# Vacuum Condensate Picture of Quantum Gravity

## Abstract

**:**

## 1. Introduction

## 2. Regularized Path Integral for Quantum Gravity

## 3. Diffeomorphism Invariant Gravitational Correlation Functions

## 4. Renormalization Group Running of Newton’s G

## 5. Gravitational Wilson Loop and Curvature Condensate

## 6. Effective Field Equations

## 7. Large Scale Curvature and Matter Density Correlations

## 8. Gravitational Slip Function with $\mathit{G}(\mathbf{\square})$

## 9. Matter Density Perturbations with $\mathit{G}(\mathbf{\square})$

## 10. Conclusions

- ○
- The vacuum condensate picture of quantum gravity contains from the start a very limited set of parameters, and is as a result strongly constrained. It involves a new, genuinely nonperturbative scale (the gravitational vacuum condensate, see Equations (58) and (59)), which relates the running of Newton’s G to the current observed value of the cosmological constant, and to the long distance behavior of physical diffeomorphism invariant curvature correlations.
- ○
- While in principle both signs could be possible, in the strong coupling limit the effective, long distance cosmological constant is positive (see Equation (58), the arguments preceding it, and more detailed discussion in [47]). The basic argument relies of the behavior of the gravitational Wilson loop: in the same strong coupling regime, it seems impossible from the lattice theory to obtain a negative value for the effective cosmological constant, irrespective of the choice of boundary conditions (which, incidentally, in the lattice context play no role in the argument).
- ○
- The theory predicts a slow increase in strength of the gravitational coupling when very large, cosmological scales are approached (see Equations (41) and (70)). In this context, the observed scaled cosmological constant $\lambda $ is seen to act as a dynamically induced infrared cutoff, similar to what happens in non-Abelian gauge theories. In principle, both the universal power and amplitude for this infrared growth are calculable from first principles in the underlying lattice theory.
- ○
- The lattice theory appears to exclude the possibility of a physically acceptable phase with gravitational screening. The perturbative, weak coupling (small G) phase is found to be inherently unstable in the lattice formulation, a consequence of the conformal instability. Thus, a genuinely semiclassical regime for quantum gravity, whereby quantum effects can be included as small perturbations, does not seem to exist. On the other hand for large enough quantum fluctuations (large G) the conformal instability is overcome, and a new stable, anti-screening phase emerges. The stability of quantum gravity can thus be viewed as an entropy effect, intimately connected to non-trivial properties of the gravitational functional measure.
- ○
- Calculations presented here give a number of specific predictions for the behavior of invariant curvature correlations as a function of geodesic distance, and specifically the powers and amplitudes involved (see Equations (91) and (93)). Perhaps the most important result is the fact that the curvature correlation function decays as the inverse distance squared ($n=1$ and thus $s=1$ and $\gamma =2$). This in turn can be used to relate in a standard way, via the quantum equations of motion, curvature correlations to matter density correlations and thus to their observed power spectrum (see Equations (102) and (107)).
- ○
- Given the exceptionally large value of the scale $\xi $ (originating from the fact that the observed $\lambda $ is very small compared to the scale associated with G), no observable deviations from classical General Relativity are expected on laboratory, solar systems and even galactic scales (see Equations (62) and (61)).

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Universal scaling exponent $\nu $ determining the running of G (see Equations (41), (70) and (62)) as a function of spacetime dimension d. Shown are the results in $2+1$ dimensions obtained from the exact solution of the lattice Wheeler–DeWitt equation [30,31], the numerical result in four dimensions [44], the $2+\u03f5$ expansion result to one [34] and two loops [35], and the large d result ${\nu}^{-1}\simeq d-1$ [53]. For actual numerical values, see Table 1.

**Figure 2.**Correlation function of two infinitesimal parallel transport loops, separated by a geodesic distance d. This correlation corresponds to the one defined in Equations (18) and (19). In the strong coupling limit, one needs, in order to get a non-zero correlation, to fully tile the minimal tube connecting the two infinitesimal initial and final loops.

