Finite Entanglement Entropy in Asymptotically Safe Quantum Gravity

Entanglement entropies calculated in the framework of quantum field theory on classical, flat or curved, spacetimes are known to show an intriguing area law in four dimensions, but they are also notorious for their quadratic ultraviolet divergences. In this paper we demonstrate that the analogous entanglement entropies when computed within the Asymptotic Safety approach to background independent quantum gravity are perfectly free from such divergences. We argue that the divergences are an artifact due to the over-idealization of a rigid, classical spacetime geometry which is insensitive to the quantum dynamics.


Introduction
One of the most remarkable, and in a way enigmatic, properties of Quantum Mechanics is the occurrence of entangled states and the possibility that local measurements instantaneously affect the result of local measurements far away. While deeply intriguing as a physical phenomenon in its own right, entanglement also received considerable attention from an "applied" perspective, being at the heart of many modern developments in quantum computation and information theory for example. An improved understanding of the entanglement structure of quantum many body systems allowed in particular developing new numerical algorithms which can help in lowering the computational effort of the simulations [1,2].
A frequently used quantity that can quantify the amount of entanglement, at least in pure quantum states, is the entanglement entropy. Let ρ = |ψ ψ| denote the density operator of an arbitrary quantum system in the pure state |ψ . We assume that the pertinent Hilbert space is a direct product H = H A ⊗ H B , and that we are only interested in predictions for measurements which affect the subspace H A alone. Such predictions are encoded in the reduced density matrix ρ A = Tr B [ρ], where Tr B denotes the partial trace over H B . Then, by definition, the entanglement entropy related to the A-B decomposition of the total system equals the von Neumann entropy of ρ A : (1.1) In practical calculation S A is often represented as the limit 2) and the replica trick is invoked in order to evaluate Tr [ρ n A ]. The latter consists in calculating Tr [ρ n A ] for positive integers n and then analytically continue it to a domain in the complex plane. If this step can be justified, calculating Tr [ρ n A ] can be seen to boil down to the evaluation of a certain partition function; it generalizes the ordinary partition function governing the quantum system considered in that it is defined over a modified, more complicated base space, a kind of Riemann surface which may cover the original base space more than once.
Consider for example a free quantum field on Minkowski space, with coordinates (t, x, y, z), and introduce a surface Σ by the condition t = 0, x = 0. This surface separates the time slice at t = 0 in two parts, x < 0 and x > 0. If we now define the subsystems A and B as comprised of the field degrees of freedom at x < 0 and x > 0, respectively, the ensuing entanglement entropy S A ≡ S is given by [3][4][5] Here Z δ is a standard partition function, albeit not on a Minkowski space but on on a conical spacetime with a deficit angle δ. Evaluating (1.3), ultraviolet (UV) divergences are encountered, and a short distance cutoff, ε, needs to be introduced, yielding In this formula, ν is a constant which depends on the type of field (ν = 1 for a scalar, for example) and A denotes the area of Σ. Hence, it is meaningful to speak of an entropy per area when Σ is infinitely extended. However, S/A suffers from an UV divergence, it diverges quadratically when the cutoff is removed (ε → 0). The result (1.4) is valid under more general conditions actually. On any (non-dynamical) curved spacetime, and for an arbitrary closed smooth surface Σ, equation (1.4) gives the leading order contribution to the entanglement entropy.
Clearly, the physical interpretation of (1.4) is hampered by its UV divergence which cannot be "renormalized away" straightforwardly. Nevertheless, ever since its first discovery [6][7][8], the fact that S is proportional to the surface area rather than the volume of the subsystem traced over has sparked considerable interest and research activities [3,[6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21][22]. One of the reasons is clearly the similarity of (1.4) and the Bekenstein-Hawking entropy in black hole thermodynamics, with A denoting the area of the horizon now [9][10][11]. This similarity inspired attempts to partially or fully explain S BH as an entanglement entropy, and thereby absorb the divergence of S in a renormalized Newton constant. (We refer to [5] for a comprehensive account.) The present paper is dedicated to the entanglement entropy (1.4) in its own right, i.e. without reference to black holes or other special systems. Trying to pin down the physical origin of its quadratic divergence, we are going to analyze what happens to the entanglement entropy when the above setting of quantum field theory on classical spacetimes is generalized to full-fledged background independent quantum gravity [23]. Concretely, we shall employ the Asymptotic Safety approach [24,25] to Quantum Einstein Gravity (QEG) [26][27][28][29][30].
