Quantum equivalence, the second law and emergent gravity
aa r X i v : . [ qu a n t - ph ] N ov Quantum equivalence, the second law and emergent gravity
Dries Sels ∗ and Michiel Wouters TQC, Universiteit Antwerpen, Universiteitsplein 1, B-2610 Antwerpen, Belgium (Dated: November 8, 2018)Since the advent of quantum mechanics we have mainly been concerned with its predictions fromthe perspective of an external observer. This is in strong contrast to the theory of general relativity,where the physics is governed by the intrinsic properties of space-time. At the same time, theprecise relation between space-time and quantum mechanics is still one of the greatest problems oftheoretical physics. This immediately raises the question on the completeness of our understandingof quantum mechanics. Here we address the problem by making an intrinsic analysis of observablesin generic quantum systems. We show that there is an extreme fine tuning problem for the emergenceof physics from the Hilbert space dynamics. However, for any initial condition and Hamiltonian,there exists a special set of observables. We show that these observables are intimately linked tothe natural configuration space in which an area law for the entanglement is inevitable. We arguethat this implies emergent gravity.
At the heart of our understanding of space-time, thetheory of general relativity, lies the equivalence principleand with it the prominent role played by the observer.The equivalence principle highlights how seemingly dif-ferent situations for an external observer are fully equiv-alent for an internal observer, a feature that is naturallyembodied in the geometry of space-time. Likewise, theobserver has played a prominent role in quantum me-chanics from the start, e.g. in Schr¨odinger’s cat experi-ment. Unfortunately, the special role of the observer hastroubled, rather than strengthened, the theory and it hasled to many different interpretations and solutions to thequantum measurement problem [1]. The lack of consen-sus on the latter [2] is a clear sign of missing insight in theemergence of physics from quantum evolution in Hilbertspace. Here, we will address this problem by making anintrinsic analysis of the observables in a generic quantumevolution.A prominent role in our study will be played by theentropy, a quantity that is central [3], but not fully un-derstood, in quantum gravity. While entropy productionin open quantum systems is well understood [4–8] on thebasis of the von Neumann entropy S = tr( ρ ln ρ ), it re-mains constant under unitary time evolution in closedsystems. For a fundamental law, that should be validfor the universe as a whole, this is not very satisfactory.The second law only appears naturally when the initialstate is taken to be separable in configuration space andwhen the entropy is computed from the local reduceddensity matrices. This indicates a deep connection be-tween space, time (entropy production) and a specific ini-tial condition. Using recent results on quantum quenches[9, 10], we will show that this scenario is universal: forevery macroscopic quantum evolution, a natural configu-ration space exists such that the initial state is separable.Moreover, the time evolution can always be described bya Hamiltonian with local couplings, for which an arealaw scaling of the entanglement entropy holds for shortevolution times. We will argue that this mathematical result implies emergent gravity. OBSERVATION AND TIME
In order to understand the physics that can emergefrom quantum mechanics, it is important to be clearon the role played by the trinity of Hamiltonian, initialcondition and observables. It is quite remarkable thatquantum mechanics is invariant under unitary transfor-mations and that the only time evolution in the theoryis a unitary transformation of the wave function. Thisimplies that it should be fundamental impossible to haveaccess to all observables. The converse immediately leadsto the conclusion that time does not exist or that timetravel is possible. In order to travel backwards for atime τ , we would just have to measure the observablesexp( − iHτ ) ˆ O exp( iHτ ) instead of the observable ˆ O .One could also say that time is meaningless, unless onehas specified the observables of interest. Indeed, if onlythe Hamiltonian is specified and no other observables,the phases of the energy eigenstates have no meaning.The Hamiltonian can be written as H = X n ǫ n | n ih n | , (1)which is invariant under the unitary transformation | n i → e iθ n | n i . But this transformation is exactly of thesame form as the time evolution. One way to break thissymmetry is by specifying an observable that does notcommute with the Hamiltonian.Any discussion of time evolution in quantum mechan-ics should therefore be based on a specific, restricted,set of observables. In an experimental context, the rel-evant observables are dictated by experimental limita-tions. These are formalised by quantum field theory,where the Hamiltonian and observables are intimatelyrelated operators, since both have simple algebraic ex-pressions in terms of the field annihilation and creationoperators. From a fundamental point of view, this tacitassumption of a simple relation between observables andkinematics however raises the question on the complete-ness of our understanding. FINE TUNING
