Localization with Auxiliary Fields

Auxiliary fields can turn an integro-differential equation into a local system that is easier to vary and integrate. They do not automatically turn the original nonlocal model into a local theory with extra freely specifiable particles. The defining Green-function prescription fixes their homogeneous solutions. Forgetting that restriction enlarges the solution space and can create spurious modes, branches, and apparent ghosts.

Required background. PDEs and Green Functions for Gravitation supplies the particular-plus-homogeneous decomposition of wave equations; Gravitational Actions and Boundary Terms supplies the integrations by parts and boundary data used in localization.

Helpful background. Einstein Equations supplies the curvature source and Bianchi consistency check; Formulations of General Relativity supplies initial-data counting and constraint propagation; Gravity Beyond General Relativity supplies the distinction between a rewriting and a new physical theory.

Localizing the RR operator

Consider the representative infrared expression

$$\Gamma_{\mathrm{RR}} =\frac{M_{\mathrm{Pl}}^2}{2}\int d^4x\sqrt{-g} \left[ R-\frac{m^2}{6}R\Box_G^{-2}R \right],$$

where $M_{\mathrm{Pl}}^{-2}=8\pi G$ and every inverse uses one declared Green function $G$. Introduce

$$U=-\Box_G^{-1}R, \qquad S=-\Box_G^{-1}U.$$

Then $S=\Box_G^{-2}R$ and

$$\Box U=-R, \qquad \Box S=-U.$$

A constrained local representation is

obtained by varying a formal self-adjoint expression and only afterward selecting the retarded solution space if causal effective evolution is intended. It is

$$\begin{aligned} \Gamma_{\mathrm{loc}} =\frac{M_{\mathrm{Pl}}^2}{2}\int d^4x\sqrt{-g} \bigg[ R\left(1-\frac{m^2}{6}S\right) &-\xi_1(\Box U+R)\\ &-\xi_2(\Box S+U) \bigg]. \end{aligned}$$

Variation with respect to $\xi_1$ and $\xi_2$ enforces the two auxiliary equations. Variation with respect to $U$ and $S$, followed by the same boundary prescription, fixes

$$\xi_1=\frac{m^2}{6}S, \qquad \xi_2=\frac{m^2}{6}U.$$

Substitution into the metric equation, after performing the variations, reproduces the prescribed nonlocal equation. Substituting the retarded solutions into an ordinary action before variation would not justify the same causal equation. This construction and the resulting metric tensor are given in Belgacem et al. 2018, § 2.4.4, equations (2.30)–(2.33). The equality is between prescribed solution spaces, not merely between the displayed differential expressions.

The missing condition in the local equations

The equation $\Box U=-R$ admits

$$U=U_G+U_{\mathrm{hom}}, \qquad \Box U_{\mathrm{hom}}=0.$$

If $U$ means $-\Box_G^{-1}R$, the Green function already fixes $U_{\mathrm{hom}}$. For a retarded inverse with zero data on an initial surface $\Sigma_*$,

$$U|_{\Sigma_*}=0, \qquad n^a\nabla_aU|_{\Sigma_*}=0,$$

and similarly for $S$. The local equations appear to require four scalar initial functions, but the original nonlocal definition supplies none beyond the metric history and its prescribed initial surface.

This point is exact in the two-dimensional Polyakov example. After localization, a field $U$ obeys $\Box U=-R$, yet the known underlying metric description selects one particular solution; the independent $U_{\mathrm{hom}}$ is absent. Belgacem and collaborators trace this example in Belgacem et al. 2018, § 2.4.4, equations (2.36)–(2.42).

For a homogeneous FLRW background,

$$\ddot U+3H\dot U=R, \qquad \ddot S+3H\dot S=U.$$

With retarded zero data during radiation domination, $R=0$ keeps both fields zero. When the trace becomes nonzero, $U$ grows from the integrated curvature history and $S$ integrates $U$ once more. Choosing nonzero radiation-era constants is not another solution of the same prescription; it selects a different inverse or promotes the local system to a different theory.

