Gas of wormholes: a possible ground state of Quantum Gravity
G. Preparata, S. Rovelli, S.S. Xue
Dipartimento di Fisica dell’Università and INFN  sezione di Milano, Via Celoria 16, Milan, Italy
(a) I.C.R.A.  International Center of Relativistic Astrophysics and INFN, La Sapienza, 00185 Rome, Italy.
Abstract
In order to gain insight into the possible Ground State of Quantized Einstein’s Gravity, we have derived a variational calculation of the energy of the quantum gravitational field in an open space, as measured by an asymptotic observer living in an asymptotically flat spacetime. We find that for Quantum Gravity (QG) it is energetically favourable to perform its quantum fluctuations not upon flat spacetime but around a “gas” of wormholes of mass , the Planck mass (GeV) and average distance , the Planck length (cm). As a result, assuming such configuration to be a good approximation to the true Ground State of Quantum Gravity, spacetime, the arena of physical reality, turns out to be well described by Wheeler’s quantum foam and adequately modeled by a spacetime lattice with lattice constant , the Planck lattice.
PACS 04.60, …(other pacs)
1 Introduction
Among the fundamental interactions of Nature, since the monumental contribution of Albert Einstein, Gravity plays the central role of determining the structure of spacetime, the arena of physical reality. As well known, in classical physics a world without matter, the Vacuum, has the simplest of all structures, it is flat (pseudoeuclidean); but in quantum physics? This is the focal question that has occupied the best theoretical minds since it became apparent, at the beginning of the 30’s, that Quantum Field Theory (QFT’) is the indispensable intellectual tool for discovering the extremely subtle ways in which the quantum world actually works. Thus the problem to solve was to find in some way or other the Ground State (GS) of Quantum Gravity (QG), which determines the dynamical behaviour of any physical system, through the nontrivial structure that spacetime acquires as a result of the quantum fluctuations that in such state the gravitational field, like all quantum fields, must experience. Of course this problem, at least in the nonperturbative regime, is a formidable one, and many physicists, J.A. Wheeler foremost among them, could but speculate about the ways in which the expected violent quantum fluctuations at the Planck distance () could change the spacetime structure of the Vacuum, from its classical, trivial (pseudoeuclidean) one. And Wheeler’s conjecture, most imaginative and intriguing, of a spacetime foam vividly expressed the intuition that at the Planck distance the fluctuations of the true QG ground state would end up in submitting the classical continuum of events to a metamorphosis into an essentially discontinuous, discrete structure^{1}^{1}1We should like to recall here that, based on Wheeler’s idea, a successful research program was initiated a few years ago to explore the consequences of the Standard Model () in a discrete spacetime, conveniently modeled by a lattice of constant , the Planck lattice (PL).
It is the purpose of this paper to give a detailed account of the results of an investigation on a possible QG ground state, which has been summarily reported in a recent letter[1]. The starting point of our attack is the realization that QG can be looked at as a nonabelian gauge theory whose gauge group is the Poincaré group. Following the analysis performed by one of us (GP)[2] of another nonabelian gauge theory QCD (whose gauge group is ), we decided to explore the possibility that the energy density (to be appropriately defined, see below) of the quantum fluctuations of the gravitational field around a nontrivial classical solution of the Einstein’s field equations for the matterless world, could be lower than the energy of the perturbative ground state (PGS), which comprises the zero point fluctuations of the gravitational field’s modes around flat spacetime. Indeed in QCD it was found that the unstable modes (imaginary frequencies) of the gauge fields around the classical constant chromomagnetic field solution of the empty space YangMills equations, in the average screen completely the classical chromomagnetic field, allowing the interaction energy between such field and the short wavelength fluctuations of the quantized gauge field, to lower the energy density of such configuration below the PGS energy density. Thus we decided to try for QG the strategy that was successful in QCD, i.e.