Some recent works on the thermalisation in closedquantum systems have come to remarkable conclusionsin this respect. It was shown that that very mild restric-tions on the precision of measurements are sufficient toprove that all observables for an overwhelmingly largefraction of time take their thermal equilibrium values[11–14]. It was also proven [15–17] that for typical ob-servables, the time scale for which large nonequilibriumfluctuations persist is extremely short, of the order of theBoltzmann time τ B = ~ /k B T , where T is the equilibriumtemperature.Those results are readily understood by inspection ofthe expectation value of an operator ˆ O for a pure state,that evolves under the Hamiltonian (1) h O ( t ) i = X n,m c n O nm c m e i ( ǫ n − ǫ m ) t + i ( φ m − φ n ) , (2)where O nm = h n | O | m i and the initial phases are denotedby φ n . A typical observable is thermalised because thereis no specific relation between the initial phases φ n . Sum-ming over all random phases gives a vanishing expecta-tion value for those observables. Even if the initial phasesare chosen appropriate, the sum (2) will in general decayquickly as a function of time, because every component inthe sum oscillates at a different frequency ω = ǫ n − ǫ m ,which causes rapid dephasing on the Boltzmann time,that is inversely proportional to the energy width of thestate τ B = 1 / ∆ E , see Fig. 1 and 2. These works solvethe problem of thermalisation in closed quantum systems,but at the same time raise the issue of understanding howa nonequilibrium state can exist for an extended periodof time, as it is the case for the observables that we areused to in physics.A first important result in this respect was the proofthat for any quantum system, there exist observables thattake a time that is exponentially large in the system sizeto thermalise [14]. The existence of such an operator canbe easily identified from expression (2). For the observ-able with matrix elements O nm = e − i ( φ n − φ m ) δ n +1 ,m + e i ( φ n − φ m ) δ n − ,m (3)which couples neighbouring energy levels, it will take atime t = 1 /δǫ for the observable to dephase, that is expo-nentially long in the system size ( δǫ is the typical energyspacing between neighbouring levels, see Fig. 2). Thisis again not a property of physical observables, with theimportant exception of spontaneous symmetry breaking. S p ace o f ob s e r v a b l e s Typical Observables F a st t h e r m a l i z a t i o n Time E xp ec t a ti on v a l u e τ b B r ok e n S y mm e t r y Figure 1. The space of all observables can be divided in (redregion) the typical observables that are thermal on the consid-ered time interval; (blue region) the fast thermalising observ-ables and (green region) slowly decaying observables, relatedto broken symmetry. An example of corresponding expecta-tion values are depicted in the same color in the left panel.
Between these two extremes, observables with intermedi-ate relaxation times exist as well, but a crucial lesson isthat the number of observables with slow relaxation timesis exponentially smaller than the number of fast relaxingones, as schematically represented in Fig. 1. Moreover,also the slow variables take on their thermal expecta-tion values for almost all times after an initial transient.Therefore the statistical probability to a system out ofequilibrium is exponentially small.All this illustrates that an extreme fine tuning betweena quantum state and observable is needed in order to ob-tain any dynamics. It is tempting to look at the so-calledfine tuning problems in cosmology and elementary par-ticle physics from this perspective. When one is puzzledby the small probability for the existence of our universe,one does so from the perspective of the observables thatwe consider to be physical (built of field operators ofstable particles). This explains the fine tuning problems:the initial condition of the universe had to be very specialwith respect to our electron and proton field operators inorder to see the present structure.The thermalisation results however offer a solution: forany initial condition and Hamiltonian, there exists a setof special observables that relax slowly. This is clear from(3): the slow observables depend on the initial condition,but for any initial condition a slow observable can be con-structed. We will argue below that those slow observablesare intimately connected to the emergence of space. Butbefore proceeding with this task, we will present our ar-gument for the equivalence of Hamiltonian evolutions ofmacroscopic systems. E n e r gy } ΔE δE }
10 spins
Figure 2. Scaling of the many-body spectrum of a typicalHamiltonian with system size. The spectrum becomes expo-nentially dense with a linear growth of the system size.
