State and Renormalization Dependence

Changing a quantum state and changing a renormalization prescription can both change the displayed stress tensor, but they represent different physics. At fixed geometry, the difference between two Hadamard states is finite and generally nonlocal. A finite scheme change is a state-independent local conserved curvature tensor and must be compensated by the gravitational couplings. Geometry is predictable only after the state, subtraction rule, scale, and finite renormalization conditions are all fixed.

Required background. Einstein Equations supplies the geometric tensors and conservation identity used to absorb local finite shifts.

Helpful background. Quantum Fields in Curved Spacetime supplies Hadamard subtraction and state dependence, while Black-Hole Geometry and Mechanics supplies the horizon-state distinctions used in applications.

State dependence is a smooth difference

For a free field, physically admissible Hadamard two-point functions share the same universal short-distance singularity. If $\omega$ and $\omega_0$ are two Hadamard states, then

$$G^+_\omega(x,y)-G^+_{\omega_0}(x,y)$$

is smooth. Acting with the point-split stress operator and taking $y\to x$ therefore gives a finite state difference,

$$\Delta_\omega\langle T_{\mu\nu}\rangle =\langle T_{\mu\nu}\rangle_\omega -\langle T_{\mu\nu}\rangle_{\omega_0}.$$

It can contain flux, thermal occupation, polarization, and dependence on global boundary or initial data. It cannot generally be written as a fixed local polynomial of the metric. Specifying only “the renormalized stress tensor” therefore leaves the physical source underdetermined.

Finite renormalization is local

In four dimensions, two covariant prescriptions satisfying the standard locality and conservation requirements may differ schematically by

$$\delta\langle T_{\mu\nu}\rangle =c_0g_{\mu\nu}+c_1G_{\mu\nu} +c_2H^{(1)}_{\mu\nu}+c_3H^{(C)}_{\mu\nu},$$

where $H^{(1)}_{\mu\nu}$ and $H^{(C)}_{\mu\nu}$ are metric variations of $\int\sqrt{-g}R^2$ and $\int\sqrt{-g}C_{\alpha\beta\gamma\delta}C^{\alpha\beta\gamma\delta}$. Every term is local, state independent, and covariantly conserved. The same coefficients shift $\Lambda$, $1/G$, and the curvature-squared couplings on the geometric side. Hu and Verdaguer show this cancellation explicitly in Hu and Verdaguer 2008, § 3.1, Eqs. (3.6)–(3.10).

Consequently, changing $\langle T_{\mu\nu}\rangle$ while holding the renormalized gravitational parameters numerically fixed is not a pure scheme change. It changes the theory’s renormalization conditions. A comparison is meaningful only when the same measured couplings at the same scale are used on both sides.

Comparison axes

Change	Depends on state?	Local in the metric?	Required companion change
Replace vacuum by a thermal or flux state	yes	generally no	recompute the self-consistent geometry
Change point-splitting scale $\mu$	no	yes	run the gravitational couplings
Add a finite $R^2$ term to the matter action	no	yes	subtract the same term from the gravitational action
Change boundary conditions or topology	usually yes	generally no	specify new state and global problem
Truncate an adiabatic or derivative series	not necessarily	approximately local	report the first omitted term and regime

Shapiro derives the local counterterm basis and its scale dependence from the vacuum effective action in Shapiro 2008, §§ 2–3. The renormalization scale $\mu$ is not an observable. A correctly combined prediction satisfies a renormalization-group identity: explicit $\ln\mu$ dependence of the quantum term cancels the running of local couplings to the retained order.

A prescription-invariant field equation

Write the renormalized equation as

$$\mathcal G_{\mu\nu}[g;c_i(\mu)] \equiv\frac{G_{\mu\nu}+\Lambda g_{\mu\nu}}{8\pi G} +2\alpha H^{(R^2)}_{\mu\nu}+2\beta H^{(C^2)}_{\mu\nu} =\langle T_{\mu\nu}\rangle_{\omega,\mu}.$$

Here $H^{(I)}_{\mu\nu}$ is the inverse-metric variation of the integrated invariant $I$, divided by $\sqrt{-g}$, as on the field-equation page. Using this unnormalized equation avoids hiding a change of $G$ in the prefactor of the source. Under a finite prescription change $\langle T_{\mu\nu}\rangle\mapsto\langle T_{\mu\nu}\rangle+K_{\mu\nu}$, shift the geometric coefficients so that $\mathcal G_{\mu\nu}\mapsto\mathcal G_{\mu\nu}+K_{\mu\nu}$. The equation’s residual is unchanged. The two actions shift oppositely, but their displayed tensors on opposite sides of this equation shift by the same $K_{\mu\nu}$. Under a state change, by contrast, $\Delta_\omega\langle T_{\mu\nu}\rangle$ remains on the right and generally produces a different solution.

Two independent checks follow. First, take a difference between states in the same scheme: all state-independent counterterms cancel. Second, differentiate a physical prediction with respect to $\ln\mu$: it must vanish up to the first omitted perturbative order. Anderson, Molina-París, and Mottola use precisely this renormalized effective-action structure before assessing response and stability Anderson, Molina-París, and Mottola 2003, § II.

Limits of the semiclassical description

The local-versus-nonlocal test is powerful but not infallible on a highly symmetric background, where distinct local tensors can become proportional. One must compare a family of geometries or impose independent renormalization conditions. State singularities can also invalidate the Hadamard premise, in which case the usual local stress tensor may not exist.

Constructing the admissible states and their fixed-background correlators remains the task of Quantum Fields in Curved Spacetime. Here they are inputs to self-consistent backreaction. Fine-grained entropy recovery and microscopic spacetime structure are separate questions; neither can be settled by moving a finite local term between the two sides of the semiclassical equation.

Exercise

Two calculations on the same geometry differ by $\delta\langle T_{\mu\nu}\rangle=cG_{\mu\nu}$. Neglect the cosmological and higher-curvature terms for this exercise. Show how to interpret this as a finite renormalization of Newton’s constant rather than a different state.

Solution

Write the original equation as $G_{\mu\nu}/(8\pi G)=\langle T_{\mu\nu}\rangle$. In the new prescription, $G_{\mu\nu}/(8\pi G')=\langle T_{\mu\nu}\rangle+cG_{\mu\nu}$ describes the same solutions precisely when

$$\frac{1}{8\pi G'}=\frac{1}{8\pi G}+c, \qquad G'=\frac{G}{1+8\pi Gc}.$$

This is a change of parameter convention with the physical renormalization condition held fixed. If one changed the source while leaving the numerical $G$ unchanged, the effective coupling in the old source convention would instead be $G/(1-8\pi Gc)$: that would change the physical equation. The two signs answer different questions.

References

• Anderson, P. R., Molina-París, C., and Mottola, E. 2003. “Linear Response, Validity of Semi-Classical Gravity, and the Stability of Flat Space.” Physical Review D 67, 024026. Open PDF.
• Hu, B. L., and Verdaguer, E. 2008. “Stochastic Gravity: Theory and Applications.” Living Reviews in Relativity 11, 3. DOI.
• Shapiro, I. L. 2008. “Effective Action of Vacuum: Semiclassical Approach.” Classical and Quantum Gravity 25, 103001. Open PDF.