Nonlocal Operators and Green-Function Prescriptions

The symbol $\Box^{-1}$ is not an algebraic reciprocal. A differential operator has a kernel, and its inverse is defined only after choosing a Green function, its domain, boundary or initial data, and the treatment of zero modes. Those choices determine whether a proposed nonlocal equation is retarded, advanced, time symmetric, or ill defined. They are part of the dynamics, not optional instructions added after a solution is found.

Required background. PDEs and Green Functions for Gravitation supplies fundamental solutions, causal support, and homogeneous solutions; Gravitational Actions and Boundary Terms supplies the variational and boundary data that define the operator domain.

Helpful background. Einstein Equations supplies the curved-spacetime wave operator and Bianchi identity; Formulations of General Relativity supplies the initial-value and constraint language; Gravity Beyond General Relativity supplies the branch, cutoff, and observable checks used throughout this chapter.

An inverse is a kernel plus a prescription

For a scalar source $F$ on a globally hyperbolic spacetime, initially take $F$ smooth with compact support so that the retarded and advanced integrals are unambiguous. Define

$$(\Box_G^{-1}F)(x) =\int_M d^4x'\sqrt{-g(x')}\,G(x,x')F(x'),$$

where

$$\Box_xG(x,x') =\frac{\delta^{(4)}(x,x')}{\sqrt{-g(x)}}.$$

The subscript $G$ is essential. Retarded, advanced, Feynman, and symmetric Green functions solve the same distributional equation but have different support and boundary conditions. If $G_1$ and $G_2$ are two such choices, then

$$\Box_x\!\left[G_1(x,x')-G_2(x,x')\right]=0.$$

Consequently, two inverses differ by a homogeneous solution. Writing only $U=\Box^{-1}F$ does not say which of those solutions has been selected. Here $\delta^{(4)}(x,x')$ denotes the coordinate delta function; the explicit $\sqrt{-g}$ supplies its covariant normalization. Sources extending to the infinite past need additional convergence or initial-surface assumptions.

Zero modes create a second ambiguity. On a compact Euclidean manifold, constants lie in the kernel of the scalar Laplacian, so an inverse exists only after imposing an orthogonality condition on $F$ and choosing how to project out the constant mode. In Lorentzian evolution the analogous information is encoded in initial data and causal support.

The retarded inverse on FLRW

With the site signature $(-,+,+,+)$, a homogeneous scalar on

$$ds^2=-dt^2+a^2(t)d\boldsymbol{x}^2$$

satisfies

$$\Box U=-\ddot U-3H\dot U, \qquad H=\frac{\dot a}{a}.$$

Define

$$U=-\Box_{\mathrm{ret}}^{-1}R$$

with vanishing data $U(t_*)=\dot U(t_*)=0$ at the initial surface. The local equation and its selected solution are

$$\ddot U+3H\dot U=R,$$ $$\boxed{ U(t)=\int_{t_*}^{t}\frac{dt'}{a^3(t')} \int_{t_*}^{t'}dt''\,a^3(t'')R(t'') }.$$

Differentiating the boxed expression twice recovers the equation and both initial conditions. This is an independent sign and normalization check. The most general solution of the differential equation would instead be

$$U(t)=U_{\mathrm{ret}}(t) +C_1+C_2\int_{t_*}^{t}\frac{dt'}{a^3(t')}.$$

The constants are not additional freely specifiable data when $U$ is defined by the displayed retarded inverse; setting them nonzero defines another operator. Belgacem and collaborators make this distinction explicit for infrared nonlocal models in Belgacem et al. 2018, §§ 2.4.4–2.4.5, equations (2.30)–(2.42).

In a radiation-dominated epoch with $R=0$, retarded zero data keep $U=0$. Entering a matter era makes $R$ nonzero and generates a history-dependent field. An advanced inverse would instead depend on future curvature. Both solve the same local inhomogeneous equation; only the prescription distinguishes them.

Inverse operators and analytic form factors

Two structures often called “nonlocal” should not be conflated. An inverse-curvature model may contain

$$R\Box_G^{-1}R \quad\text{or}\quad m^2R\Box_G^{-2}R.$$

The first term has mass dimension two, like $R$; the explicit $m^2$ gives the second the same dimension. Each inverse requires a Green-function and homogeneous-data prescription.