select a class of empty space classical solutions of Einstein’s equations that is simple and manageable;

evaluate the spectrum of the small amplitude fluctuations of the gravitational field around such solutions;

set up a variational calculation of the appropriately defined energy density in the selected background fields;

study the possible screening by the unstable modes (if any) of the classical background fields.

As for point (1) we have chosen the Schwarzschild’s wormholesolutions[3], the simplest class of solutions of Einstein’s equation after flat spacetime. In order to achieve (2) the ReggeWheeler[4] expansion has been systematically employed, yielding a well defined set of unstable modes (for Swave). This important result, already indicated in previous independent work [5], renders the development of the points (3) and (4) both relevant and meaningful, the former point yielding a lowering of the energy density due to the interaction of the shortwave length modes with the background gravitational field, the latter exhibiting the (approximate) cancellation of the independent components of the tensor of the Schwarzschild’s wormholes by the Swave unstable modes. As a result flat spacetime, like the QCD perturbative ground state, becomes “essentially unstable”, in the sense that upon it no stable quantum dynamics can be realized. On the other hand a well defined “gas” of wormholes appears as a very good candidate for the classical configuration around which the quantized modes of the gravitational field can stably fluctuate. But a discussion of the physical implications of our findings must await a more detailed description of our work, which we are now going to provide.
2 The Schrödinger functional approach
In order to develop a functional strategy aimed at determining the Ground State of Quantum Gravity, which parallels the approach developed for QCD [2], we must first identify an appropriate energy functional. In General Relativity this is a nontrivial problem for, as is well known, in the canonical quantization procedure, first envisaged by Dirac [6] and Arnowitt, Deser and Misner (ADM) [7], due to general covariance the local Hamiltonian is constrained to annihilate the physical ground state, a fact that in the Schrödinger functional approach is expressed by the celebrated WheelerDeWitt equation [8]. However we note that the problem we wish to solve concerns the minimization of the total energy of an “open space”, in which there exists a background metric field that becomes “asymptotically flat”, i.e. that for spatial infinity () behaves as
(1) 
where is the Minkowski metric. In the conventional canonical formulation, spacetime is foliated into spacelike slices with constant values of the time parameter ; the asymptotic condition (1) determines the asymptotic behaviour of the canonical variables: the spatial 3metric on , the conjugate momenta , the “lapsefunction” and the “shift vector” [7] as:
(2) 
Let us consider the ADMenergy [7], which in cartesian coordinates is given by ( is the boundary of , “” denotes partial derivative with respect to , and is the Newton constant)
(3) 
We should like to point out that is just the energy that an asymptotic observer attributes to a space region whose time foliation he is keeping anchored to his (asymptotically) flat metric: the boundary conditions select a privileged reference frame (up to Lorentz transformations) that implicitly defines the physically relevant energy. In particular, the asymptotic condition on fixes the “boundary time” unequivocally: the asymptotic observer is the only possessor of an idealized clock that allows him to describe quantities associated to the whole physical system without introducing material clocks (i.e. auxiliary fields). In this sense, he is also the only one that can really be termed as an idealized, noninterfering “observer” capable of describing geometry at the quantum level in terms of evolution, not merely in terms of correlation between variables, thus giving a full meaning to the expression “quantum geometrodynamics”[9].
At the classical level, the definition of fixes Minkowski geometry as the zero point of the total energy; the proof of its positivity [10] can be looked at as the statement that flat spacetime is the (unique) vacuum of General Relativity. This explains why the first steps towards the quantization of the theory have been based on a perturbative approach on the flat background, with the fluctuating selfinteracting field interpreted in the conventional particle view as creating and annihilating gravitons, which propagate in pseudoeuclidean space: in this sense, we call flat spacetime filled with gravitons performing zeropoint fluctuations the “perturbative ground state” (PGS).
We now know that the perturbative approach was doomed to fail: the nonrenormalizability of the theory does not allow to make any meaningful and predictive perturbative expansion. On the other hand, analyzing the theory to the lowest nontrivial order around a curved background may give us important indications of how the deadly “impasse” of the perturbative approach may be finally overcome and give back to the simplest form of QG its status and role of a “bona fide” Quantum Field Theory.
