0$), $g^{(2)}$ is real. For real and timelike momenta ($Z<0$), its value above/below the branch cut corresponds to past/future-directed momentum-vectors. There, $g^{(2)}_{\rho}$ is complex and changes to its complex conjugate across the cut. That the spectrum is different for past and future-directed momenta should come as no surprise, given that $\Box_{\rho}^{(2)}$ is retarded by definition. We will see in section~\ref{GenCB} that these features persist in all dimensions and for a much broader class of causet d'Alembertians. The infrared (IR) and ultraviolet (UV) behaviours of $g^{(2)}_{\rho}(p)$ are easily deduced from the asymptotic forms of $E_2(Z)$ (see e.g.\ (8.11.2), (8.19.1), and (8.19.8) of~\cite{DLMF}): \begin{align} \rho^{-1}g^{(2)}_{\rho}(p)&\xrightarrow{Z\to0}-Z+\cdots\\ \rho^{-1}g^{(2)}_{\rho}(p)&\xrightarrow{Z\to\infty}-2+\frac{8}{Z}+\cdots. \end{align} The first of these two equations shows that the usual d'Alembertian $\Box$ is indeed reproduced in the limit of zero non-locality. The second equation, on the other hand, reveals a UV behaviour quite unlike that of the usual d'Alembertian; in section~\ref{IRGCB} it will lead us to propose a new regularization scheme for quantum field theory. An important question is whether the evolution defined by $\Box_{\rho}^{(2)}\Phi=0$ is stable or not. To a large extent this is answered by the fact that the only zero of $g^{(2)}_{\rho}(p)$ occurs at $Z=\rho^{-1}\Lp=0$. To demonstrate this, we note that $g^{(2)}_{\rho}(p)$ has the following representation (see e.g.\ (8.19.1) and (8.6.4) of~\cite{DLMF}): \beq \rho^{-1}g^{(2)}_{\rho}(p)=-Zf(Z), \qquad f(Z)\equiv\int_{0}^{\infty}\frac{te^{-t}}{t+Z/2}\ud t. \eeq It therefore suffices to prove that $f(Z)$ has no zeros when $Z\neq0$. But the imaginary part of $f(Z)$ is \beq \text{Im}(f(Z)) = -\frac{\text{Im}(Z)}{2}\int_{0}^{\infty}\frac{te^{-t}}{\left[t+\frac{\text{Re}(Z)}{2}\right]^2+\left[\frac{\text{Im}(Z)}{2}\right]^2}\ud t. \label{eqImFz} \eeq Because the integral that multiplies $-\text{Im}(Z)/2$ in~\eqref{eqImFz} is strictly positive, $Zf(Z)$ could vanish only for real $Z$. Obviously, it does vanish for $Z=0$, but elsewhere on the real axis, it remains nonzero, as illustrated in figure~\ref{g2Dfig}. What we have just proven is that a plane wave solves the equation $\Box_{\rho}^{(2)}\Phi=0$ iff it solves the equation $\Box\Phi=0$. To the extent that the general solutions of these two wave equations can be composed of plane waves, they therefore share the same space of solutions. This, of course, is an important result in itself. But it also, a fortiori, answers the stability question in the affirmative, since we know that the evolution corresponding to $\Box$ is stable. If there remains any doubt about stability or about the fact that both $\Box\Phi=0$ and $\Box_{\rho}^{(2)}\Phi=0$ yield the same evolution, it springs from a possible uncertainty about boundary conditions. In the usual situation (that of the ordinary d'Alembertian $\Box$), one understands how to relate a general solution to its initial data on an arbitrary Cauchy surface, and when $\Phi$ falls off suitably at infinity, its total energy is defined and conserved. From energy conservation, stability also follows --- relative to the given choice of boundary conditions. On the other hand in the case of $\Box_{\rho}^{(2)}$, a connection between solutions and Cauchy data remains to be found, as does a better understanding of appropriate falloff conditions. But absent some such boundary condition there is nothing to exclude complex momenta $p$ that lead to exponential growth in time, e.g.