**3.1 Pairs in symmetric and antisymmetric states**

Consider a couple of states |1 and |2 with masses *M*1 and *M*<sup>2</sup> . We have

$$
\langle \mathbf{1} \| \mathbf{H} \| \mathbf{1} \rangle = \mathbf{c}^2 \mathbf{M}\_{1\prime} \quad \langle \mathbf{2} \| \mathbf{H} \| \mathbf{2} \rangle = \mathbf{c}^2 \mathbf{M}\_{2\prime} \quad \langle \mathbf{1} \| \mathbf{2} \rangle = 0 \tag{17}
$$

Putting now *M*1=*M*2=*M* and taking the interaction into account, the degenerate noninteracting levels are splitted. Define the symmetrical and anti-symmetrical superpositions and :

$$<\langle \boldsymbol{\nu}^{+} \rangle = \frac{1}{\sqrt{2}} (\lfloor 1 \rfloor + \lfloor 2 \rfloor) \quad \lfloor \boldsymbol{\nu}^{-} \rangle = \frac{1}{\sqrt{2}} (\lfloor 1 \rfloor - \lfloor 2 \rfloor) \tag{18}$$

Anomalous Gravitational Vacuum Fluctuations Which Act as Virtual Oscillating Dipoles 11

extension (coherence range) of the order of 10-9 m, and typical frequency 106-109 Hz. This fixes the maximum virtual mass involved, by eq. (20), to *M*10-13 kg. This is small, but definitely much larger than any atomic scale mass, and implies that also the gravitational interaction in the pairs of virtual masses is much larger than the usual gravitational

We are confronted here with a very unusual situation and we should check that our description is consistent, at least at the energy scale we are considering. (In principle the zero-mode fluctuations exist at any scale, but since they are an emergent phenomenon, computed in an effective theory, it is fair to concentrate on the scale which we deem most realistic.) First, one can easily check that the supposed localization of the zero-modes is well compatible with the Heisenberg position-momentum uncertainty principle. Second, one can prove that their interaction, though strong on the atomic scale, is much weaker than the interaction in a hypothetical gravitational bound state formed by two masses of this size. This can be easily checked, for instance, by computing the corresponding Bohr radius: this is

separated by a distance of the order of 10-9 m. So the acceleration of each zero-mode due to the presence of the other is very small, if compared to accelerations due to atomic or molecular forces. It follows that in these "weakly bound states of heavy quasi-particles" the

On a longer time scale, the interaction itself causes the zero-modes to fade out slowly as vacuum fluctuations. This is a subtle point that completes our analysis of the isolated zeromodes given in Sect. 2. As we have seen, the boundary term *<sup>M</sup> dt* in the action is constant for an isolated zero-mode, for any time interval, and therefore an isolated zero-mode will persist indefinitely in time. For interacting zero-modes the situation is more complicated, because

2. Their total ADM mass-energy is still constant, as long as radiation is negligible; this total mass-energy comprises their masses plus potential and kinetic energy. But when the emitted radiation becomes a sizeable fraction of the total mass, the ensuing change in the boundary term in the action of the zero-modes begins to cause a destructive

subsequent times. So the quantum amplitudes of these metrics tend to vanish and the result is that the zero-modes, as vacuum fluctuations, acquire a finite lifetime as they

occurs with the emission of an off-shell graviton with spin 1. This happens because the dominant graviton emission process in the decay of an excited zero-mode is oscillatingdipole emission. Quadrupolar emission, which is the only process ensuring conservation of energy, momentum and spin in the emission of on-shell gravitons, can in this case be disregarded. Since we are only interested into a lowest-order perturbative estimate (tree

and

to occur at

 *x t*, <sup>1</sup> , <sup>2</sup> *g x t*, 

. The decay of the excited state

… at

are

of the order of 2 3 19 / 10 m *Gm* , while the zero-modes in the states

distance *r* varies slowly and there is plenty of time for the transitions

1. The superposition of their metrics is not necessarily a zero-mode.

interference in the functional integral between the metrics *g*

frequency 106-109 Hz, as we shall describe in detail later.

begin to emit dipolar or quadrupolar radiation.

In this Section we compute the lifetime of an excited state

**3.2 Virtual dipole emission,** *A* **and** *B* **coefficients** 

interactions at atomic scale.

Fig. 3. Symmetric and antisymmetric bound states of zero-modes with equal mass *M*. (We assume that the wavefunction is much more localized near the masses than depicted – compared to their distance.)

The energy splitting *E* is given, as known, by

$$
\Delta E = E\_- - E\_+ = \langle \psi^- \mid H \mid \psi^- \rangle - \langle \psi^+ \mid H \mid \psi^+ \rangle = -2 \langle 1 \mid H \mid 2 \rangle \tag{19}
$$

Note that the matrix element 1| |2 *H* can be taken to be real without loss of generality. Suppose that 1| |2 *H* can be computed to a first approximation through its classical limit. The ADM energy integral at spatial infinity for the Schwarzschild-like field of two positive masses can be analytically continued to negative masses. We then obtain

$$
\Delta E = 2 \frac{GM^2}{r} \tag{20}
$$

being *r* the distance between the symmetry centers of the states |1 and |2 . This procedure reminds the computation of the bound states of two atoms in a molecule: the "internal states" of the atoms are not relevant and each atom is described by a single vector coordinate; the relevant Hamiltonian is the interaction Hamiltonian, although the full Hamiltonian of the system comprises in principle the forces inside the atoms and even inside the nuclei.

