Black-hole information puzzle: A generic string-inspired approach
aa r X i v : . [ h e p - t h ] J a n Black-hole information puzzle: A genericstring-inspired approach
Hrvoje Nikoli´cTheoretical Physics Division, Rudjer Boˇskovi´c Institute,P.O.B. 180, HR-10002 Zagreb, Croatia [email protected]
October 26, 2018
Abstract
Given the insight steming from string theory, the origin of the black-hole(BH) information puzzle is traced back to the assumption that it is physicallymeaningful to trace out the density matrix over negative-frequency Hawkingparticles. Instead, treating them as virtual particles necessarily absorbed by theBH in a manner consistent with the laws of BH thermodynamics, and tracingout the density matrix only over physical BH states, the complete evaporationbecomes compatible with unitarity.
PACS: 04.70.Dy
The semiclassical description of black-hole (BH) radiation [1] suggests that an initialpure state evolves into a final mixed thermal state [2]. A transition prom a pureto a mixed state is incompatible with unitarity of quantum mechanics (QM), whichconstitutes the famous BH information puzzle. The attempts to restore unitarity canbe divided into two types (for reviews, see, e.g., [3]). In the first type, the black holedoes not evaporate completely, but ends in a Planck-sized remnant that contains theinformation missing in the Hawking radiation. The problem is that such a light objectshould contain a huge amount of information, which seems unphysical. In particular,light objects that may exist in a huge number of different states should have a hugeprobability for creation in various physical processes, which, however, is not seen inexperiments. A variant of the remnant scenario is the creation of a baby-universenot observable from our universe, but such an idea remains rather speculative. In1he second type, the black hole evaporates completely, but the radiation is not ex-actly thermal. Instead, there are some additional subtle correlations among radiatedparticles. It is argued that this requires a sort of nonlocality not present in standardquantum field theory (QFT), suggesting that quantum gravity should contain somenew nonlocal features.The most promising candidate for a consistent theory of quantum gravity is stringtheory. Indeed, it provides new insights on BH thermodynamics (see, e.g., [4, 5] forreviews). In particular, it provides a unitary description of BH radiation and offersa microscopic explanation of the BH entropy proportional to the surface. It alsocontains some nonlocal features that might explain the desired deviation from exactthermality. Nevertheless, the theoretical description of the mechanism of BH radiationin string theory (see, e.g., [6]) seems completely different from that in the conventionalsemiclassical theory, so it remains difficult to see where exactly the semiclassicalanalysis fails. Thus, it would be desirable to understand a generic property that alarge class of models of quantum gravity, including string theory, should possess inorder to save the unitarity of BH radiation. The aim of this paper is to find sucha generic resolution of the BH information puzzle, without using any explicit modelof quantum gravity. We find that neither a new sort of nonlocality (for the caseof complete evaporation) nor a huge amount of information in a light remnant (forthe case of a remnant scenario) is needed. In fact, we find that no new unexpectedproperty of physical laws is required. Instead, the standard rules of QM applied toblack holes in a generic and intuitively appealing manner turn out to be sufficient.
First, let us observe that a thermal distribution of particles is not necessarily in-compatible with a possibility that these particles are in a pure state. For a simpleexample, consider a single quantum harmonic oscillator (with the frequency ω ) in thestate | ψ i = P n f ω,n | n i , where n = 0 , , , . . . and f ω,n = q − e − βω e − βωn/ . (1)Clearly, | ψ i is a pure state. Yet, the probabilities of different energies E = ωn are pro-portional to e − βE , which corresponds to a thermal distribution with the temperature T = 1 /β . The density matrix ρ = | ψ ih ψ | can be written as ρ = X n | f ω,n | | n ih n | + X n = n ′ f ω,n f ω,n ′ | n ih n ′ | . (2)The first (diagonal) term represents the usual mixed thermal state. The second (off-diagonal) term is responsible for the additional correlations steming from the factthat the state is pure. When a simple system (in this case, the single harmonicoscillator) interacts with an environment with a large number of unobserved degrees2f freedom, then, in practice, the presence of the second term is unobservable. Thus,for all practical purposes, the state can be described by the first term only. In QMthis is known as the phenomenon of decoherence (for a review see, e.g., [7]). Thus,decoherence provides a mechanism for an effective transition from a pure to a mixedstate | ψ ih ψ | decoher −→ X n | f ω,n | | n ih n | . (3)It does not involve any violation of unitarity at the fundamental level. As we shall see, the observations above will play a role in our resolution of the BHinformation