**5. Properties of virtual gravitons**

18 Quantum Gravity

The index "*a*" denotes the sum over particles and will be omitted in the following, considering for simplicity one single particle. The corresponding field/particle interaction

order in *p* and for fields *h* which describe a plane wave (on-shell or off-shell, see proof in

Suppose to apply eq. (36) to our case, i.e. to compute a transition probability +- due to the coupling of gravitation to single particles in ordinary matter. In this case, the particle momentum is a given numerical function of time, while (, ) *ij h t* **x** is a quantum operator which acts on the Fock vacuum creating or destroying a graviton. (In the following we shall

> 2 2 *p m*

system. This is also the magnitude order of the (*t*) term. But while the interaction Hamiltonian *H***x**, can have non-vanishing matrix elements also when acting linearly between the states + and -, because it is proportional to the non-linear function *g h* 1 Tr( ) ... , the Hamiltonian *H***x**,*particle* has non-vanishing matrix elements only to

where *Inz m* and *Outz m* denote the zero-mode components of the initial and final states, and 0 denotes the Fock vacuum, without gravitons. The matrix element is clearly zero, because it contains a single field acting between two Fock vacuum states. In other words, we can say that since neither in the initial state nor in the final state there are gravitons, the standard vertex (36) can have non-zero matrix element only when it is taken

in S.I. units, as seen, and gives a factor 10-76 after insertion in the transition probability (34).

elements already to first order in . We are not able to compute these matrix elements without a complete theory, because inside the Schwarzschild radius of the zero-modes the weak field expansion is not valid. The situation resembles that of early nuclear physics, where the nuclear matrix elements were largely unknown, apart from some general properties or magnitude orders; this did not prevent researchers from obtaining important data on the processes, based on the available information and on the crucial knowledge of

<sup>1</sup> (, ) <sup>2</sup> *<sup>i</sup> <sup>j</sup> <sup>H</sup> particle ij h t pp*

*<sup>m</sup>* **<sup>x</sup> <sup>x</sup>** (36)

*<sup>h</sup>* and omit the indices.) The numerical factor 2

ˆ ˆ 0, 0, *In h Out In h Out zm zm* (37)

*hhh* , … terms in the expansion of *H* can give non-zero matrix

<sup>2</sup> <sup>2</sup> *<sup>p</sup>* /*<sup>m</sup>* ; but this is of magnitude order 10-38

and *i*,*j* are spatial indices. This holds to lowest

, i.e. of the order of 10-19 J for an atomic

*<sup>i</sup> <sup>j</sup> p p m*

is of

,

Hamiltonian density is

(Modanese, 2011)).

second order.

On the other hand, the ˆ ˆ

the final states density.

where *m* is the particle mass, *ij ij ij h g*

often denote the field operator as ˆ

the order of the kinetic energy of the particle

Namely, we can write a matrix element of the form ˆ *In h Out* as

twice (Fig. 7) and is therefore proportional to

*hh* , ˆˆˆ

The aim of this final section is to give a simplified yet consistent physical picture of how virtual gravitons mediate the gravitational interactions. This is necessary in order to understand the link between virtual gravitons and the other kind of vacuum fluctuations studied in this paper, the zero-modes.

Note that virtual gravitons respect the usual time-energy uncertainty principle; their are not "long-lived" vacuum fluctuations like the zero-modes. This is because we consider gravitons as the particles obtained in the perturbative quantization of gravity on a flat background. It is known that the theory is not renormalizable at higher orders, but we use only tree diagrams in this work and suppose that the renormalization problem will be solved or is already solved in an effective quantum field theory of gravity (compare Sect. 1).

The concept of virtual particles mediating an interaction is not simple, and it is sometimes used improperly. In some treatments the virtual particles are seen as purely formal representations of perturbative diagrams. Instead, it is important to understand in which sense they can be regarded as particles or not.

For a real particle of given mass *m*, kinematics allows to connect the three quantities *E*, *p*, *v* through the two relations

$$E^2 - p^2c^2 = m^2c^4\tag{38}$$

$$E = \frac{mc^2}{\sqrt{1 - v^2 / c^2}}\tag{39}$$

Therefore when one of the tree quantities is known, we can find the other two. Note that from (38) and (39) one can prove the relation <sup>2</sup> *p* / / *E vc* , which connects *E* and *p* and (unlike (39)) also holds for *v*=*c*. So we can as well consider as basic relations between *E*, *p*, *v* the couple

Anomalous Gravitational Vacuum Fluctuations Which Act as Virtual Oscillating Dipoles 21

Suppose that this momentum is carried by one single virtual photon . The photon is off-

the order of 10-14 m. One can estimate, classically, that the minimum distance reached by the protons is of the order of 10-16 m. If the virtual photon is emitted at this point, its wavefunction can clearly not be regarded as a plane wave. Its propagation velocity *v* is hardly observable and relation (41) appears to suggest that *v* is very large; if we assume *v*=*c*,

The situation appears, in conclusion, to be very different from the previous example. It seems reasonable to draw a clear distinction between a scattering process, which can be described as the exchange of a single virtual particle, and the inter-particle force in static or quasi-static conditions, which is in general equivalent to the exchange of a large number of

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

/2 /2

*T T*

/2 /2

*T T Ur mm dt dt h t h t <sup>T</sup>*

1 2 1 2 00 1 1 00 2 2

**r r** (42)

1 2 /2 /2 ( ) <sup>3</sup>

=*c*=1). Finally we

(43)

**pr**

*T T i iE t t*

<sup>1</sup> ( ) lim 0 ( , ) ( , )0

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

> 1 2 1 2 2 2 /2 /2

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

*T T <sup>e</sup> U r Gm m dt dt dE d p <sup>T</sup> Epi*

gravitons have *E*0 (note that in eq.s (43) and (44) we use natural units *h/*2

<sup>2</sup>*c*<sup>2</sup> *m*

2=-*p*

/*p* 2/*c*2. The virtual photon energy

is exactly zero in this reference

, is of

2-*p*

system (it is not Lorentz-invariant). The wavelength of the photon, defined as =*h*/*p*

2*c*2=*E*

2<0: *m*

it is only by analogy with the familiar retarded classical effects.

discussed by (Clark, 2001)). The potential energy is written as

*T*

<sup>1</sup> ( ) lim

*T*

and momentum are exactly defined and their ratio *E*

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

shell, with imaginary mass *m*

virtual particles.

exchanged. One first finds

have

$$\mathbf{E}^2 - p^2 \mathbf{c}^2 = m^2 \mathbf{c}^4 \tag{40}$$

$$\frac{p}{E} = \frac{v}{c^2} \tag{41}$$

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 propagates with light speed looks reasonable.
