**5.2 Photons or gravitons vs. static force**

20 Quantum Gravity

2 *<sup>p</sup> <sup>v</sup> E c*

propagates with light speed looks reasonable.

momentum is of the order of <sup>20</sup> 2 10 *m Ep*

from the center of mass reference system.

**5.1 Example: Scattering process** 

These formulas all hold when the quantities *m*, *E*, *p*, *v* are well defined, thus for particles which are either stable or have a sufficiently long lifetime. For virtual particles the situation is more vague and one finds a range of statements in the literature. For instance, there is a simple textbook argument showing that the exchange of virtual photons gives rise to a 1/*r*<sup>2</sup> force between two charges *q*1 and *q*2. The argument is based on the time-energy uncertainty relation. One writes *Eth*, where *E* is the energy of the exchanged virtual photon and *t* its lifetime. Supposing that the virtual particle travels with light speed, its range is *r*=*ct*. Therefore if the charges *q*1 and *q*2 are at a distance *r*, the "exchanged energy" is *E*1/*r* and the corresponding force will be proportional to 1/*r*2. One must add the assumption that the number of exchanged photons is also proportional to the product *q*1*q*2 of the charges of the interacting particles. A weak point in this argument is the identification of the exchanged energy with the potential energy of the interaction. In fact, the exchanged energy depends on the velocities of the charged particles and can even be zero for static sources or in cases like that of the protons observed in their center of mass system (Fig. 8, Sect. 5.1). Apart from this, the assumption that the virtual particle has an energy uncertainty and that it

Let us consider, however, another simple example: the electromagnetic scattering of two protons with the exchange of a single virtual photon. To fix the ideas, we choose a definite energy of the two protons as seen in their center of mass system, for instance *E*=10-13 J 1 MeV. (Magnitude orders are important in these considerations, in order to estimate the wavelength and the number of the exchanged particles, as we shall see below in the case of gravitons.) In this reference system the exchanged energy is zero and the exchanged

Fig. 8. Proton-proton scattering through the exchange of a single virtual photon, as seen

kg m/s (non-relativistic approximation).

2 22 24 *E p c mc* (40)

(41)

Let us now consider a different situation (Fig. 10): a massive particle (for instance, a proton) in free fall in the gravitational field of the Earth. Suppose the particle is initially at rest. There is an exact quantum formula which allows to find the static interaction potential energy in field theories like QED, QCD etc. The generalization to quantum gravity was given by (Modanese, 1995). In this formula the graviton propagator appears explicitly, as well as the *G* constant and the masses *m*1*m*2 of the sources (showing that the amplitude of virtual gravitons generation is proportional to both these masses; this property was also discussed by (Clark, 2001)). The potential energy is written as

$$\mathcal{L}I(r) = m\_1 m\_2 \lim\_{T \to \infty} \frac{1}{T} \int\_{-T/2}^{T/2} dt\_1 \int\_{-T/2}^{T/2} dt\_2 \left\{ 0 \left| h\_{00}(t\_1, \mathbf{r}\_1) h\_{00}(t\_2, \mathbf{r}\_2) \right| 0 \right\} \tag{42}$$

This equation describes the exchange of gravitons, for an ideally infinite time, between two static masses ( 1 2 **rr r** ; see Fig. 9). In our case the masses are the Earth and the particle.

The gravitons flux is proportional to both *m*1 and *m*2 and the propagator gives the amplitude of the propagation of virtual gravitons from **r**1 to **r**2, but note that their emission and absorption probabilities are equal to 1. If we expand the Feynman propagator in four-momentum space, we can see which energies and momenta are exchanged. One first finds

$$\mathcal{KL}(r) = \text{G}m\_1m\_2 \lim\_{T \to \infty} \frac{1}{T} \int\_{-T/2}^{T/2} dt\_1 \int\_{-T/2}^{T/2} dt\_2 \int d\mathcal{E} \int d^3 p \frac{e^{i\mathbf{p}\mathbf{r} - i\mathcal{E}(t\_1 - t\_2)}}{E^2 - \vec{p}^2 - i\varepsilon} \tag{43}$$

