Geometric Entropy: Black Hole Background

Transcription

1 Geometric Entropy: Black Hole Background Frank Wilczek Center for Theoretical Physics, MIT, Cambridge MA USA March 13, 2014 Abstract I review the derivation of Hawking temperature and entropy through the consideration of Euclidean black holes. It is tempting to think that black hole entropy might arise as an entanglement entropy resulting from tracing over degrees of freedom hidden behind the horizon. Pursuing that idea leads us into adventures involving geometric entropy. While the black hole entropy as entanglement idea remains speculative, the concept of geometric entropy has found several other applications, and may have an important future. This is the first of three notes on geometric entropy. It covers some of the background motivation from black hole physics, sets up the replica trick as a geometric construction, and connects those two ideas. The second note will review the actual evaluation of geometric entropy in some simple cases, and the third will be a brief survey of ongoing projects that apply geometric entropy to physical issues. The presentation in this part is adapted from [2]. Interest in a special form of entanglement (von Neumann) entropy, socalled geometric entropy, first developed in connection with black hole physics. There is considerable though, to be sure, entirely theoretical evidence that there is no sharp distinction between black holes and other forms of matter, and in particular that black holes can come to equilibrium with appropriate heat baths, and thereby meaningfully assigned temperature and entropy 1. The interpretation of that entropy in microscopic terms remains obscure, however, both in regard to the nature the states involved and in regard to the information stored. An appealing idea, at first hearing, is 1 At the classical level of black holes appear radically different from other forms of matter, so the unification must involve quantum gravity 1

2 that the entropy is associated with information that has become hidden behind the event horizon, and perhaps, more specifically, with the inevitable entanglement of quantum states across that boundary. It is natural to define geometric entropy S(A) to be the special case of von Neumann entanglement entropy, in the situation where we take the ground state, or some other characteristic state (especially, thermal equilibrium) of a quantum system with localizable variables, and trace over the variables outside a geometric region A. More careful analysis, as we shall see, suggests that there is a contribution of this kind to the entropy, but that it is subdominant, at least for large black holes. The concept of entropy associated with carving out holes geometric entropy has, however, taken on a life of its own. It affords a viewpoint on quantum systems that is different from, and complementary to, correlation functions on the one hand or bulk thermodynamics and transport on the other. Although direct experimental manifestations have not yet been identified, valuable and unique insight into subtle forms of ordering, quantum phase transitions, and the design of algorithms for calculating properties of quantum systems has been obtained through the study of geometric entropy. 1. Black Hole Temperature and Entropy The thermal partition function Z(β) = Tr e βh (1) of most conventional (non-gravitational) quantum systems, including relativistic quantum field theories, can be represented as path integrals over imaginary-time histories where the configurations are periodic with imaginary time period β = 1/T. This follows simply by writing e βh = (e βh/n ) N for large N and repeatedly sandwiching complete sets of states 2. For most quantum field theories of interest, we can express the measure in this path integral in terms of the Euclidean i.e., imaginary time action. For example, in a relativistic scalar field theory, we have β βh = i = 0 dτ d 3 x 1 2 (( φ τ )2 + ( φ) 2 ) + V (φ)) d 4 xl E (2) 2 For anticommuting Grassmann fields, used to represent fermions, it is appropriate to apply antiperiodic boundary conditions. 2