QUANTUM EQUIVALENCE
Consider the unitary time evolution of a state | Ψ( t ) i = e − iHt | Ψ(0) i = X n c n e − iǫ n t | n i . (4)The phases of the initial condition c n have now been ab-sorbed in the definition of the eigenstates | n i : the intrin-sic quantum evolution is then clearly independent of theinitial condition. It should be understood that this is thecase, only because we have not yet specified any otherobservable of interest.The dynamics is then determined by the spectrum ofthe Hamiltonian only. The individual states of this spec-trum are not resolved as long as the time is not exponen-tially long ( t < /δǫ ). The only microscopic element thatwe remain with is the density of occupied states ρ ( E ).The following conclusion is thus unavoidable: all quan-tum systems with the same density of occupied states,on a scale set by the evolution time, are equivalent.Surprisingly, this equivalence principle leads to theconclusion that a ‘quantum theory of everything’ con-sists merely of the specification of a density of states.Studies in many body physics have shown that the den-sity of occupied states is generically of Gaussian form[18, 19]. It thus appears that all known interacting fieldtheories essentially lead to equivalent dynamics (of coursethe precise relation between different systems is typicallyenormously complicated). Under the assumption thatquantum mechanics is valid for the universe as a whole,Occam’s razor principle suggests to conjecture that it hasa Gaussian density of occupied states.We then conclude that on the level of Hilbert space dy-namics, a typical quenched many body system is equiv-alent to the whole universe. Note that in this reasoningeven the difference between systems made of fermionicand bosonic particles disappears. The reason why they appear so different to us is that we have a completely dif-ferent experimental access to them, through fermionic orbosonic field operators respectively. The way one looksat a system thus determines its properties rather thanthe intrinsic nature of the system itself. NATURAL CONFIGURATION SPACE ANDSECOND LAW
The quantum equivalence principle offers us the free-dom to choose the pair initial state/Hamiltonian thatsimplifies the identification of the observables that fea-ture a long relaxation time. In order to make con-tact with a physical configuration space, we wish towrite our Hilbert space as a tensor product of subspaces H = N j H j in such a way that the local operators relaxslowly to equilibrium.For a generic decomposition of Hilbert space, one findshowever that local observables are thermal and that theentanglement entropy is proportional to the volume of theregion [20, 21]. Note here the correspondence with thefact that typical observables are thermalised. Moreover,for a generic decomposition, the Hamiltonian is highlynon-local, such that even when initially the entanglemententropy vanishes, it will become extensive on the order ofthe Boltzmann time, together with the thermalisation ofthe local observables (see Fig. 3). Again, this is in corre-spondence with the short thermalisation time for typicalobservables. Only when the Hamiltonian is local in theconfiguration space, an initially separable state will leadto a slow growth of the entanglement entropy in time. Inthis situation, one can show that the entanglement en-tropy grows typically linear with time and quite generallyfeatures an area law [9] S ( X ) ∼ tA ( X ) . (5)Only then, the local observables coincide with the smallsubset of slowly decaying observables. Note that the sec-ond law, interpreted as the increase of entanglement en-tropy, comes out immediately, thanks to the special rela-tion between initial condition and the natural configura-tion space.Given the essential connection between spontaneoussymmetry breaking and slow thermalisation, it is betterto choose a model where one knows the broken symmetry.In that way, the physical order that will persist for longtimes is immediately apparent. One could think of aquantum spin system, starting in a separable state. Wehave thus come to a quite specific model for our universe,but thanks to the quantum equivalence principle, this isnot at the expense of generality.The celebrated Lieb-Robinson bound [22] implies a fi-nite propagation speed for the entanglement. In a cosmo-logical interpretation, the finite speed for the