paradox. Indeed, the role of decoherence in BH thermodynamics hasalready been discussed in [8]. Nevertheless, decoherence is not the main part of ourresolution. To see the true origin of the BH information puzzle, we start from the factthat standard semiclassical analysis based on the Bogoliubov transformation describesHawking radiation as particle creation in which the initial vacuum | i transforms toa squeezed state [9] | i squeeze −→ | ψ i squeeze , (4)where | ψ i squeeze = Y ω X n f ω,n ( M ) | n − ω i ⊗ | n ω i , (5)and, for massless uncharged spin-0 particles, f ω,n ( M ) are given by (1) with β ≡ πM ,where M is the BH mass. The product is taken over all possible positive values of ω . The state | n ω i represents on outgoing state containing n ω particles, each havingfrequency ω , so that their total energy is E = ωn ω . Similarly, | n − ω i represents n − ω ingoing particles, each having negative frequency − ω . In our notation, the directproduct ⊗ separates the inside states on the left from the outside states on the right.At this level the total energy is not yet conserved, as the energy of the negative-frequency states is also positive, in the sense that the sign of their energy is the sameas that of the interior matter determining the BH mass M . The conservation ofenergy is provided by another mechanism, namely by renormalization of the energy-momentum tensor implying a flux of negative energy across the horizon into the blackhole [9]. The overall effect is that the BH mass decreases, such that the total energyis conserved. However, owing to the creation of negative-frequency particles thatcarry information, the information content of the black hole increases despite thefact that its mass decreases. Does it contradict the first law of BH thermodynamics?Not necessarily, if the BH entropy proportional to the BH surface (and thus to M )is interpreted merely as the part of BH information that is available to the outsideobserver. However, string theory suggests a very different interpretation of BH en-tropy – the entropy associated with counting of the internal degrees of freedom ofthe black hole, independent on the knowledge of an outside observer. Thus, from the3tring-theory point of view, the information carried by the negative-frequency parti-cles should be unphysical . Indeed, the physical mechanism of BH radiation in stringtheory does not rest on the Bogoliubov transformation, and hence does not lead tocreation of particles in the BH interior [6]. Thus, our idea is to modify the semiclassi-cal description of particle creation, in a manner that removes the negative-frequencyparticles from physical states.For states | n − ω i we find convenient to introduce a negative effective “renormal-ized” energy E = − ωn − ω , without changing the information content of these states.This makes energy conserved already at the level of (5), making the analysis simpler.The product over ω shows that states of the form | n ω i| n ω ′ i · · · with total energies E = ωn ω + ω ′ n ω ′ + . . . also appear. Thus, it is convenient to rewrite (5) as a sumover energy eigenstates | ± E, ξ i| ψ i squeeze = X E X ξ d E,ξ ( M ) | − E, ξ i ⊗ |
E, ξ i , (6)where ξ labels different states having the same outside or inside energy ± E , and thesum is taken over non-negative values of E . The coefficients d E,ξ can be expressed interms of f ω,n , but the explicit expression will not be needed here. The squeezed state(5) is a pure state and the transition (4) is unitary [10]. Consequently, the densitymatrix constructed from (6) is pure. However, an outside observer cannot observe theinside states, so the density matrix describing the knowledge of the outside observeris given by tracing out the inside degrees of freedom of the total density matrix.Applying this to (6), one obtains ρ out = X E X ξ | d E,ξ ( M ) | | E, ξ ih E, ξ | , (7)which is a mixed state. However, we have argued that the negative-energy states arenot physical, which means that the mixed thermal state (7) is obtained by tracingout over unphysical degrees of freedom. Hence, this mixed thermal state may alsobe unphysical. A physical density matrix should be obtained by tracing out overphysical (but unobserved) degrees of freedom. The difference between unphysicaland unobserved degrees is in the fact that the former cannot be observed even inprinciple, by any observer.The unphysical negative-energy particles can be intuitively viewed as virtual par-ticles analogous to those appearing in Feynman diagrams of conventional perturbativeQFT. They cannot exist as final measurable states. Instead, they must be absorbed byphysical states. In our case, the physical object that should absorb them is the blackhole. To give a precise description of this process of absorption, one should invokea precise microscopic theory that presumably includes a quantum theory of gravityas well. Nevertheless, the essential features of such an absorption can be understoodeven without a precise microscopic theory. For simplicity, we study uncharged andunrotating black holes. Thus, we assume that a black hole with a mass M can bedescribed by a quantum state | M ; χ M > , where