**2.3 Zero-modes as quantum states**

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 corresponds to a quantum state |*i* and that <sup>2</sup> *<sup>i</sup> iHi cM* (see below for the meaning of 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 much smaller than their distance.

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:

$$\left| \begin{array}{c} \mathbf{0} \end{array} \right\rangle = \left| \mathbf{0} \right\rangle\_{\text{Fock}} + \sum\_{i} \xi\_{i} \left| \begin{array}{c} \mathbf{i} \end{array} \right\rangle \tag{15}$$

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 we shall see.

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 fullinteracting case.

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 validity of eigenvalue operator relations would lead to contradictions. For instance, by applying the full Hamiltonian to the vacuum state (15) and supposing for a moment that <sup>2</sup> *H i Mc i <sup>i</sup>* , we would obtain, *only formally*

$$\left.''H\left|0\right> = \sum\_{i} \xi\_i M\_i c^2 \left|i\right>''\tag{16}$$

From this we would conclude that |0 is not an eigenstate! Nevertheless the property 0 00 *H* is true, considering that the coefficients *<sup>i</sup>* have random phases.

We could call the states |*i* "purely gravitational, long-lived virtual particles". They are long-lived in the following sense. The classical equation for isolated zero-modes gives configurations independent from time. Adding to the pure Einstein action the boundary Gibbs-Hawking-York term, the latter takes the form *S M dt GHY* , i.e. it is a constant for any fixed time interval, and does not cause interference in the path integral. However, when the zero-modes are not isolated but interact with each other, the boundary term causes their lifetime to be finite.

In the next section we shall discuss the simplest interaction of the zero-modes (pair interaction). This displays one of the typical amazing features of virtual particles (compare Sect. 5): they are created from the vacuum "for free", but after that they follow the usual dynamical rules. When computing the amplitude of a process involving virtual particles, we do not need to take into account the initial amplitude for creating the particles at a given point of space and time, but we do compute (Sect.s 3 and 4) the amplitudes for their ensuing propagation and interaction.
