**2.2 Zero-modes in the explicit functional integral**

The zero-modes equation (plus the argument of non-interference) tell us that relevant runaway configurations of vacuum exist, in which the metric is locally very different from its classical value .We shall now consider an explicit path integral of Einstein gravitation, in order to evaluate the functional average of certain metric components and confirm this supposition.

Let us choose a spherical coordinate system. We integrate only over a sector *X* of the functional space, namely over the spherically-symmetric metric configurations with constant *g*00. If we obtain a null quadratic vacuum average in *X*, namely

$$\left\{\mathbf{g}\_{rr}(0)\right\}\_{\mathbf{X}} = \frac{\int d[\mathbf{g}] \exp\left\{\frac{i}{\hbar} S[\mathbf{g}]\right\} g\_{rr}(0)}{\int d[\mathbf{g}] \exp\left\{\frac{i}{\hbar} S[\mathbf{g}]\right\}} = 0 \tag{10}$$

this allows us to reach our conclusion: at any point there is a finite probability for a zeromode to occur.

For these metrics the Einstein action is written (Sect. 2.1)

6 Quantum Gravity

are created spontaneously and at zero energetic cost at any point of spacetime, in a homogeneous and isotropic way. Usually vacuum fluctuations have a very short life, as can be shown through the Schroedinger equation (time-energy uncertainty principle) or through a transformation to Euclidean time (when the action is positive-definite). These arguments on the lifetime of the fluctuations can not be applied here, because quantum gravity has neither a local Hamiltonian, nor a positive-definite action. Our fluctuations, if they were completely isolated, would be independent of time; in fact, their interaction causes a finite lifetime (Sect. 2.3). In Sect. 5 we shall give a comparison between this kind of vacuum fluctuations and other fluctuations present in quantum gravity, like the virtual gravitons

In order to avoid a large global curvature, the *total* average effect of the virtual masses of the zero-modes must inevitably be renormalized to zero. This is, in our view, guaranteed by the "cosmological constant paradox": nature appears to be endowed with a dynamical mechanism which relaxes to zero any constant positive or negative contributions to the vacuum energy density, coming from particle physics or even from gravity itself. So, even though such contributions are formally infinite, in the end they do not affect the curvature of spacetime. The full explanation of this mechanism can only be achieved within a complete non-perturbative theory of Quantum Gravity. Some partial evidence of the dynamical emergence of flat spacetime has been obtained in the lattice theory, and in

Therefore we shall not be concerned with the global effect of our massive vacuum fluctuations on spacetime. We shall instead consider their interactions, which result in a novel pattern of purely gravitational excited states, above a ground state in which all fluctuations pairs with equal mass are in a symmetrical superposition. Freely speaking, it's a bit like studying the local effects of pressure variations, without worrying about how the

The zero-modes equation (plus the argument of non-interference) tell us that relevant runaway configurations of vacuum exist, in which the metric is locally very different from its classical value .We shall now consider an explicit path integral of Einstein gravitation, in order to evaluate the functional average of certain metric components and confirm this

Let us choose a spherical coordinate system. We integrate only over a sector *X* of the functional space, namely over the spherically-symmetric metric configurations with

[ ]exp [ ] (0)

 

*rr*

(10)

 

(0) 0 [ ]exp [ ]

this allows us to reach our conclusion: at any point there is a finite probability for a zero-

*<sup>i</sup> dg Sg g*

*X*

*<sup>g</sup> <sup>i</sup> dg Sg*

constant *g*00. If we obtain a null quadratic vacuum average in *X*, namely

*<sup>X</sup> rr <sup>X</sup>*

which transmit the gravitational interactions.

effective field theory approaches (Hamber, 2004, Dolgov, 1997).

total force due to atmospheric pressure affects the Earth.

**2.2 Zero-modes in the explicit functional integral** 

supposition.

mode to occur.

$$S\_E = -\frac{c^4}{8\pi G} \int d^4x \sqrt{\mathbf{g}(\mathbf{x})} \mathbf{R}(\mathbf{x}) = \frac{4\pi c^4}{G} \int dt \int\_0^\nu dr \sqrt{|BA|} \left(\frac{rA'}{A^2} + 1 - \frac{1}{A}\right) \tag{11}$$

$$\mathbf{g}\_{\mu\nu}(\mathbf{x}) = \eta\_{\mu\nu}$$

where *A g rr* and *B g* 00 are functions of *r*. Define a radius *ext r* , the "external radius" of our configurations, on which we impose boundary conditions as in Sect. 2.1. This means that we integrate over configurations which outside the radius *ext r* appear like Schwarzschild metrics with mass *M*. In order to avoid singularities, we suppose 0 *M* . We can re-write the action as an integral on *r* with upper limit *r*ext, because the scalar curvature of the Schwarzschild metric is zero. We can also add the Gibbs-Hawking-York boundary term, which in this case takes the form *S M dt HGY* . For a fixed time interval, we can regard the integral *dt* as a constant.

