How unitary cosmology generalizes thermodynamics and solves the inflationary entropy problem

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Citation Tegmark, Max. “How Unitary Cosmology Generalizes Thermodynamics and Solves the Inflationary Entropy Problem.” Physical Review D 85.12 (2012): 123517. © 2012 American Physical Society.

As Published http://dx.doi.org/10.1103/PhysRevD.85.123517

Publisher American Physical Society

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Citable link http://hdl.handle.net/1721.1/72144

Terms of Use Article is made available in accordance with the publisher's policy and may be subject to US copyright law. Please refer to the publisher's site for terms of use. PHYSICAL REVIEW D 85, 123517 (2012) How unitary cosmology generalizes thermodynamics and solves the inflationary entropy problem

Max Tegmark Department of Physics and MIT Kavli Institute, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA (Received 27 August 2011; published 11 June 2012) We analyze cosmology assuming unitary quantum mechanics, using a tripartite partition into system, observer, and environment degrees of freedom. This generalizes the second law of thermodynamics to ‘‘The system’s entropy cannot decrease unless it interacts with the observer, and it cannot increase unless it interacts with the environment.’’ The former follows from the quantum Bayes theorem we derive. We show that because of the long-range entanglement created by cosmological inflation, the cosmic entropy decreases exponentially rather than linearly with the number of bits of information observed, so that a given observer can reduce entropy by much more than the amount of information her brain can store. Indeed, we argue that as long as inflation has occurred in a non-negligible fraction of the volume, almost all sentient observers will find themselves in a post-inflationary low-entropy Hubble volume, and we humans have no reason to be surprised that we do so as well, which solves the so-called inflationary entropy problem. An arguably worse problem for unitary cosmology involves gamma-ray-burst con- straints on the ‘‘big snap,’’ a fourth cosmic doomsday scenario alongside the ‘‘big crunch,’’ ‘‘big chill,’’ and ‘‘big rip,’’ where an increasingly granular nature of expanding space modifies our life-supporting laws of physics. Our tripartite framework also clarifies when the popular quantum gravity approximation G 8GhTi is valid, and how problems with recent attempts to explain dark energy as gravitational backreaction from superhorizon scale fluctuations can be understood as a failure of this approximation.

DOI: 10.1103/PhysRevD.85.123517 PACS numbers: 98.80.Cq, 04.70.Dy

and has recently been clarified in an important body of I. INTRODUCTION work by Carroll and collaborators [35,36]. The basic prob- The spectacular progress in observational cosmology lem is to explain why our early Universe had such low over the past decade has established cosmological inflation entropy, with its matter highly uniform rather than clumped [1–4] as the most popular theory for what happened early into huge black holes. The conventional answer holds that on. Its popularity stems from the perception that it ele- inflation is an attractor solution, such that a broad class of gantly explains certain observed properties of our Universe initial conditions lead to essentially the same inflationary that would otherwise constitute extremely unlikely fluke outcome, thus replacing the embarrassing need to explain coincidences, such as why it is so flat and uniform, and extremely unusual initial conditions by the less embarrass- why there are 105-level density fluctuations which appear ing need to explain why our initial conditions were in the adiabatic, Gaussian, and almost scale-invariant [5–7]. broad class supporting inflation. However, [36] argues that If a scientific theory predicts a certain outcome with the entropy must have been at least as low before inflation probability below 106, say, then we say that the theory as after it ended, so that inflation fails to make our state is ruled out at 99.9999% confidence if we nonetheless seem less unnatural or fine-tuned. This follows from the observe this outcome. In this sense, the classic big bang mapping between initial states and final states being inver- model without inflation is arguably ruled out at extremely tible, corresponding to Liouville’s theorem in classical high significance. For example, generic initial conditions mechanics and unitarity in quantum mechanics. consistent with our existence 13.7 billion years later predict The main goal of this paper is to investigate the entropy observed cosmic background fluctuations that are about problem in unitary quantum mechanics more thoroughly. 105 times larger than we actually observe [8]—the We will see that this fundamentally transforms the prob- so-called horizon problem [1]. In other words, without lem, strengthening the case for inflation. A secondary goal inflation, the initial conditions would have to be highly is to explore other implications of unitary cosmology, for fine-tuned to match our observations. example, by clarifying when the popular approximation However, the case for inflation is not yet closed, even G 8GhTi is and is not valid. The rest of this paper aside from issues to do with measurements [9], competing is organized as follows. In Sec. II, we describe a quantita- theories [10–12], and the so-called measure problem tive formalism for computing the quantum state and its [8,13–33]. In particular, it has been argued that the so- entropy in unitary cosmology. We apply this formalism to called ‘‘entropy problem’’ invalidates claims that inflation the inflationary entropy problem in Sec. III and discuss is a successful theory. This entropy problem was articu- implications in Sec. IV. Details regarding the big snap lated by Penrose even before inflation was invented [34], scenario are covered in Appendix B.

1550-7998=2012=85(12)=123517(19) 123517-1 Ó 2012 American Physical Society MAX TEGMARK PHYSICAL REVIEW D 85, 123517 (2012)

II. SUBJECT, OBJECT, AND ENVIRONMENT A. Unitary cosmology The key assumption underlying the entropy problem is that quantum mechanics is unitary, so we will make this assumption throughout the present paper.1 As described in [40], this suggests the history schematically illustrated in Fig. 1: a wave function describing an early Universe quan- tum state (illustrated by the fuzz at the far left) will evolve deterministically according to the Schro¨dinger equation into a quantum superposition of not one but many macro- scopically different states, some of which correspond to large semiclassical post-inflationary universes like ours, and others which do not and completely lack observers. The argument of [40] basically went as follows: (1) By the Heisenberg uncertainty principle, any initial state must involve micro-superpositions, micro- scopic quantum fluctuations in the various fields. (2) Because the ensuing time evolution involves insta- bilities (such as the well-known gravitational insta- bilities that lead to the formation of cosmic FIG. 1 (color online). Because of chaotic dynamics, a single large-scale structure), some of these micro- early-Universe quantum state jc i typically evolves into a quan- superpositions are amplified into macro- tum superposition of many macroscopically different states, superpositions, much like in Schro¨dinger’s cat some of which correspond to a large semiclassical post- example [41]. More generally, this happens for any inflationary universe like ours (each with its galaxies, etc. in chaotic dynamics, where positive Lyapunov expo- different places), and others which do not and completely lack observers. nents make the outcome exponentially sensitive to initial conditions. (3) The current quantum state of the Universe is thus a is unitary is that of the total quantum state of the entire superposition of a large number of states that are Universe. We unfortunately have no observational infor- macroscopically different (Earth forms here, Earth mation about this total entropy, and what we casually refer forms 1 m further north, etc.), as well as states that to as ‘‘the’’ entropy is instead the entropy we observe for failed to inflate. our particular branch of the wave function in Fig. 1.We (4) Since macroscopic objects inevitably interact with should generally expect these two entropies to be quite their surroundings, the well-known effects of deco- different—indeed, the entropy of the entire Universe may herence will keep observers such as us unaware of well equal zero, since if it started in a pure state, unitarity such macro-superpositions. ensures that it is still in a pure state.

This shows that with unitary quantum mechanics, the B. Deconstructing the Universe conventional phrasing of the entropy problem is too sim- It is therefore interesting to investigate the cosmological plistic, since a single preinflationary quantum state evolves entropy problem more thoroughly in the context of unitary into a superposition of many different semiclassical post- quantum mechanics, which we will now do. inflationary states. The careful and detailed analysis of the Most discussions of quantum statistical mechanics split entropy problem in [36] is mainly performed within the the Universe into two subsystems [42]: the object under context of classical physics, and quantum mechanics is consideration and everything else (referred to as the envi- only briefly mentioned, when correctly stating that ronment). At a physical level, this ‘‘splitting’’ is simply a Liouville’s theorem holds quantum mechanically too as matter of accounting, grouping the degrees of freedom into long as the evolution is unitary. However, the evolution that two sets: those of the object and the rest. At a mathematical level, this corresponds to a choice of factorization of the 1The forms of nonunitarity historically invoked to address the Hilbert space. quantum measurement problem tend to make the entropy prob- As discussed in [43], unitary quantum mechanics can be lem worse rather than better: both Copenhagen-style wave- even better understood if we include a third subsystem as function collapse [37,38] and proposed dynamical reduction mechanisms [39] arguably tend to increase the entropy, trans- well, the subject, thus decomposing the total system (the forming pure (zero-entropy) quantum states into mixed states, entire Universe) into three subsystems, as illustrated in akin to a form of diffusion process in phase space. Fig. 2:

