Remarks on the quantum gravity interpretation of 4D dynamical triangulation

TL;DR

This paper reviews 4D dynamical triangulation phenomenology and discusses its interpretation through a Euclidean effective action resembling continuum quantum gravity, highlighting potential insights into quantum gravity models.

Contribution

It offers a reinterpretation of 4D dynamical triangulation phenomenology within a continuum Euclidean effective action framework, bridging discrete models and continuum theories.

Findings

Connection between dynamical triangulation and Euclidean effective action.

Insights into quantum gravity interpretation of triangulation models.

Potential implications for continuum quantum gravity theories.

Abstract

We review some of the phenomenology in 4D dynamical triangulation and explore its interpretation in terms of a euclidean effective action of the continuum form $\intx \sqrt{g} [\mu -\frac{1}{16\pi G} R + \cdots]$.

Equations

$$Z({\kappa }_2,N)=\sum _{\mathcal{T},N_4=N}{e}^{{\kappa }_2{N}_2},$$

$$Z$$

$$S_{\mathrm{eff}}[g]$$

$${\mu }_c=\frac{\Lambda }{8\pi G}.$$

$$N^{\prime }(r)=⟨\sum _y{\delta }_{d_{xy},r}⟩,$$

$$\frac{S_{\mathrm{eff}}}{2{\pi }^2}={\int }_{r_1}^{r_2}dr{a}^3\left[\mu -\frac{1}{16\pi G}\left(\frac{{\dot{a}}^2}{a^2}+\frac{1}{a^2}\right)\right].$$

$$a(r)=r_0\sinh \frac{r}{r_0},R=-\frac{12}{r_0^2}=-32\pi |G|\mu ,$$

$$a(r)=r_0\sin \frac{r}{r_0},R=+\frac{12}{r_0^2}=32\pi G\mu ,$$

$$(\Delta +m_0^2{)}_{xy}{G}_{yz}={\delta }_{xz},$$

$$g_{\mu \nu }={\Omega }^2{\bar{g}}_{\mu \nu },,R={\Omega }^{-2}\bar{R}-6{\Omega }^{-3}\bar{\Delta }\Omega ,$$

$$S_{\mathrm{eff}}$$

Full text

4D Dynamical Triangulation (DT) is a formulation of purely euclidean geometrodynamics [1]. Its customary canonical partition function reads

[TABLE]

where the summation is over all distinct triangulations ${\cal T}$ (describing simplicial manifolds) with $S^{4}$ topology and $N_{k}$, $k=0,1,\ldots,4$ is the number of $k$-simplices. The way the model is derived shows that $\kappa_{2}\propto 1/G_{0}$, with $G_{0}$ the bare Newton constant. In this talk I explore a tentative continuum interpretation of the model, described by the effective theory

[TABLE]

Here the path integral is over real metrics modulo coordinate transformations, $G$ denotes a renormalized Newton constant and the $\cdots$ indicate higher derivative terms like $R^{2}$, etc. There may also be nonlocal terms related to the conformal anomaly [2]. The integral over $\mu$ produces the volume fixing delta function $\delta(\int d^{4}x\,\sqrt{g}-V)$. If this integral were done in the saddle point approximation, the saddle point value $\mu_{c}$ would be related to a renormalized cosmological constant by

[TABLE]

In general the higher order terms may be present in an effective action. Here we expect them to regulate the unboundedness of the Einstein- Hilbert part in the ultraviolet. The underlying DT model is finite and the nontrivial results of numerical simulations show that the action is not stuck at its minimum value in the relevant $\kappa_{2}$ range. So entropy effects somehow provide a regulating effect which can be implemented in $S_{\rm eff}$ through the higher order terms.

Let us now go through some of the DT phenomenology found in numerical simulations and compare this with the effective action (3).

1. There is a crumpled phase ($\kappa_{2}<\kappa_{2}^{c}$) and an elongated phase ($\kappa_{2}>\kappa_{2}^{c}$), which has characteristics of a branched polymer [3]. Since $\kappa_{2}\propto 1/G_{0}$ this suggests that the effective theory also has a transition at some value of $1/G$, presumably near $1/G=0$.

