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

In this Section we compute the lifetime of an excited state . The decay of the excited state 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

Anomalous Gravitational Vacuum Fluctuations Which Act as Virtual Oscillating Dipoles 13

1 1 ˆ ˆ | | 1| 2| 1 2 2 2

in the end are taken to be equal. The origin of the coordinate system is in the center of mass. This mass dipole moment has purely quantum origin, because in our system there are no masses of different signs, and it is known that in this case the classical mass dipole moment computed with respect to the center of mass is zero. We could say that the non-zero matrix element (23) is due to the quantum tunnelling between the states and |2 . This corresponds

values of *M* and *r* found in Section 3.1 supposing an excitation frequency of the order of 1

<sup>1</sup> <sup>3</sup> *B r* 

Real particles with negative mass cannot exist, because they would make the world terribly unstable, popping up spontaneously from the vacuum with production of energy. In this work, however, we hypothesize the existence of long-lived virtual particles with negative mass, whose creation from the vacuum does not require or generate any energy. We recognize that these virtual particles have negative mass by looking at their metric at infinity, which is Schwarzschild-like, but with negative *M* and negative ADM energy. We know that the dynamics of virtual particles, after their creation, is similar to that of real

We do not know any general principle about the "classical" dynamics of virtual particles with negative mass. Actually, virtual particles of this kind are an emergent phenomenon guessed from the path integral and can only be observed in a very indirect way. It is interesting, nonetheless, to make some reasonable hypothesis and check the consequences. Our basic assumption will be the following: for an isolated system comprising particles with

> *CM i i i*

11 22

of the excited level

(Modanese, 2011) and Sect. 5). The general dependence of *B* on the frequency

**3.3 Digression: Elementary dynamics of virtual particles with negative mass** 

particles, and we have computed quantum amplitudes involving them.

positive and negative mass, the position of the center of mass, defined by

length *r* of the dipoles is easily obtained from eq.s (20), (22) and (23):

MHz, one finds <sup>12</sup> *B* 10 m3/Js2 for the stimulated emission coefficient and <sup>1</sup>

**d d**

Note that *B* is independent from the Newton constant *G*.

the lifetime for spontaneous emission (taking

1 2

where 1

1 2 **r r** , 2

to a mass oscillation.

Eq. (22) gives the lifetime

 

1 1 1| 2| |1 |2 2 2

(23)

**rr r**

**r r** ; here **r** is the displacement between the masses *M*1 and *M*2, which

by spontaneous emission. With the

*f* 1 m/s: compare discussion in

(24)

**r r** *<sup>M</sup>* (25)

*A* 1 s for

and on the

*MM M*

diagrams) we can use the linearized Einstein theory in the form of the "Maxwell-Einstein" equations

$$\begin{aligned} \nabla \cdot \mathbf{E}\_G &= -4 \,\pi \mathbf{G} \,\rho\_m \\ \nabla \times \mathbf{E}\_G &= -\frac{\partial \mathbf{B}\_G}{\partial t} \\ \nabla \cdot \mathbf{B}\_G &= 0 \\ \nabla \times \mathbf{B}\_G &= -\frac{4 \pi \mathbf{G}}{c^2} \mathbf{j}\_m + \frac{1}{c^2} \frac{\partial \mathbf{E}\_G}{\partial t} \end{aligned} \tag{21}$$

Here **E***G* is the gravito-electric (Newtonian) field, **B***G* is the gravito-magnetic field, and **j***<sup>m</sup>* , *<sup>m</sup>* are the mass-energy current and density. The elementary quantization of the field modes in a finite volume *V* follows the familiar scheme used for the computation of spontaneous and stimulated electromagnetic emission of atoms in a cavity. We have discussed in (Modanese, 2011) the conditions for applicability of the Einstein-Maxwell equations to plane waves in vacuum.