3 where the Euclidean action density L E is obtained through substituting τ = it in the standard Lagrangian density. The factor i, together with the factor i that appears accompanying the Lagrangian density L in the path integral for real-time evolution indicates, at least heuristically, that e βh is the analytic continuation of e i L, thus linking thermal to real-time physics. In his book with Hibbs [3], Feynman expressed the thought that there ought to be a more direct way to forge this formal connection. That would be desirable, not only as a matter of intellectual hygiene, but also to better ground the sort of manipulations in which we are about to indulge, where we extrapolate that formal connection to quantum gravity, and guesswork supplants deduction. Inspired by the significance of Euclidean fields, we consider the Euclidean version of black holes. Here I will consider only Schwarzschild black holes, but similar results hold for holes with angular momentum (Kerr) and/or electric charge (Reissner-Nordstrom). The Euclidean Schwarzschild metric is ds 1 = (1 2M r ) dτ 2 + (1 2M r ) 1 dr 2 + r 2 dω 2 (3) where dω 2 = dθ 2 +sin 2 θdφ 2 is the unit sphere metric. (We have chosen units with G = c = 1.) In Lorentz-Minkowski signature the horizon at r = 2M is, famously, a coordinate singularity. It can be removed by an appropriate choice of coordinates, and does not represent any non-smoothness in geometrical quantities. In Euclidean signature the same cancellations do not occur, and we must investigate the geometry near r = 2M anew. Near r = 2M, we can approximate the metric as ds 2 u 2M dτ 2 + 2M u du2 + M 2 dω 2 u r 2M (4) Introducing this becomes v = 8Mu (5) ds 2 v2 16M 2 dτ 2 + dv 2 + M 2 dω 2 (6) 3

4 We see that the near-horizon geometry factorizes into the product of a sphere of radius M and a τ r structure which looks like a plane parameterized in polar coordinates, with τ/4m playing the role of the angular variable. If the periodicity of that variable is 2π, the geometry will be regular; otherwise it will exhibit a conical singularity. Note that in either case, the region behind the horizon has disappeared. The regular geometry occurs when τ is periodic with period 8πM. Identifying this periodicity with an inverse temperature, we are led to β = 8πM T = 1 8πM This T matches the Hawking temperature of the black hole, which originally was calculated in quite a different way. Restoring units, we find T = hc3 (8) 8πGM Given that result for the temperature, we can calculate a thermodynamic entropy (7) S(M) = M 0 dm T (m) = 4πM 2 = 1 4 4π(2M)2 = 1 4 A (9) where A is the area of the two-sphere over the horizon. Restoring units, we find S(M) = c3 4G h A (10) One can also attempt to calculate the entropy directly from the partition function, as S = ( β + 1) ln Z (11) β Here a striking, and possibly profound, result emerges. When we evaluate our functional integral expression for Z in the black hole background, we find that the classical Euclidean action S(M) [4] and thus ln Z ln e S(M) (12) 4

5 and finally S = S(M)! Deriving that answer 3 requires some care. Indeed, on the face of it we would expect the Euclidean action gr? 0 (13) since R = 0 for a solution of the vacuum Einstein equations. There is the subtlety, however, that R contains second derivatives of the metric (as opposed to powers of the first derivatives), and the proper action is the expression we get formally integrating by parts, so that we get only first derivatives. The difference is a surface term, which supplies the entire non-zero answer. Reviewing the details would entail a significant digression, and so I refer you to [4]. It is impressive and remarkable that closely results hold for all black holes whether spinning or charged, and including truly thermal (non-extremal) cases. In the present context, the most salient point is perhaps disappointing: The calculation appears to make no reference whatsoever to entanglement entropy. Indeed, there is a difficulty even at the level of dimensional analysis. Entanglement entropy, in its definition, contains no reference to h, while S(M) h 1, as is typical for classical entropy. Now in an interacting system it is possible that the coupling-constant dependence of entanglement entropy brings in powers of h, so some reformulation of the problem might embody the vision of black hole entropy as entanglement, but no such formulation is presently available 4. It is certainly suggestive, at an extremely heuristic level, that the entropy emerges from a boundary term, and that we have, by going to periodic imaginary time, changed the boundary conditions at the horizon. Perhaps an appropriate surface theory, with a classical phase space whose volume is proportional to the horizon area, underlies the state counting. On the positive side, there is a straightforward connection between entanglement entropy and quantum corrections to black hole entropy, as we will now discuss. 2. Geometric Entropy: Replica Trick The replica trick was first developed in connection with the treatment of polymer folding problems and is an important tool in the theory 3 Deriving it, that is, formally. 4 There are relevant, but complex and scattered, results on model black holes in string theory, which I will not review here. 5

