**6. Conclusions**

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 kind ("zero-modes").

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 splitting *E*=*E- -E*+=*GM*2/*r*. The excited state - can decay into the state + by emitting a 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* coefficient depends on the wavelength; for 1 m/s one has *A* 1 s-1.

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

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 absorption process.