As we are going to argue, the divergence present in the standard result (1.4) originates from the fact that it answers, or tries to answer, an unphysical question that could never arise in a real physical experiment. The over-idealization consists in considering "test fields" on an externally prescribed classical spacetime. Instead, if the entanglement is studied in a universe where the geometry is free to adjust itself dynamically according to the gravitational dynamics implied by Asymptotic Safety, the corresponding entropy turns out to be finite.
This is even the more remarkable as a number of quantum gravity models are known to fail in rendering the entropy finite [31]. It should be also emphasized that the proposed non-perturbative mechanism for achieving a finite entanglement entropy does not rely on "hiding" its divergences in Newton's constant or similar couplings which parametrize the action functional.
The rest of this paper is organized as follows. As a preparation we briefly recall in section 2 the derivation of equation (1.4) for classical spacetimes. We also show how it relates in a natural way to the Effective Average Action (EAA), the scale dependent functional that is used in the Asymptotic Safety program. Then, in section 3, we proceed to QEG and analyze the entanglement entropy in a universe with a scale dependent spacetime geometry which is governed by an asymptotically safe renormalization group flow.
2 Entanglement entropy on a rigid background (A) For any free matter field Φ, governed by a quadratic action S [Φ], the evaluation of the entropy by means of equation (1.3) consists in computing a one-loop determinant on a locally flat spacetime with a conical defect, log Z δ = − 1 2 log det S (2) . Here S (2) denotes the Hessian operator of S. For a scalar, say, S (2) = − + m 2 . Standard manipulations lead to the regularized proper time representation (2) .
Here the length parameter ε is introduced as a short distance cutoff in order to cure the divergence of the t-integral at the lower limit. So the essential ingredient we need is the heat kernel K δ (t) as a function of the deficit angle δ, For a real, massless, minimally coupled scalar, the relevant part of K δ (t) can be found to be [32,33]: Here we set AL 2 ≡ d 4 x for the 4D Euclidean volume. Using (2.3) in (2.2) one obtains exactly the anticipated result for the entanglement entropy, equation (1.4), with ν = 1 for the real scalar. Other systems of (higher spin) free fields lead to an analogous formula with other values of the finite constant ν, see [5] for a detailed discussion.
(B) As a further preparation for the case of quantum gravity let us explain how the above standard calculation should be interpreted within the general framework of the EAA and the functional renormalization group [34].
The EAA for a scalar on a classical spacetime, Γ k [Φ], can be seen as the ordinary effective action for a field whose bare action under the functional integral has been augmented by a mode cutoff term: implements an infrared (IR) cutoff by giving a non-zero mass square R k = k 2 the low momentum modes contained in Φ, while annihilating the others, R k = 0. This modification leads to the following variant of the partition function on the cone: A simple way of analyzing (2.4) is to exploit that the precise form of R k is largely arbitrary. In any case R k will leave the contribution to the trace coming from the high momentum modes untouched, while that of the low momentum modes receives an additional factor ρ (t) ∼ e −k 2 t , or a qualitatively similar one, which then suppresses the integrand of the t-integral at large t 1/k 2 . Hence, rather than choosing a specific R k and computing the resulting ρ (t), we may equally well select right away a suitable function ρ (t) with the correct properties, ρ (t 1/k 2 ) ≈ 0 and ρ (t 1/k 2 ) ≈ 1. The simplest choice is the step function ρ (t) = θ (k −2 − t) which, of course, amounts to a version of the Schwinger's proper time regularization [35,36]. Applied to (2.4) it yields with the same kernel K δ (t) as in (2.1).
By taking the k-derivative of (2.5) we can get rid of the UV cutoff ε at this point: which evaluates to the following simple RG equation for the, now scale dependent, entanglement entropy: Here we wrote A ≡ A [ḡ] to emphasize the fact that A is a proper area with respect to a classical, externally prescribed metric,ḡ αβ . By adopting the discussion in [5] it is easy to see that (2.6) holds not only in flat space but also yields the leading scale dependence on curved classical spacetimes with any metricḡ αβ . Furthermore, equation (2.6) is equivalent to the RG equation discussed in [21] which employs a more general cutoff.