spreadingof entanglement implies that a given point in space onlyhas information about a finite part of the universe. Thiscorresponds with an observable universe that is only apart of the whole. Initially, all points in space are identi-cal and they do not share any information. Consequentlythere is no reasonable definition of space. It has further-more been shown by Van Raamsdonk [23] that, in thecontext of gauge-gravity correspondence [24, 25], separa-ble quantum states lead to disconnected space-times.With our condensed matter construction, we come toa universe that is homogeneous in space, in analogy toVolovik’s Helium droplet universe [26]. Actually, our onlyadditional ingredient to Volovik’s universe is the identi-fication of the initial condition as a separable state. Itis this initial condition that allows for the creation ofquasi-particles ex nihilo . As time goes on, entanglementspreads, which can be described as the creation and prop-agation of quasi-particles. GRAVITY FROM ENTANGLEMENT SCALING
We highlighted above the connection between a slowrelaxation of local observables and an area law for the en-tanglement entropy. From a physical point of view, thisis actually a rather surprising situation, because it doesnot correspond to the scaling of a collection of indepen-dent quasi-particles, that is extensive. We are thus ledto the conclusion that the emergent quasi-partices shouldbe correlated. In our previous work, we have shown thatcorrelations due to an initial condition can be describedin a statistical generalised Gibbs description [27, 28] asdue to fictitious interactions [29]. The area law for theentropy and universality of this interaction (it should acton all types of quasi-particles) leads us to the conjecturethat this ‘spooky’ interaction is gravity.It was actually the suggestive relation between arealaws for the ground state of condensed matter systemsand black holes [30, 31] that formed a major motivationfor the study of entanglement entropy. We wish to stressthat the area law is here the consequence of the shorttime evolution (times for which the observable universeis smaller than the whole system) and not of the factthat the system is in the ground state. In addition, thereare thermodynamic indications that gravity is related toa negative contribution to the entropy. It has long beenknown that gravitational systems have a negative specificheat, both for systems of Newtonian gravitating masses[32] and for black holes [3]. The idea that gravity andmore generally Einstein relativity has an entropic ori-gin was already introduced two decades ago by Jacobson[33]. More recently, gravity was argued to be an entropy-related force on the basis of holographic arguments byVerlinde [34]. Inspired by Verlinde’s work, there havebeen several other works that speculate on the connec-tion between entanglement entropy and gravity [35–37].In contrast to the previous works on entropic gravity, our analysis did not require any assumptions in addi-tion to unitary quantum evolution. Verlinde’s analogywith colloid and bio-physical systems is rather confusingin this respect, because it suggests the association deco-herence effects to gravitational interactions [38]. In ourview, gravity is rather a consequence of missing entropythan due to an additional entropic process. It is becauseof the slow growth of entropy according to an area lawthat the quasi-particles have to be correlated in space,which is perceived as a gravitational interaction.
CONCLUSIONS
Let us recapitulate the main results of our analysis.We argued that the time evolution in Hilbert space ofmacroscopic quantum systems is equivalent. The factthat most observables are most of the time equal to theirthermal expectation value was interpreted as a fine tun-ing problem: a precise relation is required between thewave function and an observable in order to show devia-tions from the ergodic average.We used the quantum equivalence principle in orderto construct a model where the local observables relaxslowly. It was argued that the initial state should be sep-arable in the natural configuration space and that theHamiltonian is local. An area law for the entanglementis then inevitable. The usual second law of thermody-namics is a direct consequence of the separability of theinitial state. 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