Adiabatic Subtraction for Local Observables

Adiabatic subtraction removes the state-independent large-momentum terms required to define a specified local observable. It does not prepare an adiabatic vacuum and it does not turn an instantaneous particle number into a stress tensor. In four dimensions, the ordinary scalar variance requires terms through second adiabatic order, whereas the stress tensor requires the complete fourth-order subtraction.

Required background. FLRW mode quantization fixes exact modes; adiabatic states fixes derivative counting; the renormalized-stress axioms fix allowed finite ambiguities; and scheme comparison fixes translations. Helpful background. Mode-sum renormalization treats cancellation and cutoff errors.

Exact energy and pressure mode integrands

Work first with a massive minimally coupled scalar, $m>0$ and $\xi=0$, so every term can be displayed compactly. With $\chi=a\phi$ and $\mathcal H=a'/a$,

$$v_k''+\left(\omega_k^2-\frac{a''}{a}\right)v_k=0, \qquad \omega_k^2=k^2+m^2a^2,$$

and define the physical-derivative combination

$$D_k=v_k'-\mathcal Hv_k.$$

Computing $T_{00}$ and the spatial trace independently gives

$$\rho_{\rm bare} =\frac1{4\pi^2a^4}\int_0^\infty dk\,k^2\mathcal E_k, \qquad \mathcal E_k=|D_k|^2+\omega_k^2|v_k|^2,$$

and

$$p_{\rm bare} =\frac1{4\pi^2a^4}\int_0^\infty dk\,k^2\mathcal P_k, \qquad \mathcal P_k=|D_k|^2 -\left(\frac{k^2}{3}+m^2a^2\right)|v_k|^2.$$

These formulas expose two frequent mistakes: using $v_k'$ instead of $D_k$, and obtaining pressure by differentiating a numerically integrated energy rather than by evaluating the spatial stress.

The exact mode equation implies

$$D_k'=-\mathcal H D_k-\omega_k^2v_k.$$

It follows directly that each mode satisfies

$$\mathcal E_k' -\mathcal H\mathcal E_k +3\mathcal H\mathcal P_k=0.$$

After restoring the factor $a^{-4}$, this is

$$\rho_{\rm bare}' +3\mathcal H(\rho_{\rm bare}+p_{\rm bare})=0$$

before regularization. The identity is the benchmark that the subtraction terms must preserve.

Fourth-order counterterms without a hidden expansion

Count $a$ as adiabatic order zero and each conformal-time derivative as one order. Write

$$\Omega_k^2=\omega_k^2+\sigma, \qquad \sigma=-\frac{a''}{a},$$

where $\sigma$ is second order. Expand the WKB frequency as

$$W_k=\omega_k+W_{k,2}+W_{k,4}+O(\partial^6).$$

Substitution into

$$W_k^2=\Omega_k^2 -\frac12\frac{W_k''}{W_k} +\frac34\left(\frac{W_k'}{W_k}\right)^2$$

gives a reproducible recursion:

$$W_{k,2} =\frac1{2\omega_k} \left[ \sigma -\frac{\omega_k''}{2\omega_k} +\frac{3\omega_k'^2}{4\omega_k^2} \right].$$

Let $r_{k,2}=W_{k,2}/\omega_k$. The fourth-order term is

$$W_{k,4} =-\frac{W_{k,2}^2}{2\omega_k} -\frac{r_{k,2}''}{4\omega_k} +\frac{\omega_k'r_{k,2}'}{4\omega_k^2}.$$

This form is shorter and less error-prone than a fully expanded polynomial in $a,a',\ldots,a^{(4)}$, while determining that polynomial uniquely.

