Quantum-only metrics in spherically symmetric gravity

The Einstein action for the gravitational field has some properties which make of it, after quantization, a rare prototype of systems with quantum configurations that do not have a classical analogue. Assuming spherical symmetry in order to reduce the effective dimensionality, we have performed a Monte Carlo simulation of the path integral with transition probability $e^{-\beta |S|}$. Although this choice does not allow to reproduce the full dynamics, it does lead us to find a large ensemble of metric configurations having action $|S|\ll \hbar$ by several magnitude orders. These vacuum fluctuations are strong deformations of the flat space metric (for which $S=0$ exactly). They exhibit a strong polarization in the scalar curvature $R$. In the simulation we fix a length scale $L$ and divide it into $N$ sub-intervals. The continuum limit is investigated by increasing $N$ up to $\sim 10^6$; the average squared action $\langle S^2 \rangle$ is found to scale as $1/N^2$. This makes possible in principle to obtain large metric fluctuations also at scales much larger than the Planck scale.

The Einstein action for the gravitational field has some properties which make of it, after quantization, a rare prototype of systems with quantum configurations that do not have a classical analogue. Assuming spherical symmetry in order to reduce the effective dimensionality, we have performed a Monte Carlo simulation of the path integral with transition probability e −β|S| . Although this choice does not allow to reproduce the full dynamics, it does lead us to find a large ensemble of metric configurations having action |S| by several magnitude orders. These vacuum fluctuations are strong deformations of the flat space metric (for which S = 0 exactly). They exhibit a strong polarization in the scalar curvature R. In the simulation we fix a length scale L and divide it into N sub-intervals. The continuum limit is investigated by increasing N up to ∼ 10 6 ; the average squared action S 2 is found to scale as 1/N 2 . This makes possible in principle to obtain large metric fluctuations also at scales much larger than the Planck scale.

I. INTRODUCTION
Efforts towards the unification of General Relativity and Quantum Mechanics into a coherent theory of Quantum Gravity were started long ago and have been intensifying in the last decades. In spite of big progress in loop Quantum Gravity [1,2], asymptotic safety [3] and discrete spacetime models [4][5][6][7], "the revolution is still unfinished", in the words of C. Rovelli. The spin-offs of this research work, however, are manifold and remarkable in their own right. In general, it is fair to say that the quest for unification has led to a better comprehension of both General Relativity and Quantum Mechanics.
The present author made some early contributions to Quantum Gravity in the covariant formulation by showing that the Wilson loop vanishes to leading order [8] and proposing an alternative expression for the static potential of two sources [9]. This formula was used by Muzinich and Vokos [10] and by Hamber and Liu [11], respectively in perturbation theory and in non-perturbative Regge calculus, to give an estimate of quantum corrections to the Newton potential. The vanishing of the Wilson loop (and of curvature correlations [12]) to leading order is only one of the peculiar aspects of the quantum field theory of gravity, that sets it apart from other successful quantum field theories like QED and QCD.
Another well known problem is the fact that Einstein's action is not positive definite.
Although many possible extensions and generalizations of the Einstein action have been proposed [13], which could help in addressing open issues in cosmology, the Einstein action is the natural action at intermediate energies and arises directly from the quantization of massless spin 2 fields.
In our opinion, the indefinite sign of the Einstein action gives us a chance to explore a phenomenon that is otherwise unknown in quantum field theory and more generally in Quantum Mechanics, namely the existence of configurations for which the action is zero, like for the classical vacuum, but not a stationary point. We call them zero modes of the action and we have proven analytically their existence for the Einstein action as well as for a peculiar elementary quantum system (the massless harmonic oscillator [14,15]).
The purpose of this work is to show that if the condition S = 0 defining the zero modes is relaxed to S/ 1, then a large ensemble of these modes can be numerically constructed via a Metropolis -Monte Carlo algorithm. The condition S/ 1 implies that these modes can play an important role in the path integral, although they are very different from classical solutions. In this sense the Einstein action offers an example of a dynamical system with unique quantum properties and a possible prototype for similar systems in other branches of physics.
The algorithm for the generation of the zero modes ensemble has been presented in [16], but the application was limited to metric configurations at the Planck scale, and accordingly the discretization limited to N = 10 2 space sub-intervals. Several authors have found, with various techniques, that the vacuum state of quantum gravity has a non-trivial structure at that scale. In particular, field configurations with spherical symmetry have been considered by [17,18]. One may wonder how this structure scales up to larger distances, and that is the main purpose of this work. After returning to more transparent physical units, we have performed several simulations looking for quantum zero modes with S/ 1 at a scale L L P and in the continuum limit N → ∞. It turns out that just the continuum limit allows to obtain such modes. The exact scaling dependence on L and N is reported in Sect.

