A propagator with positive spectral weight has spectral dimension at least D

Several approaches to quantum gravity report a spectral dimension, the dimension seen by a diffusing probe, that falls from four at long distances to about two at short distances. We ask whether such a fall can occur for the diffusion generated by the inverse of a propagator with a positive Källén–Lehmann representation. Let $G(p^2)$ be a rotation-invariant Euclidean propagator with a massless pole, nonnegative spectral weight, no constant term and an inverse that grows at least as a power of $p^2$ , and let a particle diffuse on flat momentum space $\mathbb R^D$ with the inverse propagator as a time-independent generator. Then the spectral dimension is at least $D$ at every diffusion time, and equals $D$ at all times only for the free propagator (Theorem 2).

Positive weight is a property of the Euclidean propagator; whether it coincides with unitarity depends on how a ghost pole is quantised (section 4). A spectral dimension below $D$ , defined as in (1) or with another measure or diffusion time, therefore needs one of four things. The first is negative weight, for which Proposition 4 gives a sum rule when the weight has finite total variation. The others are a generator that is not rotation invariant, as in Hořava gravity, a measure on momentum space other than the flat one, and a diffusion time rescaled by the scale. A constant term leaves the spectral dimension undefined (section 7). For asymptotic safety the consequence is that the inverse of a fluctuation propagator of this form cannot give the value two (section 5).

Sotiriou, Visser and Weinfurtner [SVW] show how a dispersion relation determines the spectral dimension and how to invert the relation. The value two in asymptotic safety was obtained by Lauscher and Reuter [LR] from a generator built from the running cosmological constant. Calcagni, Eichhorn and Saueressig [CES] replaced the identification of the scale in that construction by one with the diffusion time, and observed that heat kernels of such operators need not be positive. Hořava [Hořava, spectral dimension] found $d_s=1+(D-1)/z$ for anisotropic scaling with exponent $z$ in $D-1$ space dimensions. Theorem 2 adds the inequality at every scale, proved from a virial identity (Proposition 1) that holds for any symbol.

1. The spectral dimension

Consider a particle diffusing on $\mathbb R^D$ with generator $-\psi(-i\nabla)$ , where $\psi(\xi)\ge0$ is the symbol: for the ordinary Laplacian $\psi(\xi)=\lvert\xi\rvert^2$ . The probability density to be back at the starting point after diffusion time $\sigma$ is the return amplitude

P(\sigma)=\int_{\mathbb R^D}d^D\xi\;e^{-\sigma\psi(\xi)},\qquad d_s(\sigma)=-2\,\frac{d\ln P}{d\ln\sigma}=2\sigma\langle\psi\rangle_\sigma, \tag{1}

where $\langle g\rangle_\sigma=\int g\,e^{-\sigma\psi}d^D\xi\big/P(\sigma)$ . The factor $2$ is chosen so that the free case gives the dimension: $P\propto\sigma^{-D/2}$ and $d_s=D$ . The quantity $d_s(\sigma)$ is the spectral dimension at the scale $\sigma^{1/2}$ .

For a field theory, $\psi$ is the inverse Euclidean propagator. If the propagator is $G(p^2)$ we take $\psi(\xi)=F(\lvert\xi\rvert^2)$ with $F(x)=1/G(x)$ , $x=p^2$ . This is the usual definition of the spectral dimension of a quantum field theory from its two-point function. We use four hypotheses.

(H1) The measure is flat: momentum space is $\mathbb R^D$ with Lebesgue measure, with no reduced density of states.

(H2) The diffusion is autonomous: $\psi$ does not depend on $\sigma$ .

(H3) The symbol is rotation invariant, $\psi(\xi)=F(\lvert\xi\rvert^2)$ . For a Lorentz-invariant theory this is a property of the Euclidean continuation.

(H4) The propagator has the Stieltjes form with a massless pole, a nonnegative weight and no constant term,

G(x)=\frac ax+\int_{(0,\infty)}\frac{\rho(d\mu)}{x+\mu},\qquad a>0,\quad\rho\ge0,\quad\int\frac{\rho(d\mu)}{1+\mu}<\infty, \tag{2}

and $F$ satisfies the growth condition $F(x)\ge c\,x^{\varepsilon}-C$ for some $c,\varepsilon>0$ , which makes $P(\sigma)$ finite for every $\sigma$ .