123517-2 HOW UNITARY COSMOLOGY GENERALIZES ... PHYSICAL REVIEW D 85, 123517 (2012) (entropy collapse” “wave-function observation, Measurement, freedom corresponding to the angle of the pointer, and 27 decrease excludes all of the other 10 degrees of freedom asso- ciated with the atoms in the multimeter. Similarly, the

SUBJECT) OBJECT 28 Hso ‘‘subject’’ excludes most of the 10 degrees of freedom Hs Ho associated with the elementary particles in your brain. The (Always conditioned on) term ‘‘perception’’ is used in a broad sense in item 1, including thoughts, emotions, and any other attributes of Object Hse Hoe decoher the subjectively perceived state of the observer. (ent ropy Just as with the currently standard bipartite decomposition Subject increaseence decoherence, into object and environment, this tripartite decomposition is finalizing ) decisions different for each observer and situation: the subject degrees ENVIRONMENT of freedom depend on which of the many observers in our He Universe is doing the observing, the object degrees of (Always traced over) freedom reflect which physical system this observer chooses to study, and the environment degrees of freedom correspond FIG. 2 (color online). An observer can always decompose the to everything else. For example, if you are studying an world into three subsystems: the degrees of freedom correspond- electron double-slit experiment, electron positions would ing to her subjective perceptions (the subject), the degrees of constitute your object and decoherence-inducing photons freedom being studied (the object), and everything else (the would be part of your environment, whereas in many quan- environment). As indicated, the subsystem Hamiltonians Hs, tum optics experiments, photon degrees of freedom are the Ho, He and the interaction Hamiltonians Hso, Hoe, Hse can object while electrons are part of the environment. cause qualitatively very different effects, providing a unified picture including both decoherence and apparent wave-function This subject-object-environment decomposition of the degrees of freedom allows a corresponding decomposition collapse. Generally, Hoe increases entropy and Hso decreases entropy. of the Hamiltonian:

H ¼ Hs þ Ho þ He þ Hso þ Hse þ Hoe þ Hsoe; (1) (1) The subject consists of the degrees of freedom where the first three terms operate only within one sub- associated with the subjective perceptions of the system, the second three terms represent pairwise interac- observer. This does not include any other degrees tions between subsystems, and the third term represents of freedom associated with the brain or other parts any irreducible three-way interaction. The practical use- of the body. fulness of this tripartite decomposition lies in that one can (2) The object consists of the degrees of freedom that often neglect everything except the object and its internal the observer is interested in studying, e.g., the dynamics (given by Ho) to first order, using simple pre- pointer position on a measurement apparatus. scriptions to correct for the interactions with the subject (3) The environment consists of everything else, i.e., all and the environment, as summarized in Table I. The effects the degrees of freedom that the observer is not of both Hso and Hoe have been extensively studied in the paying attention to. By definition, these are the literature. Hso involves quantum measurement, and gives degrees of freedom that we always perform a partial rise to the usual interpretation of the diagonal elements of trace over. the object density matrix as probabilities. Hoe produces A related framework is presented in [43,44]. Note that decoherence, selecting a preferred basis and making the the first two definitions are very restrictive. Suppose, for object act classically under appropriate conditions. Hse, example, that you are measuring a voltage using one of causes decoherence directly in the subject system. For those old-fashioned multimeters that has an analog pointer. example, [43] showed that any qualia or other subjective Then, the ‘‘object’’ consists merely of the single degree of perceptions that are related to neurons firing in a human

TABLE I. Summary of three basic quantum processes discussed in the text.

Interaction Dynamics Example Effect Entropy ! 10 1 1 Object-object ° UUy ° 2 2 Unitary evolution Unchanged 00 1 1 ! 2 2 ! P 1 1 1 ° h j i 2 2 ° 2 0 Object-environment ijPiPj j i 1 1 1 Decoherence Increases 2 2 0 2 ! y P 1 ii 2 0 10 Object-subject ° y , ¼ hs j iP ° Observation Decreases tr i j i j j 1 i i 0 2 00

123517-3 MAX TEGMARK PHYSICAL REVIEW D 85, 123517 (2012) Object Object brain will decohere extremely rapidly, typically on a time evolution decohe- scale of order 1020 seconds, ensuring that our subjective rence perceptions will appear classical. In other words, it is Ho Hoe (Entropy (Entropy useful to split the Schro¨dinger equation into pieces: three constant) increases) governing the three parts of our Universe (subject, object, Observation/Measurement and environment), and additional pieces governing the (Entropy decreases) Hso interactions between these parts. Analyzing the effects of these different parts of the equation, the Ho part gives most of the effects that our textbooks cover, the H part gives _1 _1 so 2 2 Everett’s many worlds (spreading superpositions from the =+ object to you, the subject), the Hoe part gives traditional { { decoherence, the Hse part gives subject decoherence. ρ ρ

C. Entropy in quantum cosmology FIG. 3 (color online). Time evolution of the 6 6 density In the context of unitary cosmology, this tripartite de- matrix for the basis states j"i€ , j#i€ , j^€ "i, j^€ #i, j_€ "i, j_€ #i composition is useful not merely as a framework for clas- as the object evolves in isolation, then decoheres, then gets sifying and unifying different quantum effects, but also as a observed by the subject. The final result is a statistical mixture j "i j #i framework for understanding entropy and its evolution. In of the states ^€ and _€ , simple zero-entropy states like the one we started with. short, Hoe increases entropy while Hso decreases entropy, in the sense defined below. the subject, 2 for the object). For definiteness, we denote To avoid confusion, it is crucial that we never talk of the ji j i j i entropy without being clear on which entropy we are the three subject states € , ^€ , and _€ , and interpret them as the observer feeling neutral, happy, and sad, referring to. With three subsystems, there are many inter- j"i j#i esting entropies to discuss, for example, that of the subject, respectively. We denote the two object states and , that of the object, that of the environment, and that of the and interpret them as the spin component (‘‘up’’ or whole system, all of which will generally be different from ‘‘down’’) in the z direction of a spin-1=2 system, say a silver atom. The joint system consisting of subject and one another. Any given observer can describe the state ¼ j"i of an object of interest by a density matrix which is object therefore has only 2 3 6 basis states: € , o j#i€ , j^€ "i, j^€ #i, j_€ "i, j_€ #i. In Fig. 3, we have therefore computed from the full density matrix in two steps: (1) Tracing: Partially trace over all environment de- plotted as a 6 6 matrix consisting of nine two-by-two grees of freedom. blocks. (2) Conditioning: Condition on all subject degrees of freedom. 1. Effect of Ho: Constant entropy If the object were to evolve during a time interval t In practice, step 2 often reduces to what textbooks call without interacting with the subject or the environment ‘‘state preparation,’’ as explained below. When we say ‘‘the (Hso ¼ Hoe ¼ Hsoe ¼ 0), then its reduced density matrix y entropy’’ without further qualification, we will refer to the o would evolve into UoU with the same entropy, since object entropy So: the standard von Neumann entropy of the time-evolution operator U e iHot is unitary. this object density matrix o, i.e., Suppose the subject stays in the state ji€ and the object starts out in the pure state j"i. Let the object Hamiltonian So tro logo: (2) Ho correspond to a magnetic field in the y direction Throughout this paper, we use logarithms of base two so causing the spinp toffiffiffi precess to the x direction, i.e., to the that the entropy has units of bits. Below, when we speak of state ðj"i þ j#iÞ= 2. The object density matrix o then the information (in bits) that one system (say the environ- evolves into ment) has about another (say the object), we will refer to the quantum mutual information given by the standard ¼ Uj"ih"jUy ¼ 1ðj"i þ j#iÞðh"j þ h#jÞ definition [45–47] o 2 ¼ 1ðj"ih"j þ j"ih#j þ j#ih"j þ j#ih#jÞ; (4) þ 2 I12 S1 S2 S12; (3) where S12 is the joint system, while S1 and S1 are the corresponding to the four entries of 1=2 in the second entropies of each subsystem when tracing over the degrees matrix of Fig. 3. of freedom of the other. This is quite typical of pure quantum evolution: a basis Let us illustrate all this with a simple example in Fig. 3, state eventually evolves into a superposition of basis states, where both the subject and object have only a single degree and the quantum nature of this superposition is manifested of freedom that can take on only a few distinct values (3 for by off-diagonal elements in o. Another familiar example

