**4. Interaction of the zero-modes with a variable -term**

14 Quantum Gravity

is invariant in time. From this assumption one can prove in a straightforward way several strange properties of particles with negative mass. These properties can be summarized by saying that in the usual dynamical rules their mass really behaves like a negative number, namely: (a) The acceleration of the virtual particle is opposite to the applied force. (b) The momentum is opposite to the velocity. (c) The kinetic energy is negative. The kinetic energy is defined as usual through the work of the applied force, in such a way that the sum

Applying these rules one obtains a bizarre behaviour in the scattering processes and in the dynamics. For instance, although the gravitational potential energy of two virtual particles with negative mass is negative, *E*pot =-*GM*1*M*2/*r* (compare Sect. 3.1), the two particles experience a repulsion, due to Property (a). They tend to run away from each other; while their distance increases, their *E*pot decreases in absolute value, and their (negative) *E*kin increases in absolute value. If the particles were initially at rest at some distance *r*0 (Fig. 5),

when their distance goes to infinity they gain a *E*kin equal to their initial *E*pot.

Fig. 5. "Classical" motion of two virtual particles with negative mass initially at rest at distance *r*0. Although their potential energy is negative, they feel a repulsion and their (negative) kinetic energy increases in absolute value as their distance goes to infinity.

recoil of the zero-modes (in the same direction of the emission). The conservation equations

2 0 *r g r g MvE E Mv p*

where *E* is the energy gap, *E*g and *p*g are the graviton energy and momentum, *v*r is the recoil velocity of the zero-mode and 2*M*-10-13 kg is the zero-mode mass. After replacing

<sup>1</sup> <sup>0</sup> *EE E g g <sup>M</sup>*

2

2 2

(Sect. 3.2) the momentum of the emitted graviton is balanced by the

(27)

(26)

*E*kin+*E*pot is conserved.

In the decay

give

 *p*g=*E*g, the system (26) leads to the equation

In Sect. 3 we have computed the probability of the decay process with emission of a virtual graviton. The excitation process (transition of a zero-modes pair from a symmetric to an anti-symmetric state) can occur by absorption of a virtual graviton or by coupling to an external source. It is easy to show (Sect. 4.3) that the coupling of zero-modes

to "ordinary" matter with energy-momentum *dx dx T m d d* is exceedingly weak.

(Note that certain interactions between zero-modes and massive particles vanish exactly for symmetry reasons. For instance, a particle in uniform motion can never "lose energy in collisions with the zero-modes", because in its rest reference system the particle will see the vacuum, zero-modes included, as homogeneous and isotropic. There are possible exceptions to this argument: accelerated particles, or particles in states with large *p* uncertainty.)

The coupling to a (*t*) term, or local time-dependent vacuum energy density, can lead to a significant transition probability. This is due to the presence of the non-linear *g* factor in the coupling, and corresponds physically to the fact that such a term does not describe isolated particles, but coherent, delocalized matter.