The representation (2) is the Källén–Lehmann form of the Euclidean propagator in the standard quantisation of a unitary theory, with $\rho(\mu)$ the weight of states of mass squared $\mu$ ; we call $\rho\ge0$ positive weight. A constant term $b>0$ added to (2) makes $F$ bounded by $1/b$ and $P$ infinite, so $d_s$ is undefined. This is why the constant is excluded.

2. A virial identity for any symbol

Dropping (H3) and (H4) leaves an exact formula for any symbol.

Proposition 1 (virial identity). Let $\psi:\mathbb R^D\to[0,\infty)$ be locally Lipschitz with $\psi(\xi)\ge c\lvert\xi\rvert^{\varepsilon}-C$ for some $c,\varepsilon>0$ , and $\lvert\psi\rvert+\lvert\xi\cdot\nabla\psi\rvert\le C'(1+\lvert\xi\rvert)^n$ almost everywhere (so that the integrations by parts below have no boundary term). Then $P(\sigma)<\infty$ for all $\sigma>0$ and

d_s(\sigma)=D+2\sigma\,\Big\langle\psi-\tfrac12\,\xi\cdot\nabla\psi\Big\rangle_\sigma. \tag{3}

Consequently, if $t\mapsto\psi(t\xi)/t^2$ is nonincreasing on $t>0$ for every $\xi$ , then $d_s(\sigma)\ge D$ for all $\sigma$ ; if it is nondecreasing, then $d_s(\sigma)\le D$ for all $\sigma$ .

In (3), $2\sigma\langle\psi\rangle_\sigma$ is the spectral dimension itself, while $\sigma\langle\xi\cdot\nabla\psi\rangle_\sigma=D$ by an integration by parts (shown below). The integrand $\psi-\frac12\xi\cdot\nabla\psi$ vanishes for a quadratic symbol and has the sign of the excess or deficit of growth along rays relative to $\lvert\xi\rvert^2$ .

Proof. Finiteness of $P$ follows from the lower bound on $\psi$ , and differentiating under the integral gives $d_s=2\sigma\langle\psi\rangle_\sigma$ , as in (1), by dominated convergence with the polynomial bounds. The vector field $\xi\,e^{-\sigma\psi}$ is Lipschitz with divergence $(D-\sigma\,\xi\cdot\nabla\psi)e^{-\sigma\psi}$ , and its flux through the sphere of radius $R$ is at most $R^{D}e^{-\sigma(cR^\varepsilon-C)}\to0$ . Hence $\int(D-\sigma\xi\cdot\nabla\psi)e^{-\sigma\psi}d^D\xi=0$ , that is $D=\sigma\langle\xi\cdot\nabla\psi\rangle_\sigma$ . Subtracting this from $d_s=2\sigma\langle\psi\rangle_\sigma$ gives (3). The derivative of $\psi(t\xi)/t^2$ at $t=1$ is $\xi\cdot\nabla\psi-2\psi$ , and applying this at the point $t\xi$ shows that monotonicity for all $t$ fixes the sign of $\psi-\frac12\xi\cdot\nabla\psi$ at every point.

For a rotation-invariant symbol $\psi=F(\lvert\xi\rvert^2)$ the integrand is $F-xF'=-x^2(F/x)'$ , so $d_s\ge D$ whenever $F(x)/x$ is nonincreasing. Neither concavity nor monotonicity of $F$ is needed. A concave $F$ with $F(0)=0$ , which is the case for the reciprocal of a Stieltjes function, is a special case.

3. Positive spectral weight

Theorem 2 (positivity and the spectral dimension). Under (H1) to (H4), $d_s(\sigma)\ge D$ for every $\sigma>0$ . More precisely,

d_s(\sigma)-D=2\sigma\Big\langle F(x)^2\int\rho(d\mu)\frac{\mu}{(x+\mu)^2}\Big\rangle_\sigma\ \ge0, \tag{4}

so equality holds at some $\sigma$ , and then at all $\sigma$ , exactly when $\rho=0$ , that is for the free propagator $G=a/x$ .