For a normalized WKB mode

$$w_k=\frac{e^{-i\int W_kd\eta}}{\sqrt{2W_k}},$$

one has

$$|w_k|^2=\frac1{2W_k}, \qquad |w_k'-\mathcal Hw_k|^2 =\frac{W_k}{2} +\frac1{2W_k} \left(\mathcal H+\frac{W_k'}{2W_k}\right)^2.$$

The energy and pressure counterterms are therefore explicitly

$$\mathcal E_{k,\rm ad}^{(0-4)} =\left[ \frac{W_k}{2} +\frac{ \left(\mathcal H+W_k'/(2W_k)\right)^2+\omega_k^2 }{2W_k} \right]_{0-4},$$ $$\mathcal P_{k,\rm ad}^{(0-4)} =\left[ \frac{W_k}{2} +\frac{ \left(\mathcal H+W_k'/(2W_k)\right)^2 -k^2/3-m^2a^2 }{2W_k} \right]_{0-4}.$$

The bracket means: insert $W_k=\omega_k+W_{k,2}+W_{k,4}$, expand algebraically, and discard derivative order six and higher. Thus the prescription states every input needed to regenerate the full fourth-order integrands; it does not hide them behind “subtract the WKB terms.”

The renormalized quantities are

$$\rho_{\rm ren} =\frac1{4\pi^2a^4} \lim_{K\to\infty}\int_0^Kdk\,k^2 \left(\mathcal E_k-\mathcal E_{k,\rm ad}^{(0-4)}\right) +\rho_{\rm fin},$$ $$p_{\rm ren} =\frac1{4\pi^2a^4} \lim_{K\to\infty}\int_0^Kdk\,k^2 \left(\mathcal P_k-\mathcal P_{k,\rm ad}^{(0-4)}\right) +p_{\rm fin}.$$

The zeroth-, second-, and fourth-order pieces remove quartic, quadratic, and logarithmic ultraviolet behavior, respectively. Parker and Fulling established this conserved mode-by-mode construction for a positive-mass minimally coupled scalar in Robertson–Walker spacetime Parker and Fulling 1974. Fulling, Parker, and Hu extended the analysis to conformal coupling Fulling, Parker, and Hu 1974, and Bunch treated arbitrary scalar curvature coupling Bunch 1980.

Conservation and cutoff verification

Pressure must be integrated independently from its own counterterm. For each cutoff $K$, record

$$\mathcal C_K(\eta)= \rho_{\rm ren,K}' +3\mathcal H(\rho_{\rm ren,K}+p_{\rm ren,K}).$$

The bare identity proved above and the complete $0+2+4$ counterterm set make $\mathcal C_K\to0$ as the cutoff and numerical resolution are increased. Subtracting fourth order from the energy while stopping at second order in the pressure leaves a finite conservation failure even if both integrals appear ultraviolet stable.

A useful verification table is:

Check	Quantity varied	Acceptance criterion
Mode evolution	time step and precision	Wronskian and equation residuals converge independently
Ultraviolet tail	$K$ and the fitting window	subtracted $k^2\mathcal E_k$ and $k^2\mathcal P_k$ are integrable with stable fitted powers
Conservation	derivative and quadrature resolution	$\mathcal C_K$ tends to zero with the same continuum limit
Scheme translation	finite local couplings	the difference is state independent, local, and conserved

Because the bare and subtraction terms nearly cancel, increasing $K$ at fixed arithmetic precision can make the answer worse. Precision, time resolution, and the momentum cutoff must be varied separately.

For the scalar variance,

$$\langle\phi^2\rangle_{\rm ren} =\frac1{2\pi^2a^2} \lim_{K\to\infty}\int_0^Kdk\,k^2 \left[ |v_k|^2-\left(\frac1{2W_k}\right)_{0-2} \right] +c_m m^2+c_RR.$$

Second order suffices for this observable in four dimensions. The constants $c_m,c_R$ are the ordinary local Wick-square ambiguity; they cannot reproduce arbitrary fourth-order terms Hollands and Wald 2001, Theorem 5.1 and the following remark. Subtracting the complete fourth-order stress is therefore not the same instruction as subtracting fourth order from every composite.

State data and finite renormalization remain separate

The exact modes $v_k$ encode the state. The counterterms above depend only on $a,m,\xi$ and the local derivative expansion. Changing the state changes the smooth remainder; it must not change the subtraction terms.

