**7. References**

24 Quantum Gravity

Supposing that the real particle is a proton, it is easy to check that conservation requires *v*p 1 m/s, 10-7 m. If the distance between the real particle and the graviton source is much larger than , then the wavefunction of the virtual graviton can be properly described as a plane wave. If it is legitimate to apply the kinematical relations of Sect. 5.1 to this plane wave, it follows that the virtual graviton propagates like a tachyon (Recami et al., 2000). This does not violate the causal principles of special relativity, because the propagation of a single virtual particle cannot be modulated to obtain a signal. The existence of such tachyonic virtual gravitons would be a consequence of the unique features of their source

In quantum gravity the vacuum fluctuations have a more complex structure than in other field theories with positive-definite action. In particular, there are vacuum fluctuations which in the non-interacting approximation have infinite lifetime, and seen from the outside appear as Schwarzschild metrics with negative mass. These vacuum fluctuations behave as pseudo-particles which are created "for free" from the vacuum at any point in spacetime. The non-interacting vacuum can in fact be described as an incoherent, homogeneous and isotropic superposition of a Fock vacuum plus infinite states of this

When the interaction is taken into account, one finds that each pair of zero-modes with equal virtual mass *M* and distance *r* can be in two states, denoted by + and -, with energy

virtual off-shell graviton with spin 1. The energy-momentum ratio *E*/*p* of the virtual graviton can take in principle any value, being the total momentum preserved by the recoil of the zero-modes pair. The *A* and *B* Einstein coefficients of spontaneous and stimulated emission have been computed in weak-field approximation. The *B* coefficient turns out to be of the order of *r*2/2*h*, where is the frequency corresponding to the gap *E*. The *A*

The excitation process + - cannot occur by interaction with single incoherent particles, because the relative amplitude is exceedingly small, involving a double elementary particle/graviton vertex. Instead, a sizeable excitation amplitude is obtained in the interaction with an external source of the form *dxg*(*t*) (local vacuum energy density term, due to the presence of condensed matter in a coherent state). By taking into account the density of final states one finds, for a length scale of the -term of the order of 10-9 m, an

The virtual gravitons emitted in the decay - + are very different from those exchanged in the usual gravitational interactions. Consider, for instance, a nucleon in free fall near the surface of the Earth. If it was initially at rest, it reaches a velocity of 1 m/s in approximately 0.1 s, absorbing 1014 virtual gravitons of very low frequency and large wavelength. For comparison, a single virtual graviton of frequency 107 Hz emitted in a vacuum decay - + can transfer the same momentum to the nucleon in a single quick

coefficient depends on the wavelength; for 1 m/s one has *A* 1 s-1.

excitation time + - of the order of 10-23 s.

*-E*+=*GM*2/*r*. The excited state - can decay into the state + by emitting a

(virtual decay

**6. Conclusions** 

kind ("zero-modes").

splitting *E*=*E-*

absorption process.

).


**1. Introduction**

invariance.

In Ref. [1], Weinberg suggested that the general theory of relativity may have a non-trivial UV fixed point, with a finite dimensional critical surface in the UV limit, so that it would be asymptotically safe with an S-matrix that depends on only a finite number of observable parameters. In Refs. [2–4], strong evidence has been calculated using Wilsonian [5–8] field-space exact renormalization group methods to support asymptotic safety for the Einstein-Hilbert theory. We have shown in Refs. [9–19] that the extension of the amplitude-based, exact resummation theory of Ref. [20] to the Einstein-Hilbert theory (we call the extension resummed quantum gravity) leads to UV fixed-point behavior for the dimensionless gravitational and cosmological constants, but with the bonus that the resummed theory is actually UV finite. More evidence for asymptotic safety for quantum gravity has been calculated using causal dynamical triangulated lattice methods in Ref. [21]1. There is no known inconsistency between our analysis and Refs. [2–4, 21]. Our results are also consistent with the results on leg renormalizability of quantum gravity in Refs. [23, 24].

**Planck Scale Cosmology and Asymptotic Safety** 

**in Resummed Quantum Gravity: An Estimate of**/

**2**

B.F.L. Ward\* *Dept. of Physics,* 

> *1USA 2Switzerland*

*1Baylor University, Waco,* 

*2TH Physics Unit, CERN, Geneva* 

The reader unfamiliar with the methods of Wilson in the context of the renormalization group may consult Refs. [2, 5–8] for the details of the approach. Here we stress that in the Wilsonian formulation of the renormalization group, it does not matter whether the theory under study is actually renormalizable because the idea is to thin the degrees of freedom to those relevant to the momentum scale *k* under study. When one does this, the operators in the theory then fall into the classes of relevant, marginal and irrelevant operators as one studies the response of the theory to changes in the value of *k*. If the theory is renormalizable, then as *k* → ∞ there will be a finite number of relevant or marginal operators in the effective action, yielding an S-matrix that depends on only a finite number of parameters. If the theory is non-renormalizable, there will be an infinite number of relevant or marginal operators in the effective action as *k* → ∞. It was for this reason that the authors

<sup>1</sup> We also note that the model in Ref. [22] realizes many aspects of the effective field theory implied by the anomalous dimension of 2 at the UV-fixed point but it does so at the expense of violating Lorentz

\*Work supported in part by NATO grant PST.CLG.980342.

Wetterich, C. (1998). Effective Nonlocal Euclidean Gravity, *General Relativity and Gravitation*, Vol.30, pp. 159-172