\ an imaginary multiple of a real lightlike vector. For these reasons, we would like to discuss stability from a slightly different angle, which also will be helpful when we come to deal with the 4D case. Quite generally, instabilities tend to be associated with exponentially growing ``modes" (in this case plane waves). Let us then \emph{assume} that we can take this as our criterion of (in)stability. And to exclude the kind of ``fake instability'' mentioned above, let us also require any putative unstable mode, $\Phi(x)=e^{ip\cdot x}$, to be bounded at spatial infinity in at least one Lorentz frame. (Unfortunately we cannot say ``in all Lorentz frames'', since for a plane wave, exponential growth in time induces exponential growth in space via a Lorentz boost.) We might hope that the condition just formulated is equivalent to the following more natural one: consider only solutions of $\Box_{\rho}^{(2)}\Phi(x)=0$ which have compact support on every Cauchy hypersurface (compact spatial support in every frame.) Be that as it may, if this criterion is accepted, then we can establish stability very simply in the present case, because an unstable mode, $\Phi(x)=e^{ip\cdot x}$, is then precisely one such that $p$ possesses a future-directed timelike imaginary part: $p=p_R+ip_I$ with $p_I\cdot p_I<0$ and $p_I^0>0$. This, however, is impossible for $Z=0$, as one sees from the equation $0=p\,\cdot\,p=p_R\cdot p_R-p_I\cdot p_I+2ip_R\cdot p_I$, whose right-hand side has a strictly positive real part when $p_I$ is timelike. For logical completeness, we should also observe that~\eqref{g2Dexact} is valid for all complex $p$ whose imaginary parts are timelike and future-directed. (For more general complex momenta, the integral defining $\Box_{\rho}^{(2)}\Phi$ might not converge, a circumstance that, depending once again on the choice of falloff conditions, might or might not impinge on the claimed identity between our solutions and those of the ordinary wave equation.) ]]>

0$, where $p=p_R+ip_I$ and the Lorentzian norm is given by $p\cdot p=p_R\cdot p_R-p_I\cdot p_I+2ip_R\cdot p_I$. For momenta satisfying these conditions, the integral that defines $\chi(p,\rho)$, and consequently $\Box^{(D)}_{\rho}e^{ip\cdot x}$, is absolutely convergent. Plugging~\eqref{chiP} into~\eqref{gGen} we find \beq \rho^{-2/D}g^{(D)}_{\rho}(p)=a+2(2\pi)^{D/2-1}Z^{\frac{2-D}{4}}\sum_{n=0}^{\Lmax}\frac{b_n}{n!}C_D^n \int_{0}^{\infty} s^{D(n+1/2)}e^{-C_Ds^D}K_{\frac{D}{2}-1}(Z^{1/2}s)\ud s, \label{geng} \eeq where $Z$ is a dimensionless quantity defined by \beq Z\equiv\rho^{-\frac{2}{D}}\Lp. \eeq For real $p=p_R$, $g^{(D)}_{\rho}(p)$ can be defined by first adding a small future-pointing and timelike imaginary part $p_I^{\epsilon}$ to $p_R$, and then taking the limit as $p_I^{\epsilon}$ shrinks: \beq g^{(D)}_{\rho}(p_R):=\lim_{\epsilon\to0^{+}}g^{(D)}_{\rho}(p_R+ip_I^{\epsilon}), \qquad p_I^{\epsilon}\cdot p_I^{\epsilon}=-\epsilon^2. \eeq When $p_R$ is timelike, this amounts to changing $Z=\rho^{-\frac{2}{D}}p_R\cdot p_R$ on the right hand side of~\eqref{geng} to $Z+ i\epsilon$ for past-directed, and $Z- i\epsilon$ for future-directed $p_R$. This is illustrated in figure~\ref{branchCut}. Because of the appearance of $Z^{1/2}$ in~\eqref{geng} and the fact that $K_{\nu}(\bar{z})=\overline{K_{\nu}(z)}$, it follows for timelike $p$ that \beq g^{(D)}_{\rho}(-p)=\overline{g^{(D)}_{\rho}(p)}. \label{cmplxConj} \eeq Therefore, \emph{$g^{(D)}_{\rho}(p)$ differs for past- and future-directed timelike $p$}. This is to be expected, since requiring $\Box^{(D)}_{\rho}$ to be retarded builds in a direction of time. For spacelike momenta ($Z>0$), $g^{(D)}_{\rho}(p)$ is real, as follows from the fact that $K_{\nu}(z)$ is real when $\nu$ is real and $\text{ph}(z)=0$~\cite{DLMF}. ]]>

