. It does not establish that two nonperturbative Hilbert-space theories are unitarily equivalent, nor does it make the metric dynamical.

Exercises

1. Retarded inverse. Verify $(1-\Delta_1^RQ)(1+\Delta_0^RQ)=1$ on compact sources, using the resolvent identity.

Solution

The retarded resolvent identity is $\Delta_1^R=\Delta_0^R-\Delta_0^RQ\Delta_1^R$. Expanding the product leaves $\Delta_0^RQ-\Delta_1^RQ-\Delta_1^RQ\Delta_0^RQ$, which vanishes after the equivalent rearrangement of that identity. Proper support makes every composition defined.

2. Linear-functional check. Explain why the quantum contraction comparison first appears for functionals of degree at least two.

Solution

$\alpha_{d_Q}$ is generated by a second functional derivative. It annihilates constants and linear functionals, while for a quadratic functional the second derivative is nonzero and the first contraction term contributes.

References

• Drago, Nicolò, Thomas-Paul Hack, and Nicola Pinamonti. “The Generalised Principle of Perturbative Agreement and the Thermal Mass.” Annales Henri Poincaré 18 (2017): 807–868. DOI; Open preprint.
• Hollands, Stefan, and Robert M. Wald. “Conservation of the Stress Tensor in Perturbative Interacting Quantum Field Theory in Curved Spacetimes.” Reviews in Mathematical Physics 17 (2005): 227–312. DOI; Open preprint.