123517-4 HOW UNITARY COSMOLOGY GENERALIZES ... PHYSICAL REVIEW D 85, 123517 (2012) of this is the familiar spreading out of the wave packet of a corresponding to the so-called von Neumann reduction [53] free particle. which was postulated long before the discovery of decoher- ence; we can interpret it as an object having been measured 2. Effect of Hoe: Increasing entropy by something (the environment) that refuses to tell us what the outcome was.2 his was the effect of Ho alone. In contrast, Hoe will generally cause decoherence and increase the entropy of This suppression of the off-diagonal elements of the the object. Although decoherence is now well understood object density matrix is illustrated in Fig. 3. In this ex- j i¼j"i [48–52], we will briefly review some core results here ample, we have only two object states o1 and j i¼j#i that will be needed for the subsequent section about o2 , two environment states, and an interaction h j i¼ measurement. such that 1 2 0, giving Let jo i and je i denote basis states of the object and the ° 1ðj"ih"j þ j#ih#j i i o 2 : (9) environment, respectively. As discussed in detail in This new final state corresponds to the two entries of 1=2 in [50,52], decoherence (due to Hoe) tends to occur on time scales much faster than those on which macroscopic ob- the third matrix of Fig. 3. In short, when the environment finds out about the system state, it decoheres. jects evolve (due to Ho), making it a good approximation to assume that the unitary dynamics is U e iHoet on the decoherence time scale and leaves the object state un- 3. Effect of Hso: Decreasing entropy changed, merely changing the environment state in a way Whereas H typically causes the apparent entropy of j i oe that depends on the object state oi , say from an initial the object to increase, H typically causes it to decrease. j i j i so state e0 into some final state i : Figure 3 illustrates the case of an ideal measurement, where the subject starts out in the state ji€ and Hso is of such a Uje ijoii¼jiijoii: (5) 0 form that the subject gets perfectly correlated with the ¼j ih j This means that the initial density matrix e0 e0 o object. In the language of Eq. (3), an ideal measurement ¼ Pof the object-environment system, where o is a type of communication where the mutual information h j j ij ih j ij oi o oj oi oj , will evolve as Iso between the subject and object systems is increased to y y its maximum possible value [46]. Suppose that the mea- ° UU ¼ Uje0ihe0joU X surement is caused by Hso becoming large during a time y ¼ ho j jo iUje ijo ihe jho jU interval so brief that we can neglect the effects of Hs and i o j 0 i 0 j þ ij Ho. The joint subject object density matrix Rso then X ° y ½ ¼ h j j ij ij ih jh j evolves as so UsoU , where U exp i Hsodt . oi o oj i oi j oj : (6) j"i j#i ij If observing makes the subject happy and makes the subject sad, then we have Uj"i¼j€ ^€ "i and Uj#i¼€ The reduced density matrix for the object is this object- j_€ #i. The state given by Eq. (9) would therefore evolve environment density matrix partial-traced over the environ- into ment, so it evolves as X ¼ 1Uðjih€ jÞ€ ðj"ih"j þ j#ih#jÞUy ° h j j i o 2 0 tre ek ek ¼ 1ðUj€ "ih"j€ Uy þ Uj€ #ih#j€ Uy Xk 2 ¼ h j j ih j ih j ij ih j ¼ 1ðj "ih "j þ j #ih #jÞ ¼ 1ð þ Þ oi o oj j ek ek i oi oj 2 ^€ ^€ _€ _€ 2 ^€ _€ ; (10) ijk X j "ih "j ¼ j ih j j ih jh j i as illustrated in Fig. 3, where ^€ ^€ ^€ and _€ oi oi o oj oj j i j #ih #j ij _€ _€ . This final state contains a mixture of two sub- X jects, corresponding to definite but opposite knowledge of ¼ h j i PioPj j i ; (7) the object state. According to both of them, the entropy of ij P the object has decreased from 1 bit to zero bits. As men- j ih j¼ where we used the identity k ek ek I in the penulti- tioned above, there is a separate object density matrix o mate step and defined the projection operators Pi joiihoij that project onto the ith eigenstate of the object. This well- 2Equation (34) is known as the Lu¨ders projection [54] for the known result implies that if the environment can tell more general case where the Pi are moreP general projection ¼ ¼ whether the object is in state i or j, i.e., if the environment operators that still satisfy PiPj ijPi, Pi I. This form reacts differently in these two cases by ending up in two also follows from the decoherence formula (7) for the more general case where the environment can only tell which group of orthogonal states, hjjii¼0, then the corresponding ði; jÞ states the object is in (because the eigenvalues of Hoe are element of the object density matrix gets replaced by zero: degenerate within each group), so that hjjii¼1 if i and j X belong to the same group and vanishes otherwise. One then ° 0 PioPi; (8) obtains an equation of the same form as Eq. (8), but where each i projection operator projects onto one of the groups.

123517-5 MAX TEGMARK PHYSICAL REVIEW D 85, 123517 (2012) ¼j ih j corresponding to each of these two observers. Each of these This means that an initial density matrix s0 s0 o ¼ two observers picks out her density matrix by conditioning Pof the subject-object system, where o h j j ij ih j the density matrix of Eq. (10) on her subject degrees ij oi o oj oi oj , will evolve as of freedom, i.e., the density matrix of the happy one is ° UUy ¼ Ujs ihs j Uy ^€ and that of the other one is _€ . These are what Everett X 0 0 o termed the ‘‘relative states’’ [46], except that we are ¼ h j j i j ij ih jh j y oi o oj U s0 oi s0 oj U expressing them in terms of density matrices rather than Xij wave functions. In other words, a subject by definition ¼ h j j ij ij ih jh j has zero entropy at all times, subjectively knowing her oi o oj i oi j oj : (12) state perfectly. Related discussion of the conditioning ij operation is given in [43,44]. Since the subject will decohere rapidly, on a time scale In many experimental situations, this projection step in much shorter than that on which subjective perceptions defining the object density matrix corresponds to the change, we can apply the decoherence formula (8) to this familiar textbook process of quantum state preparation. expression with Pi ¼jsiihsij, which gives For example, suppose an observer wants to perform a X X ° ¼ j ih j j ih j quantum measurement on a spin 1=2 silver atom in the state PkPk sk sk sk sk j"i. To obtain a silver atom prepared in this state, she can k Xk ¼ h j j ih j ih j ij i simply perform the measurement of one atom, introspect, oi o oj sk i j sk sk and if she finds that she is in state j^€ i, then she know that her ijk j"i atom is prepared in the desired state —otherwise, she hskjjoiihojj discards it and tries again with other atom until she suc- X ¼ j ih j ðkÞ ceeds. Now, she is ready to perform her experiment. sk sk o ; (13) In cosmology, this state preparation step is often so k obvious that it is easy to overlook. Consider, for example, where the state illustrated in Fig. 1 and ask yourself what density X ðkÞ h j j ih j ih j ij ih j matrix you should use to make predictions for your own o oi o oj sk i j sk oi oj ij future cosmological observations. All experiments you can X ever perform are preceded by you introspecting and im- ¼ PioPjhskjiihskjji (14) plicitly confirming that you are not in one of the stillborn ij galaxy-free wave-function branches that failed to inflate. Since those dead branches are thoroughly decohered from is the (unnormalized) density matrix that the subject per- j i the branch that you are in, they are completely irrelevant to ceiving sk will experience. Equation (13) thus describes a predicting your future, and it would be a serious mistake sum of decohered components, each of which contains the j i not to discard their contribution to the density matrix of subject in a pure state sk . For the version of the subject j i your Universe. This conditionalization is analogous to the perceiving sk , the correct object density matrix to use ðkÞ use of conditional probabilities when making predictions for all its future predictions is therefore o appropriately in classical physics. If you are playing cards, for example, renormalized to have unit trace: the probabilistic model that you make for your opponent’s ð Þ P y k P P hs j ihs j i hidden cards reflects your knowledge of your own cards; ° o ¼ ij Pi o j k i k j ¼ k k ; o ðkÞ P jhs j ij2 y you do not consider shuffling outcomes where you were tro i tr o i k i trkk dealt different cards than those you observe. (15) Just as decoherence can be partial, when hjjii 0,so can measurement, so let us now derive how observation where changes the density matrix also in the most general case. X ¼ hs j iP : (16) Let js i denote the basis states that the subject can per- k k i i i i ceive—as discussed above, these must be robust to deco- herence, and will for the case of a human observer This can be thought of as a generalization of Everett’s so- correspond to ‘‘pointer states’’ [55] of certain degrees of called relative state from wave functions to density matri- freedom of her brain. Just as in the decoherence section ces and from complete to partial measurements. It can also above, let us consider general interactions that leave the be thought of as a generalization of Bayes’ theorem from object unchanged, i.e., such that the unitary dynamics is the classical to the quantum case: just like the classical U e iHsot during the observation and merely changes the Bayes’ theorem shows how to update an observer’s classi- subject state in a way that depends on the object state joii, cal description of something (a probability distribution) in j i j i say from an initial state s0 into some final state i : response to new information, the quantum Bayes’ theorem shows how to update an observer’s quantum description of j ij i¼j ij i U s0 oi i oi : (11) something (a density matrix).