0$ and $p^0<0$. For a fixed sign of $p^0$, $g_{\rho}^{(D)}(p)$ is only a function of $p\cdot p$, making~\eqref{meh6} the Laplace transform of a Lorentz-invariant function. Similarly to how we derived~\eqref{chiP}, we use the result of~\cite{LTRLI} to compute $G(x-y)$: \begin{align} \rho^{2/D}G(x-y)\overset{x\succ y}{=}&\frac{2}{\pi(2\pi)^{D/2}}\int_0^{\infty}d\xi~\xi^{D/2}\notag\\ &\times\text{Re}\left[\left(\sqrt{s_{\epsilon}^2}\right)^{1-\frac{D}{2}} K_{\frac{D}{2}-1}(\sqrt{s_{\epsilon}^2}\,\xi) \left(\frac{1}{\tilde g(-\xi^2+i\epsilon)}-\frac{1}{\overline{\tilde g(-\xi^2+i\epsilon)}}\right)\right], \label{meh7} \end{align} where $s_{\epsilon}^2=-(t_x-t_y+i\epsilon)^2+|\vec r_x-\vec r_y|^2$ and $\epsilon$ is a small positive number which should be taken to zero at the end of calculations. When $x-y$ is timelike and future-directed, we can let $\sqrt{s_{\epsilon}^2}=-i\tau_{xy}$ where $\tau_{xy}>0$. Using properties of Bessel functions (see e.g.\ (10.27.9) of~\cite{DLMF}),~\eqref{meh7} can be simplified into the following form for even D: \begin{align} \rho^{2/D}G(x-y) &\overset{x\succ y}{=} \frac{-i(-1)^{\frac{D}{2}}\tau_{xy}^{1-\frac{D}{2}}}{(2\pi)^{D/2}} \int_0^{\infty}d\xi~\xi^{D/2} \left(\frac{1}{\tilde g(-\xi^2+i\epsilon)}-\frac{1}{\overline{\tilde g(-\xi^2+i\epsilon)}}\right)J_{\frac{D}{2}-1}(\tau_{xy}\xi)\notag\\ &=\frac{2(-1)^{1+\frac{D}{2}}\tau_{xy}^{1-\frac{D}{2}}}{(2\pi)^{D/2}} \int_0^{\infty}d\xi~\xi^{D/2} \frac{\text{Im}\left[\tilde g(-\xi^2+i\epsilon)\right]}{\left|\tilde g(-\xi^2+i\epsilon)\right|^2}J_{\frac{D}{2}-1}(\tau_{xy}\xi). \end{align} Using $(x/2)^{1-D/2}J_{\frac{D}{2}-1}(x)\xrightarrow{x\to 0}\Gamma(D/2)^{-1}$ (see e.g.\ (10.2.2) of~\cite{DLMF}) and the fact that $\text{Im}\left[\tilde{g}(-\xi^2+i\epsilon) \right]$ is exponentially damped for large $\xi$ (see appendix~\ref{UVdetEven}), it can be verified that \beq \lim_{x\to y}\rho^{2/D}G(x-y)=\frac{2^{2-\frac{D}{2}}(-1)^{1+\frac{D}{2}}}{(2\pi)^{\frac{D}{2}}\Gamma(\frac{D}{2})}\int_0^{\infty}d \xi~ \xi^{D-1}\frac{\text{Im}\left[\tilde g(-\xi^2+i\epsilon) \right]}{\left|\tilde g(-\xi^2+i\epsilon)\right|^2}. \eeq Thus $G(x-y)$ approaches a constant in the coincidence limit.\footnote{One can understand intuitively why $G_R(x-y)$ is the sum of a $\delta$-function with a bounded remainder by noticing that (up to an overall numerical factor) our nonlocal d'Alembertian operator has the form $1-S$, where the `$1$' corresponds to the first term in~\eqref{twopointthree} or~\eqref{tatu2} and the remainder $S$ is given by an integral-kernel which is both bounded and retarded. The inverse operator $G_R$ would then be $G_R=(1+S)^{-1}=1+SG_R=1+S+S^2+S^3\cdots$, a series that should converge sufficiently near to $x=y$. Since the operator $1$ is represented by a term of $\delta(x-y)$ in $G_R(x-y)$, one sees that $G_R(x-y)$ is the sum of a $\delta$-function with a term involving only smooth bounded functions. } Strictly speaking, the discussion above only analyzes the behavior of $G(x-y)$ as $\tau_{xy}$ \emph{approaches} $0$, and consequently it does not exclude the presence of terms which blow up discontinuously on the light cone, such as $\delta(\tau_{xy}^2)$. However, a similar treatment for the case where $x-y\not=0$ is null rather than timelike removes this loophole. ]]>

