**2. Isolated zero-modes: Non trivial static metrics with null action**

Our starting point is a very general property of Einstein gravity: it has a non-positivedefinite action density. As a consequence, some non trivial static field configurations (metrics) exist, which have zero action. We call these configurations zero-modes of the

action. The Einstein action is 4 4 8 *<sup>E</sup> <sup>c</sup> S d x gR <sup>G</sup>* (plus boundary term; see Sect. 3) and the zero-mode condition is

$$\int d^4x \sqrt{\mathcal{g}} \mathcal{R} = 0 \tag{1}$$

This condition is, of course, satisfied by any metric with *R*(*x*)=0 everywhere (vacuum solutions of the Einstein equations (28), like for instance gravitational waves). But since the density *gR* is not positive-definite, the condition can also be satisfied by metrics which do not have *R*(*x*)=0 everywhere, but regions of positive and negative scalar curvature. The nonpositivity of the Einstein action has been studied by Hawking, Greensite, Mazur and Mottola and others (Greensite, 1992; Mazur & Mottola, 1990). Wetterich later found that also the effective action is always un-defined in sign (Wetterich, 1998).

We are interested into these zero-action configurations because, in the Feynman path integral, field configurations with the same action tend to interfere constructively and so to give a contribution to the integral distinct from the usual classical contribution of the configurations near the stationary point of the action. Let us write the Feynman path integral on the metrics ( ) *g x* as

$$I = \int d[\,\mathbf{g}] \exp\left(\frac{i}{\hbar} S\_E[\,\mathbf{g}]\right) \tag{2}$$

Suppose there is a subspace *X* of metrics with constant action. The contribution to the integral from this subspace is simply

2 Quantum Gravity

The outline of the work is the following. In Section 2 we show the existence of the zeromodes and discuss their main features, using their classical equation and the path integral. This Section contains some definitely mathematical parts, but we have made an effort to translate all the concepts in physical terms along the way. Section 3 is about the pair interactions of zero-modes: symmetric and antisymmetric states, transitions between these states, virtual dipole emission and its *A* and *B* coefficients. Section 3.3 contains a digression on the elementary dynamics of virtual particles with negative mass. Section 4 is devoted to the interaction of the zero-modes with a time-variable -term. We discuss in detail the motivations behind the introduction of such a term and compare its effect to that of "regular" incoherent matter by evaluating their respective transition rates. Finally, in Section 5 we discuss in a simplified way the properties of virtual gravitons; the virtual gravitons exchanged in a quasi-static interaction are compared to virtual particles exchanged in a scattering process and to virtual gravitons emitted in the decay of an excited

**2. Isolated zero-modes: Non trivial static metrics with null action** 

4

*<sup>c</sup> S d x gR* 

8 *<sup>E</sup>*

the effective action is always un-defined in sign (Wetterich, 1998).

as

Our starting point is a very general property of Einstein gravity: it has a non-positivedefinite action density. As a consequence, some non trivial static field configurations (metrics) exist, which have zero action. We call these configurations zero-modes of the

This condition is, of course, satisfied by any metric with *R*(*x*)=0 everywhere (vacuum solutions of the Einstein equations (28), like for instance gravitational waves). But since the density *gR* is not positive-definite, the condition can also be satisfied by metrics which do not have *R*(*x*)=0 everywhere, but regions of positive and negative scalar curvature. The nonpositivity of the Einstein action has been studied by Hawking, Greensite, Mazur and Mottola and others (Greensite, 1992; Mazur & Mottola, 1990). Wetterich later found that also

We are interested into these zero-action configurations because, in the Feynman path integral, field configurations with the same action tend to interfere constructively and so to give a contribution to the integral distinct from the usual classical contribution of the configurations near the stationary point of the action. Let us write the Feynman path

> exp *<sup>E</sup> <sup>i</sup> I d <sup>g</sup> <sup>S</sup> <sup>g</sup>*

Suppose there is a subspace *X* of metrics with constant action. The contribution to the

*<sup>G</sup>* (plus boundary term; see Sect. 3) and the

<sup>4</sup> *d x gR* <sup>0</sup> (1)

(2)

4

zero-mode.

action. The Einstein action is

integral on the metrics ( ) *g x*

integral from this subspace is simply

zero-mode condition is

$$I\_X = \exp\left(\frac{i}{\hbar}\hat{S}\_E\right)\Big|\_X d\left[\lg\right] = \exp\left(\frac{i}{\hbar}\hat{S}\_E\right)\mu(X) \tag{3}$$

where ˆ *SE* is the constant value of the action in the subspace and *X* its measure. The case <sup>ˆ</sup> <sup>0</sup> *SE* is a special case of this.

$$\left. \left. \left. \varepsilon\_{\mathsf{E}} \left[ \mathcal{g} \right] \right| \right|\_{\mathsf{X}} = \widehat{\left. \left. \widehat{\mathsf{S}} \right|}\_{\mathsf{E}}$$

Fig. 1. Subspace *X* of metrics with constant action. All the metrics (spacetime configurations) in *X* have the same action ˆ *SE* . In particular, there exist a subspace whose metrics all have zero action.

The zero-modes can only give a significant contribution to the path integral if they are not isolated configurations (like a line in 2D, which has measure zero), but a whole fulldimensional subset of all the possible configurations. They are "classical" fields, not in the sense of being solutions of the Einstein equations in vacuum, but in the sense of being functions of spacetime coordinates which are weighed in the functional integral with nonvanishing measure.