123517-6 HOW UNITARY COSMOLOGY GENERALIZES ... PHYSICAL REVIEW D 85, 123517 (2012) ðkÞ ¼ P We recognize the denominator tro entropy-reducing object-environment interactions include h j j ijh j ij2 i oi o oi sk i as the standard expression for the dissipation (which can in some cases purify a high-energy probability that the subject will perceive jski. Note that mixed state to closely approach a pure ground state) and the same final result in Eq. (15) can also be computed error correction (for example, where a living organism directly from Eq. (12) without invoking decoherence, as reduces its own entropy by absorbing particles with low o ° hskjjski=trhskjjski, so the role of decoherence lies entropy from the environment, performing unitary error merely in clarifying why this is the correct way to compute correction that moves part of its own entropy to these the new o. particles, and finally dumping these particles back into To better understand Eq. (15), let us consider some the environment). simple examples: In regard to the other law of thermodynamics above, (1) If hsijji¼ij, then we have a perfect measure- note that it is merely on average that interactions with the ment in the sense that the subject learns the exact object cannot increase the entropy (because of Shannon’s famous result that the entropy gives the average number of object state, and Eq. (15) reduces to o ° Pk, i.e., bits required to specify an outcome). For certain individual the observer perceiving jski knows that the object is in its kth eigenstate. measurement outcomes, observation can sometimes in- crease entropy—we will see an explicit example of this (2) If jii is independent of i, then no information whatsoever has been transmitted to the subject, in the next section. For a less cosmological example, consider helium gas in and Eq. (15) reduces to o ° o, i.e., nothing changes. a thermally insulated box, starting off with the gas particles in a zero-entropy coherent state, where each atom is in a (3) If for some subject state k we have hsijji¼1 for some group of j values, vanishing otherwise, then rather well-defined position. There are positive Lyapunov the observer knows only that the object state is in exponents in this system because the momentum transfer in atomic collisions is sensitive to the impact parameter, so this group (this can happen if Hso has degenerate eigenvalues). Equation (15) then reduces to kk , before long, chaotic dynamics has placed every gas particle trk in a superposition of being everywhere in a box—indeed, where k is the projection operator onto this group in a superposition of being all over phase space, with a of states. Maxwell-Boltzmann distribution. If we define the object to be some small subset of the helium atoms and call the rest 4. Entropy and information of the atoms the environment, then the object entropy So In summary, we see that the object decreases its entropy will be high (corresponding to a roughly thermal density H=kT when it exchanges information with the subject and in- matrix o / e ) even though the total entropy Soe creases it when it exchanges information with the environ- remains zero; the difference between these two entropies ment. Since the standard phrasing of the second law of reflects the information that the environment has about the thermodynamics is focused on the case where interactions object via quantum entanglement as per Eq. (3). In classi- with the observer are unimportant, we can rephrase it in a cal thermodynamics, the only way to reduce the entropy of more nuanced way that explicitly acknowledges this a gas is to invoke Maxwell’s demon. Our formalism caveat: provides a different way to understand this: the entropy Second law of thermodynamics: The object’s entropy decreases if you yourself are the demon, obtaining infor- cannot decrease unless it interacts with the subject. mation about the individual atoms that constitute the We can also formulate an analogous law that focuses on object. decoherence and ignores the observation process: Another law of thermodynamics: The object’s entropy III. APPLICATION TO THE INFLATIONARY cannot increase unless it interacts with the environment. ENTROPY PROBLEM In Appendix A, we prove the first version and clarify the mathematical status and content of the second version. A. A classical toy model Note that for the above version of the second law, we are To build intuition for the effect of observation on en- restricting the interaction with the environment to be of a tropy in inflationary cosmology, let us consider the simple form of Eq. (5), i.e., to be such that it does not alter the state toy model illustrated in Fig. 4. This model is purely clas- of the system, merely transfers information about it to the sical, but we will show below how the basic conclusions environment. In contrast, if general object-environment generalize to the quantum case as well. We will also see interactions Hoe are allowed, then there are no restrictions that the qualitative conclusions remain valid when this on how the object entropy can change: for example, there is unphysical toy model is replaced by realistic inflation always an interaction that simply exchanges the state of the scenarios. object with the state of part of the environment, and if the Let us imagine an infinite space pixelized into discrete latter is pure, this interaction will decrease the object voxels of finite size, each of which can be in only two entropy to zero. More physically interesting examples of states. We will refer to these two states as habitable and

123517-7 MAX TEGMARK PHYSICAL REVIEW D 85, 123517 (2012)  f þð1 fÞ2n if i ¼ 0; p PðA Þ¼ (17) i i ð1 fÞ2n if i>0; i.e., there is a probability f of being in the i ¼ 0 state because inflation happened in our region, plus a small probability 2n of being in any state in case inflation did not happen here. Now, suppose that we decide to measure b bits of information by observing the state of the first b voxels. The probability PðHÞ that they are all habitable is simply the total probability of the first 2nb states, i.e.,

2nXb1 PðHÞ¼ pi i¼0 ¼ f þð1 fÞ2n þð2nb 1Þð1 fÞ2n ¼ f þð1 fÞ2b; (18) independent of the number of voxels n in our region. This result is easy to interpret: either we are in an inflationary region (with probability f), in which case these b voxels FIG. 4 (color online). Our toy model involves a pixelized are all habitable, or we are not (with probability 1 f), in space where pixels are habitable (green/light grey) or uninhabit- which case they are all habitable with probability 2b. able (red/dark grey) at random with probability 50%, except If we find that these b voxels are indeed all habitable, inside large contiguous inflationary patches where all pixels are then using the standard formula for conditional probabil- habitable. ities, we obtain the following revised probability dis- tribution for the state of our region: uninhabitable, and in Fig. 4, they are colored green/light ð Þ PðA &HÞ p b PðA jHÞ¼ i grey and red/dark grey, respectively. We assume that some i i PðHÞ inflationlike process has created large habitable patches in 8 > fþð1fÞ2n ¼ > b if i 0; this space, which fill a fraction f of the total volume, and <> fþð1fÞ2 that the rest of space has a completely random state where ¼ ð1fÞ2 n ¼ nb (19) > b if i 1; ...; 2 1; each voxel is habitable with 50% probability, indepen- > fþð1fÞ2 :> dently of the other voxels. 0 if i ¼ 2nb; ...; 2n 1: Now, consider a randomly selected region (which we will refer to as a ‘‘universe’’ by analogy with our Hubble We are now ready to compute the entropy S of our region volume) of this space, lying either completely inside an given various degrees of knowledge about it, which is inflationary patch or completely outside the inflationary defined by the standard Shannon formula patches—almost all regions much smaller than the typical inflationary patch will have this property. Let us number 2Xn1 ðbÞ ½ ðbÞ ð Þ the voxels in our region in some order 1; 2; 3; ..., and S h pi ;hp p logp; (20) ¼ let us represent each state by a string of zeroes and ones i 0 denoting habitable and uninhabitable, where a 0 in the where, as mentioned, we use logarithms of base two so that ith position means that the ith voxel is habitable. For the entropy has units of bits. Consider first the simple case example, if our region contains 30 voxels, then of no inflation, f ¼ 0. Then, all nonvanishing probabilities ð Þ ‘‘000000000000000000000000000000’’ denotes the state reduce to p b ¼ 2bn and the entropy is simply where the whole region is habitable, whereas i ‘‘101101010001111010001100101001’’ represents a SðbÞ ¼ n b: (21) rather typical noninflationary state. Finally, we label each state by an integer i which is simply its bit string inter- In other words, the state initially requires n bits to describe, preted as a binary number. one per voxel, and whenever we observe one more voxel, Letting n denote the number of voxels in our region, the entropy drops by 1 bit: the 1 bit of information we gain there are clearly 2n possible states i ¼ 0; ...; 2n 1 that it tells us merely about the state of the observed voxel, and can be in. By our assumptions, the probability pi that our tells us nothing about the rest of space since the other region is in the ith state (denoted Ai)is voxels are statistically independent.