Supposing *B* constant ( *B M* 1 ), the path integral over these field modes is written

$$\begin{aligned} &\int d[A] \exp\left\{\frac{i}{\hbar} (S\_E + S\_{HGY})\right\} = \\ &= \int d[A] \exp\left\{\frac{i}{\hbar} \frac{4\pi\sqrt{|B|}}{G} \int dt \right\} \int\_0 ds \sqrt{|A|} \left(\frac{sA'}{A^2} + 1 - \frac{1}{A}\right) \left| \exp\left\{-\frac{i}{\hbar} M \int dt \right\} \right| \end{aligned} \tag{12}$$

The second exponential can be disregarded in the functional averages, because it cancels with the normalization factor in the denominator. In the first exponential, let us define a constant factor <sup>1</sup> <sup>4</sup> *<sup>B</sup> dt G* and discretize the integral in *ds*. We divide the integration interval [0,1] in ( 1) *N* small intervals of length and replace the integral with a sum, where the derivative is written as a finite variation. We obtain

$$\int d[A]e^{\frac{i}{\hbar}S\_{\mathbb{E}}} = \left| \prod\_{j=0}^{N+1} dA\_i \exp\left\{ i\alpha \delta \sum\_{j=0}^{N} \sqrt{|A\_j|} \left( \frac{j\delta \left(A\_{j+1} - A\_j\right)}{\delta A\_j^2} + 1 - \frac{1}{A\_j} \right) \right\} \right| \tag{13}$$

The presence of the square root and of the fractions with *Aj* makes the integrals very complicated. Let us change variables. Suppose 0 *A* , which is physically a widely justified assumption (and remember we are looking for a sufficient condition, i.e. we want to show that there exist a set of gravitational configurations for which the functional average of a quadratic quantity is different from the classical value). Define 1 / *A* . This gives the new path integral

$$\left| \bigcap\_{j=0}^{+\infty} \prod\_{j=0}^{N+1} d\boldsymbol{\gamma}\_{j} \frac{2}{\boldsymbol{\gamma}\_{j}^{3}} \exp \left| -i\alpha \boldsymbol{\delta} \sum\_{j=0}^{N} \left( 2j \boldsymbol{\delta} \frac{\left(\boldsymbol{\gamma}\_{j+1} - \boldsymbol{\gamma}\_{j}\right)}{\boldsymbol{\delta}} - \frac{1}{\boldsymbol{\gamma}\_{j}} + \boldsymbol{\gamma}\_{j} \right) \right| \tag{14}$$

Anomalous Gravitational Vacuum Fluctuations Which Act as Virtual Oscillating Dipoles 9

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

> *i H Mc i*

From this we would conclude that |0 is not an eigenstate! Nevertheless the property

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

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

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).

" (16)

have random phases.

<sup>2</sup> *H i Mc i <sup>i</sup>* , we would obtain, *only formally*

lifetime to be finite.

 and :

propagation and interaction.

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

**3.1 Pairs in symmetric and antisymmetric states** 

Consider a couple of states |1 and |2 with masses *M*1 and *M*<sup>2</sup> . We have

2 2

Putting now *M*1=*M*2=*M* and taking the interaction into account, the degenerate noninteracting levels are splitted. Define the symmetrical and anti-symmetrical superpositions

1 1 | (|1 |2 ) | (|1 |2 ) 2 2

 

1 2 1|H|1 c , 2|H M M |2 c , 1|2 0 (17)

(18)

" <sup>2</sup> 0 *i i*

0 00 *H* is true, considering that the coefficients *<sup>i</sup>*

(Note that <sup>1</sup> 2 *jj j* in the continuum limit.) We want to use this to compute the average <sup>2</sup> *<sup>m</sup>* , where *m* is a fixed intermediate index. This is the average of the squared field 2 at the point *s m* , therefore in the continuum limit it gives the average of <sup>2</sup> at the origin. We know that the system has zero-modes for which 0 *A* at the origin, and therefore . So we would like to show that <sup>2</sup> *<sup>m</sup>* for 0 . This can indeed be done (Modanese, 2011), and implies in turn that (10) is true. One can also check that this is not an artefact of the continuum limit.
