The Geometrical Origin of Dark Energy

The geometrical formulation of the quantum Hamilton-Jacobi theory shows that the quantum potential is never vanishing, so that it plays the role of intrinsic energy. Such a key property selects the Wheeler-DeWitt (WDW) quantum potential $Q[g_{jk}]$ as the natural candidate for the dark energy. This leads to the WDW Hamilton-Jacobi equation with a vanishing kinetic term, and with the identification $$ \Lambda=-\frac{\kappa^2}{\sqrt{\bar g}}Q[g_{jk}] \ . $$ This shows that the cosmological constant is a quantum correction of the Einstein tensor, reminiscent of the von Weizs\"acker correction to the kinetic term of the Thomas-Fermi theory. The quantum potential also defines the Madelung pressure tensor. Such a geometrical origin of the vacuum energy density, a strictly non-perturbative phenomenon, provides strong evidence that it is due to a graviton condensate. Time independence of the WDW wave-functional then would imply that the ratio between the Planck length and the Hubble radius is a time constant, providing an infrared/ultraviolet duality. This indicates that the structure of the Universe is crucial for a formulation of Quantum Gravity.


Introduction
In spite of the tremendous efforts, understanding the origin of the cosmological constant [1] [2] [3] is still an open question. In this paper we show that the cosmological constant is naturally interpreted in terms of the quantum potential associated to the spatial metric tensor. The starting point concerns the geometrical derivation of the Quantum Hamilton-Jacobi Equation (QHJE), suggested by the x−ψ duality observed in [4] and formulated in [5] (see [6] for a short review). Such a formulation reproduces the main results of quantum mechanics, including energy quantization and tunneling, without using any probabilistic interpretation of the wave function, which is one of the problems in formulating a consistent theory of quantum gravity. Furthermore, it has been shown that if space is compact, then there is no notion of particle trajectory [7]. The idea underlying the derivation of the QHJE is that, like general relativity, even quantum mechanics has a geometrical interpretation. This is done by imposing the existence of point transformations connecting different states, which, in turn, leads to a cocycle condition that uniquely fixes the structure of the QHJE. In such a formulation, it has been shown that the quantum Hamilton characteristic function S is non-trivial even in the case of the free particle with vanishing energy. Such a result is deeply related to the solution of Einstein's paradox, discussed later, and concerning the classical limit of bound states in the de Broglie-Bohm theory.
In the present paper we are interested in the fact that, unlike in the de Broglie-Bohm theory, even in the case of a free particle with vanishing energy, the quantum potential is non-trivial [5]. It is just such a property that led in [8] to the proposal that there is a deep relation between quantum mechanics and gravity. In particular, it was emphasized that the characteristic property of the quantum potential is its universal nature, which is, like gravity, a property possessed by all forms of matter. Subsequently, the deep relation between gravity and quantum mechanics was also stressed by Susskind in his GR=QM paper [9] and where it is emphasized that where there is quantum mechanics there is also gravity . An explicit relation between quantum mechanics and gravity arises in the case of the free particle with vanishing energy, whose quantum potential includes the Planck is the Schwarzian derivative of f . Such a result follows by requiring that the free particle of energy E consistently reproduces both the → 0 and E → 0 limits. On the other hand, since in the problem there are no scales, one is forced to use universal constants. It turns out that the Planck length is the only candidate satisfying the consistency conditions. The main feature of (1.1) is the appearance of the Planck length that provides a link between quantum mechanics and gravity, which is related to the invariance of the quantum potential under Möbius transformations of S. Since E = 0 corresponds to the ground state, such a non-trivial Q can be considered as an intrinsic energy. The above mechanism suggests that in the case of general relativity such an intrinsic energy is at the origin of the cosmological constant, namely where Q[g jk ] is the quantum potential coming from the Wheeler-DeWitt (WDW) equation [11][12] without matter and with vanishing spatial curvature. This is reminiscent of the von Weizsäcker correction to the kinetic term of the Thomas-Fermi theory [10]. It is worth mentioning that also the Madelung pressure tensor is defined in terms of the quantum potential. Inspired by (1.1), we propose that Eq.(1.2) corresponds to the quantum potential in the vacuum, where, besides ℓ P , there is another natural length, the Hubble radius R H = c/H 0 = 1.36·10 26 m. The cosmological constant would then be fixed by suitable conditions, just like a particle in a box. We will see that the energy density of the vacuum is naturally interpreted in terms of a graviton condensate. Our investigation uses a basic property of the WDW equation, namely, the time independence of the WDW wave-functional. This indicates that, like the Hubble radius, even the Planck length is time-dependent. In particular, time independence of the WDW wave-functional, suggests that the ratio is a space-time constant. This provides an exact infrared/ultraviolet duality. The paper is organized as follows. In sect. 2 we shortly review the derivation of the WDW Hamilton-Jacobi equation. In sect. 3 we discuss the main points of the quantum Hamilton-Jacobi theory formulated in [5]. In particular, we will consider its geometrical origin and focus on the solution of Einstein's paradox, which in turn is related to the non-triviality of the QHJE for the free particle with vanishing energy. In sect. 4 we show that, contrarily to the de Broglie-Bohm formulation, the quantum potential is not trivial even in the case of the WDW Hamilton-Jacobi equation with 3 R = 0 and vanishing cosmological constant. In sect. 5 we show that the cosmological constant is interpreted in terms of the WDW quantum potential and show that the result is naturally interpreted in terms of the WDW equation in the vacuum with boundary conditions defined by R H . Time independence of the WDW wave-functional then implies that K is a space-time constant.