III.
The outline of the paper is the following. In Sect. II we recall the form of the Einstein action reduced for spherically symmetric and time-independent metrics, first in the continuum version and then in the discretized version. We also recall our previous results from simulations at the Planck scale, made using a small number N of sub-intervals. In Sect. III we analyze the scaling properties of the discretized action with respect to N , up to N ∼ 10 6 , and also the scaling with respect to the inverse temperature β of the Metropolis algorithm and the length scale L of the vacuum fluctuations. In Sect. III C the observed polarization patterns of the metrics are reported and discussed. Sect. IV A offers for illustration purposes a mathematical example of zero modes in an oscillating 2D integral. Sect. IV B considers a possible extension to higher dimensions. Finally, Sect. IV C briefly summarizes our conclusions.

II. THE DISCRETIZED ACTION
Our physical model [16] is defined by the Einstein action of the gravitational field, computed for a metric with spherical symmetry and independent from time. The only field variable is the metric component g rr (r) = A(r), with 0 ≤ r < +∞. This kind of dimensional reduction of gravity has been already employed in several classical and quantum models.
The continuum action is This is derived from the usual expression of the Einstein action √ gRd 4 x in units such that c = 1, and in which the integral over time has been replaced by a factor τ , meaning that the metrics we are considering are stationary but have a limited duration τ . This is clearly an approximation, and eventually we should introduce a function of time describing an adiabatic switch on/off; we expect the corresponding time derivatives to give a negligible contributions to the curvature.
In principle it is possible to improve the model by increasing the number of degrees of freedom, making the algorithms more complicated but still manageable, at least in two ways: (1) besides the component g rr , consider as variable also the component g 00 (r), which at the moment is taken constant and equal to 1; (2) introduce a dependence on an angle θ. The full expression of R in this case is given in [16] and refs.
In the discretized version of the action the variable r runs on an interval (0, L) divided into N parts. After defining δ = L/N , we can say that r takes the values 0, δ, 2δ, ... , N δ, The boundary condition on the right end of the interval is A(L) = A N = 1, while on the left, for A(0), we do not set any constraint. We suppose that A(r) = 1 for r ≥ L. As a consequence, we are not considering metric perturbations extended to infinity, but only fields different from flat space in (0, L) (localized fluctuations). Clearly, L must be regarded as one of the parameters for which we will need a scaling analysis.
Upon quantization the variables A 0 , A 1 , ... A N become the integration variables of a path integral. Since the action is not positive definite, we suppose at the beginning that this path integral is of the Lorentzian kind, with weight e iS/ .
In the continuum action (1) we replace the integral with a sum and so we obtain the discretized action We are looking for anomalous fluctuations with respect to the trivial classical solution A(r) = 1 everywhere, which gives S = 0 (flat space). Our idea is to use the path integral as follows: if there is an ensemble of non-trivial metrics such that S/ 1, we suppose that they may describe important vacuum fluctuations. They do not need to be stationary points of the action like the classical configurations; they can also be exact zero modes of the action (for example, those we have found already with analytical techniques [CQG]) or modes with almost-zero action. An important requirement to make them relevant is that they must have a large volume in configuration space: the Montecarlo simulations will tell us if this is the case, and we may also expect (as confirmed in [16]) that according to the same simulations certain exact analytical zero modes will turn out to be too little probable to be physically relevant. A further discussion of this idea in relation to the stationary phase principle can be found in Sect. IV.