123517-8 HOW UNITARY COSMOLOGY GENERALIZES ... PHYSICAL REVIEW D 85, 123517 (2012) More generally, substituting Eq. (19) into Eq. (20)gives     f þð1fÞ2n ð1fÞ2n SðbÞ ¼ h þð2nb 1Þh : f þð1fÞ2b f þð1fÞ2b (22) As long as the number of voxels is large (n b) and the Non-inflationary region inflated fraction f is non-negligible (f 2n), this en- tropy is accurately approximated by

    Inflationary region f ð1fÞ2n SðbÞ h þ2nbh þð Þ b þð Þ b f 1 f 2 f 1 f 2 in bits Entropy n hðfÞþ2bhð1fÞ ¼ þ þlog½f þð1fÞ2b: 2bf þ f þð1fÞ2b 1f 1 (23)

The sum of the last two terms is merely an n-independent constant of order unity which approaches zero as we b observe more voxels (as increases), so in this limit, Number of voxels observed Eq. (23) reduces to simply ð 1 Þ FIG. 5 (color online). How observations change the entropy ðbÞ f 1 n S : (24) for an inflationary fraction f ¼ 0:5. If successive voxels are all 2b observed to be habitable, the entropy drops roughly exponen- For the special case f ¼ 1=2 where half the volume is tially in accordance with Eq. (25) (green/grey dots). If the first inflationary, Eq. (23) reduces to the more accurate result voxel is observed to be uninhabitable, thus establishing that we are in a noninflationary region, then the entropy instead shoots ð Þ n S b þ log½1 þ 2 b (25) up to the line of slope 1 given by Eq. (21) (grey/red squares). 2b þ 1 More generally, we observe b habitable voxels and then one without approximations. uninhabitable one, the entropy first follows the dots, then jumps Comparing Eq. (21) with either of the last two equations, up to the squares, then follows the squares downward regardless we notice quite a remarkable difference, which is illus- of what is observed thereafter. This figure illustrates the case ¼ 120 trated in Fig. 5: in the inflationary case, the entropy de- with n 50 voxels—although n 10 is more relevant to our creases not linearly (by 1 bit for every bit observed), but actual Universe, the drop toward zero of the green curve would be too fast to be visible in the such a plot. exponentially. This means that in our toy inflationary uni- verse model, if an observer looks around and finds that even a tiny nearby volume is habitable, this dramatically ¼ bn reduces the entropy of her universe. For example, if ing probabilities pi 2 , and we recover Eq. (21)even f ¼ 0:5 and there are 10120 voxels, then the initial entropy when f 0. Thus, observing merely the first voxel to be is about 10120 bits, and observing merely 400 voxels (less uninhabitable causes the entropy to dramatically increase, ð Þ ¼ than a fraction 10 117 of the volume) to be habitable brings from 1 f n to n 1, roughly doubling if f 0:5.We this huge entropy down to less than 1 bit. can understand all this by recalling Shannon’s famous How can observing a single voxel have such a large result that the entropy gives the average number of bits effect on the entropy? The answer clearly involves the required to specify an outcome. If we know that our long-range correlations induced by inflation, whereby Universe is not inflationary, then we need a full n bits of this single voxel carries information about whether infla- information to specify the state of the n voxels, since they tion occurred or not in all the other voxels in our Universe. are all independent. If we know that our Universe is infla- If we observe b logf habitable voxels, it is exponen- tionary, on the other hand, then we know that all voxels are tially unlikely that we are not in an inflationary region. We habitable, and we need no further information. Since a therefore know with virtual certainty that the voxels that fraction (1 f) of the universes are noninflationary, we we will observe in the future are also habitable. Since our thus need ð1 fÞn bits on average. Finally, to specify uncertainty about the state of these voxels has largely gone whether it is inflationary or not, we need 1 bit of informa- away, the entropy must have decreased dramatically, as tion if f ¼ 1=2 and more generally the slightly smaller Eq. (24) confirms. amount hðfÞþhð1 fÞ, which is the entropy of a two- To gain more intuition for how this works, consider what outcome distribution with probabilities f and 1 f. The happens if we instead observe the first b voxels to be average number of bits needed to specify a universe is uninhabitable. Then, Eq. (19) instead makes all nonvanish- therefore

123517-9 MAX TEGMARK PHYSICAL REVIEW D 85, 123517 (2012) Sð0Þ ð1 fÞn þ hðfÞþhð1 fÞ; (26) Now, suppose that we, just as in the previous section, decide to measure b bits of information by observing the ¼ which indeed agrees with Eq. (23) when setting b 0. state of the first b voxels and find them all to be habitable. In other words, the entropy of our Universe before we To compute the resulting density matrix ðbÞ, we thus have made any observations is the average of a very large condition on our observational results using Eq. (15) with number and a very small number, corresponding to infla- the projection matrix P ¼j0...0ih0...0j, with b occur- tionary and noninflationary regions. As soon as we start rences of 0 inside each of the two brackets, obtaining observing, this entropy starts leaping toward one of these PP two numbers, reflecting our increased knowledge of which ðbÞ ¼ : (32) of the two types of region we inhabit. trPP Finally, we note that the success in this inflationary Substituting Eq. (31) into this expression and performing explanation of low entropy does not require an extreme some straightforward algebra gives anthropic selection effect where life is a priori highly unlikely; contrariwise, the probability that our entire 2Xn1 ðbÞ ¼ ðbÞjc ihc j Universe is habitable is simply f, and the effect works pi i i ; (33) fine also when f is of order unity. i¼0 ðbÞ where pi are the probabilities given by Eq. (19). We can B. The quantum case now compute the quantum entropy S of our region, which To build further intuition for the effect of observation on is defined by the standard von Neuman formula entropy, let us generalize our toy model to include quantum SðbÞ trh½ðbÞ;hðÞ log: (34) mechanics. We thus upgrade each voxel to a 2-state quan- tum system, with two orthogonal basis states denoted j0i This trace is conveniently evaluated in the jc ii basis where (‘‘habitable’’) and j1i (‘‘uninhabitable’’). The Hilbert ð Þ Eq. (33) shows that the density matrix b is diagonal, space describing the quantum state of an n-voxel region reducing the entropy to the sum thus has 2n dimensions. We label our 2n basis states by the same bit strings as earlier, so the state of the 30-voxel 2Xn1 ðbÞ ½ ðbÞ example given in Sec. III A above would be written S h pi : (35) i¼0 jc ii¼j101101010001111010001100101001i; (27) Comparing this with Eq. (20), we see that this result is corresponding to basis state i ¼ 759 669 545. If the region identical to the one we derived for the classical case. In is inflationary, all its voxels are habitable, so its density other words, all conclusions we drew in the previous matrix is section generalize to the quantum-mechanical case as well. ¼j ih j yes 000...0 000...0: (28) If it is not inflationary, then we take each voxel to be in the C. Real-world issues mixed state Although we repeatedly used words like ‘‘inflation’’ and 1 ‘‘inflationary’’ above, our toy models of course contained ¼ ½j0ih0jþj1ih1j; (29) 2 no inflationary physics whatsoever. For example, real eter- independently of all the other voxels, and the density nal inflation tends to produce a messy spacetime with matrix no of the whole region is simply a tensor product significant curvature on scales far beyond the cosmic par- of n such single-voxel density matrices. In the general case ticle horizon, not simply large uniform patches embedded that we wish to consider, there is a probability f that the in Euclidean space,3 and real inflation has quantum field region is inflationary, so the full density matrix is degrees of freedom that are continuous rather than simple qubits. However, it is also clear that our central result ¼ þð Þ fyes 1 f no regarding exponential entropy reduction has a very ¼ fj000...0ih000...0jþð1 fÞ ...: simple origin that is independent of such physical details: long-range entanglement. In other words, the key was (30) simply that the state of a small region could sometimes Expanding the tensor products, it is easy to show that we reveal the state of a much larger region around it (in our get 2n different terms, and that this full density matrix can case, local smoothness implied large-scale smoothness). be rewritten in the form 3 2Xn1 It is challenging to quantify the inflationary volume fraction f ¼ pijc iihc ij; (31) in such a messy spacetime, but as we saw above, this does not i¼0 affect the qualitative conclusions as long as f is not exponen- tially small—which appears unlikely given the tendency of where pi are the probabilities given by Eq. (17). eternal inflation to dominate the total volume produced.

