Adiabatic Subtraction for Local Observables

Adiabatic subtraction renormalizes a specified local FLRW observable by removing its state-independent large-$k$ WKB terms. It does not prepare an adiabatic vacuum. In four dimensions, $\phi^2$ normally requires subtraction through second adiabatic order, while the stress tensor requires fourth order.

Required background. FLRW mode quantization fixes the modes; adiabatic states fixes the expansion; the renormalized-stress axioms fix allowed ambiguities; and scheme comparison fixes translations. Helpful background. See mode-sum renormalization and point splitting.

Observable-specific subtraction

For a homogeneous Gaussian scalar state,

$$\langle\phi^2\rangle_{\rm ren} =\frac{1}{2\pi^2a^2} \int_0^\infty dk\,k^2 \left(|v_k|^2-|v_k|^2_{\rm ad,0-2}\right) +C_{\phi^2},$$

where $C_{\phi^2}$ is an allowed finite local term. The energy density and pressure are mode integrals built from $v_k,v_k'$, $m,\xi,a$, and derivatives of $a$. Their fourth-order subtraction has the schematic form

$$\rho_{\rm ren} =\frac{1}{4\pi^2a^4}\int_0^\infty dk\,k^2 \left(\rho_k-\rho_{k,\rm ad}^{(0-4)}\right) +\rho_{\rm local}^{\rm fin},$$

with an analogous $p_{\rm ren}$. Zeroth order removes quartic divergences, second order renormalizes $G$ and mass-dependent local terms, and fourth order renormalizes curvature-squared terms. Subtracting unnecessary higher orders is not an allowed ambiguity of the fixed renormalizable stress prescription. It becomes an EFT redefinition only if the corresponding higher-derivative operators, finite couplings, and truncation error are explicitly added.

Parker and Fulling constructed the conserved adiabatic stress subtraction for homogeneous spacetimes Parker and Fulling 1974, §§II–IV, pp. 344–352.

First application: conserved scalar stress

Choose one fourth-order adiabatic/Hadamard state and evolve exact modes. Compute $\rho_k$ and $p_k$ independently, derive both subtraction integrands from the same WKB expansion, and verify

$$\rho_{\rm ren}'+3\mathcal H \left(\rho_{\rm ren}+p_{\rm ren}\right)=0.$$

This identity is more discriminating than ultraviolet convergence: subtracting the density to fourth order while using an unmatched pressure leaves a finite violation. Also monitor the Wronskian and cutoff tails. After subtraction, the residual integrands should fall rapidly enough that increasing $k_{\max}$ changes the answer below the declared numerical tolerance.

State and scheme variations are distinct tests. Holding exact $v_k$ fixed while changing a valid scheme shifts the result only by a local conserved curvature tensor. Changing the initial $v_k$ changes the smooth state-dependent part and is not absorbed into gravitational couplings.

Local ambiguity and tail diagnostics

In four dimensions the allowed finite stress ambiguity is a linear combination of conserved tensors obtained by varying the cosmological, Einstein–Hilbert, and curvature-squared terms. Its coefficients are spacetime-independent constants fixed by renormalization conditions. A proposed “scheme change” that depends on the state’s occupation, on an earlier value of $a$, or on a comoving cutoff is therefore not an allowed local ambiguity.

The subtraction integrand is usefully checked before integration. Expand the exact large-$k$ density and pressure through fourth adiabatic order and confirm coefficient-by-coefficient cancellation against the counterterms. The remaining tails must be integrable in the same physical state and must satisfy continuity after integration. Because large bare and subtraction pieces nearly cancel, raising arithmetic precision can be as important as extending $k_{\max}$; apparent plateaus at fixed precision can be subtraction loss rather than continuum convergence.

For $\phi^2$, fourth-order terms may be retained only if the observable definition and its finite local ambiguity are translated accordingly; second order removes the divergences required by the ordinary four-dimensional prescription. For $T_{\mu\nu}$, stopping at second order leaves curvature-squared divergences and cannot source gravity. Conversely, subtracting sixth or higher order without adding and matching the associated higher-derivative EFT operators changes the problem rather than improving the fixed stress definition.

The structure map places local subtraction between state-dependent modes and the backreaction source, bypassing the particle-number branch.

Adiabatic subtraction renormalizes each local composite to its required order; conservation links density and pressure counterterms. Schematic; not to scale.

Domain and failure conditions

The canonical domain table records the second- versus fourth-order distinction. The displayed integrals assume a smooth spatially flat FLRW geometry and a state with sufficient ultraviolet falloff.

Adversarial test. Keep the state and exact modes fixed, alter subtraction order and finite $R^2$-type couplings, and compare. A valid difference is a permitted local conserved tensor with corresponding coupling translation. A nonlocal time-history term or violation of continuity signals inconsistent subtraction, not new particles.

The failure map treats residual cutoff dependence and nonconservation as hard failures for backreaction.

Ultraviolet finiteness is necessary but not sufficient: the renormalized stress must be conserved and scheme differences must be local. Schematic; not to scale.

References

• Parker, L., and S. A. Fulling, “Adiabatic Regularization of the Energy-Momentum Tensor of a Quantized Field in Homogeneous Spaces,” Physical Review D 9, 341–354 (1974), doi:10.1103/PhysRevD.9.341.