2. Recent evidence [4] indicates strongly that the transition between the two phases is first order. This is often seen as a disaster for a continuum interpretation. However, we have suggested earlier [5] that continuum behavior may be automatic without tuning to a second order phase transition. This happens in DT in two dimensions. A field theoretic example is provided by the purely discrete $Z(n)$ gauge-Higgs systems. For sufficiently large but finite $n$ these models have a Coulomb phase with massless photons and a Higgs phase separated by a first order transition [6]. The models can approximate the continuum abelian Higgs model arbitrarily well. Analysis of the continuum model shows the possibility of a first order phase transition.

3. Continuum behavior is supported by evidence for scaling [5]. This can be seen in the observable

[TABLE]

the volume at geodesic distance $r$. It is maximal at some distance $r_{m}$, which can be used to set a distance scale. Plotting $r_{m}N^{\prime}(r)/N$ for various $N$ and suitably adjusted $\kappa_{2}$ we find that it scales approximately in the sense that it approaches a continuous function $\rho(r/r_{m},\tau)$ as $N$ increases. Here $\tau$ distinguishes different shapes of $\rho$; for given $N$ we may think of $\tau=\kappa_{2}$. Hence, measuring distances in units of $r_{m}$ we can let the lattice distance go to zero as $N\rightarrow\infty$, while keeping the shape of $\rho$ fixed. This scaling analysis needs to be redone for larger lattices, especially in the light of the first order nature of the phase transition.

4. ‘Physical’ scalar curvature observables can be obtained from $N^{\prime}(r)$: an average curvature $R_{V}$ and a ‘running’ curvature $R_{\rm eff}$ [5]. These are negative in the crumpled phase, positive at the transition, and appear to be ill defined in the elongated phase. This can be compared with predictions from (3) as follows. Assume that for slow variations the $R/G$ term in (3) dominates and that for intermediate distances spacetimes are homogenous and isotropic on the average, as described by a euclidean FRW scale factor $a(r)$, with effective action

[TABLE]

Identifying $v_{\rm eff}N^{\prime}(r)$ with $2\pi^{2}a(r)^{3}$, where $v_{\rm eff}$ is an effective volume, this becomes an effective action for $N^{\prime}(r)$. Here $r_{1}$ should not be too small for the higher order terms in (3) to be neglected, and $r_{2}$ should not be too large to avoid strong fluctuations at large distances. We get the following stationary points:

[TABLE]

for $G<0$ and

[TABLE]

for $G>0$, assuming $\mu>0$. The first case corresponds to the negative curvatures $R_{V}$, $R_{\rm eff}$ found in the crumpled phase, the second corresponds to the approximate four-sphere behavior $N^{\prime}(r)\propto\sin^{3}r/r_{0}$ found in the transition region [5].

Further support for the negative curvature interpretation of the crumpled phase comes from probing the DT euclidean spacetimes with scalar test particles [7]. Solving the equation

[TABLE]

on every configuration (where $\Delta_{xy}$ is the lattice Laplace-Beltrami operator) and averaging this over the configurations at fixed geodesic distance $d_{xy}=r$ gives a propagator $G(r)$. We interprete this propagator as corresponding to an average background geometry. The exponential fall-off of $G(r)$ at large distances determines an effective mass $m$. In the continuum, a space of constant negative curvature with curvature radius $r_{0}$ as in (6) gives a nonzero $m$ even if $m_{0}=0$, namely $m=3/r_{0}$. We have measured $G(r)$ and $m$ in the crumpled phase and found indeed nonzero effective masses, correlated with the curvature radii $r_{0}\equiv\sqrt{-12/R_{V}}$ obtained from $N^{\prime}(r)$.