We thus study the quantum fluctuations of the gravitational field upon a generic, asymptotically flat stationary background geometry, solution of the sourceless Einstein’s equations; in particular, for the background metric we can choose a foliation orthogonal to Killing timelike vectors and put it in static form. On a given slice , the 3metric is thus given by
(4) 
where is the spatial background metric and the fluctuation to be quantized (). We can now expand the total ( the sum of the Hamiltonian and ) energy of space in powers of the fluctuations (the number denotes the order of the expansion):
(5) 
and are the superhamiltonian and super momentum, as defined by ADM [7]:
(6) 
where is the determinant of 3metric and the corresponding curvature scalar, with the “supermetric” given by
(7) 
while ( is the covariant derivative with respect to )
(8) 
We note that, a priori, also and can be expanded in (5); while the background terms and are fixed functions, subject to the asymptotic conditions (2), the higher order terms represent true fluctuations in the lapse function and the shift vector: variations of and yield, at the classical level, the constraints:
(9) 
leaving in the expression of the classical energy only the term.
Since , and form together a solution of matterless Einstein’s equations, the linear term in the canonical Lagrangian density must be a total divergence. In our case, where is static, this is a purely spatial divergence and, keeping the asymptotic flatness of background in mind, it must necessarily coincide with . Thus, we have:
(10) 
that, together with the constraint (9) allows us to rewrite the energy (5) as
(11) 
Note the survival of only the classical order terms in and .
The quantization of the theory promotes the canonical pair on to operators obeying the commutation rules:
(12) 
and acting on a Hilbert space of functionals that are annihilated by the constraints (9). The evolution of the physical states is governed by the “Schrödinger equation”
(13) 
where the Hamiltonian operator is given by (11). Note that this is just the description of the quantum dynamics made by the asymptotic observer at infinity.
We point out that our definition of the Hilbert space is truly consistent within our restriction of phase space to two pairs of canonical operators, obtained by the gauge conditions (that respect (9)). This does not mean a loss of invariance (and of physical reality) at all: despite its look, the total energy (11) is nothing but a rearrangement of the invariant ADMenergy.
We should also be aware that the configuration space representation of the canonical operators
(14) 
acting on state functionals
(15) 
is not easily manageable beyond the 1loop level, where connected ghost terms appear. Beside that, for the expansion of the Hamiltonian operator (11) contains products of conjugate operators, thus posing an ordering problem. These problems are related to the bad ultraviolet divergences that would still yield a nonrenormalizable behaviour, the possible solution of which emerges from the results of Section 6, which show that the structure of the vacuum is, with good probability, essentially discontinuous at the Planck scale . Thus, in the rest of our analysis the QFT we shall work with will be cutoff at the Planck scale, having clearly in mind that our results will only be meaningful if consistent with this fundamental assumption (see Section 6).
As for the constraints (9), the problems are easier to solve. In fact we first notice that at the lowest order, the Hamiltonian operator retains only quadratic terms in the fields, on which we have to impose consistently first order constraints, that do not annihilate the quantum energy. The following terms in the expansions (9) can be enforced through a systematic correction of the state functional that readapts nonphysical degrees of freedom order by order, thus not affecting the dynamics based on the degrees of freedom (two for each space point) isolated at the lowest level. ^{2}^{2}2See for example the procedure followed in Appendix B of ref [2]. Thus in spite of the problems typical of Quantum Gravity, the parallelism with the situation in QCD [2] is fully regained.