6 of spin glasses and systems with random disorder more generally. In those situations, one is faced with the problem of forming thermal averages over a fixed sample of the random disorder, which is then subject to averaging. To get thermodynamic quantities, such as the free energy, one must evaluate an average of ln Trρ, where ρ is the thermal density matrix. Formally, as we have seen, it is often simpler to evaluate Trρ and its powers, which are conveniently expressed as functional integrals. If we can manage the evaluation of such powers, then we can get at the logarithm through ln Trρ = lim n 0 d dn Trρn (14) The n th power represents n copies, or replicas, of the system. When we are dealing with quenched disorder, averaging over the disorder induces a kind of coupling among the replicas, and there is even the possibility of replica symmetry breaking, which plays an important role in the theory of spin glasses [1]. Our use of the trick, though inspired by the same relationship between ln and n, is a bit different. We are primarily interested in evaluating von Neumann entropy, so we shall use Tr ρ ln ρ = lim( d + 1) ln Tr ρn n 1 dn ρ ρ Tr ρ (15) where we have allowed for use of unnormalized density matrices ρ at intermediate stages. Now let us return to the idea of carving out a hole, in the simplest possible context of scalar field theory on a line, and a hole which is the negative real axis. We want to consider the density matrix of the ground state, tracing over degrees of freedom on the negative real axis. Wave functions Ψ of states are functionals of spatial field configurations φ(x) of the scalar field, Ψ = Ψ(φ(x)). We can represent the wave function of the ground state by a functional integral over imaginary time according to Ψ 0 (φ(x)) = Dψ(z) e 1 2 d 2 z (( ψ τ )2 +( ψ x )2 +m 2 ψ 2 ) (16) with the following understandings: 6

7 z = (x, τ) τ runs over τ 0 The field ψ(z) is integrated subject to the boundary conditions that it vanishes as τ and ψ(x, τ = 0 ) = φ(x) (17) We can consider this functional integral as implementing evolution of a configuration-independent state at τ = to the minimum energy state at τ = 0. Since that evolution is governed by the Euclidean action e Hτ, the ground state emerges. We can get another copy of the ground state wave functional by a similar expression, where now we integrate over positive values of τ and impose boundary conditions at τ = 0 +. Thus the density matrix for two functional φ 1 (x + ), φ 2 (x + ), tracing over the degrees of freedom on the negative x axis, is given by a functional integral ρ(φ 2, φ 1 ) = = Dψ(z) e 1 2 d 2 z (( ψ τ )2 +( ψ x )2 +m 2 ψ 2 ) (18) with the modified understandings: τ is unrestricted The field ψ(z) is integrated subject to the boundary conditions that it vanishes as τ and ψ(x +, τ = 0 ) = φ 1 (x) ψ(x +, τ = 0 + ) = φ 2 (x) (19) The same reasoning allows us to express powers of the density matrix! To get ρ n, we take the functional integral over an n-sheeted covering of the plane, with a discontinuity of the form Eqn. (19) across the final copy (only) of the positive axis. The n-sheeted covering can be considered as a deficit angle 3. Comparison δ = 2π(1 n) (20) Returning to the black hole situation: If we impose the wrong periodicity β on our solution, instead of the Hawking value β H, we have a deficit angle δ = 2π β H β β H (21) 7

8 Now we can compare the operations that lead from partition function to entropy in the two cases lim β β H and observe that they are the same. β β + 1 = lim δ 0 2π d dδ + 1 lim d n 1 dn + 1 = lim 2π d δ 0 dδ + 1 (22) In the Euclidean Schwarzschild background the radial coordinate extends only over positive values, just as in the geometric entropy. The geometry is not that of flat space, but in principle we could (and should) do a corresponding calculation for the realistic geometry. We see that entanglement entropy does contribute to the black hole entropy, when we consider the contribution of additional quantum fields, besides the metric field. Their contribution, however, appears to supplement, rather than to replace, the semi-classical S(M). References [1] parisi [2] C. Callan and F. Wilczek Phys. Lett. B (1994). [3] R. Feynman and A. Hibbs Quantum Mechanics and Path Integrals, section 10.5, and especially page 296 (McGraw-Hill, 1965). [4] G. Gibbons and S. Hawking Phys. Rev. D (1977). 8