Positivity of the weight makes $xG(x)$ nondecreasing, which is the statement that $F(x)/x=1/(xG)$ is nonincreasing; what remains is to check the hypotheses of Proposition 1.

Proof. From (2), $xG(x)=a+\int\rho(d\mu)\,x/(x+\mu)$ , which is nondecreasing, with derivative $(xG)'=\int\rho(d\mu)\,\mu/(x+\mu)^2\ge0$ . Hence $F/x=1/(xG)$ is nonincreasing, and $F-xF'=(G+xG')/G^2=(xG)'F^2\ge0$ . This gives $F'\le F/x=1/(xG)\le1/a$ , the last step because $xG\ge a$ . The function $F$ is therefore increasing with $0<F(x)\le x/a$ , and $\psi(\xi)=F(\lvert\xi\rvert^2)$ is locally Lipschitz with $0\le\psi\le\lvert\xi\rvert^2/a$ and $0\le\xi\cdot\nabla\psi=2xF'\le2x/a$ . Together with the growth condition of (H4), this verifies the hypotheses of Proposition 1, and (3) reduces to (4). If $d_s(\sigma)=D$ at some $\sigma$ , the nonnegative integrand vanishes almost everywhere, so $(xG)'=0$ and $\rho=0$ .

The pole enters through $a>0$ . It bounds $F/x$ by $1/a$ , which gives the bounds above, it fixes the equality case, and it determines the infrared limit in Proposition 3(b). The reciprocal of a Stieltjes function is a complete Bernstein function [Schilling–Song–Vondraček], hence concave with $F(0)=0$ , which gives the same inequality through $F\ge xF'$ ; (4) shows that the weaker monotonicity of $xG$ is enough. The inequality is strict away from the free case, and a positive propagator can overshoot $D$ , as the second row of the table shows.

Examples

The table below gives $d_s(\sigma)$ in $D=4$ from $d_s=2\sigma\langle F\rangle_\sigma$ . The mass scale of the massive pole is $M=1$ , so $\sigma$ is in units of $M^{-2}$ .

$G(x)$	$\sigma=10^{-8}$	$10^{-4}$	$10^{-2}$	$1$	$10$	$10^{2}$	$10^{4}$
$1/x$	4.0000	4.0000	4.0000	4.0000	4.0000	4.0000	4.0000
$1/x+1/(x+1)$	4.0000	4.0000	4.0050	4.3523	4.6784	4.1169	4.0012
$1/x-1/(x+1)$	2.0001	2.0088	2.0865	2.7009	3.4290	3.8941	3.9988
$e^{-x}/x$	0.2464	0.5037	0.9282	2.2596	3.3375	3.8905	3.9988

The second row has positive weight, and $d_s\ge4$ everywhere with a maximum of $4.737$ at $\sigma\approx5.9$ , so a positive massive pole raises the dimension above $D$ at intermediate scales. The third row has a ghost of weight $-1$ and goes from $2$ at short distances to $4$ . The fourth row, a massless pole times an entire form factor, has no Stieltjes representation and goes to values below one without a plateau at two.

Limits

Proposition 3 (limits). (a) Let $F\ge0$ be locally bounded with $P(\sigma)<\infty$ for all $\sigma$ , and suppose $F(x)/x^\gamma\to\kappa\in(0,\infty)$ as $x\to\infty$ for some $\gamma>0$ . Then $d_s(\sigma)\to D/\gamma$ as $\sigma\to0^+$ . (b) Under the hypotheses of Theorem 2, $d_s(\sigma)\to D$ as $\sigma\to\infty$ . (c) Under (H4), $\gamma\le1$ whenever the limit in (a) exists, so $D/\gamma\ge D$ .