WDW Hamilton-Jacobi equation
In the ADM formulation space-time is foliated into a family of closed 3-dimensional hypersurfaces indexed by the time parameter that we fix at the present time. We choose the signature (−, +, +, +). Denote by g ij = 4 g ij the metric tensor of the three dimensional spatial slices. Let N = (− 4 g 00 ) −1/2 be the lapse and N k = 4 g 0k the shift vector. We then have the standard 3+1 decomposition (2.1) Setḡ = det g ij and κ 2 = 8πG/c 4 . The Einstein-Hilbert Lagrangian density can be equivalently expressed in the form where 3 R is the intrinsic spatial scalar curvature, Λ the cosmological constant, K the trace of the extrinsic curvature and D j denotes the j component of the covariant derivative. Let π 0 and π k be the momenta conjugate to N and N k respectively. Since L is independent of both ∂ x 0 N and ∂ x 0 N k , we have the primary constraints Time conservation of the primary constraints implies secondary constraints, given by the weak vanishing of the super-momentum, and of the super-Hamiltonian, where π jk is the momentum canonically conjugated to g jk , that is and is the DeWitt supermetric. The conservation in time of the secondary constraints do not imply further constraints. By a Legendre transform one gets the Hamiltonian showing that N and N k are the Lagrange multipliers of H and H k respectively. The implementation of the primary constraints at the quantum level is obtained by settinĝ meaning that Ψ does not depend on any of the non-dynamical variables. At the quantum level the conjugate momenta of a field φ would correspond to −i δ φ , so that, since [δ (3) On the other hand, by (2.7) we have [π ij ] = MT −2 , which is different from the dimension of the canonical choice ofπ jk , namely which also fixes the normalization of the classical relation where S is the functional analogue of Hamilton's principal function. By (2.12), the supermomentum constraint readsĤ which is satisfied if Ψ is invariant under diffeomorphisms of the hypersurface. The other secondary constraint, that isĤΨ = 0, is the WDW equation Let us now consider the key identity 1 Ae βS which holds for any complex constant β. Setting β = i/ and in (2.15) gives the WDW Hamilton-Jacobi equation, corresponding to the following quantum deformation of the Hamilton-Jacobi equation together with the continuity equation The last term in (2.18), that is is called the quantum potential.

QHJE and Einstein paradox
To understand why even in the case 3 R = 0 there are non-trivial S and Q, it is useful to recall Einstein's paradox (see e.g. Ref. [13] pg. 243). This concerns the issue in Bohmian mechanics when considering the classical limit in the QHJE, in the case of a particle in an infinite potential well. More generally, the problem holds for all states described by a wave-function corresponding to Hamiltonian eigenstates of any one-dimensional bound state. In this case one can easily show that ψ E ∈ L 2 (R) is proportional to a real function. Therefore, if one sets, as in Bohm theory, ψ E = Re i S , then S is a constant. On the other hand, in the Bohmian formulation, p = ∂ x S is identified with the mechanical momentum mẋ, so that, quantum mechanically, one would have p = 0. Therefore, as in the case of the harmonic oscillator, a quantum particle at rest should start moving in the classical limit, where S and p are non-trivial. In other words, it is clear that it is not possible to get a non-trivial S as the → 0 limit of a constant function. The resolution of the paradox is that the quantum analogue of S is not necessarily the phase of the wave function. As we will show, this in fact also underlies the WKB approximation that even if one starts with the identification ψ = exp(iS W KB / ), with S W KB complex, then real wave functions are identified with a linear combination of in and out waves. In our formulation, such a choice is not ad hoc, rather it follows from the request that the cocycle condition is always satisfied [5]. In particular, note that if Re i S is a solution of the stationary Schrödinger equation, then, this is also the case of Re − i S . This is the key to introduce the so-called bipolar decomposition As a result, in the case of a real ψ E , the only constraint is just |A| = |B| and one gets a non-trivial S with a well-defined classical limit. Such a solution of Einstein's paradox is a consequence of the geometric derivation of the QHJE, that excludes in a natural way, and from the very beginning, the existence of states with a constant S [5]. The use of the bipolar decomposition was previously discussed by Floyd [14]. Later we will see that in the functional case of the WDW Hamilton-Jacobi equation, the corresponding S and the quantum potential assume a non-trivial role even when 3 R = 0. This is just the functional analogue of basic properties of the quantum potential that we now discuss. The main point that characterizes the non-trivial properties of the quantum potential is its connection with the Möbius invariance of the Schwarzian derivative {f, x}, that, in order to be well defined, requires that f ∈ C 2 (R) and ∂ 2 x f differentiable on R. The continuity equation ∂ x (R 2 ∂ x S) = 0 implies that R is proportional to (∂ x S) −1/2 , so that the quantum potential can be expressed in terms of S only {S, x} , where β is a constant with the dimension of an action. Such an identity implies that the QHJE (3.5) can be also expressed in the form The solution of this non-linear differential equation is we see that there is a non-trivial mixing between the kinetic term and the quantum potential in (3.3).