**Figure 3.**Correlation function for two large parallel transport loops of size ${r}_{c}$ and orientation ${\omega}_{c}$, separated by a geodesic distance d. This correlation corresponds to the one defined in Equation (49). In the strong coupling limit, one needs, in order to get a non-zero correlation, to fully tile the tube connecting the two large initial and final loops.

**Figure 4.**Running gravitational coupling $G\left(r\right)$ versus r, obtained from $G\left(q\right)$ in Equation (41) by setting $q\sim 1/r$ with an exponent $\nu =1/3$. In view of Equation (42), lattice quantum gravity calculations imply a slow rise of G with distance scale, with roughly a 5% effect on scales of ≈790 Mpc, and a 10% effect on scales of ≈990 Mpc. In this plot, ${G}_{c}$, the short distance fixed point value for Newton’s constant, corresponds quite closely to the known laboratory value.

**Figure 5.**Running gravitational coupling $G\left(r\right)$ versus r, obtained from $G\left(q\right)$ in Equation (41) by setting $q\sim 1/r$ with an exponent $\nu =1/3$. Note that the approximate Hartree–Fock analytical result of Equation (63) (red line) initially rises more rapidly for small r. The nonperturbative scale $\xi $ is related to the gravitational vacuum condensate, as in Equation (42).

**Figure 6.**Qualitative behavior of the (gravitational quantum fluctuation-induced) matter density power spectrum $P\left(q\right)$ with a running Newton’s $G\left(q\right)$, as given explicitly in Equation (114). Here, it is compared to the results of Equations (99) and (103) for a constant G. In addition, the spectral exponent is $s=1$ and the amplitude is ${a}_{0}$, as discussed in the text (see Equations (101) and (113)); in the plot, the q wave vector is measured for convenience in units of $\xi $. Note the rather marked turnover for small $q\approx 4.20/\xi $ due to the running of G, as discussed in the text. The nonperturbative scale $\xi $ is related to the gravitational vacuum condensate, as in Equations (59) and (58).

**Figure 7.**Effective spectral index $s\left(q\right)$ as defined in Equations (99) and (115). The horizontal line at the top is the $s=1$ value, corresponding to a constant (scale-independent) Newton’s G. The spectral index approaches the value $s=1$ for large $q\gg 1/\xi $, but dips below zero for $q\simeq 1/\xi $.

**Table 1.**Comparison of estimates for the universal gravitational scaling exponent $\nu $, based on a variety of different analytical and numerical methods. These include the numerical results of [44], The $2+\u03f5$ expansion for pure gravity carried out to one and two loops [34,35], an estimate for the leading exponent in a truncated renormalization group expansion [54,55], a simple argument based on geometric features of the quantum vacuum polarization cloud for gravity [53], and finally the value obtained from consistency of the exact solution to the nonlocal field equation with a $G(\square )$ for the case of the static isotropic metric [56,57].

Method Used to Compute the Exponent $\mathit{\nu}$ in d = 4 | Universal Exponent $\mathit{\nu}$ |
---|---|

Euclidean Lattice Quantum Gravity [44] | ${\nu}^{-1}=2.997\left(9\right)$ |

Perturbative $2+\u03f5$ expansion to one loop [34] | ${\nu}^{-1}=2$ |

Perturbative $2+\u03f5$ expansion to two loops [35] | ${\nu}^{-1}=22/5=4.40$ |

Einstein–Hilbert RG truncation [54] | ${\nu}^{-1}\approx 2.80$ |

Recent improved Einstein–Hilbert RG truncation [55] | ${\nu}^{-1}\approx 3.0$ |

Geometric argument [53] ${\rho}_{vac\phantom{\rule{0.277778em}{0ex}}pol}\left(r\right)\sim {r}^{d-1}$ | ${\nu}^{-1}=d-1=3$ |

Lowest order strong coupling (large G) expansion [47] | ${\nu}^{-1}=2$ |

Nonlocal field equations with $G(\square )$ for the static metric [56] | ${\nu}^{-1}=d-1$ for $d\ge 4$ |

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https://doi.org/10.3390/sym11010087