According to our fundamental assumption to cut the theory at the Planck scale, we shall perform a 1loop calculation, with the Hamiltonian operator truncated at . We simply adopt the representation (14) and (15), thus obtaining the Schrödinger equation
(16) 
where
(17) 
with annihilated by the first order constraints
(18) 
Setting, as usual,
(19) 
the problem can be reduced to the eigenvalue equation
(20) 
We can now investigate the ground state of the theory. Instead of solving directly the eigenvalue equation, we look for the minimization of the expectation value of on a class of gaussian wavefunctionals:
(21) 
If the background is stable under the action of quantum fluctuations, at the 1loop level this result coincides with the solution of (20); if, on the contrary, simple minimization leads to an imaginary part in , then we have discovered an unstable configuration, whose physical meaning must be investigated. We demonstrate in the next section that the latter case occurs when fluctuate around the “wormhole solution” discovered by Schwarzschild in 1916 [3], whose line elements in polar coordinates are given by ()
(22) 
and depend on the single parameter , the ADMmass, such that
(23) 
3 Quantum Fluctuations on a Schwarzschild Background
We shall now address the problem to evaluate the expectation value on a gaussian trial functional of the Hamiltonian (17) where, according to our fundamental hypothesis (to be checked for consistency at the end of the calculation), we keep only the quadratic terms in the field quantum fluctuations . This truncation corresponds to the oneloop approximation. From a classical standpoint this amounts to a calculation of the energy carried by the quantized gravitational waves propagating on a fixed background, in the weak field approximation.
In our analysis we shall follow closely the steps of the ref.[2], where a similar calculation was carried out for a YangMills theory. We begin by constructing the Hilbert space of the states of the gravitational field, introducing the following scalar product:
(24) 
where denotes the measure of the functional space and represents the FadeevPopov determinant, depending on the gauge adopted, necessary to recuperate the gaugeinvariance, i.e. the general covariance of QG. The Hilbert space will thus be the space of the statevectors that with the metric (24) are normalizable. We note that for an infinitesimal coordinate transformation the quantum field gets transformed as:
(25) 
just like a weak classical field. And in our approximation, being the gaugeconditions (9) linear in the field , the determinant does not depend on and can be therefore neglected. For a generic operator , the expectation value on a state can be defined as:
(26) 
Let us consider now a hypersurface at a fixed time . We wish to compute the expectation value of the (truncated) Hamiltonian on the gaussian trial functional:
(27) 
where and is the one half Schwarzschild radius). In order to get a normalizable we require that be real and positive, symmetric under the exchanges . (
The second order Hamiltonian density is given by
(28) 
with
(29) 
(30) 
We observe that for a Gaussian wavefunctional the expectation value of two fields is given by:
(31) 
where satisfies the relationship:
(32) 
In this way one gets for the expectation value of the Hamiltonian^{3}^{3}3In order to clearly separate the classical from the quantum (oneloop) contributions for the rest of this Section we shall keep the Planck constant , instead of putting it equal to one, as done in the natural unit system.
(33) 
where
(34) 
and the differential operator is defined by
(35) 
and represents the “potential” contribution to the quadratic Hamiltonian (17). In order to guarantee the general covariance of our calculation, it is necessary to impose on the physical state the quantum constraints, which in our approximation are
(36) 
(37) 
which are obeyed provided,
(38) 
where denotes the covariant derivative with respect to the background field; and, fixing the lapse function and the shift vector as
(39) 
we have
(40) 
By consistency with the trace condition we must also impose , which yields the further constraint:
(41) 
The elements of the Hilbert space of the physical modes of the gravitational field are thus the symmetric tensors of rank 2 , defined in , obeying the boundary conditions (2) and the gauge conditions:
(42) 
normalizable with respect to the scalar product:
(43) 
In this way we may construct in our Hilbert space a complete orthogonal system, by making use of the spectral decomposition of the operator or, better,. of the operator defined as:
(44) 
where in order to go from to a total divergence has been added to the integrand, without changing the “potential” contribution to the Hamiltonian. Thus the operator becomes in our Hilbert space a selfadjoint hermitian operator, whose eigenfunctions ( denotes a complete set of indices),
(45) 
build the sought complete orthonormal basis. In this basis the “propagator” has the simple form;
(46) 
where denotes a set of variational parameters to be determined by the minimization of the expectation value (33).
