6 Discussion

The model that we have developed in section 2-3 leads to some interesting conjectures and speculations, that go beyond its immediate practical value. Indeed, a would-be quantum theory of gravity should lead to two clear-cut limiting cases: when gravitational interactions are negligible one should recover the ordinary Quantum Mechanics, while when quantum effects can be neglected one should get back General Relativity. Since our heuristic model is essentially built up on the uncertainty principle and on relativistic causality, which are the essential ingredients of the limiting cases, we expect it to be a useful tool for understanding the basic observational aspects of a more general and deeper theory.

Thus we conjecture that this simple approach may actually give some useful hints for the solution of outstanding problems, such as the origin of a nonzero cosmological constant: just as it happens for other kinds of quantum fluctuations [16] the gravitational energy density 2 gives a contribution to the effective cosmological constant:

$$\Delta \Lambda =\frac{8\pi}{c^{4}} G\rho _{tot}= \frac{8\pi}... ...{c^{4}} G\hbar n_{0} \ln {\frac{{c}}{{l_{\mathrm{P}} H(t)}}},$$ (12)

where $H(t)$ is Hubble's constant, with its time dependence explicitly spelled out. Recent observations favour a nonzero and positive cosmological constant [17,18,19], such that $\Omega_{\Lambda}\approx 0.7$ and $\Lambda = 3\Omega_{\Lambda}H^{2}/c^{2}$; this means that we can use 12 to estimate the order of magnitude of the free parameter $n_{0}$ if we assume that most of the effective cosmological constant is due to the fluctuations studied in this paper. In particular

$$n_{0}\sim \frac{3\Omega_{\Lambda}H^{2} c^{2}} {4\pi G\hbar \... ...right]} \sim 3 \cdot 10^{22} \mathrm{m}^{-3} \mathrm{s}^{-1} ,$$

so that the spectral density of the mean square fluctuation of the gravitational potential 8 evaluates to

$$\left\langle {\left\vert \Delta \Phi (f)\right\vert }^{2}\ri... ...\frac{(2 \cdot 10^{-91} \mathrm{m}^{4} \mathrm{s}^{-4})}{{f}.}$$ (13)

Obviously, the fluctuation rate $n_{0}$ might actually be calculated in the framework of a fundamental theory, like Quantum Gravity.

We wish to stress that the energy density that we obtain in the model is a weakly time dependent quantity and that it might actually justify the various scalar fields recently introduced to explain the measured acceleration of the expansion of the universe and the ``why now?'' problem (often called quintessence in the current literature, see, e.g., [20]).

Another point, which we probably should comment about⁷is the application of the time-energy uncertainty principle to our framework. It might, in principle, be disputable, since it is indeed very well known that there is a ``time problem'' in quantum gravity. This can be traced back to the very distinctive role that time has in quantum mechanics, which can be easily seen already from the structure of the Schrödinger equation and at a deeper level can be recognized as related to the fact that, unlike the case for position and momentum, in the quantization process time is not represented by any quantum operator. Thus whereas the position-momentum uncertainty principle can be directly related to the non-commutativity of the corresponding quantum operators, the situation for the time-energy uncertainty principle seems radically different. It is mainly this particular character of the time parameter in quantum mechanics that raises a problem when we try to merge the quantum picture with the relativistic one. In a relativistic theory, general relativity say, time runs not any more as an external universal regular clock but critically depends on the state of the observer: this non-uniqueness in the choice of the time parameter is also well witnessed by the vanishing of the (super)hamiltonian in general relativity and in the ADM formalism it is clearly seen in the possibility of choosing freely the lapse function. The often called timelessnes of general relativity and the consequently supposed timelessness of quantum gravity (which finds a still stronger confirmation in the Wheeler-De Witt equation) has been discussed many times. In our opinion it is probably not out of place to imagine that in a full theory of quantum gravity, time would perhaps be only a derived concept, maybe not applicable to all physical systems and, maybe, only to situations involving something that could be understood in terms of a classical measurment process. In particular it seems conceivable that the time concept could emerge only for systems of high complexity (since if we think at a very elementary system, it turns out nearly impossible to see how a coherent operative, i.e. involving a measurement process description, could be given) and probably could find its final definition only in a coherent solution to the measurement problem in the framework of quantum gravity.

If indeed time does not exist at all at the microscopic level, or at the Planck scale, if the most fundamental laws of nature are truly timeless, being quantum and covariant in nature, it is perhaps not out of place to imagine that the emergence of time could definitely be related to the emergence of classical behaviour in a system. Since, at the end, to have a physical concept of time, we need a measurement process for the concept of time. At some scale, maybe unknown (as many times pointed out by John Bell), our apparatus will necessarily perform a ``classical'' reduction and connect the wave function to a state of a classical pointer for some definite observable. Even if we do not really know at which scale this reduction may happen, it is appealing to imagine that at the same level while we are performing a classical (i.e. human scale) measurement, our classical (i.e. intuitive) notion of time also enters the picture. A picture in which even quantum mechanics, and thus also the energy-time uncertainty principle, is just our best approximation in the gravitationally weak regime for the properties fulfilled by a measurement process involved with some deeper (maybe timeless) set of laws of nature (quantum gravity). In this perspective and from the point of view that we had in mind to set up in this paper, which is concerned with the outcome of an interferometric detection of the effects that quantum fluctuations of the gravitational field could have on the propagation of light, we thus feel more confident that the energy-time uncertainty principle can be considered satisfied by the fluctuations of the energy density, if not in their deepest (yet unknown) quantum gravitational nature, at least in the aspects that they show us in the detection (reduction) process.