χ M labels different BH states havingthe same mass M . We assume that the number of different states increases with M and that there is only one state with mass M = 0, i.e., that | χ > = | > . In4articular, such an assumption is consistent with string theory asserting that entropyof the internal BH degrees of freedom is proportional to the surface, i.e., to M . It isalso consistent with a more naive possibility that the entropy is proportional to thevolume, i.e., to M . In fact, proportionality of entropy to the surface rests on thevalidity on the Einstein equation, while thermal particle creation from a horizon isa much more general phenomenon [11]. As our analysis will not depend on validityof the Einstein equation, we will not be able to specify the exact number of stateswith mass M . For our purposes, it is sufficient to assume that the absorption ofnegative-energy particles takes a generic form | M ; χ M > | − E, ξ i absorp −→ | M − E ; χ M − E > . (8)Such a form is dictated by energy conservation, which, indeed, is consistent with thefirst law of BH thermodynamics. Note that the left-hand side of (8) has a largernumber of different states than the right-hand side. Consequently, the operator gov-erning the absorption (8) is not invertible, and thus cannot be unitary. Nevertheless,the overall unitarity is not necessarily violated. To see why, note that, although thesqueezing (4) is described by a formally unitary operator, it is not unitary on the phys-ical Hilbert space (because the physical Hilbert space does not contain the unphysicalnegative-energy particles). Thus, neither the squeezing (4) nor the absorption (8) arephysical processes by themselves. What is physical is their composition | M ; χ M > → X E X ξ d E,ξ ( M ) | M − E ; χ M − E > ⊗| E, ξ i . (9)Thus, if the initial state is | Ψ i = | M ; χ M > , then we have a physical transition | Ψ i → | Ψ i , where | Ψ i is the right-hand side of (9). The physical process (9) isexpected to be unitary. (An explicit verification of unitarity requires a more specificmodel of quantum gravity.) In fact, one may forget about the virtual subprocesses(4) and (8) and consider (9) as the only directly relevant physical process. Indeed,the process of BH radiation in a more advanced theory of quantum gravity may notbe based on a Bogoliubov transformation at all, so it may not be formulated in termsof creation of virtual negative-energy particles appearing in (4), but directly in termsof physical processes of the form of (9). In fact, this is exactly what occurs in stringtheory [6].Note also that in (8) we assume that the right-hand side does not depend on ξ .This reflects on the right-hand side of (9) in the fact that the new BH state does notdepend on the state of radiation ξ . This means that there is no correlation betweenradiated particles and BH interior, except for the trivial correlation expressing the factthat total energy must be conserved. The absence of such correlations is expectedalso from a more general view of the semiclassical description of particle creation[12]. As we shall see, this destruction of the (unphysical) information contained inthe negative-energy particles on the left-hand side of (8) makes the remnant scenarioviable, by removing the unwanted huge information that otherwise would have tobe be present in a light remnant. Nevertheless, later we also discuss a possibility torelax the assumption that the nontrivial correlation between exterior radiation andBH interior is completely absent. 5 The process of radiation – unitary evolution andthe role of wave-function collapse
Now the analysis of further steps of the process of BH radiation is mainly technical.After (9), the remaining BH state radiates again, now at a new larger temperaturecorresponding to the new smaller BH mass M − E . Thus, the next step | Ψ i → | Ψ i is based on a process analogous to (9) | M − E ; χ M − E > → X E ′ X ξ ′ d E ′ ,ξ ′ ( M − E ) ×| M − E − E ′ ; χ M − E − E ′ > ⊗| E ′ , ξ ′ i , (10)so | Ψ i = X E X E ′ X ξ X ξ ′ d E,ξ ( M ) d E ′ ,ξ ′ ( M − E ) ×| M − E − E ′ ; χ M − E − E ′ > ⊗| E, ξ i| E ′ , ξ ′ i . (11)Repeating the same process t times, we obtain | Ψ t i = X E · · · X E t X ξ · · · X ξ t × d E ,ξ ( M ) · · · d E t ,ξ t ( M − E − · · · − E t − ) × | M − E ; χ M −E > ⊗| E , ξ i · · · | E t , ξ t i , (12)where E = P tt ′ =1 E t ′ . (A continuous description of evolution labeled by a continuoustime parameter t is also possible, but this does not change our main conclusions.)States with the same energy E can be grouped together, so we can write | Ψ t i = X E X Ξ | M − E ; χ M −E > ⊗ D ( t ) E , Ξ |E , Ξ i , (13)where Ξ = { ξ , . . . , ξ t } and the coefficients D ( t ) E , Ξ can be expressed in terms of d E t ′ ,ξ t ′ .Note that, for any finite t , | Ψ t i contains contributions from all possible BH masses M ′ = M − E . At first sight, it seems to imply that the unitary evolution (13) preventsthe black hole from evaporating completely during a finite time t . Nevertheless,this is not really true. To see why, it is instructive to consider a simpler quantumdecay | a i → | b i in which the unitary evolution usually implies