An infinite-derivative model instead uses an analytic form factor,

$$\mathcal F\!\left(\frac{\Box}{M^2}\right) =\sum_{n=0}^{\infty}f_n\frac{\Box^n}{M^{2n}}.$$

This series involves no inverse at the expansion point, but its action on a field still requires a functional domain and a definition of the convergent operator. In Fourier space around Minkowski, $\Box\to-k^2$, so the analytic zeros of the kinetic form factor determine propagator poles. On a curved or time-dependent background, a flat-space Fourier definition is not a global prescription.

Logarithmic form factors provide a third example. A spectral representation such as

$$\log\!\left(\frac{-\Box}{\mu^2}\right) =\int_0^\infty dm^2 \left[ \frac{1}{m^2+\mu^2}-\frac{1}{m^2-\Box} \right]$$

inherits the pole prescription of every resolvent. This representation and its effective-action origin are reviewed in Belgacem et al. 2018, § 2.2, equations (2.10)–(2.15).

Variation and causal equations are different questions

A bilocal classical functional has the schematic form

$$S_{\mathrm{nl}} =\frac12\int d^4x\,d^4x'\, \Phi(x)K(x,x')\Phi(x').$$

Because the two field variations are interchangeable, its Euler–Lagrange equation contains the symmetric part

$$K_{\mathrm{sym}}(x,x') =\frac12\left[K(x,x')+K(x',x)\right].$$

A purely retarded kernel is not symmetric: transposition turns it into an advanced kernel. Therefore ordinary variation of a fundamental classical bilocal action does not by itself produce a purely retarded equation.

Causal effective equations can arise from an in-in, or Schwinger–Keldysh, effective-action construction. Alternatively, a phenomenological model may first vary a formal expression and then declare every inverse in the equations to be retarded. These are different foundations and should be labeled as such. The in-out versus in-in distinction and the symmetry obstruction are explained in Belgacem et al. 2018, §§ 2.1 and 2.4.1, Eqs. (2.1)–(2.9) and (2.28).

Controlled limits and failure tests

Removing the coefficient of a nonlocal term recovers GR only when the chosen solutions remain on the same Green-function branch and their history integrals stay bounded. A small coefficient multiplying a secularly growing inverse can produce an order-one late-time effect. The expansion parameter must therefore include both the coupling and the accumulated kernel.

Three tests should accompany every inverse operator:

1. apply $\Box$ and recover the source with the declared sign;
2. evaluate the initial or boundary data and verify the selected homogeneous solution;
3. check a zero-mode or long-time limit in which the inverse can fail to exist or the perturbative expansion becomes nonuniform.

Common pitfalls

Cancel $\Box\Box^{-1}=1$ without a domain. The identity holds only on the subspace selected by the Green function and zero-mode projection.

Localize first and choose initial data later. Free auxiliary homogeneous data change the original nonlocal theory.

Call a retarded classical action variational. A retarded kernel becomes symmetrized under ordinary variation. Causal effective equations require an in-in construction or an explicitly imposed equation-level prescription.

Exercises

In Minkowski space let a homogeneous test source be $F(t)=F_0$ for $t_*\le t\le t_1$ and zero afterward. Compute $U=-\Box_{\mathrm{ret}}^{-1}F$ with zero data at $t_*$, and show how adding a homogeneous solution changes the prescription. The source is a scalar test function, not the Ricci scalar of Minkowski space, which vanishes.

Solution

Here $a=1$ and the defining equation is $\ddot U=F$. For $t_*\le t\le t_1$,

$$U(t)=\frac{F_0}{2}(t-t_*)^2, \qquad \dot U(t)=F_0(t-t_*).$$

For $t\gt t_1$, continuity of $U$ and $\dot U$ gives

$$U(t) =\frac{F_0}{2}(t_1-t_*)^2 +F_0(t_1-t_*)(t-t_1).$$

Direct differentiation returns $F_0$ inside the source interval and zero afterward, while $U(t_*)=\dot U(t_*)=0$. Adding $A+B(t-t_*)$ still solves the same inhomogeneous equation, but changes the initial data to $U(t_*)=A$ and $\dot U(t_*)=B$. It is therefore not the declared retarded inverse. An advanced solution would instead be fixed by future data and would generally be nonzero before $t_*$.

References

• Belgacem, Enis, Yves Dirian, Stefano Foffa, and Michele Maggiore (2018), “Nonlocal Gravity: Conceptual Aspects and Cosmological Predictions,” Journal of Cosmology and Astroparticle Physics 2018(03), 002. DOI; Open PDF.
• Maggiore, Michele (2017), “Nonlocal Infrared Modifications of Gravity: A Review,” Fundamental Theories of Physics 187, 221–281. DOI; Open PDF.