The allowed finite stress ambiguity has the general locally covariant form

$$\delta T_{\mu\nu} =c_0m^4g_{\mu\nu} +c_1m^2G_{\mu\nu} +c_2I_{\mu\nu} +c_3J_{\mu\nu},$$

where $I_{\mu\nu}$ and $J_{\mu\nu}$ arise from curvature-squared actions and the $c_i$ are spacetime-independent constants. On four-dimensional conformally flat FLRW, curvature identities make the curvature-squared basis degenerate; one must fit only independent combinations. A shift depending on occupation number, an initial slice, or a comoving cutoff is not a renormalization-scheme ambiguity.

Adiabatic and DeWitt–Schwinger subtraction agree after the same finite local conditions are matched for the scalar case; del Río and Navarro-Salas give a direct comparison and extend the analysis to spinors del Río and Navarro-Salas 2015. Equality of divergent terms alone is not enough to compare two quoted finite answers.

Common pitfalls

Writing “fourth-order adiabatic/Hadamard state.” Finite fourth adiabatic order and the Hadamard condition are different regularity claims. Choose a state with the regularity required by the calculation, then apply the observable-specific subtraction separately.

Subtracting more orders to obtain a smoother plot. Sixth and higher orders are not free improvements of the fixed renormalizable stress prescription. They change the definition unless the corresponding higher-derivative EFT operators, finite couplings, and truncation error are included.

Exercises

1. Derive the second-order WKB frequency

Insert $W_k=\omega_k+W_{k,2}+O(\partial^4)$ into the nonlinear WKB equation and recover the displayed formula for $W_{k,2}$.

Solution

At second order,

$$W_k^2=\omega_k^2+2\omega_kW_{k,2},$$

while derivatives of $W_{k,2}$ begin at third order and do not contribute. The right side is

$$\omega_k^2+\sigma -\frac12\frac{\omega_k''}{\omega_k} +\frac34\frac{\omega_k'^2}{\omega_k^2}.$$

Equating the second-order parts and dividing by $2\omega_k$ gives

$$W_{k,2}=\frac1{2\omega_k} \left[ \sigma-\frac{\omega_k''}{2\omega_k} +\frac{3\omega_k'^2}{4\omega_k^2} \right].$$

2. Reject a state-dependent “scheme”

Suppose two calculations differ by $\Delta T_{\mu\nu}=F[N_k]G_{\mu\nu}$, where $F$ changes when the occupation function $N_k$ changes. Can this be an allowed finite renormalization?

Solution

No. The coefficient of $G_{\mu\nu}$ renormalizes a gravitational coupling and must be a spacetime-independent, state-independent constant fixed by a renormalization condition. Dependence on $N_k$ changes when the quantum state changes, so it belongs to the smooth state-dependent expectation value, not to the local scheme ambiguity. The strongest valid comparison is obtained only after holding the state fixed and translating constant local couplings.

References

• Bunch, T. S. “Adiabatic Regularization for Scalar Fields with Arbitrary Coupling to the Scalar Curvature.” Journal of Physics A: Mathematical and General 13 (1980): 1297–1310. DOI.
• del Río, Adrián, and José Navarro-Salas. “Equivalence of Adiabatic and DeWitt–Schwinger Renormalization Schemes.” Physical Review D 91 (2015): 064031. DOI. Open PDF.
• Fulling, Stephen A., Leonard Parker, and B. L. Hu. “Conformal Energy-Momentum Tensor in Curved Spacetime: Adiabatic Regularization and Renormalization.” Physical Review D 10 (1974): 3905–3924; erratum 11 (1975): 1714. DOI.
• Hollands, Stefan, and Robert M. Wald. “Local Wick Polynomials and Time Ordered Products of Quantum Fields in Curved Spacetime.” Communications in Mathematical Physics 223 (2001): 289–326. DOI.
• Parker, Leonard, and Stephen A. Fulling. “Adiabatic Regularization of the Energy-Momentum Tensor of a Quantized Field in Homogeneous Spaces.” Physical Review D 9 (1974): 341–354. DOI.