123517-10 HOW UNITARY COSMOLOGY GENERALIZES ... PHYSICAL REVIEW D 85, 123517 (2012) This allowed a handful of measurements in that small separated by regions that inflate forever [56–58]. These region to, with a non-negligible probability, provide a pockets together contain an infinite volume and infinitely massive entropy reduction by revealing that the larger many particles, stars, and planets. Moreover, certain region was in a very simple state. We saw that the result observable quantities like the density fluctuation ampli- was so robust that it did not even matter whether this long- tude that we have observed to be Q 2 105 in our range entanglement was classical or quantum mechanical. part of spacetime [7,59] take different values in different It is not merely inflation that produces such long-range places.4 Taken together, these two facts create what has entanglement, but any process that spreads rapidly outward become known as the inflationary ‘‘measure problem’’ from scattered starting points. To illustrate this robustness [8,13–33]: the predictions of inflation for certain observ- to physics details, consider the alternative example where able quantities are not definite numbers, merely probability Fig. 4 is a picture of bacterial colonies growing in a Petri distributions, and we do not yet know how to compute dish: the contiguous spread of colonies creates long-range these distributions. entanglement, so that observing a small patch to be colon- The failure to predict more than probability distributions ized makes it overwhelmingly likely that a much larger is of course not a problem per se, as long as we know how region around it is colonized. Similarly, if you discover that to compute them (as in quantum mechanics). In inflation, a drop of milk tastes sour, it is extremely likely that a much however, there is still no consensus around any unique and larger volume (your entire milk carton) is sour. A random well-motivated framework for computing such probability bacterium in a milk carton should thus expect the entire distributions despite a major community effort in recent carton to be sour just like a random cosmologist in a years. The crux of the problem is that when we have a habitable post-inflationary patch of space should expect messy spacetime with infinitely many observers who sub- her entire Hubble volume to be post-inflationary. jectively feel like you, any procedure to compute the fraction of them who will measure say one Q value rather IV. DISCUSSION than another will depend on the order in which you count them, just as the fraction of the integers that are even In the context of unitary cosmology, we have investi- depends on the order in which you count them [8]. There gated the time evolution of the density matrix with which are infinitely many such observer ordering choices, many an observer describes a quantum system, focusing on the of which appear reasonable yet give manifestly incorrect processes of decoherence and observation and how they change entropy. Let us now discuss some implications of predictions [8,20,25,31,33], and despite promising devel- our results for inflation and quantum gravity research. opments, the measure problem remains open. A popular approach is to count only the finite number of observers existing before a certain time t and then letting t !1,but A. Implications for inflation this procedure has turned out to be extremely sensitive to the Although inflation has emerged as the most popular choice of time variable t in the spacetime manifold, with no theory for what happened early on, bolstered by improved obviously correct choice [8,20,25,31,33], The measure measurements involving the cosmic microwave back- problem has eclipsed and subsumed the so-called fine- ground and other cosmological probes, the case for infla- tuning problem, in the sense that even the rather special tion is certainly not closed. Aside from issues to do with inflaton potential shapes that are required to match obser- measurements [9] and competing theories [10–12], there vation can be found in many parts of the messy multidimen- are at least four potentially serious problems with its sional inflationary potential suggested by the string theoretical foundations, which are arguably interrelated: landscape scenario with its 10500 or more distinct minima (1) The entropy problem; [60–64], so the question shifts from asking why our inflaton (2) The measure problem; potential is the way it is to asking what the probability is of (3) The start problem; finding yourself in different parts of the landscape. (4) The degree-of-freedom problem. In summary, until the measure problem is solved, infla- tion strictly speaking cannot make any testable predictions Since we described the entropy problem in the at all, thus failing to qualify as a scientific theory in Introduction, let us now briefly discuss the other three. Popper’s sense. Please note that we will not aim or claim to solve any of these three additional problems in the present paper, 4 merely to highlight them and describe additional difficul- Q depends on how the inflaton field rolled down its potential, so for a one-dimensional potential with a single minimum, Q is ties related to the degree-of-freedom problem. generically different in regions where the field rolled from the left and from the right. If there potential has more than one 1. The measure problem dimension, there is a continuum of options, and if there are multiple minima, there is even the possibility that other effective Inflation is generically eternal, producing a messy parameters (physical ‘‘constants’’) may differ between different spacetime with infinitely many post-inflationary pockets minima, as in the string theory landscape scenario [60–64].

123517-11 MAX TEGMARK PHYSICAL REVIEW D 85, 123517 (2012) 2. The start problem mary, none of the three logical possibilities for the number Whereas the measure problem stems from the end of of quantum degrees of freedom N (it is infinite, it changes, inflation (or rather the lack thereof), a second problem it stays constant) is problem free. stems from the beginning of inflation. As shown by Borde, Guth, and Vilenkin [65], inflation must have had a 4. The case for inflation: The bottom line beginning, i.e., cannot be eternal to the past (except for In summary, the case for inflation will continue to lack a the loophole described in [66,67]), so inflation fails to rigorous foundation until the measure problem, the start provide a complete theory of our origins, and needs to be problem, and the degree-of-freedom problem have been supplemented with a theory of what came before. (The solved, so until then, we cannot say for sure whether same applies to various ekpyrotic and cyclic universe inflation solves the entropy problem and adequately ex- scenarios [65].) plains our low observed entropy. However, our results have The question of what preceded inflation is wide open, shown that inflation certainly makes things better. We have with proposed answers including quantum tunneling seen that claims to the contrary are based on an unjustified from nothing [56,68], quantum tunneling from a neglect of the density matrix conditioning requirement (the ‘‘pre-big-bang’’ string perturbative vacuum [69,70], and third dynamical equation in Table I), thus conflating the quantum tunneling from some other noninflationary state. entropy of the full quantum state with the entropy of Whereas some authors have argued that eternal inflation subsystems. makes predictions that are essentially independent of how Specifically, we have showed that by producing a quan- inflation started, others have argued that this is not the case tum state with long-range entanglement, inflation creates a [71–73]. Moreover, there is no quantitative agreement situation where observations can cause an exponential between the probabilities predicted by different scenarios, decrease in entropy, so that merely a handful of quantum some of which even differ over the sign of a huge exponent. measurements can bring the entropy for our observable The lack of consensus about the start of inflation not Universe down into the low range that we in fact observe. only undermines claims that inflation provides a final This means that if we assume that sentient observers answer, but also calls into question whether some of the require at least a small volume (say enough to fit a few claimed successes of inflation really are successes. In the atoms) of low temperature (1016 GeV), then almost context of the above-mentioned entropy problem, some all sentient observers will find themselves in a post- have argued that tunneling into the state needed to start inflationary low-entropy universe, and we humans have inflation is just as unlikely as tunneling straight into the no reason to be surprised that we do so as well. current state of our Universe [35,36], whereas others have argued that inflation still helps by reducing the amount of mass that the quantum tunneling event needs B. Implications for quantum gravity to generate [74]. We saw above that unjustified neglect of the density matrix conditioning requirement (the third dynamical 3. The degree-of-freedom problem equation in Table I) can lead to incorrect conclusions about inflation. The bottom line is that we must not conflate the A third problem facing inflation is to quantum- total density matrix with the density matrix relevant to us. mechanically understand what happens when a region of Interestingly, as we will now describe, this exact same space is expanded indefinitely. We discuss this issue in conflation has led to various incorrect claims in the litera- detail in Appendix B below, and provide merely a brief ture about quantum gravity and dark energy, for example, summary here. Quantum gravity considerations suggest that dark energy is simply backreaction from superhorizon that the number of quantum degrees of freedom N in a quantum fluctuations. comoving volume V is finite. If N increases as this volume expands, then we need an additional law of physics that h i specifies when and where new degrees of freedom are 1. Is G 8G T ? created, and into what quantum states they are born. If N Since we lack a complete theory of quantum gravity, we does not increase, on the other hand, life as we know it may need some approximation in the interim for how quantum eventually be destroyed in a ‘‘big snap’’ when the increas- systems gravitate, generalizing the Einstein equation ingly granular nature of space begins to alter our effective G ¼ 8GT of general relativity. A common assump- laws of particle physics, much like a rubber band cannot be tion in the literature is that to a good approximation, stretched indefinitely before the granular nature of its atoms cause our continuum description of it to break G ¼ 8GhTi; (36) down. Moreover, in the simplest scenarios where the num- ber of observers is proportional to post-inflationary vol- where G on the left-hand side is the usual classical ume, such big snap scenarios are already ruled out by Einstein tensor specifying spacetime curvature, while dispersion measurements using gamma-ray bursts. In sum- hTi on the right-hand side denotes the expectation value