0$). The necessary {and} sufficient condition for avoiding unstable modes is then \beq \tilde{g}(Z)\neq0 \,,\qquad \forall\ Z\neq0, \eeq where $\tilde{g}(Z)$ is defined in~\eqref{tildeg}. Let us argue why this is the case. First observe that plane solutions of our wave-equation correspond exactly with zeros of $\tilde{g}(Z)$. If the above condition is verified, then the only such zero is at $Z=0$, just as for the usual d'Alembertian. But we know (as is also easy to demonstrate ab initio) that there are no unstable modes in the usual case. Conversely, when the above condition is violated for some complex ${Z}\neq0$, it is always possible to find a corresponding ${p}$ with a timelike and future-directed imaginary part which satisfies ${p}\cdot {p}=\rho^{\frac{2}{D}}Z$. To see this, we let ${p}={p}_R+i{p}_I$ and take ${p}_R=<\pi^0_R,\vec{\pi}_R>$ and $p_I=<\pi_I,\vec{0}>$ with $\pi_I>0$. This is always possible because $p_I$ is timelike and future-directed. The equations that need to be satisfied are \beq {p}_R\cdot {p}_R-{p}_I\cdot {p}_I=\rho^{\frac{2}{D}}\text{Re}(Z), \qquad 2{p}_R\cdot {p}_I=\rho^{\frac{2}{D}}\text{Im}(Z). \qquad \eeq Substituting for ${p}_I$ leads to \beq \pi^0_R = \frac{\rho^{\frac{2}{D}}\text{Im}({Z})}{-2\pi_I}, \qquad |\vec{\pi}_R|^2=\rho^{\frac{2}{D}}\text{Re}({Z})+\frac{\rho^{\frac{4}{D}}\text{Im}({Z})^2}{4\pi_I^2} - \pi_I^2. \eeq This system of equations always has a solution. In fact, there is a whole family of such unstable modes parametrized by $\pi_I$. Note however that the condition $|\vec{\pi}_R|^2>0$ puts an upper bound on the value of $\pi_I$, and therefore on the growth rate of such an instability. We have thus reduced the question of whether or not $\Box_{\rho}^{(D)}$ has unstable modes to the question of whether $\tilde{g}(Z)$ has zeros other than $Z=0$ in the complex plane. We can answer this question by counting the zeros of $\tilde{g}(Z)$ with the aid of the ``argument principle" of complex analysis: \beq \frac{1}{2\pi i}\oint_{C}\frac{\tilde{g}'(Z)}{\tilde{g}(Z)}dZ=N-P, \eeq where $N$ and $P$ are the number of zeros and poles, respectively, inside of the closed contour C, which we choose as shown in figure~\ref{stabilityCont}. The number of poles inside $C$ is zero because all terms appearing in $\tilde{g}(Z)$ are finite in that region (at least when $\Lmax$ is finite). As shown in figure~\ref{stabilityCont}, the path of integration $C$ comprises four pieces: $C_2$ and $C_4$ run from $-\infty$ to 0 a distance $\epsilon$ above and below the negative real axis respectively, $C_3$ is a semicircle of radius $\epsilon$ about the origin, and $C_1$ is (almost) a circle whose radius should be taken to infinity. For large $Z$ we have from~\eqref{uvGen}, \beq \frac{\tilde{g}'(Z)}{\tilde{g}(Z)}\xrightarrow{Z\to\infty}\frac{-D2^{D-1}\pi^{\frac{D}{2}-1}\Gamma(D/2)}{2a}b_0Z^{-\frac{D}{2}-1}+\cdots, \eeq and it follows that \beq \int_{C_1}\frac{\tilde{g}'(Z)}{\tilde{g}(Z)}dZ=0. \eeq (We remind the reader of our standing assumption that $a\neq0$. See the remarks following~\eqref{meh1}.) On the other hand the IR behaviour, $\tilde{g}(Z)\xrightarrow{Z\to0}-Z$, leads to \beq \int_{C_3}\frac{\tilde{g}'(Z)}{\tilde{g}(Z)}dZ=i\pi. \eeq Also, because $\tilde{g}(x+i\epsilon)=\overline{\tilde{g}(x-i\epsilon)}$ for $x<0$: \beq \int_{C_2+C_4}\frac{\tilde{g}'(Z)}{\tilde{g}(Z)}dZ=2i\int_{C_2}\text{Im}\left[\frac{\tilde{g}'(Z)}{\tilde{g}(Z)}\right]dZ. \eeq Performing this last integral will allow us to determine whether $\Box_{\rho}^{(D)}$ has unstable modes or not. Given a choice of the parameters $a$ and $b_n$, the last integral can be computed numerically. In the minimal 4D case discussed in section~\ref{org4Dsec}, we find that $\Box_{\rho}^{(4)}$ has precisely two ``unstable zeros''. (Notice that because $\tilde{g}(\bar{Z})=\overline{\tilde{g}(Z)}$, if $Z$ is a zero of $\tilde{g}(Z)$, so also is $\bar{Z}$.) We have located these zeros numerically, as shown in figure~\ref{stability4D}. With different choices of the parameters $\{a, b_n\}$, the number of zeros can change, but we have not been able to find any choice that would make $\Box_{\rho}^{(4)}$ stable. It would be interesting to find an analytical method to check for stability. ]]>