Changing variables to (*t*1+*t*2), (*t*1-*t*2) we find that the integral in (*t*1+*t*2) cancels the factor 1/*T*. By integrating exp[-*iE*(*t*1-*t*2)] one obtains (*E*): this selects the static limit, i.e. the exchanged gravitons have *E*0 (note that in eq.s (43) and (44) we use natural units *h/*2=*c*=1). Finally we have

Anomalous Gravitational Vacuum Fluctuations Which Act as Virtual Oscillating Dipoles 23

the force can only be understood if we consider the details of the wavefunction (Baez, 1995).

Fig. 10. The wavefunction of virtual gravitons exchanged in a quasi-static interaction is not a

Let us estimate how many gravitons are exchanges between the Earth and a free-falling proton or nucleon. To stay close to the static limit, consider a short interval *t*. The proton, initially at rest, acquires in this time a momentum *p*=*mgt*. The absorbed gravitons have, on the average, individual momentum <sup>41</sup> *p r* / 10 kg m/s. The number of absorbed

We have seen that the virtual gravitons exchanged in a quasi-static attractive gravitational interaction have very small energy and momentum. Their wavefunctions do not resemble plane waves. The propagation of this "stream" composed of a large number of virtual

features. Their energy is much larger (10-27 J). Their momentum is not fixed by the emission process, since the recoil of the emitting zero-modes can balance it in any case. One of these gravitons can be individually absorbed by a real particle at rest (for instance a proton), in such a way to conserve energy and momentum, provided the product *f* of the graviton frequency and wavelength is equal to half the final velocity of the real particle. In fact the

1 <sup>2</sup> 1 2

*<sup>p</sup> g p*

*mv hf E E <sup>v</sup> <sup>h</sup> p p mv*

*gp p*

where the suffix "*g*" denotes the virtual graviton and "*p*" the real particle. This is a quantum process that satisfies the conservation balance, thus it can happen and will in fact happen, with a certain amplitude. The amplitude for the final step (absorption by the real particle) is

2

*p*

(45)

(Sect. 3.2) have completely different

plane wave: for most of them, the wavelength is larger than the traveled distance.

A naïve particle exchange can only explain repulsive forces.

gravitons is then of the order of *N*/*t* 1015 s-1.

gravitons is a collective phenomenon occurring at light velocity.

unitary, by analogy with the similar process in the static exchange.

**5.3 Virtual gravitons emitted in the decay** 

The virtual gravitons emitted in the decay

balance equations are

$$ML(r) = -Gm\_1m\_2 \int d^3 p \frac{e^{i\mathbf{p}\cdot\mathbf{r}}}{p^2} = -\frac{2Gm\_1m\_2}{\pi r} \int\_0 dp' \frac{\sin p'}{p'} \tag{44}$$

with *p*' **p** *r* . (A similar reasoning also applies to quantum electrodynamics.) The last integral is equal to / 2 and the main contribution to the integration comes from the momentum region *p*' , i.e. *pr* . This means that in the classical interaction of two masses at distance *r,* the majority of the exchanged virtual gravitons have momentum *p* /*r* (restoring ), or wavelength *r* .

Fig. 9. Static potential energy of two masses *m*1, *m*2 as the outcome of graviton exchange. Virtual gravitons are emitted and absorbed at all possible times *t*1, *t*2; the final result is obtained by integration over *t*1 and *t*2.

The propagation velocity is not the same for all virtual gravitons, as is seen from the fact that their emission/absorption times vary from - to +; correspondingly, their invariant masses also vary. Being a static formula, eq. (42) cannot show that the propagation velocity of the force is *c*. For this we need some generalization to moving sources; the formalism of Quantum Field Theory will ensure that the retardation effects are accounted for.