Proof. (a) Substitute $x=\sigma^{-1/\gamma}y$ in the radial form $P=\frac12\Omega_D\int x^{D/2-1}e^{-\sigma F(x)}dx$ and in $\sigma\langle F\rangle$ , with $\Omega_D$ the area of the unit sphere. Pointwise $\sigma F(\sigma^{-1/\gamma}y)\to\kappa y^\gamma$ . Choose $x_1$ with $F(x)\ge\frac\kappa2x^\gamma$ for $x\ge x_1$ . For $\sigma\le1$ and $y\ge x_1$ both the denominator and the numerator integrand are bounded by a constant times $y^{D/2-1}e^{-\kappa y^\gamma/4}$ , using $te^{-t}\le e^{-t/2}$ ; on $y<x_1$ they are bounded by a constant times $y^{D/2-1}$ . All are integrable, so by dominated convergence $2\sigma\langle F\rangle\to2\langle\kappa y^\gamma\rangle$ in the limit measure $y^{D/2-1}e^{-\kappa y^\gamma}dy$ . With $u=\kappa y^\gamma$ the limit measure becomes $u^{D/(2\gamma)-1}e^{-u}du$ , a Gamma law of shape $D/(2\gamma)$ , whose mean is $D/(2\gamma)$ ; hence $d_s\to D/\gamma$ . (b) Here $F(x)=x/a\,(1+o(1))$ as $x\to0$ ; choose $x_1$ with $F\ge x/(2a)$ on $(0,x_1)$ . For $\sigma\ge1$ split the integrals at $x_1$ . On $x>x_1$ , $F\ge F(x_1)$ because $F$ is increasing, so $e^{-\sigma F}\le e^{-\sigma F(x_1)/2}e^{-F/2}$ , and this part is bounded by $e^{-\sigma F(x_1)/2}$ times a constant, exponentially small against $P\sim\sigma^{-D/2}$ ; the same holds for the numerator of $\langle F\rangle$ . On $x<x_1$ substitute $x=y/\sigma$ : $\sigma F(y/\sigma)\to y/a$ , and the integrands are dominated by $y^{D/2-1}e^{-y/(2a)}$ times a constant. In the limit $2\sigma\langle F\rangle\to2\langle y/a\rangle=D$ , the mean of $y/a$ under $y^{D/2-1}e^{-y/a}dy$ being $D/2$ . (c) Since $F/x$ is nonincreasing, $F(x)\le xF(1)$ for $x\ge1$ , so $\gamma\le1$ .

Part (c) concerns the ultraviolet value: a positive-weight propagator has $G\gtrsim1/x$ at large $x$ , so it cannot fall off like $x^{-2}$ . It can fall off more slowly, $G\sim x^{-1+\eta/2}$ with $0<\eta<2$ , which needs infinite total weight and gives $\gamma=1-\eta/2<1$ , that is $d_s=D/\gamma>D$ in the ultraviolet.

4. Where a reduction can come from

Giving up rotation invariance (H3), positivity in (H4), the flat measure (H1) or autonomy (H2) each makes $d_s<D$ possible. The pole and the growth condition in (H4) are kept in all cases.

Anisotropic symbols

If the symbol is not rotation invariant, (3) allows $d_s<D$ only if the symbol grows faster than quadratically along some ray, and gives it when that growth dominates the average, as in the following example. Hořava’s gravity at a Lifshitz point [Hořava, Lifshitz point] has $D=4$ Euclidean dimensions $(\omega,\vec q)$ and, in the ultraviolet, the dispersion $\psi=\omega^2+q^2+q^6$ with anisotropic scaling $z=3$ . Then $\psi-\frac12\xi\cdot\nabla\psi=-2q^6<0$ , and (3) reads $d_s=4-4\sigma\langle q^6\rangle_\sigma$ . The values are

$\sigma$	$10^{-8}$	$10^{-4}$	$10^{-2}$	$1$	$10$	$10^{2}$	$10^{4}$
$d_s$	2.0000	2.0018	2.0391	2.7038	3.7282	3.9948	4.0000

and the limits follow by scaling. In the ultraviolet the $q^6$ term dominates and $P\propto\sigma^{-1/2}\sigma^{-1/2}$ (one factor from $\omega$ , one from the three momenta $q$ with $\psi\sim q^6$ ), so $d_s=2$ . In the infrared $P\propto\sigma^{-2}$ . For each fixed $q$ the propagator $1/(\omega^2+q^2+q^6)$ is a Stieltjes function of $\omega^2$ with positive weight. What the example gives up is rotation invariance, that is, it introduces a preferred frame, whose consequences for low-energy physics are discussed on the page frame leakage.