123517-12 HOW UNITARY COSMOLOGY GENERALIZES ... PHYSICAL REVIEW D 85, 123517 (2012) of the quantum field theory operator T, i.e., hTi details of where the mass is located. For a detailed modern tr½T, where is the density matrix. Indeed, this as- treatment of small-scale averaging and its interpretation as sumption is often (as in some of the examples cited below) ‘‘integrating out’’ UV degrees of freedom, see [76]. Since made without explicitly stating it, as if its validity were very large scales tend to be observable and very small scales self-evident. tend to be unobservable, a useful rule of thumb in many So, is the approximation of Eq. (36) valid? It clearly situations is ‘‘condition on large scales, trace out small works well in many cases, which is why it continues to scales.’’ be used. Yet, it is equally obvious that it cannot be univer- In summary, the popular approximation of Eq. (36)is sally valid. Consider the simple example of inflation accurate if both of these conditions hold: with a quadratic potential starting out in a homogeneous (1) The spatially fluctuating stress-energy tensor for a and isotropic quantum state. This state will qualitatively generic branch of the wave function can be approxi- evolve as in Fig. 1, into a quantum superposition of many mated by its spatial average. macroscopically different states, some of which corre- (2) The quantum ensemble average can be approxi- spond to a large semiclassical post-inflationary universe mated by a spatial average for a generic branch of like ours (each with its planets, etc. in different places). the wave function. Yet, since both the initial quantum state and the evolution equations have translational and rotational invariance, the 2. Dark energy from superhorizon final quantum state will too, which means that hTi is quantum fluctuations? homogeneous and isotropic. But, Eq. (36) then implies that The discovery that our cosmic expansion is accelerating G is homogeneous and isotropic as well, i.e., that space- has triggered a flurry of proposed theoretical explanations, time is exactly described by the Friedmann-Robertson- most of which involve some form of substance or vacuum Walker metric. The easiest way to experimentally rule density dubbed dark energy. An alternative proposal that this out is to stand on your bathroom scale and note the has garnered significant interest is that there is no dark gravitational force pulling you down. In this particular energy, and that the accelerated expansion is instead due to branch of the wave function, there is a planet beneath gravitational backreaction from inflationary density pertur- you, pulling you downward, and it is irrelevant that there bations on scales much larger than our cosmic particle are other decohered branches of the wave function where horizon [77,78]. This was rapidly refuted by a number of the planet is instead above you, to your left, to your right, groups [79–82], and a related claim that superhorizon h i etc., giving an average force of zero. T is position- perturbations can explain away dark energy [83] was independent for the quantum field density matrix corre- rebutted by [84]. sponding to the total state, whereas the relevant density Although these papers mention quantum mechanics matrix is the one that is conditioned on your perceptions perfunctorily at best (which is unfortunate given that thus far, which include the observation that there is a planet the origin of inflationary perturbations is a purely beneath you. quantum-mechanical phenomenon), a core issue in these The interesting question regarding Eq. (36) thus be- refuted models is precisely the one we have emphasized comes more nuanced: when exactly is it a good approxi- in this paper: the importance of using the correct mation? In this spirit, [75] poses two questions: How density matrix, conditioned on our observations, rather unreliable are expectation values? and How much spatial than a total density matrix that implicitly involves incorrect variation should one expect? We have seen above that the averaging—either quantum ‘‘ensemble’’ averaging as in first step toward a correct treatment is to compute the Eq. (36) or spatial averaging. For example, as explained in density matrix conditioned on our observations (the third [84], a problem with the effective stress-energy tensor dynamic process in Table I) and use this density matrix to hTi of [83] is that it involves averaging over regions of describe the quantum state. Having done this, the question space beyond our cosmological particle horizon, even of whether Eq. (36) is accurate basically boils down to the though our observations are limited to our backward light question of whether the quantum state is roughly ergodic, cone. i.e., whether a small-scale spatial average of a typical Such unjustified spatial averaging is the classical phys- classical realization is well approximated by the quantum ics equivalent of unjustified use of the full density matrix in ensemble average hTitr½T. This ergodicity tends quantum mechanics: in both cases, we get correct statisti- to hold for many important cases, including the inflationary cal predictions only if we predict the future given what we case where the quantum wave functional for the primordial know about the present. Classically, this corresponds to fields in our Hubble volume is roughly Gaussian, homoge- using conditional probabilities, and quantum mechanically neous, and isotropic [14]. Spatial averaging on small scales this corresponds to conditioning the density matrix using is relevant because it tends to have little effect on the the bottom equation of Table I—neither is optional. In gravitational field on larger spatial scales, which depends classical physics, you should not expect to feel comfortable mainly on the large-scale mass distribution, not on the fine in boiling water full of ice chunks just because the spatially

123517-13 MAX TEGMARK PHYSICAL REVIEW D 85, 123517 (2012) averaged temperature is lukewarm. In quantum mechanics, off of it.5 In other words, observation and decoherence you should not expect to feel good when entering water both share the same first step, with another system obtain- that is in a superposition of very hot and very cold. ing information about the object—the only difference is Similarly, if there is no dark energy and the total quantum whether that system is the subject or the environment, i.e., state of our spacetime corresponds to a superposition of whether the last step is conditioning or partial tracing: states with different amplitudes for superhorizon modes, (i) observation ¼ entanglement þ conditioning; then we should not expect to perceive a single semiclassi- (ii) decoherence ¼ entanglement þ partial tracing: cal spacetime that accelerates (as claimed for some models Our formalism assumes only that quantum mechanics is [77,78]), but rather to perceive one of many semiclassical unitary and applies even to observers—i.e., we assume that spacetimes from a decohered superposition, all of which observers are physical systems too, whose constituent par- decelerate. ticles obey the same laws of physics as other particles. The Dark energy researchers have also devoted significant issue of how to derive Born rule probabilities in such a interest to so-called phantom dark energy, which has an unitary world has been extensively discussed in the litera- equation of state w<1 and can lead to a big rip a finite ture [45,46,89–92]—for thorough criticism and defense of time from now, when the dark energy density and the these derivations, see [93,94], and for a subsequent deriva- cosmic expansion rate becomes infinite, ripping apart tion using inflationary cosmology, see [95]. The key point of everything we know. The same logical flaw that we high- the derivations is that in unitary cosmology, a given quan- lighted above would apply to all attempts to derive such tum measurement tends to have multiple outcomes as illus- results by exploiting infrared logarithms in the equations trated in Fig. 1, and that a generic rational observer can for density and pressure [85] if they give w<1 on scales fruitfully act as if some nonunitary random process (‘‘wave- much larger than our cosmic horizon, or more generally to function collapse’’) realizes only one of these outcomes at talking about ‘‘the equation of state of a superhorizon the moment of measurement, with probabilities given by the mode’’ without carefully spelling out and justifying any Born rule. This means that in the context of unitary cos- averaging assumptions made. mology, what is traditionally called the Copenhagen inter- pretation is more aptly termed the Copenhagen C. Unitary thermodynamics and the approximation: an observer can make the convenient Copenhagen approximation approximation of pretending that the other decohered wave-function branches do not exist and that wave-function In summary, we have analyzed cosmology assuming collapse does exist. In other words, the approximation is unitary quantum mechanics, using a tripartite partition that apparent randomness is fundamental randomness. into system, observer, and environment degrees of free- In summary, if you are one of the many observers in dom. We have seen that this generalizes the second law of Fig. 1, you compute the density matrix with which to best thermodynamics to The system’s entropy cannot decrease predict your future from the full density matrix by perform- unless it interacts with the observer, and it cannot increase ing the two complementary operations summarized in unless it interacts with the environment. Quantitatively, the Table I: conditioning on your knowledge (generalized system (object) density matrix evolves according to one of ‘‘state preparation’’) and partial tracing over the the three equations listed in Table I depending on whether environment.6 the main interaction of the system is with itself, with the environment, or with the observer. The key results in this paper follow from the third equation of Table I, which 5As described in detail, e.g., [48–52], decoherence is not gives the evolution of the quantum state under an arbitrary simply the suppression of off-diagonal density matrix elements measurement or state preparation, and can be thought of as in general, but rather the occurrence of this in the particular basis a generalization of the positive operator valued measure of relevance to the observer. This basis is in turn determined formalism [86,87]. dynamically by decoherence of both the object [48–52] and the subject [43,88]. Informally speaking, the entropy of an object decreases 6Note that the factorization of the Hilbert space into subject, while you look at it and increases while you do not [43]. The object, and environment subspaces is different for different common claim that entropy cannot decrease simply corre- branches of the wave function, and that generally no global sponds to the approximation of ignoring the subject in factorization exists. If you designate the spin of a particular silver atom to be your object degree of freedom in this branch of Fig. 2, i.e., ignoring measurement. Decoherence is simply the wave function, then a copy of you in a branch where planet a measurement that you do not know the outcome of, and Earth (including you, your lab, and said silver atom) is a light measurement is simply entanglement, a transfer of quantum year further north will settle on a different tripartite partition into information about the system: the decoherence effect on the subject, object, and environment degrees of freedom. object density matrix (and hence the entropy) is identical Fortunately, all observers here on Earth here in this wave- function branch agree on essentially the same entropy for our regardless of whether this measurement is performed by observable Universe, which is why we tend to get a bit sloppy another person, a mouse, a computer, or a single particle and hubristically start talking about the entropy, as if there were that encodes information about the system by bouncing such a thing.