Let us consider the transitions between and . We shall see that they are mainly of two types: (a) excitation due to the interaction with a local -term dependent on time (variable vacuum energy density, associated with coherent matter - compare Sect. 4); (b) decay with emission of a virtual graviton. We look for a relation between the frequency of the transition and the virtual mass of the excited states. In the ground state, all couples with equal mass will be in their symmetric superposition state. Any transition of one couple from its symmetric to its antisymmetric state gives an excited state with energy (20). Since there exist zero-modes with any (negative) mass, at any distance, there is actually a continuum of excited states.

For the same energy, in principle, there are transitions to excited levels involving different masses at different distances, provided the ratio <sup>2</sup> *M r* / is the same. In practice, however, there is an upper limit on the scale *r*, because the time-variable -term has a typical spatial 10 Quantum Gravity

Fig. 3. Symmetric and antisymmetric bound states of zero-modes with equal mass *M*. (We assume that the wavefunction is much more localized near the masses than depicted –

 | | | | 2 1| |2 *EE E H H H* 

masses can be analytically continued to negative masses. We then obtain

 

Note that the matrix element 1| |2 *H* can be taken to be real without loss of generality. Suppose that 1| |2 *H* can be computed to a first approximation through its classical limit. The ADM energy integral at spatial infinity for the Schwarzschild-like field of two positive

> <sup>2</sup> *GM <sup>E</sup> r*

being *r* the distance between the symmetry centers of the states |1 and |2 . This procedure reminds the computation of the bound states of two atoms in a molecule: the "internal states" of the atoms are not relevant and each atom is described by a single vector coordinate; the relevant Hamiltonian is the interaction Hamiltonian, although the full Hamiltonian of the system comprises in principle the forces inside the atoms and even

time (variable vacuum energy density, associated with coherent matter - compare Sect. 4);

frequency of the transition and the virtual mass of the excited states. In the ground state, all couples with equal mass will be in their symmetric superposition state. Any transition of one couple from its symmetric to its antisymmetric state gives an excited state with energy (20). Since there exist zero-modes with any (negative) mass, at any distance, there is actually

For the same energy, in principle, there are transitions to excited levels involving different masses at different distances, provided the ratio <sup>2</sup> *M r* / is the same. In practice, however, there is an upper limit on the scale *r*, because the time-variable -term has a typical spatial

with emission of a virtual graviton. We look for a relation between the

 and 

2

 (19)

(20)

due to the interaction with a local -term dependent on

. We shall see that they are mainly of

compared to their distance.)

inside the nuclei.

(b) decay

two types: (a) excitation

Let us consider the transitions between

a continuum of excited states.

 The energy splitting *E* is given, as known, by

extension (coherence range) of the order of 10-9 m, and typical frequency 106-109 Hz. This fixes the maximum virtual mass involved, by eq. (20), to *M*10-13 kg. This is small, but definitely much larger than any atomic scale mass, and implies that also the gravitational interaction in the pairs of virtual masses is much larger than the usual gravitational interactions at atomic scale.

We are confronted here with a very unusual situation and we should check that our description is consistent, at least at the energy scale we are considering. (In principle the zero-mode fluctuations exist at any scale, but since they are an emergent phenomenon, computed in an effective theory, it is fair to concentrate on the scale which we deem most realistic.) First, one can easily check that the supposed localization of the zero-modes is well compatible with the Heisenberg position-momentum uncertainty principle. Second, one can prove that their interaction, though strong on the atomic scale, is much weaker than the interaction in a hypothetical gravitational bound state formed by two masses of this size. This can be easily checked, for instance, by computing the corresponding Bohr radius: this is of the order of 2 3 19 / 10 m *Gm* , while the zero-modes in the states and are separated by a distance of the order of 10-9 m. So the acceleration of each zero-mode due to the presence of the other is very small, if compared to accelerations due to atomic or molecular forces. It follows that in these "weakly bound states of heavy quasi-particles" the distance *r* varies slowly and there is plenty of time for the transitions to occur at frequency 106-109 Hz, as we shall describe in detail later.

On a longer time scale, the interaction itself causes the zero-modes to fade out slowly as vacuum fluctuations. This is a subtle point that completes our analysis of the isolated zeromodes given in Sect. 2. As we have seen, the boundary term *<sup>M</sup> dt* in the action is constant for an isolated zero-mode, for any time interval, and therefore an isolated zero-mode will persist indefinitely in time. For interacting zero-modes the situation is more complicated, because