In [5] the QHJE was derived by a slight modification of the way one gets the classical Hamilton-Jacobi equation. Namely, instead of looking for maps from (x, p) to (X, P ), seen as independent variables, such that the new Hamiltonian is the trivial one,H = 0, we looked for transformations x →x such thatṼ −Ẽ = 0, but with the transformation of p fixed by imposing that S(x) transforms as a scalar function. That is holding for any pair of physical systems, including the one with V − E = 0.
A key consequence of (3.10) is that S(x) can never be a constant. In particular, imposing that (3.10) holds even when the coordinate x refers to the state with V − E = 0, forces the introduction of an additional term in the classical Hamilton-Jacobi equation. Then, one considers three arbitrary states, denoted by A, B and C, and imposes the condition coming from the commutative diagram of maps Implementation of such a consistency condition is equivalent to a cocycle condition that fixes the additional term to be the quantum potential [5]. The outcome is just the QHJE. Another feature of the above formulation is that the quantum potential is never trivial even in the case V − E = 0. In particular, a careful analysis of the quantum potential for a free particle with vanishing energy shows that the → 0 and E → 0 limits in the case of the free particle of energy E, leads to the appearance of the Planck length in the expression for the quantum potential Q of a free particle with E = 0 (1.1). It should be stressed that the present formulation leads to a well defined power expansion in for S. This is different with respect to the WKB approximation since S WKB is defined by 11) so that, in general, S WKB takes complex values. The formulation is also different with respect to the de Broglie-Bohm theory. Besides the case of real wave-functions illustrated above, also the quantum potential (1.1) turns out to be different. The difference also appears in the case of the free particle of energy E. Indeed, the solution of Eq.
Here the constants are chosen in such a way that S = ± √ 2mEx. Such a choice, fixed by the consistency condition that the non-trivial S 0 is obtained from S in the E → 0 limit, relates p-x duality, also called Legendre duality, and Möbius invariance of the Schwarzian derivative [5]. Another consistency condition comes from the classical limit. Since S cl = ± √ 2mEx, we have implying that the constants A, B, C and D depend on [5]. The above analysis shows that S is the natural quantum analog of the classical action. In particular, the formulation solves Einstein's paradox and the power expansion of S in is completely under control. Furthermore, it leads to a dependence of S on the fundamental constants, shedding light on the quantum origin of interactions. It also implies that if space is compact, then time parametrization cannot be defined [7]. The formulation, that follows from the simple geometrical principle (3.10), extends to arbitrary dimensions and to the relativistic case as well [15]. It reproduces, together with other features, such as energy quantization, the non existence of trajectories of the Copenhagen interpretation, without assuming any interpretation of the wave-function. 4 The WDW Hamilton-Jacobi equation with 3 R = 0 and Λ = 0.
Let us go back to the WDW Hamilton-Jacobi equation by considering the case 3 R = 0, Λ = 0, so that the WDW equation reduces to Setting Ψ = Ae i S , the WDW Hamilton-Jacobi equation reads Note that in this case the formulation does not suffer the well known problem of the WDW equation, due to the presence of the order two functional derivative at the same point: such an operator is in general ill-defined since it may lead to δ(0) singularities. On the other hand, the wave functional Ψ[g ij ] now depends linearly on g ij , so that the action of the second-order functional derivative on Ψ[g ij ] is well defined. We then have where with T jk (x) an arbitrary complex tensor density field of weight 1 and C a complex constant. The general expression of S is and for A we have A = |T g + C| .