(C) At this point we want to emphasize that in the EAA framework one usually regards the RG equations, requiring no UV cutoff, as having a more fundamental status than the functional integral from which they are derived in a formal way only. 1 In particular this is the stance taken in the Asymptotic Safety program. This concerns not only the RG equations for the running couplings which parametrize the action functional Γ k itself, but also the RG equations for the co-evolving running parameters appearing, for example, in composite operators or observables that do not correspond to terms in Γ k [37][38][39]. In this sense, the entanglement entropy S (k) is an example of the latter case. Conceptually speaking, it is an "observable" that has a scale dependence in its own right, its RG running depends on the EAA, Γ k , at least in sufficiently complex truncations.
Like the EAA itself, the co-evolving quantities, too, are defined in the "continuum limit" on the basis of their RG flow. Hence the UV renormalization problem translates into the task of finding complete, i.e. fully extended, solutions (trajectories) to all RG equations, those of the co-evolving quantities included [28].
Let us illustrate this shifted viewpoint by the example of S (k). While, conceptually speaking, we consider S (k) a co-evolving quantity with respect to some trajectory of Γ k , equation (2.6) happens to be simple enough to require no input from Γ k to be integrated: 1 That is, in presence of a UV regulator.
This difference of two entropies is the contribution of the field modes with momenta in the interval [k 1 , k 2 ]. We are particularly interested in the limits k 1 → 0 and k 2 → ∞. The first limit is unproblematic, yielding Obviously k 2 plays the role of the UV cutoff here. In the jargon of standard field theory one would refer to S (k 2 ) as the "bare", and to S (0) as the "renormalized" or "physical" quantity. From the EAA perspective, equation (2.8) corresponds to a finite segment of the RG trajectory, {Γ k , k ∈ [0, k 2 ]}, whose lower endpoint Γ 0 = Γ equals the standard effective action (with a UV cutoff), having an associated entropy S (0). It remains to take the second limit, k 2 → ∞, in which Γ k→∞ is known to approach the classical (bare) action, S, essentially. The natural value of the associated entropy is S (k 2 → ∞) = 0 since Γ k→∞ defines the limiting case of the effective field theory with no quantum fluctuations integrated out yet. Now, ideally, we would let k 2 → ∞ in equation (2.8), keeping S (k 2 ) = 0 fixed, and thereby obtain a finite physical value for the entropy, S (0). But clearly this is thwarted by the k 2 2 -dependence on the RHS of (2.8) which causes S (0) ∝ k 2 2 to diverge. In this manner we re-discover the quadratic divergence of the entanglement entropy in the framework of the EAA. It is signalled by the non-existence of an RG trajectory that extends to all k ∈ [0, ∞).
Next let us see how the situation changes in quantum gravity.

Entanglement entropy in QEG
Up to now we considered matter fields in a prescribed classical background spacetime. Now we go on to Quantum Einstein Gravity (QEG) as defined by a complete, asymptotically free RG trajectory Γ k [h αβ , Φ;ḡ αβ ], k ∈ [0, ∞). As usual, h αβ andḡ αβ denote the metric fluctuation and the background metric, respectively. 2 While our arguments are general, in explicit calculations we will employ the single-metric Einstein-Hilbert truncation coupled to the matter fields Φ; the only running gravitational couplings are the Newton's constant G (k) and the cosmological constant Λ (k) then [25,40]. It is assumed that the matter fields combined in Φ are such that they do not destroy the Asymptotic Safety of pure gravity [96]. We may also assume that the reconstruction problem [41] has been solved within the truncation considered. As a result, we have a regularized functional integral at our disposal, 2 We suppress the Faddeev-Popov ghosts here. which approaches a well defined limit when its UV regulator is removed (Λ → ∞) and which reproduces the asymptotically safe RG trajectory. (See [41] for a detailed discussion.) (A) What is the meaning of the calculation in section 2 in the Asymptotic Safety context, if any? First of all, as it stands the result for the entanglement entropy refers to a free field. So let us assume that among the matter fields Φ there is at least one that appears quadratically in the fixed point action 3 , and let us compute its contribution to the entanglement entropy.