For the gravity interpretation of DT it is of course essential to exhibit its ability to attract. We have investigated if there is a binding energy between two scalar test particles, by comparing the two-particle mass $M$ extracted from $\langle G_{xy}^{2}\rangle$ with $2m$. The computation was done in the transition region $\kappa_{2}\approx\kappa_{2}^{c}$, because there the average spacetimes as seen through $N^{\prime}(r)$ resemble most closely the classical $S^{4}$. In this case $m$ (interpreted as the constituent mass) is roughly proportional to $m_{0}$, although the ratio $m/m_{0}$ increases as $m_{0}$ gets smaller. The results show that there is indeed a positive binding energy $E_{b}=2m-M$. We tried to see if the nonrelativistic formula $E_{b}=G^{2}m^{5}/4$ could be used to define a renormalized Newton constant $G$, but the binding energy did not seem to behave like $m^{5}$. The reason for this may be strong finite size effects: using $G^{2}=4E_{b}/m^{5}$ for the smallest mass suggested a Planck length $\sqrt{G}$ of only a third of the typical length scale of the configurations, $r_{m}$ [8].

We now venture into some strong speculations about the transition. In the infrared we expect the Einstein-Hilbert part of the effective action (3) to dominate, because it has fewer derivatives. So the unboundedness of this term is still relevant for large volumes, $V/G^{2}\gg 1$ (this ignores the nonlocality of possible AM terms [2]). We may follow ref. [9] and introduce a conformal factor $\Omega$ by

[TABLE]

where $\bar{R}$ and $\bar{\Delta}$ are evaluated with the metric $\bar{g}_{\mu\nu}$. The condition fixing $\Omega$ is that $\bar{R}$ is constant, independent of $x$. Making a partial integration the action takes the form

[TABLE]

The unboundedness is clear from the negative sign in front of the derivative term. Performing the integration over $\Omega$ along the imaginary direction effectively changes this sign to positive, which is proposed to cure the unboundendness problem [9]. However, the same effect is obtained by choosing $G<0$, keeping $\Omega$ and therefore also the original metric real. If we assume that the saddle point value $\mu_{c}$ is positive, then a possibly negative $\bar{R}\Omega^{2}$ term would be subdominant to the $\mu\Omega^{4}$ term. So for negative $G$ the euclidean theory may also be well defined in the infrared. This appears to contradict perturbation theory where changing the sign of $G$ does not cure the unboundedness problem.

Let us ignore this difficulty and continue very schematically, concentrating on the $\Omega$ modes. We lump all the higher order terms into $\zeta R^{2}$ and assume $\zeta>0$ for stabilization. For small fluctations about some background we expect a propagator of the schematic form $(-G^{-1}p^{2}+\zeta p^{4})^{-1}$. For negative $G$ this is stable but for positive $G$ we expect condensation of some nonzero momentum modes. Suppose $L$ represents the size of the system, then $p\propto n/L$, where $n$ is an integer. The modes with $-G^{-1}p^{2}+\zeta p^{4}<0$ are unstable, i.e. $|n|=1,2,\dots,n_{m}$, with $n_{m}$ the largest positive integer smaller than $L/\sqrt{\zeta G}$. Such a condensation of nonzero momentum modes may describe the branched polymer behavior of the DT model in the elongated phase. The modes with $|n|>n_{m}$ remain uncondensed. For $G^{-1}$ sufficiently small, $1>L/\sqrt{\zeta G}$, $n_{m}=0$ and there is still no condensation. This may correspond to the DT transition region, on the crumpled side, where we found $S^{4}$ behavior. If this schematic reasoning makes sense then the maximum size of the $S^{4}$-like ‘universe’ can only be of order of $\sqrt{\zeta G}$, i.e. the Planck length.

Finally we recall the possibility that the DT theory may be ‘trivial’ [5]. For negative $G$, writing $\Omega=\sqrt{16\pi|G|/12}\,\phi$ gives a standard gradient term for $\phi$, a $\phi^{4}$-like theory with coupling $\lambda=2\pi|G\Lambda|/3$ (cf (4,10)). If, because of triviality, this $\lambda$ approaches zero as $N\rightarrow\infty$ (while tuning $\kappa_{2}$ such that we stay on a scaling curve $\rho$), then we might still obtain a large size ‘universe’ (of order $\Lambda^{-1/2})$) in relation to $\sqrt{|G|}$, which itself is arbitrarily large compared to the lattice distance.

Acknowledgements The DT phenomenology reviewed above was obtained in collaboration with B.V. de Bakker. We thank P. Białas, G. ’t Hooft and M. Visser for useful discussions. This work is supported by FOM.