(4.6) By (2.13) and (4.5), it follows that at the quantum level the momentum conjugate to g jk is so that the kinetic term in the WDW Hamilton-Jacobi equation reads (4.8) Note that, by (4.2), this also corresponds to −Q[g jk ]. Furthermore, one may easily check that such an expression of Q[g jk ] is just the functional analogue of the quantum potential of the free particle of vanishing energy (1.1).

Cosmological constant from the quantum potential
The discrepancy between the measured value of the cosmological constant and the theoretical prediction follows by considering Λ/κ 2 as a contribution to the effective vacuum energy density ρ ef f = ρ+Λ/κ 2 , where T µν = ρg µν . Considering the QFT vacuum energy density as due to infinitely many zero-point energy of harmonic oscillators, we get (here = c = 1) where Λ U V is the Planck mass. A result which is in complete disagreement with the estimation, based on experimental data, ρ ef f ≈ 10 −47 GeV 4 . A problem with the above derivation is that it is based on the perturbative formulation of QFT. This corresponds to use the canonical commutation relations of the free theory that selects the vacuum of the free theory. On the other hand, the true vacuum of nontrivial QFT's is highly non-perturbative and is not unitarily equivalent to the free one. As a matter of fact, perturbation theory erroneously treats the quantum fields evolving as the free ones between point-like interaction events. From the physical point of view, the role of renormalization is to iteratively change the parameters of the theory, that then will depend on the physical scale. In other words, perturbation theory is a way to mimic the interacting theory by a free one, with the parameters becoming scale dependent. It has been observed in [16] that the cutoff corresponding to the value of the cosmological constant may be related to an infrared/ultraviolet duality. In particular, the authors of [16], inspired by the Bekenstein bound S πM 2 P L 2 for the total entropy in a volume of size L 3 , proposed the following relation between the infrared cutoff 1/L and Λ U V An estimation of the infrared scale of QFT can be derived by considering the precision tests of the electron's anomalous magnetic moment a e . In this respect, as observed in [17], an estimate of the correction to the usual calculation imposed by the IR scale µ is Requiring that such an indeterminacy be smaller than the uncertainty of the theoretical prediction for a e gives µ ≤ 10 −2 eV , (5.4) which is the value corresponding to the cutoff that leads to the same order of magnitude of the experimental value of ρ.
The above analysis indicates that the cosmological constant is related to the infrared problem, a non-perturbative phenomenon concerning the structure of the vacuum which has physically measured consequences. For example QED finite transition amplitudes are obtained by summing over states with infinitely many soft photons. We saw that the quantum potential plays the role of intrinsic energy. In the case of the WDW equation such physical modes are naturally interpreted as a graviton condensate. Such a contribution should be identified with non-propagating degrees of freedom and without any reference to the matter content. This leads to the identification ρ vac = Λ/κ 2 . In this context, we stress that the vacuum energy is a purely quantum property and the absence of the kinetic term does not imply, as in the de Broglie-Bohm theory, the Einstein's paradox. The fact that the cosmological constant is a quantum correction to the Einstein tensor given in terms of the quantum potential is reminiscent of the von Weizsäcker correction to the kinetic term of the Thomas-Fermi theory. Furthermore, we note that the quantum potential also defines the Madelung pressure tensor. Now observe that the absence of propagating degrees of freedom implies that the quantum potential in (5.5) corresponds to the one of the WDW Hamilton-Jacobi equation without the kinetic term, that is S = 0 . We now adapt the analysis that led to (1.1) to Eq.(5.7). The main difference is that now the problem both the Planck length and the typical size of the observed universe. The latter should appear as a boundary condition for the WDW equation in the vacuum. By dimensional analysis, it follows that A should be a function of the ratio with L U a fundamental length describing the geometry of the Universe. The obvious candidate is the Hubble radius R H = c/H 0 = 1.36 · 10 26 m, whose size is of the same order of the radius of the observable universe and that, besides Λ, is the only quantity which is spatially constant. Nevertheless, since in the WDW equation there is no notion of time, it follows that A cannot depend on any time-dependent quantities, so that K = ✁ ✁ ℓ P R H = 5.96 · 10 −61 , (5.9) should be a space-time constant. Time variation of fundamental constants is a crucial and widely investigated subject [18][19] [20]. Eq.(5.9) implies an infrared/ultravilet duality suggesting that the geometry of the Universe is crucial for a formulation of Quantum Gravity.
We conclude by observing that very recently, in [21], it has been argued by a different perspective, that the formulation of the quantum Hamilton-Jacobi theory introduced in [5], could in fact be at the origin of the cosmological constant.