A. Results at the Planck scale
In [16] we chose units such that c = = G = 1, and we chose to explore a duration and length scale τ = 1, L = 1 (Planck scale). We took N = 100 in order to have a meaningful but "quick" discretization and we run a Metropolis algorithm [19] with a return probability exp(−β 2 S 2 ) or exp(−β|S|) in order to avoid the instability problems related to the indefinite sign of the action.
The result of the simulations is that for suitable values of the inverse temperature β one finds an ensemble of equilibrium configurations in which S ∼ δ · 10 −7 ∼ 10 −9 and S 2 ∼ δ 2 · 10 −14 or less. The sum N h=0 S h is found to oscillate around zero with an amplitude ∼ 10 −7 .
In other words, after starting formally with a Lorentzian weight e iS/ in order to skip the instability problems, and after realizing that the weight e iS/ cannot be implemented numerically, the trick of using a weight exp(−β 2 S 2 ) or exp(−β|S|) in the algorithm is not meant as a solution of the instability or a way to study the full dynamics, but only as a way to obtain explicitly a set of fields with almost-zero action and a large volume in configuration space.
The {A h } configurations of the equilibrium ensemble have a peculiar dependence on the coordinate r (r = δh): a sort of polarization with a step in the middle of the interval and A < 1 in the inner region, A > 1 in the outer region. Intuitively this matches the expectation of a cancellation between contributions to the integral of R over different regions, which is also a typical feature of some of the analytical zero modes [20]. One can check that the inner region has always negative R, while the opposite holds for the outer region. We shall see below that when the number N of sub-intervals in the discretized action grows (continuum limit), this simple pattern of "polarization into two regions" changes. It follows that as magnitude order we can rewrite the discretized action (2) as Due to the factor 10 66 , it seems very difficult to obtain S/ 1 in the discretized model, as soon as L and τ are larger than the Planck scale 10 −33 cm. If, however, we consider the In order to make contact with our previous data, let us start with a number of subintervals N = 100 and proceed by repeatedly multiplying N by 2. In this first series of trials we change β in inverse proportion to N , in such a way that the factor βL/N in the exponent of exp(−β|Ŝ|) stays constant and optimal thermalization is achieved. The variation inŜ is due to the factor 1/N and to the fact that the sum h S h has more terms. The good news N β MC steps e −β|Ŝ| Ŝ Ŝ 2 100 10 7 2 · 10 9 0.17 4.3 · 10 −9 2.0 · 10 −14 200 2 · 10 7 2 · 10 9 0.11 6.4 · 10 −9 5.1 · 10 −15 400 4 · 10 7 4 · 10 9 0.022 5.4 · 10 −9 1.3 · 10 −15 800 8 · 10 7 8 · 10 9 0.024 1.4 · 10 −9 3.2 · 10 −16 1600 16 · 10 7 8 · 10 9 0.055 3.0 · 10 −10 7.8 · 10 −17 3200 32 · 10 7 8 · 10 9 0.14 1.2 · 10 −9 2.3 · 10 −17 6400 64 · 10 7 8 · 10 9 0.27 1.5 · 10 −9 9.9 · 10 −18 12800 128 · 10 7 16 · 10 9 0.25 8.7 · 10 −10 2.8 · 10 −18 25600 256 · 10 7 16 · 10 9 0.29 3.4 · 10 −10 5.8 · 10 −19 204800 2048 · 10 7 16 · 10 10 3.3 · 10 −11 5.3 · 10 −21 409600 4096 · 10 7 16 · 10 10 1.5 · 10 −11 1.3 · 10 −21 TABLE I: Average values of the return probability e −βŜ , the action Ŝ and the squared action Ŝ 2 in dependence on N (number of sub-intervals of (0, L)). Here L = 10 cm. The inverse temperature β changes in proportion to N , in order to maintain the factor β/N constant. The discretized field components A h are randomly increased in the Montecarlo steps by ±ε, with ε = 10 −6 . For the last two values of N the calculation of e −βŜ (and of A h ) was omitted in order to speed-up the algorithm and increase the precision in Ŝ 2 . See also plot of Ŝ 2 in Fig. 1. is that this finer subdivision leaves h S h almost unchanged and soŜ scales as 1/N (Tab.

I).
When N is increased, we also need to increase the number of Montecarlo steps in order to obtain precise results, because at each step the algorithm changes at random one value of A h in the range h = 0, 1, ..., N , by an amount ±ε. The averages in the table are computed well after thermalization (after 50% of the steps for N up to 25600, and then after 75% of the steps). The quantity e −βŜ is the average of the return probability for the steps in which |Ŝ| increases. This probability increases with N (except for N = 100 and N = 200, where the polarization pattern is changing, see Sect. III C). Ŝ and Ŝ 2 are the averages of the actionŜ and of its square. The most important quantity is Ŝ 2 , which tells us how much the action oscillates about its average. The dependence of Ŝ 2 on N reported in Tab. I is also plotted in Fig. 1, from which a scaling very close to 1/N 2 can be deduced. This means that in the continuum limit N → ∞ the phase S/ , eq. 128 · 10 7 12800 16 · 10 9 0.25 8.7 · 10 −10 2.8 · 10 −18 256 · 10 7 12800 16 · 10 9 0.16 5.0 · 10 −10 8.2 · 10 −19 512 · 10 7 12800 16 · 10 9 0.095 3.3 · 10 −10 2.8 · 10 −19 1024 · 10 7 12800 16 · 10 9 0.055 2.2 · 10 −10 1.0 · 10 − 19   TABLE II: Scaling of e −βŜ , Ŝ and Ŝ 2 in dependence on β with N fixed, L = 10, ε = 10 −6 . Note the decrease of the average return probability e −βŜ , coherent with the role of the inverse temperature β in the thermalization process. The first value of β is chosen, to ensure thermalization, in such a way that the product Lβ is the same as for data with L = 10, β = 128 · 10 7 , N = 12800; this implies that β must now be equal to 128 · 10 21 .