Negative weight

Drop positivity but keep the other hypotheses. Let $a>0$ and let $\rho$ be a signed measure of finite total variation on $(0,\infty)$ , so that $G(x)=a/x+\int\rho(d\mu)/(x+\mu)$ has a massless pole of residue $a$ . Put $W=a+\rho((0,\infty))$ , the total weight.

Proposition 4 (sum rule). With $G$ as above, $xG(x)\to W$ as $x\to\infty$ . If $W>0$ and $G>0$ on $(0,\infty)$ , then $F=1/G$ satisfies $F(x)\sim x/W$ , $P(\sigma)$ is finite, and $d_s(\sigma)\to D$ as $\sigma\to0$ . If $W<0$ , then $G<0$ at large $x$ and $P=\infty$ for every $\sigma$ . Hence, when $G>0$ on $(0,\infty)$ , a spectral dimension in the ultraviolet that is defined and different from $D$ forces $W=0$ , so that the negative part of $\rho$ has total mass at least $a$ . For $\rho=\sum_ir_i\delta_{m_i^2}$ , a sum of point masses at $m_i^2$ with residues $r_i$ , this reads $\sum_ir_i=-a$ , the sum running over all massive poles, positive and negative. If $W=0$ , $G>0$ on $(0,\infty)$ and $\int\mu\lvert\rho\rvert<\infty$ , then $x^2G(x)\to-\int\mu\,\rho(d\mu)$ , and when this limit is positive $F\sim\kappa x^2$ with $\kappa=-1/\int\mu\rho(d\mu)$ and $d_s\to D/2$ , which is $2$ in $D=4$ .

Proof. $xG(x)=a+\int\rho(d\mu)\,x/(x+\mu)$ with $\lvert x/(x+\mu)\rvert\le1$ and pointwise limit $1$ , so $xG\to W$ by dominated convergence. If $W>0$ and $G>0$ , then $F$ is continuous and positive on $(0,\infty)$ , $F\sim x/a$ at the origin and $F\sim x/W$ at infinity; the growth condition holds, so $P$ is finite, and Proposition 3(a) with $\gamma=1$ gives the limit. If $W<0$ , then $G<0$ for large $x$ , $F\to-\infty$ like $x/\lvert W\rvert$ , and $e^{-\sigma F}$ grows too fast for $P$ to converge. So $W=0$ , that is $\rho((0,\infty))=-a$ , and the negative part of $\rho$ has mass at least $a$ . For $W=0$ , $x^2G=x\,(xG)=-\int\rho\,\mu x/(x+\mu)\to-\int\mu\rho$ when $\int\mu\lvert\rho\rvert<\infty$ . The last statement is Proposition 3(a) with $\gamma=2$ .

The model case is the propagator of quadratic gravity [Stelle 1977] restricted to the massive spin-two ghost, $G=1/x-1/(x+M^2)=M^2/(x(x+M^2))$ of the third row of the table, with $F=x(x+M^2)/M^2$ , $\gamma=2$ and $\kappa=1/M^2$ : it has $a=1$ , $r_1=-1$ , $W=0$ and $\int\mu\rho=-M^2$ , consistent with $d_s=2$ in the ultraviolet. As a second example, $G=1/x+\tfrac12/(x+1)-\tfrac32/(x+2)=(2.5x+2)/(x(x+1)(x+2))$ has a positive and a negative pole, $W=0$ and $\int\mu\rho=-2.5$ , so $\kappa=1/2.5$ and $d_s\to2$ in the ultraviolet by Proposition 4.

Prescriptions for the ghost pole that act only in the Lorentzian continuation leave the Euclidean $G$ , hence $d_s$ , unchanged. The fakeon prescription of Anselmi [Anselmi], applied to quantum gravity in [Anselmi–Piva 2018], is of this kind, and the Lee–Wick formulation of [Anselmi–Piva 2017] defines the theory from a Euclidean one by a nonanalytic Wick rotation, so that the Euclidean propagator is the one with the ghost pole. Positivity of $\rho$ in (2) and unitarity of the Lorentzian theory are therefore different statements.