Apparent and genuine degrees of freedom

After integration by parts, the auxiliary action contains scalar kinetic mixing. Diagonalizing that local kinetic matrix can expose a combination with a wrong-sign term. Its interpretation depends on the theory being defined.

• If the nonlocal expression is an effective equation with fixed auxiliary homogeneous data, the wrong-sign auxiliary combination has no independent creation operator or external state. It can still drive a classical instability, so its time evolution must be checked.
• If the localized action is declared fundamental and $U$, $S$, $\xi_1$, and $\xi_2$ receive arbitrary canonical data, the solution space has been enlarged. A wrong-sign propagating combination is then a genuine ghost unless an additional constraint removes it.

One cannot use the first interpretation’s restricted data together with the second interpretation’s claim of a fundamental local action. Maggiore and Mancarella emphasize the effective, prescribed character of the auxiliary fields for the RR model in Maggiore and Mancarella 2014, §§ II–III, Eqs. (2)–(7).

A solution-space equivalence test

To verify a localization, perform four checks.

1. Vary the constrained action before imposing the auxiliary equations and retain the boundary terms that define the operator domain.
2. Solve the multiplier equations with the same Green function as the original inverse.
3. Substitute the solution into the metric equation and recover the nonlocal expression, including signs and coefficients.
4. Add one nonzero homogeneous auxiliary solution. If it changes the metric equation while the original metric history is fixed, it is outside the original theory.

The fourth test is decisive. The RR metric correction contains terms such as $SG_{ab}$, $\nabla_a\nabla_bS$, gradients of $S$ and $U$, and $U^2$. A homogeneous shift that solves the auxiliary wave equations generally changes these terms. It is therefore not a redundant description of the same metric solution.

Boundary and causality qualifications

Integrating $\xi\Box U$ by parts assumes a boundary condition. On a finite region, discarded normal derivatives can alter the variational problem. On an initial-value domain, the retarded prescription fixes data on the past surface. Those two statements must be made compatible.

Ordinary variation of a bilocal classical action symmetrizes its kernel and does not produce a purely retarded equation by itself. A causal in-in effective equation can carry a retarded inverse, while a phenomenological model may impose it at equation level. Localization preserves whichever foundation was declared; it does not solve the causality question.

Common pitfalls

Count every auxiliary datum as physical. The original inverse fixes the homogeneous solution. Count data in the prescribed solution space.

Dismiss every wrong-sign auxiliary term as harmless. It is not an independent quantum ghost under restricted effective data, but it can still produce a classical instability.

Use local equations without the original Green function. That is an enlarged theory, not a computational implementation of the nonlocal model.

Exercises

Let $U=-\Box_{\mathrm{ret}}^{-1}F$ on homogeneous Minkowski space with $U(t_*)=\dot U(t_*)=0$. Localize the definition, count the apparent initial data, and show explicitly why $U+A+B(t-t_*)$ is spurious for fixed $F$. Here $F(t)$ is a prescribed test source; the Ricci scalar of the fixed Minkowski metric itself is zero.

Solution

With $\Box U=-\ddot U$, the defining equation is

$$\ddot U=F.$$

Its local general solution is

$$U(t) =\int_{t_*}^{t}dt'\,(t-t')F(t') +A+B(t-t_*).$$

As a second-order local equation it appears to carry the two data $A=U(t_*)$ and $B=\dot U(t_*)$. The retarded inverse with zero initial data selects $A=B=0$; hence

$$U_{\mathrm{ret}}(t) =\int_{t_*}^{t}dt'\,(t-t')F(t').$$

Adding $A+B(t-t_*)$ leaves $\ddot U=F$ unchanged but changes the initial data. It also changes any localized metric correction containing $U$, $U^2$, or its first derivative. Because the original operator already selected $U_{\mathrm{ret}}$, the shifted field is not another physical excitation of that model. If $A$ and $B$ are instead declared free, one has defined a different local theory and must analyze its Hamiltonian and kinetic signs afresh.

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, and Michele Mancarella (2014), “Nonlocal Gravity and Dark Energy,” Physical Review D 90, 023005. DOI; Open PDF.