Infinite-Derivative and Entire-Function Gravity

An infinite series of derivatives need not introduce an infinite tower of propagating modes. Around a fixed background, the poles of the complete propagator—not the number of terms in a Taylor expansion—control the linear spectrum. A zero-free entire kinetic form factor can soften large Euclidean momenta while leaving only the massless graviton pole. This is a background-dependent quadratic statement, not a proof of nonlinear health or causality.

Required background. PDEs and Green Functions for Gravitation supplies operator domains, Fourier symbols, and pole prescriptions; Gravitational Actions and Boundary Terms supplies the variational problem for curvature form factors.

Helpful background. Einstein Equations supplies the massless spin-2 reference theory; Formulations of General Relativity supplies gauge reduction and initial-data counting; Gravity Beyond General Relativity supplies the cutoff and branch qualifications used below.

The quadratic propagator question

Consider a covariant action expanded about Minkowski spacetime,

$$\begin{aligned} S=\frac12\int d^4x\sqrt{-g}\bigg[ M_{\mathrm{Pl}}^2R &+R\,\mathcal F_1(\Box)R\\ &+R_{ab}\mathcal F_2(\Box)R^{ab} +C_{abcd}\mathcal F_3(\Box)C^{abcd} \bigg]. \end{aligned}$$

After gauge reduction, its source-saturated quadratic propagator can be organized with the spin-2 and scalar projectors:

$$\Pi(k) =\frac{P^{(2)}}{M_{\mathrm{Pl}}^2k^2a(-k^2)} +\frac{P_s^{(0)}}{M_{\mathrm{Pl}}^2k^2 \left[a(-k^2)-3c(-k^2)\right]}.$$

The functions $a$ and $c$ are definite linear combinations of the $\mathcal F_i$ after expansion about the chosen background. General relativity has $a=c=1$, giving

$$\Pi_{\mathrm{GR}} =\frac{1}{M_{\mathrm{Pl}}^2k^2} \left(P^{(2)}-\frac12P_s^{(0)}\right).$$

The GR expression contains no independent scalar excitation in addition to the graviton. The negative coefficient of $P_s^{(0)}$ in the GR expression is part of the constrained massless spin-2 propagator; it is not by itself a propagating scalar ghost. An extra pole with a negative residue is a different issue. A simple no-extra-pole class is defined by

$$a(z)=c(z)=e^{\gamma(z)}, \qquad \gamma(0)=0,$$

with $\gamma$ entire and real on the relevant real axis. Then $e^{\gamma(z)}$ has no finite complex zeros, and

$$\Pi(k) =\frac{e^{-\gamma(-k^2)}}{M_{\mathrm{Pl}}^2k^2} \left(P^{(2)}-\frac12P_s^{(0)}\right).$$

The only pole is the GR pole at $k^2=0$. The normalization $a(0)=1$ recovers the same infrared residue. The general quadratic construction and its spin-projector conditions are derived in Biswas et al. 2012, Eqs. (10)–(13).

The analytic property doing the work is therefore zero-free entirety of the complete kinetic multiplier, together with the corresponding scalar-sector relation. Merely choosing each coefficient function to be analytic is not enough: an entire function such as $1+z$ has a zero and creates an additional pole.

Ultraviolet softening without a new pole

Choose

$$a(\Box)=e^{-\Box/M^2}.$$

After Wick rotation, $\Box\to -k_E^2$ in the convention used for the kinetic symbol, and the Euclidean propagator scales as

$$\Pi_E(k_E)\sim \frac{e^{-k_E^2/M^2}}{k_E^2}.$$

Large Euclidean momenta are exponentially suppressed, while for $k_E^2\ll M^2$,

$$e^{-k_E^2/M^2} =1-\frac{k_E^2}{M^2}+O\!\left(\frac{k_E^4}{M^4}\right).$$

The weak-curvature, low-momentum limit thus approaches GR. This does not make $M$ a hard Wilsonian cutoff by itself; it is the nonlocality scale of the chosen form factor, and the quantum completion and loop expansion must be specified separately.

Here $M\gt0$ is an inverse length. For Fourier expressions use $e^{-i\omega t+i\boldsymbol k\cdot\boldsymbol x}$, $k^2=-\omega^2+\lvert\boldsymbol k\rvert^2$, and $\Box\mapsto-k^2$. Euclidean damping does not justify an arbitrary Lorentzian Wick rotation: the exponential grows in some complex directions, so a contour and a function space are still required.

A static point source gives an instructive transform. The modified Poisson equation

$$e^{-\nabla^2/M^2}\nabla^2\Phi =4\pi GM_s\delta^{(3)}(\boldsymbol{x})$$

has momentum-space solution proportional to $-e^{-k^2/M^2}/k^2$. The angular integral yields

$$\boxed{ \Phi(r)=-\frac{GM_s}{r} \operatorname{erf}\!\left(\frac{Mr}{2}\right) }.$$

At large radius the error function tends to one and Newton’s potential is recovered. At the origin,

$$\Phi(0)=-\frac{GM_sM}{\sqrt{\pi}},$$

which is finite. The result checks dimensions and shows spatial smearing, but a finite linear potential is not a theorem that the full curvature is nonsingular for every nonlinear solution. The weak-field calculation is self-consistent near the origin only when $GM_sM\ll1$. In the static transform use $\Phi(\boldsymbol x)=\int d^3k\, e^{i\boldsymbol k\cdot\boldsymbol x}\widetilde\Phi(\boldsymbol k)/(2\pi)^3$.