Density of states and the diffusion clock

Dropping (H1) or (H2) changes the definition of $P$ . With a measure $\nu(dx)\propto x^{D_{\rm eff}/2-1}dx$ in place of the flat one and $F=x$ , $d_s=D_{\rm eff}$ . If the diffusion time is replaced by a function $\sigma\to s(\sigma)$ of itself, which is what happens when a scale is identified with an inverse diffusion time in a generator that depends on the scale, $d_s$ is multiplied by $d\ln s/d\ln\sigma$ ; for $s=\sigma^{\gamma}$ the factor is $\gamma$ . In both cases $d_s<D$ is possible with a positive propagator, and the reduction is carried by the measure or the clock. A reported reduction $d_s\to D'<D$ is therefore a statement about a propagator together with a measure and a clock.

5. The asymptotic-safety value two

Lauscher and Reuter [LR] obtain $d_s=2$ in four dimensions from a generator built from the running of the cosmological constant. The eigenvalue $x=p^2$ of a mode is identified with the scale $k^2$ of the effective average action, which gives the generator $\psi(\xi)=\lvert\xi\rvert^2f(\lvert\xi\rvert^2)$ , with $f(k^2)$ the ratio of the running cosmological constant at scale $k$ to its macroscopic value. This is autonomous, on a flat measure and rotation invariant, so (H1) to (H3) hold. At the fixed point the cosmological constant runs as $k^2$ , hence $f\propto x$ , $\psi\propto\lvert\xi\rvert^4$ , and Proposition 3(a) with $\gamma=2$ gives $d_s=D/2$ in every dimension $D$ (equation (3.15) of [LR]); the value two is special to $D=4$ . The generator is defined by the running of a coupling and is not the inverse of a propagator, so (H4) does not apply.

The same paper also gives the dressed graviton propagator of the fixed-point regime, $G^{-1}\propto(p^2)^{1-\eta/2}$ with the anomalous dimension $\eta=2-D$ (their equation (2.6)), which in $D=4$ is $G\propto x^{-2}$ . Taking the inverse of this propagator as the generator is a different construction, which we adopt here as the one to which Theorem 2 applies. With $F=1/G\sim x^{\gamma}$ and $\gamma=1-\eta/2$ , Proposition 3(a) gives $d_s=D/\gamma=2D/(2-\eta)$ , which is $2$ for $\eta=2-D$ in every $D$ and agrees with the generator above only in $D=4$ . By Proposition 3(c) a positive-weight propagator with a massless pole has $\gamma\le1$ , that is $\eta\ge0$ , so in $D>2$ the value $2$ cannot be reached by such a propagator (in $D=2$ the free propagator has $d_s=2=D$ ).

The Lorentzian graviton spectral function has been computed non-perturbatively and is reported positive, with a massless one-graviton peak and a multi-graviton continuum [FLPR]; the self-consistent computation in an on-shell scheme has unit total spectral weight and a close-to-quadratic decay of the continuum [PRW]. If the spectral representation of the transverse-traceless fluctuation propagator is the representation (2), with the pole and without a constant term, Theorem 2 gives $d_s\ge4$ at every scale for the diffusion it generates. With unit total weight, $W=1$ , so $F\sim x$ in the ultraviolet and $d_s\to4$ at both ends (Proposition 3). The value two is therefore not reached by this propagator used as the generator of an autonomous diffusion on a flat measure.

Calcagni, Eichhorn and Saueressig [CES] use a different route. They identify the scale with the inverse diffusion time, $k\sim\sigma^{-1/(\theta+2)}$ for a metric running as $k^\theta$ (their equation (13), where the exponent is called $\delta$ ), so that the effective generator depends on $\sigma$ and $d_s=2D/(2+\theta)$ (their equation (17)). This is the clock change of section 4 and lies outside (H2).