The Einstein *A*-coefficient of spontaneous emission turns out to be related to the *B*coefficient and to the mass dipole moment by the relation

$$A = \left( B \frac{8\pi\hbar}{\lambda^3} \right) = \left( \frac{G}{\hbar^2} \left| \left< \hat{\mathbf{d}} \right> \right|^2 \frac{8\pi\hbar}{\lambda^3} \right) \tag{22}$$

where the electromagnetic coupling constants have been replaced, up to an irrelevant adimensional factor, by the gravitational constants, according to eq.s (21). The operator ˆ **d** is the mass-dipole moment and the matrix element is taken between the initial and final state of interest.

Fig. 4. Emission of a virtual graviton with spin 1 in the spontaneous decay . The matrix element of the mass-dipole moment operator between and has module *Mr*/2.

It is straightforward to check that there is an oscillating mass dipole between the states and :

12 Quantum Gravity

diagrams) we can use the linearized Einstein theory in the form of the "Maxwell-Einstein"

*G*

*t*

*G*

· 4

 

**E**

*G*

*G m*

**<sup>B</sup> <sup>E</sup>**

 

*G m*

· 0

*G*

**B**

2 2

*c c t*

*G*

(21)

**d** is

2

 

**d** (22)

 and 

. The

has module *Mr*/2.

 

32 3

 

where the electromagnetic coupling constants have been replaced, up to an irrelevant adimensional factor, by the gravitational constants, according to eq.s (21). The operator ˆ

the mass-dipole moment and the matrix element is taken between the initial and final state

4 1

*G*

Here **E***G* is the gravito-electric (Newtonian) field, **B***G* is the gravito-magnetic field, and **j***<sup>m</sup>* ,

 *<sup>m</sup>* are the mass-energy current and density. The elementary quantization of the field modes in a finite volume *V* follows the familiar scheme used for the computation of spontaneous and stimulated electromagnetic emission of atoms in a cavity. We have discussed in (Modanese, 2011) the conditions for applicability of the Einstein-Maxwell

The Einstein *A*-coefficient of spontaneous emission turns out to be related to the *B*-

8 8 <sup>ˆ</sup> | | *<sup>G</sup> A B* 

Fig. 4. Emission of a virtual graviton with spin 1 in the spontaneous decay

It is straightforward to check that there is an oscillating mass dipole between the states

matrix element of the mass-dipole moment operator between

**<sup>E</sup> B j**

equations

of interest.

and :

equations to plane waves in vacuum.

coefficient and to the mass dipole moment by the relation

$$\begin{aligned} \langle \boldsymbol{\psi}^{+} | \hat{\mathbf{d}} | \boldsymbol{\psi}^{-} \rangle &= \frac{1}{\sqrt{2}} (\langle 1 | + \langle 2 |) \hat{\mathbf{d}} \frac{1}{\sqrt{2}} (| 1 \rangle - | 2 \rangle) = \\ \mathbf{f} &= \frac{1}{2} (\langle 1 | + \langle 2 |) (M\_1 \mathbf{r}\_1 | 1 \rangle - M\_2 \mathbf{r}\_2 | 2 \rangle) = \frac{1}{2} M \mathbf{r} \end{aligned} \tag{23}$$

where 1 1 2 **r r** , 2 1 2 **r r** ; here **r** is the displacement between the masses *M*1 and *M*2, which

in the end are taken to be equal. The origin of the coordinate system is in the center of mass.

This mass dipole moment has purely quantum origin, because in our system there are no masses of different signs, and it is known that in this case the classical mass dipole moment computed with respect to the center of mass is zero. We could say that the non-zero matrix element (23) is due to the quantum tunnelling between the states and |2 . This corresponds to a mass oscillation.

Eq. (22) gives the lifetime of the excited level by spontaneous emission. With the values of *M* and *r* found in Section 3.1 supposing an excitation frequency of the order of 1 MHz, one finds <sup>12</sup> *B* 10 m3/Js2 for the stimulated emission coefficient and <sup>1</sup> *A* 1 s for the lifetime for spontaneous emission (taking *f* 1 m/s: compare discussion in (Modanese, 2011) and Sect. 5). The general dependence of *B* on the frequency and on the length *r* of the dipoles is easily obtained from eq.s (20), (22) and (23):

$$B \approx \frac{1}{\hbar} \alpha r^3 \tag{24}$$

Note that *B* is independent from the Newton constant *G*.