C. Polarization pattern
The simple bipolar polarization pattern observed in the averaged field values A h for N = 100 [16] changes when N increases. Multiple oscillations begin to appear, with an envelope changing with N (see an example in Fig. 2, (a)), until for N approximately greater than 3200 the situation stabilizes and all the oscillations have almost exactly the same amplitude (Fig. 2, (b)). The number of oscillations does not depend on any of the physical parameters N , L, β and ε. It appears to be a general "mathematical" feature of the minimum configuration of the discretized action h S h . Fig. 3 shows two details of Fig. 2 (b), namely on sub-intervals with 1600 and 400 values of h. From these details we can see that the fixed total number of oscillations in the interval (0, L) is approximately equal to 10 2 , even though, as mentioned, there appears to be no relation between this number and the physical parameters.
As discussed in [16], A key concept of this work, already discussed analytically in [14,15,20], is that of zero modes of the action. This relates to a peculiar property of the gravitational field, not easily found in other physical systems: the non-positivity of the action in the path integral A simple mathematical example can help to elucidate the idea of zero modes. Consider a 2D integral with oscillating integrand, of the form where f (x, y) is a smooth function, and suppose that 1. We expect the main contribution to the integral to come from the region near the origin x = 0, y = 0, where the phase of The contributions near the origin come from the stationary point of the phase, those along the diagonal from the zero mode y = x. In (b) the region near the stationary point is smaller because is smaller, but the length of the zero mode is unaffected. In (c) the contributions of the disconnected zero modes y = x ± 2π also appear, because the temperature is much higher.
is s ij , the integral is approximated by

Fig. 4 represents with a density plot the contributions of the individual cells in a case
where the function is simply f (x, y) = 1 (see caption for details). If we compute instead the average r cos φ over all sampling points, namely with f (x, y) = x 2 + y 2 , we obtain at low temperature approximately 0.7 (half the diagonal), showing that the regions which contribute to the integral are in fact spread along the zero mode. However, when the temperature is increased (Fig. 4, (c)) the strong destructive interference along the zero modes tend to cancel their contributions, leaving only the contribution near the origin. This can also be seen from the fact that the average r cos φ decreases.
In the simple case of a 2D integral in x, y all these properties can be easily predicted, because we can plot the integrand and we know that the main contributions arise in the regions where the integrand is large and are directly proportional just to the area of these regions. One can also predict that being zero modes 1-dimensional, in the limit of small they do not contribute to the 2D integral.

B. Extension to higher dimension
For a path integral in infinite dimensions, with a non-polynomial action, all this a priori information is not available. Even if we are able to solve the exact equation for the zero modes (analogue of x 2 − y 2 = 0 in the 2D example) [16,20], it is hard to assess the "volume" of the solutions in the functional space, and even harder to asses this volume for the weaker but crucial condition S .
An higher-dimensional extension of the polynomial example above could be in principle the following: consider the integral The dimension of these modes is (m + n − 1) (for example, for a phase proportional to (x 2 1 + x 2 2 − y 2 1 ) the zero mode is a conical surface), so for m, n → ∞ they might indeed contribute to the integral.
To complete the analogy, note that in the gravitational case the contribution to the adaptive Monte Carlo coming from the configurations which make the action stationary appears to be actually negligible.

C. Conclusion
In conclusion, a non-perturbative Monte Carlo algorithm for the discretized action like that employed in this work (and in much more complete form by Hamber, Ambjørn and coworkers [4][5][6][7]) seems to be at present the only tool available for exploring quantum metrics closely connected to the classical vacuum state like the polarized configurations we have found in this work. The astounding detailed structure of these configurations (Sect. III C) and their stability and reproducibility are intriguing, and possibly part of more general patterns valid beyond the approximations made here (spherical symmetry, g 00 = 1, modes almost stationary in time).
We have shown that the average squared action Ŝ 2 of the polarized configurations scales as 1/N 2 up to a number N of sub-intervals of the order of 10 6 , for any length scale L. If this behavior can be extrapolated to larger N , their adimensional action S/ 10 66 τŜ can be 1 also at scales much larger than the Planck scale.
Future work should be devoted to an extension of the simulations to the case with angular and time dependence, and to a phenomenological comparison with observational constraints on gravitational vacuum fluctuations [21,22].