The condition *p* /*r* or *r* shows that the wavefunctions of the exchanged virtual gravitons are very different from plane waves: these functions do not even make a complete oscillation over a distance equal to the Earth radius! Such virtual gravitons can hardly be regarded as "particles". This should actually be expected, because the attractive character of 22 Quantum Gravity

<sup>2</sup> sin ' ( ) ' '

with *p*' **p** *r* . (A similar reasoning also applies to quantum electrodynamics.) The last

masses at distance *r,* the majority of the exchanged virtual gravitons have momentum

Fig. 9. Static potential energy of two masses *m*1, *m*2 as the outcome of graviton exchange. Virtual gravitons are emitted and absorbed at all possible times *t*1, *t*2; the final result is

Quantum Field Theory will ensure that the retardation effects are accounted for.

The propagation velocity is not the same for all virtual gravitons, as is seen from the fact that their emission/absorption times vary from - to +; correspondingly, their invariant masses also vary. Being a static formula, eq. (42) cannot show that the propagation velocity of the force is *c*. For this we need some generalization to moving sources; the formalism of

gravitons are very different from plane waves: these functions do not even make a complete oscillation over a distance equal to the Earth radius! Such virtual gravitons can hardly be regarded as "particles". This should actually be expected, because the attractive character of

*r* shows that the wavefunctions of the exchanged virtual

*i e Gm m <sup>p</sup> U r Gm m d p dp p*

**pr**

1 2 2

*r* .

integral is equal to / 2

momentum region *p*'

*p* /*r* (restoring ), or wavelength

obtained by integration over *t*1 and *t*2.

The condition *p* /*r* or

, i.e. *pr*

3 1 2

0

and the main contribution to the integration comes from the

. This means that in the classical interaction of two

*r p* 

(44)

the force can only be understood if we consider the details of the wavefunction (Baez, 1995). A naïve particle exchange can only explain repulsive forces.

Fig. 10. The wavefunction of virtual gravitons exchanged in a quasi-static interaction is not a plane wave: for most of them, the wavelength is larger than the traveled distance.

Let us estimate how many gravitons are exchanges between the Earth and a free-falling proton or nucleon. To stay close to the static limit, consider a short interval *t*. The proton, initially at rest, acquires in this time a momentum *p*=*mgt*. The absorbed gravitons have, on the average, individual momentum <sup>41</sup> *p r* / 10 kg m/s. The number of absorbed gravitons is then of the order of *N*/*t* 1015 s-1.

### **5.3 Virtual gravitons emitted in the decay**

We have seen that the virtual gravitons exchanged in a quasi-static attractive gravitational interaction have very small energy and momentum. Their wavefunctions do not resemble plane waves. The propagation of this "stream" composed of a large number of virtual gravitons is a collective phenomenon occurring at light velocity.

The virtual gravitons emitted in the decay (Sect. 3.2) have completely different features. Their energy is much larger (10-27 J). Their momentum is not fixed by the emission process, since the recoil of the emitting zero-modes can balance it in any case. One of these gravitons can be individually absorbed by a real particle at rest (for instance a proton), in such a way to conserve energy and momentum, provided the product *f* of the graviton frequency and wavelength is equal to half the final velocity of the real particle. In fact the balance equations are

$$\frac{\frac{\text{hf}}{\text{h}}}{\frac{\text{h}}{\text{A}}} = \frac{E\_{\text{g}}}{p\_{\text{g}}} = \frac{E\_{p}}{p\_{p}} = \frac{\frac{1}{2}mv\_{p}^{2}}{mv\_{p}} = \frac{1}{2}v\_{p} \tag{45}$$

where the suffix "*g*" denotes the virtual graviton and "*p*" the real particle. This is a quantum process that satisfies the conservation balance, thus it can happen and will in fact happen, with a certain amplitude. The amplitude for the final step (absorption by the real particle) is unitary, by analogy with the similar process in the static exchange.

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**7. References** 

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 (virtual decay ).