123517-14 HOW UNITARY COSMOLOGY GENERALIZES ... PHYSICAL REVIEW D 85, 123517 (2012) D. Outlook Because E also has the property that all its diagonal h j i¼ Using our tripartite decomposition formalism, we elements are unity (Eii i i 1), it is conveniently showed that because of the long-range entanglement cre- thought of as a (complex) correlation matrix. ated by cosmological inflation, the cosmic entropy de- We wish to prove that decoherence always increases the ð Þ creases exponentially rather than linearly with the entropy S tr log of the density matrix, i.e., that number of bits of information observed, so that a given Sð EÞSðÞ; (A2) observer can produce much more negentropy than her brain can store. Using this result, we argued that as long for any two positive semidefinite matrices and E such as inflation has occurred in a non-negligible fraction of the tr ¼ 1 and Eii ¼ 1, with equality only for the trivial volume, almost all sentient observers will find themselves case where E ¼ , corresponding to the object- in a post-inflationary low-entropy Hubble volume, and we environment interaction having no effect. Since I have humans have no reason to be surprised that we do so as been unable to find a proof of this in the literature, I will well, which solves the so-called inflationary entropy prob- provide a short proof here. lem. As detailed in Appendix B, an arguably worse prob- A useful starting point is the Corrollary J.2.a in [96] lem for unitary cosmology involves gamma-ray-burst [their Eq. (7)], which follows from a 1985 theorem by constraints on the big snap, a fourth cosmic doomsday Bapat and Sunder. It states that scenario alongside the big crunch, big chill, and big ðÞð EÞ; (A3) rip, where an increasingly granular nature of expanding space modifies our effective laws of physics, ultimately where ðÞ denotes the vector of eigenvalues of a matrix , killing us. arranged in decreasing order, and the symbol denotes Our tripartite framework also clarifies when the popular majorization. A vector with components 1; ...;n major- quantum gravity approximation G 8GhT i is valid, izes another vector with components 1; ...;n if they and how problems with recent attempts to explain dark have the same sum and energy as gravitational backreaction from superhorizon Xj Xj scale fluctuations can be understood as a failure of this ¼ approximation. In the future, it can hopefully shed light i i for j 1; ...;n; (A4) i¼1 i¼1 also on other thorny issues involving quantum mechanics and macroscopic systems. i.e., if the partial sums of the latter never beat the former: þ þ 1 1, 1 2 1 2, etc. In other words, the ACKNOWLEDGMENTS eigenvalues of the density matrix before decoherence ma- jorize the eigenvalues after decoherence. The author wishes to thank Anthony Aguirre and Ben Given any two numbers a and b where a>b, let us Freivogel for helpful discussions about the degree-of- define a Robin Hood transformation as one that brings freedom problem, Philip Helbig for helpful comments, them closer together while keeping their sum constant: Harold Shapiro for help proving that Sð EÞ> SðÞ, a ° a c and b ° a þ c for some constant 0

123517-15 MAX TEGMARK PHYSICAL REVIEW D 85, 123517 (2012) where the function hðxÞ¼x logx is concave. This con- (3) N is infinite, so we do not need to give a yes or no cludes the proof of Eq. (A2), i.e., of the theorem that answer. decoherence increases entropy. By making other concave Option 3 has been called into doubt by quantum gravity choices of h, we can analogously obtain other theorems considerations. First, the fact that our classical notion of about the effects of decoherence. For example, choosing space appears to break down below the Planck scale ð Þ¼ 2 34 h x x proves that decoherence also increases the rpl 10 m calls into question whether N can signifi- 2 ð Þ¼ 3 linear entropy 1 tr . Choosing h x logx proves cantly exceed V=rpl, the volume V that we are considering, that decoherence increasesP the determinant of the density measured in Planck units. Second, some versions of the ¼ matrix, since log det i logi. so-called holographic principle [97] suggest that N may be 3 2=3 2 smaller still, bounded not by the V=rpl but by V =rpl, 2. Conjecture that observation reduces roughly the area of our volume in Planck units. Let us expected entropy therefore explore the other two options: 1 and 2. The The observation formula from Table I can be thought of hypothesis that degrees of freedom are neither created as the quantum Bayes theorem. It says that observing nor destroyed underlies not only quantum mechanics (in subject state i changes the object density matrix to both its standard form and with nonunitary Ghirardi- Rimini-Weber–like modifications [39]), but classical me- S S ðiÞ ¼ jk ij ik chanics as well. Although quantum degrees of freedom can jk ; (A6) pi freeze out at low temperatures, reducing the ‘‘effective’’ number, this does not change the actual number, which is where Sij hsijji and X simply the dimensionality of the Hilbert space. 2 pi ¼ jjjSijj (A7) j 1. Creating degrees of freedom can be interpreted as the probability that the subject will The holographic principle in its original form [97] sug- ðiÞ perceive state i. The resulting entropy Sð Þ can be both gests option 2, changing N.7 Let us take our comoving smaller and larger than the initial entropy SðÞ, as simple volume V to be our current cosmological particle horizon examples show. However, I conjecture that observation volume, also known as our ‘‘observable Universe,’’ always decreases entropyX on average, specifically, that of radius 1026 m, giving a holographic bound of ðiÞ 120 piSð Þ

Let N denote the number of degrees of freedom in a 7More recent versions of the holographic principle have fo- finite comoving volume V of space. Does N stay constant cused on the entropy of three-dimensional light sheets rather over time, as our Universe expands? There are three logi- than three-dimensional volumes, evading the implications below cally possible answers to this questions, none of which [98,99]. 8An even more extreme example occurs if a Planck-scale appears problem free: region with a mere handful of degrees of freedom generates a (1) Yes whole new universe with say 10120 degrees of freedom via the (2) No Farhi-Guth-Guven mechanism [100].

123517-16 HOW UNITARY COSMOLOGY GENERALIZES ... PHYSICAL REVIEW D 85, 123517 (2012) 2. The big snap neously a cosmological distance c=H away, where 1 17 This leaves option 1, constant N. It too has received H 10 s is the Hubble time, they will reach us indirect support from quantum gravity research, in this separated in time by an amount case the AdS/CFT correspondence, which suggests that v aE 2 t H1 H1 (B2) quantum gravity is not merely degree-of-freedom preserv- v ℏc ing but even unitary. This option suffers from a different j j problem which I have emphasized to colleagues for some if the energy difference E E2 E1 is of the same 4 time, and which I will call the big snap. order as E1. Structure on a time scale of 10 s has been If N remains constant as our comoving volume expands reported in the gamma-ray burst GRB 910711 [106]in indefinitely, then the number of degrees of freedom per unit multiple energy bands, which [107] interpret as a lower volume drops toward zero9 as N=V. Since a rubber band bound t & 0:01 s for E ¼ 200 keV. Substituting this consists of a finite number of atoms, it will snap if you into Eq. (B2) therefore gives the constraint stretch it too much. Similarly, if our space has a finite ℏc ð Þ1=2 21 number of degrees of freedom N and is stretched indef- ascale factor, pushing this granularity to expand by different amounts before inflation ends, so we larger scales. It is hard to imagine business as usual once should expect the probability to find ourselves in a given a * 1026 m so that the number of degrees of freedom in region 1017 seconds after the end of inflation to be 3 our Hubble volume has dropped below 1. However, it is proportional to ða=aÞ as long as a

123517-17 MAX TEGMARK PHYSICAL REVIEW D 85, 123517 (2012) unchanging number of degrees of freedom N must repeat effects from these faraway regions do not propagate into this calculation using its own formalism and successfully the galaxies. Note, however, that this scenario saves only explain why we do not observer greater time dispersion in the observers, not the underlying theory. Indeed, the dis- gamma-ray bursts. crepancy between theory and observation merely gets Another important caveat is that our space is not ex- worse: repeating the above volume weighting argument panding uniformly: indeed, gravitationally bound regions now predicts that we are most likely to find ourselves alive like our Galaxy are not expanding at all. In specific models and well in a galaxy after the big snap has taken place where the degrees of freedom are localized on spatial throughout most of space, so the lack of any strange scales smaller than galaxies, one could imagine galaxy- observed signatures in light from distant extragalactic dwelling observers happily surviving long after intergalac- sources (energy-dependent arrival time differences for tic space has undergone a big snap, as long as deleterious gamma-ray bursts, say) becomes even harder to explain.

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