Three assumptions enter this reading. The scalar function $G(p^2)$ stands for the transverse-traceless part of the graviton propagator, which has a definite spectral representation; the spin-zero part and the gauge dependence are not covered. Positivity is a hypothesis taken from the cited computations. The generator of [LR] is built from the background-field running of the cosmological constant, and its relation to the positivity of the fluctuation propagator is not established.

6. A single ghost and a laboratory bound

Under its hypotheses Proposition 4 requires negative weight of total size at least $a$ for a reduction in the ultraviolet, and models of Stelle type supply it with massive ghosts, whose Newtonian potential is finite at the origin through cancellation between scalar and tensor massive modes [MNS]. Take a single ghost with $G=1/x-1/(x+M^2)$ . For this propagator the third row of the table shows the transition at $\sigma$ of order $M^{-2}$ ( $d_s=3.43$ at $\sigma=10$ and $2.70$ at $\sigma=1$ in units $M=1$ ), that is at lengths of order $1/M$ . The static potential is the Fourier transform of $1/k^2-1/(k^2+M^2)$ , so the Newtonian potential is multiplied by $1-e^{-Mr}$ : finite at $r=0$ , with a Yukawa correction of relative strength $\alpha_Y=-1$ and range $\lambda=1/M$ . A torsion-balance test of the inverse-square law with detector-attractor separations down to $52\ \mu$ m limits, at $95\%$ confidence, gravitational-strength Yukawa interactions to ranges below $38.6\ \mu$ m [Lee et al.]. The published constraint is on $\lvert\alpha_Y\rvert$ , and we apply it at $\lvert\alpha_Y\rvert=1$ ; the curves for each sign are in the supplementary material of that paper and may differ slightly. This gives $M\ge\hbar c/38.6\ \mu\text{m}=5.11$ meV (from $\hbar c=197.327$ eV nm), so a reduction carried by one such ghost cannot set in at lengths above $38.6\ \mu$ m. A ghost at $M=2.24$ meV, the fourth root of the dark-energy density in order of magnitude, has $\lambda=88\ \mu$ m and is excluded under these assumptions (a single Yukawa term and the limit at $\lvert\alpha_Y\rvert=1$ ).

The bound is an estimate within a model with one Yukawa term of strength $-1$ . With several massive poles and the massless pole normalised to unit residue, the residues sum to $-1$ when the ultraviolet value differs from $D$ and $G>0$ (Proposition 4), and the bound on each depends on its own strength. For quadratic gravity the static potential is $V=-(G_{\rm N}M_\odot/r)\,(1+\tfrac13e^{-m_0r}-\tfrac43e^{-m_2r})$ [Stelle 1978], with a spin-two ghost of strength $-4/3$ and mass $m_2$ and a scalar of strength $+1/3$ and mass $m_0$ . For the spin-two term alone, the limit at $\lvert\alpha_Y\rvert=4/3$ excludes at least the ranges excluded at $\lvert\alpha_Y\rvert=1$ , so $m_2$ is bounded at least as strongly as above; the published curves are for one Yukawa term, so the effect of the scalar term, which matters when $m_0$ and $m_2$ are close, is not covered by them.

7. Constant terms, tensor propagators and jump generators

The page The four-dimensional causal-set d’Alembertian has no positive spectral representation gives the two-dimensional Benincasa–Dowker propagator a constant term of size $1/2$ in its normalisation; the page proves that its spectral density is non-negative and supports the Stieltjes form numerically. Its symbol is therefore bounded, $P$ is infinite, and its spectral dimension needs a regularised definition. Whether the inequality of Theorem 2 survives such a definition is open.

A spin-two propagator with several tensor structures and gauge parameters needs its own derivation of the corresponding statement, which has not been done. Non-perturbative information on the Lorentzian spectral function beyond the transverse-traceless mode would show whether the conclusion of section 5 changes.

Proposition 1 gives a criterion for non-radial generators with jump (Lévy) parts. Whether every symmetric Lévy exponent with a nondegenerate Gaussian part has $d_s\ge D$ is open. The spectral dimension of an ensemble of geometries or of a causal-set random walk is an average of positive Markov chains and is not governed by a single autonomous generator, so Theorem 2 does not apply to it. Whether a clock identification of the kind used in [CES] can be phrased as a property of a measurable observable is also open.

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