(47) 
We may now easily compute the expectation value (33), and obtain:
(48) 
and minimizing with respect to the variational function , i.e. imposing
(49) 
we readily get
(50) 
which inserted in (48) finally yields:
(51) 
All the above makes sense if and only if
(52) 
i.e. the eigenvalue of the “potential” operator are positive definite. If, instead, for some the oneloop approximation, yielding imaginary contributions to the energy of the ground state, breaks down, showing that the Perturbative Ground State (PGS, is essentially unstable.
This is precisely the situation found in the study of YangMills theories[2] where, going beyond the oneloop approximation, one could easily check that the modes belonging to the sector where did not contribute to the energy of the state terms of ^{4}^{4}4Like it happens, according to (51), to the modes belonging to the “stable sector”, for which ., but rather of , just like the classical term . This “promotion” of a quantum contribution to a classical one, can be understood when we realize that the amplitude of the modes with (the “unstable modes”)[see Eqs.(31) and (46)] is only prevented from becoming infinite by the neglected positive terms of . In this way becomes and the negative contribution from the “unstable modes” is just classical, i.e. .
In the calculation of Ref.[2] one could explicitly prove that this “promoted” quantum contribution completely screens the classical positive term (such as, in our case ), thus realizing a “vacuum” state whose energy density is way that of below the PGS, which as a result becomes unstable at all spacetime scales. In the case of QG the problem of going beyond the oneloop approximation is formidable, utterly beyond our present means of analysis, however, as shall be discussed below, to figure out the contributions to the energy of the trial states of possible “unstable modes” with appears reasonably doable.
In order to precede further we must first compute the operator and then diagonalize it, which we shall do next. Defining , we have:
(53) 
and for the perturbative calculations it is useful to introduce:
(54) 
which turns out to be tensor.
Observing that (“” denotes the full covariant derivative, while “” is the covariant derivative with respect to the Schwarzschild background)
(55) 
we may write the Riemann and the Ricci tensors as:
furthermore:
(56) 
We may now compute the curvature scalar , which for convenience we decompose as the sum of three terms:
In this way to first order in we have:
(57) 
while to second order we obtain:
(58) 
As for we have
(59) 
and
(60)  
respectively. While for the third term one has:
(61) 
By summing the different terms we obtain the following expansion of the scalar curvature:
(62) 
(63) 
(64)  
But we are not done yet, we must expand the square root of the determinant of the metric . Due to we may stop at the first order, obtaining:
(65) 
In this way for we get the following expansion:
(66)  
We are now ready to compute the operator . Let us fix the gauge as: and , the latter condition by use of the former giving
(67) 
We also note that given a vector , one has
(68) 
Thus we obtain for the integral of the second order potential,
(69) 
Recalling Eq.(44), Eq.(45) becomes:
(70) 
where we have added the terms in e in order to remain inside the Hilbert space of the tensors , which obey the conditions [see Eq.(42)]
(71)  
By taking the trace of eq.(70) with we get immediately that
(72) 
By consistency, we must require that the terms in do not contribute to the potential energy, this obviously implies that
(73) 
which requires that for the following conditions be satisfied:
(74) 
and
(75) 
4 The eigenvalues and eigenmodes of the second order potential
Due to the spherical symmetry of the Schwarzschild background a particularly suitable method to obtain the solutions of the eigenvalue problem posed by Eq.(70) is the one devised by T. Regge and J.A. Wheeler [4], for the study of the small (classical ) fluctuations around the Schwarzschild solution. By making use of this method the eigenfunction are separated in two classes (see appendix B), the “even” solutions with parity , equal to the parity of spherical harmonics , and the “odd” solutions with opposite parity.For the “even” solutions, if we set
the eigenvalue equations will turn out to be, as we shall see in a moment, completely factorized. As for the vector (70), factorization is achieved if we set,
Substituting (4) and (4) in (70) (see appendix C) we obtain a system of differential equations for the radial functions only:
And by substituting the same expansions in the constraint Eq.(42), we obtain,
(76) 
(77) 
(78)  