In the gravitational EAA approach, Background Independence is established by studying the dynamics of the metric fluctuation h αβ = g αβ −ḡ αβ and matter fields on all backgrounds simultaneously, i.e.ḡ αβ should be left completely arbitrary in the calculation of Γ k and the concomitant running quantities.
In this spirit, we now interpret equation (2.6) as the result of a calculation in an arbitrary but fixed, classical background metric,ḡ αβ . So at this point k∂ k S (k) should be understood as a functional ofḡ αβ . (B) Quantum gravity, and specifically QEG, differs most fundamentally from any standard quantum field theory in that it must dynamically generate the spacetime geometry in which all other physics is going to take place then. In particular the theory should be able to distinguish physically realistic, stable states |ψ from unstable or impossible ones that would never be seen in Nature. The EAA encodes information about physically acceptable states via the metric expectation value it gives rise to, ψ|ĝ αβ |ψ = g αβ .
In the background field formalism, knowing Γ k , we can search for self-consistent background metricsḡ αβ = (ḡ sc k ) αβ . By definition, when the h αβ fluctuations (and the matter fields) are quantized in a self-consistent background,ĥ αβ has vanishing expectation value, ĥ αβ ≡ h αβ = g αβ −ḡ αβ = 0, and so g αβ =ḡ αβ . In these special backgrounds the quantum fluctuations are particularly tame, and we may regard g αβ =ḡ αβ = (ḡ sc k ) αβ as the expectation value of the metric operator in a physically realistic state. 4 Self-consistent backgrounds are found by solving the tadpole equation [44]: Moreover, thanks to the tadpole equation (3.1), the self-consistent background can also be employed to compute the partition function of the system. A detail discussion regarding the properties of such a partition function can be found in [44]. The so computed partition function is then a functional of the self-consistent background: Z [ḡ sc k ]. According to the discussion of section 2, in order to compute the entanglement entropy via the replica trick, one must introduce a deficit angle in the geometry of the system and remove the singularity at the end of the calculation. Namely, one must evaluate the quantity Z ḡ sc k,δ , where δ is the deficit angle. Note that in general there is no reason for Z ḡ sc k,δ to be determined directly by Z [ḡ sc k ] since the introduction of the deficit angle changes the topology of the spacetime and a new calculation is required. 5 Focusing on the Einstein-Hilbert truncation now, the tadpole equation happens to have the same structure as the classical Einstein equation: Since under rescalings of the metric the Ricci tensor behaves as R µ ν (c 2ḡsc k ) = c −2 R µ ν (ḡ sc k ), it follows that solutions to (3.2) respond to changes of the cosmological constant in such a way that Λ (k) (ḡ sc k ) αβ = const. It proves convenient to introduce an arbitrary normalization scale µ in order to write this relation as Λ (k) (ḡ sc k ) αβ = Λ (µ) ḡ sc µ αβ , or as In the second equality we inserted the dimensionless cosmological constant λ (k) = Λ (k) /k 2 , and correspondingly for k = µ. Likewise we redefine the field variables by writing them as dimensionless multiples of the cutoff, or appropriate powers thereof. The dimensionless metric coefficients are theñ g αβ ≡ k 2 g αβ andg αβ ≡ k 2ḡ αβ , and so, from (3.3): Let us recall that in the case of an asymptotically safe UV limit it is the dimensionless couplings that assume fixed point values. For instance, λ (k) approaches a finite number lim k→∞ λ (k) = λ * . Accordingly, it is the dimensionless form of the tadpole equation that continues to be meaningful in the limit of k → ∞, admitting a finite solution (C) After these preparations we return to the entanglement entropy and reconsider the calculation of section 2 within QEG. In order to obtain the corresponding entropy S QEG (k) we chooseḡ αβ in the final result for a rigid background, equation (2.8), to be a self-consistent one for the corresponding scale, (ḡ sc k ) αβ . In this manner we obtain the entanglement entropy related to a system of fields inhabiting a spacetime which is indeed physically realizable, or, at the very least, is much closer to a realizable one than it would be on a generic background. Clearly this is a necessary prerequisite if the entropy computed is to have a physical meaning, and hence a reason to be finite.