an exponential law | ψ ( t ) i = √ − e − Γ t | b i + √ e − Γ t | a i . For any finite t , there is a finite probability e − Γ t that the decay has not yet occurred. Nevertheless, a wave-function collapse associatedwith an appropriate quantum measurement implies that at each time the particle willbe found either in the state | a i or | b i . Analogously, if the BH mass M ′ is measuredat time t , the wave-function collapse implies | Ψ t i measure −→ | M − E ; χ M −E > ⊗ N E X Ξ D ( t ) E , Ξ |E , Ξ i , (14)6here N E is the normalization factor, N − E = P Ξ | D ( t ) E , Ξ | . Now the black hole isin a definite pure state | M − E ; χ M −E > and the outside particles are in a definitepure state N E P Ξ D ( t ) E , Ξ |E , Ξ i . (More realistically, the measurement uncertainty ∆ M ′ is smaller for smaller M ′ , so the outside particles are closer to a pure state when M ′ issmaller.) For example, it is conceivable that some quantum mechanism might preventtransitions (9) for M − E < M min (where M min is a hypothetic minimal possible BHmass). In this case, (14) may correspond to a transition to a BH remnant with a mass M − E = M min . Such a BH remnant is not correlated with the radiated particles (ex-cept for the correlation implied by energy conservation) and the information contentof the remnant is determined only by its mass. The absence of such correlations isa consequence of the assumption that the right hand-side of (8) does not depend on ξ . This assumption could also be relaxed by allowing that at least some different ξ ’smay correspond to different BH states. In this case, the BH state in (14) would alsodepend on Ξ, so it would not sit in front of the sum over Ξ, which would imply thatneither the black hole nor the radiation is in an exactly pure state, but that thereis a small correlation between them. Nevertheless, the maximal amount of possiblecorrelation is restricted by the smallness of the BH mass. In particular, if M min = 0,then (14) may correspond to a complete evaporation of the black hole, in which casethe BH state | > must be unique, implying that the final state of radiation must bea pure state N M P Ξ D ( t ) M, Ξ | M, Ξ i . We have seen that, under reasonable assumptions, the BH radiation is in a pure statewhenever the BH mass is measured exactly. Does it mean that the BH radiationis not really thermal? Actually not. Instead, the situation is analogous to thatin the discussion around Eqs. (1)-(3). For example, if the BH mass is measuredafter the first step (9), then the radiation collapses to a pure state equal (up toan overall normalization factor) to P ξ d E,ξ ( M ) | E, ξ i . This state is obtained from Q ω P n f ω,n ( M ) | n ω i by rewriting it as a sum of products and retaining only thosestates the total energy of which is equal to E . The density matrix of such a pure statetakes a form analogous to (2). Due to the decoherence induced by the interaction withthe environment, in practice such a state can be effectively described by a mixed stateanalogous to (3). From (1) we see that it is a thermal mixed state. More precisely, asthe total energy E is exactly specified, while the number of particles is specified onlyin average, this is a thermal state corresponding to a grand microcanonical ensemble.By contrast, the thermal state (7) (in which both total energy and number of particlesare specified only in average) corresponds to a grand canonical ensemble.At the end, let us recall that our resolution of the BH information puzzle involves4 different types of seemingly nonunitary evolutions. The process of squeezing (4) isformally unitary [10], but it is not unitary on the physical space. It is allways accom-panied with another nonunitary virtual process (the absorption of negative-energy7articles) Eq. (8), which together are combined into a physical unitary process (9).This represents the core of our resolution of the BH information puzzle. The thirdnonunitary process is the wave-function collapse (14). The exact meaning of the col-lapse depends on the general interpretation of QM that one adopts. In particular,in some interpretations (e.g., many-world interpretation and the Bohmian interpreta-tion) a true collapse does not really exist, making QM fully consistent with unitarity.Finally, the fourth nonunitary process is the phenomenon of decoherence (3), whichcorresponds only to an effective violation of unitarity, not a fundamental one.Finally note that, although our analysis allows a complete BH evaporation withouta true violation of unitarity, no new nonlocal mechanism has been involved. The onlynew mechanism is the absorption (8), which, however, occurs only inside the blackhole, thus not violating locality. Some nonlocal mechanisms are involved in ouranalysis, namely quantum entanglement and quantum wave-function collapse, butthese are standard nonlocal aspects of QM. Acknowledgements
The author is grateful to T. Jacobson for valuable discussions on Hawking radiationand BH information. This work was supported by the Ministry of Science of theRepublic of Croatia under Contract No. 098-0982930-2864.
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