**3. Pair interactions of zero-modes**

8 Quantum Gravity

origin. We know that the system has zero-modes for which 0 *A* at the origin, and

done (Modanese, 2011), and implies in turn that (10) is true. One can also check that this is

The explicit calculation of the average (0) *rr <sup>X</sup> g* in a sector of the functional integral is conceptually important, but in practice it does not help much in giving a quantum representation of the zero-modes and their interactions. The properties of the zero-modes as "classical" metrics are more useful for that purpose. We shall suppose that each zero-mode

the gravitational Hamiltonian *H* in this context). The states |*i* are localized and mutually orthogonal. Different |*i* correspond to field configurations centered at different points. In the following we shall also suppose for simplicity that their Schwarzschild radii are always

According to this line of thought, the "true non-interacting ground state" of the gravitational vacuum is obtained in principle as the limit of an infinite incoherent

superposition of flat spacetime (Fock vacuum) plus single zero-mode wavefunctions:


This definition of the ground state is clearly difficult to put on a rigorous basis. We are mainly interested, however, into the *excitations* with respect to this ground state. The most relevant among these excitations are those resulting from pair interactions of zero-modes, as

Note that fixing *iHi* amounts to a much weaker statement than giving a gravitational quantum Hamiltonian operator *H*, because *iHi* is only a matrix element and a classical limit of the total energy for an asymptotically flat configuration (ADM energy (Murchada & York, 1974)). So whenever we write here the full gravitational Hamiltonian *H,* in fact we only exploit some properties of its matrix elements, like in a Heisenberg representation of quantum mechanics. This is consistent with our path integral approach to the full-

In other words, in the following we use neither the "full" gravitational Hamiltonian operator *H*, nor eigenvalue relations. (Interaction Hamiltonians on a background metric like that employed in Sect. 4 do not suffer from these limitations.) In fact, the Hamiltonian *H* is very difficult to define in quantum gravity. Even classically, there exists no generally accepted expression for the gravitational energy density. Furthermore, assuming the

*i* 

in the continuum limit.) We want to use this to compute the

, therefore in the continuum limit it gives the average of <sup>2</sup>

*<sup>m</sup>* for 0

*<sup>i</sup> iHi cM* (see below for the meaning of

*i* (15)

. This can indeed be

at the

*<sup>m</sup>* , where *m* is a fixed intermediate index. This is the average of the squared field

(Note that <sup>1</sup> 2 *jj j* 

at the point *s m*

average <sup>2</sup> 

therefore

we shall see.

interacting case.

2 

 

. So we would like to show that <sup>2</sup>

corresponds to a quantum state |*i* and that <sup>2</sup>

not an artefact of the continuum limit.

**2.3 Zero-modes as quantum states** 

much smaller than their distance.

We have introduced the concept of ground state in an effective theory of Quantum Gravity as given by the Fock vacuum plus a random superposition of zero-modes. In this Section we show that non-interacting zero-modes with equal mass are coupled in degenerate symmetric and anti-symmetric wavefunctions. The introduction of interaction removes the degeneration. The excited states form a continuum and the interaction of the vacuum with an external coherent oscillating source leads to transitions, with a probability which we shall compute in Sect. 4. As in Sect. 2, we denote with a capital *M* a zero-mode mass (virtual and negative).