Thus we obtain from equation (2.8), writing k = k 2 from now on, Here we also exploited the fact that the area scales as A [c 2 g αβ ] = c 2 A [g αβ ]. As a result, the entropy difference (3.6) depends on the scale k only via the dimensionless self-consistent metric, that is (ḡ sc k ): Remarkably enough, when we let k → ∞ the quantity (3.7) approaches a well defined limit S QEG (0) − S QEG (∞) ≡ ∆S QEG : This perfectly finite result for the entanglement entropy in QEG is our main result.
(D) Using (3.5) we may rewrite (3.8) in the more practically applicable forms is the dimensionful proper area measured with the background metric at the normalization point µ.
(E) Let us consider the following thought experiment to determine the entanglement entropy related to a given surface Σ. In order to measure the area of Σ we must choose a specific "yard stick" (or "microscope"); it is characterized by a certain minimal length which it is able to resolve, ℓ. The actual measurement consists in using this yard stick to partition Σ in little squares of side length ℓ, and counting the resulting "pixels"; let N (ℓ) denote their total number. We may assume that the best possible effective field theory description of this measuring procedure is obtained from that EAA which has its scale k, or in the present case µ, adapted to the scale of the experiment, µ ≈ ℓ −1 . Hence the measurement is described as taking place in the classical spacetime geometry with g sc µ µ=ℓ −1 . Recall also [45] that the length scale k −1 pertaining to Γ k [h;ḡ] is a proper length with respect to its second argument,ḡ αβ . As a consequence, we can say that the little squares we counted have the proper area ℓ 2 = µ −2 with respect to the optimum self-consistent background metric g sc ℓ −1 . Hence the result of counting pixels, N (ℓ), has the following interpretation within the effective field theory: (3.11) If we accept this interpretation, along with equation (3.9), we can deduce the desired entropy from our pixel count: Again, N (ℓ) will not be independent of ℓ in general, but the product λ (1/ℓ) N (ℓ), and hence the entropy, are ℓ-independent. This has an important consequence: If the RG trajectory, and in particular the function λ (k) are known, we can determine the entropy on the basis of the formula (3.9) by performing the experiment on any scale we like. This may lead to different numbers of pixels, but the resulting entropy is always the same provided the running of the cosmological constant is taken into account properly.
For example, we could decrease ℓ to the point that µ = 1/ℓ → ∞ enters the scaling regime of the UV fixed point so that λ (1/ℓ) → λ * . This limit gives rise to the following representation of the entropy: As soon as the RG trajectory reaches the fixed point regime, λ (µ) stops running. Hence, by (3.10), the area scales as A (µ) ∝ 1/µ 2 so that N (ℓ) becomes independent of ℓ; it no longer increases when ℓ is decreased even further.

Summary
When defined on a rigid classical spacetime geometry, quantized matter fields are known to give rise to an entanglement entropy which is proportional to the area of the entangling surface, with a factor of proportionality which is quadratically divergent though. In this paper we employed instead a background independent approach to quantum gravity and regarded the entanglement entropy as a scale dependent quantity which RG-evolves in parallel with the Effective Average Action. The latter controls the geometry of spacetime at the mean field level, among other things, and in particular it determines the self-consistent background geometries for each scale. The leading term of the entanglement entropy in those geometries turned out to be perfectly finite. The cutoff dependence of the entropy is precisely cancelled by the RG running of the metric in the infinite cutoff limit.
While, for illustrative purposes, we considered the Einstein-Hilbert truncation here, the finiteness of the entropy is a direct consequence of Asymptotic Safety as such and it applies also to more refined truncations schemes.
All that is required is the scaling behaviour of the metric corresponding to a non-Gaussian UV fixed point. Hence the finiteness of the leading entropy term is obtained analogously in d spacetime dimensions for surfaces Σ of co-dimension two.
From the perspective of Asymptotic Safety, the notorious quadratic UV divergence seems to occur because one is asking an unphysical question, and tries to compute a quantity that never could be measured in Nature, not even in principle. The divergence disappears as soon as we admit that, at asymptotically high scales, spacetime is actually fractal like [117], and carries a metric which strongly depends on the "length of the yard stick" that is used to probe the spacetime.