**4.3 Comparison with the effect of incoherent matter**

The action of free incoherent particles is

$$S = \sum\_{a} m\_{a} \int \sqrt{g\_{\mu\nu}(\mathbf{x}\_{a}) d\mathbf{x}\_{a}^{\mu} d\mathbf{x}\_{a}^{\nu}} \tag{35}$$

Anomalous Gravitational Vacuum Fluctuations Which Act as Virtual Oscillating Dipoles 19

Fig. 7. Zero-mode excitation by double interaction with incoherent matter. A single massive particle can cause such an excitation by emitting and re-absorbing a virtual graviton, but the

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

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

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

*mc <sup>E</sup>*

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*

2 2 2 1 /

*v c*

2 22 24 *E p c mc* (38)

(39)

probability of this process is very small.

**5. Properties of virtual gravitons** 

studied in this paper, the zero-modes.

sense they can be regarded as particles or not.

through the two relations

the couple

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 Hamiltonian density is

$$H\_{\mathbf{x},particle} = \frac{1}{2m} h\_{ij}(t, \mathbf{x}) p^i p^j \tag{36}$$

where *m* is the particle mass, *ij ij ij h g* and *i*,*j* are spatial indices. This holds to lowest order in *p* and for fields *h* which describe a plane wave (on-shell or off-shell, see proof in (Modanese, 2011)).

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 often denote the field operator as ˆ *<sup>h</sup>* and omit the indices.) The numerical factor 2 *<sup>i</sup> <sup>j</sup> p p m* is of the order of the kinetic energy of the particle 2 2 *p m* , i.e. of the order of 10-19 J for an atomic 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

second order.

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

$$\left\langle \text{In} \middle| \hat{h} \middle| \text{Out} \right\rangle = \left\langle \text{O}\_{\prime} \text{In}\_{z-m} \middle| \hat{h} \middle| \text{O}\_{\prime} \text{Out}\_{z-m} \right\rangle \tag{37}$$

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 twice (Fig. 7) and is therefore proportional to <sup>2</sup> <sup>2</sup> *<sup>p</sup>* /*<sup>m</sup>* ; but this is of magnitude order 10-38 in S.I. units, as seen, and gives a factor 10-76 after insertion in the transition probability (34).

On the other hand, the ˆ ˆ *hh* , ˆˆˆ *hhh* , … terms in the expansion of *H* can give non-zero matrix 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 the final states density.

Fig. 7. Zero-mode excitation by double interaction with incoherent matter. A single massive particle can cause such an excitation by emitting and re-absorbing a virtual graviton, but the probability of this process is very small.