Why finite truncation changes the spectrum

Expanding the exponential and stopping after one correction gives

$$a(-k^2)\simeq1+\frac{k^2}{M^2}.$$

The corresponding scalar denominator factorizes:

$$\frac{1}{k^2(1+k^2/M^2)} =\frac{1}{k^2}-\frac{1}{k^2+M^2}.$$

The truncated theory contains an extra pole with the opposite residue. That pole is absent from the exact exponential. A finite higher-derivative truncation can be a controlled low-momentum expansion when treated perturbatively and order reduced, but solving its polynomial equations nonperturbatively changes the degrees of freedom.

This is also a useful adverse check on numerical approximations. Replacing an entire operator by a high-degree polynomial and evolving all polynomial modes can populate artifacts that do not converge to solutions of the exact operator.

Initial data are counted by poles

For a scalar analogue,

$$e^{-\Box/M^2}\Box\varphi=0,$$

the exponential is invertible on a function space where the operator is defined. Applying its inverse leaves $\Box\varphi=0$, with the usual two initial functions. An additional simple zero of the kinetic form factor instead gives a pole of the propagator as a function of invariant momentum squared. At a generic fixed spatial momentum it supplies a pair of frequency roots and corresponding additional initial data if the defining contour retains those roots. Zeros of the inverse propagator, rather than poles of the inverse propagator, identify these modes.

Barnaby and Kamran formulate this result using contour and Laplace-transform methods: the number of initial data is tied to propagator poles enclosed by the chosen contour, not mechanically to “infinitely many derivatives.” Their initial-value construction and qualifications for analytic data appear in Barnaby and Kamran 2008, §§ 2.1–2.3, 3.1, and 5.3. The contour is itself part of the operator definition; changing it can include or exclude modes.

What pole avoidance does not prove

The derivation is about quadratic perturbations around one background. It does not establish:

• absence of extra modes on curved or time-dependent backgrounds;
• positive kinetic and gradient matrices after background-dependent mixing;
• causal support of the Lorentzian kernel;
• a well-posed nonlinear initial-value problem;
• radiative stability of the zero-free relation; or
• convergence of the entire-operator expansion on the fields being used.

Zeros can move when the background changes the kinetic operator, and interactions can lower the strong-coupling scale. These are separate tests.

Common pitfalls

Entire means zero free. Entire functions may have zeros. The safe quadratic multiplier is usually written as the exponential of an entire function precisely to exclude them.

A Taylor truncation preserves the exact spectrum. A finite polynomial generically introduces roots and must be treated as a perturbative approximation, not a new exact kinetic operator.

No extra Minkowski pole proves nonlinear ghost freedom. It proves only the linear pole statement on that background and branch.

Exercises

Starting from the static propagator $-4\pi GM_s e^{-k^2/M^2}/k^2$, derive the real-space potential and its two limiting forms. Then repeat the pole count after truncating the exponential at first order.

Solution

Spherical Fourier transformation gives

$$\Phi(r) =-\frac{2GM_s}{\pi r} \int_0^\infty dk\,e^{-k^2/M^2}\frac{\sin kr}{k}.$$

Differentiate the integral with respect to $r$:

$$\frac{d}{dr}\int_0^\infty dk\, e^{-k^2/M^2}\frac{\sin kr}{k} =\int_0^\infty dk\,e^{-k^2/M^2}\cos kr =\frac{\sqrt{\pi}M}{2}e^{-M^2r^2/4}.$$

Integrating from $r=0$ yields

$$\int_0^\infty dk\,e^{-k^2/M^2}\frac{\sin kr}{k} =\frac{\pi}{2}\operatorname{erf}\!\left(\frac{Mr}{2}\right),$$

and therefore

$$\Phi(r) =-\frac{GM_s}{r} \operatorname{erf}\!\left(\frac{Mr}{2}\right).$$

For $Mr\gg1$, $\Phi\to-GM_s/r$. For $Mr\ll1$, $\operatorname{erf}(Mr/2)=Mr/\sqrt{\pi}+O(r^3)$, so $\Phi(0)=-GM_sM/\sqrt{\pi}$.

The first-order kinetic truncation has denominator $k^2(1+k^2/M^2)$. Partial fractions reveal a massless pole and an additional pole at $k^2=-M^2$ with opposite residue. The exact exponential has no such zero, demonstrating that finite truncation and exact resummation have different nonperturbative spectra.

References

• Barnaby, Neil, and Niky Kamran (2008), “Dynamics with Infinitely Many Derivatives: The Initial Value Problem,” Journal of High Energy Physics 2008(02), 008. DOI; Open PDF.
• Biswas, Tirthabir, Erik Gerwick, Tomi Koivisto, and Anupam Mazumdar (2012), “Towards Singularity- and Ghost-Free Theories of Gravity,” Physical Review Letters 108, 031101. DOI; Open PDF.