1 Introduction and Summary

UCB-PTH-11/02


Physical Theories, Eternal Inflation, and Quantum Universe


Yasunori Nomura


Berkeley Center for Theoretical Physics, Department of Physics,

University of California, Berkeley, CA 94720, USA

Theoretical Physics Group, Lawrence Berkeley National Laboratory, CA 94720, USA


1 Introduction and Summary

A combination of eternal inflation [1] and the string landscape [2] provides a theoretically well-motivated framework in which we might understand how nature selects our local universe to take the form as we see today. On one hand, the string landscape provides an enormous number of quantum field theories, of or more, as consistent vacuum solutions to string theory. On the other hand, eternal inflation physically realizes all these vacua in spacetime, allowing us to have an anthropic understanding for the structure of physical theories [3], including the value of the cosmological constant [4].

This elegant picture, however, suffers from the issue of predictivity arising from infinities associated with eternally inflating spacetime, known as the measure problem [5]. In fact, it is often said that “In an eternally inflating universe, anything that can happen will happen; in fact, it will happen an infinite number of times” [6]. While this sentence captures well the origin of the issue, its precise meaning is not as clear as one might naively think. Consider the part “anything that can happen will happen.” One might think it to mean that for a given initial state, anything that is allowed to happen in the landscape will happen. However, any single “observer” (geodesic) traveling in eternally inflating spacetime will see only a finite number of vacua before he/she hits a (big crunch or black hole) singularity or is dissipated into time-like infinity of Minkowski space. Namely, for this observer, something that can happen may not happen. We are thus led to interpret the phrase to mean either “anything that can happen will happen with a nonzero probability” or “anything that can happen will happen at least to somebody.” The former interpretation, however, leads to the question of what probability we are talking about, and the latter the question of whether considering the histories of all different observers physically makes sense despite the fact that they can never communicate with each other at late times. Similar questions also apply to the part “it will happen an infinite number of times.” Indeed, any randomly selected observer will see a particular event only a finite number of times.

The first aim of this paper is to present a view on these and related issues in eternal inflation, by providing a well-defined framework for prediction in eternally inflating spacetime, i.e. in the eternally inflating multiverse. Our framework is based on the following three basic observations:

  • In any physical theory, predicting (or postdicting) something means that, given what we “already know,” we obtain information about something we do not know. In fact, there are two aspects in this: one is the issue of dynamical evolution and the other about probabilities. Suppose one wants to predict the future (or explore the past) in a classical world, given perfect knowledge about the present. This requires one to know the underlying, dynamical evolution law, but only that. On the other hand, if one’s future is determined only probabilistically, or if one’s current knowledge is imperfect, then making predictions/postdictions requires a definition of the probabilities, in addition to knowing the evolution law. The measure problem in eternal inflation is of this second kind.

  • Let us assume that the underlying evolution law is known (e.g. the landscape scalar potential in string theory). Making predictions/postdictions is then equivalent to providing relative weights among all physical possibilities that are consistent with the information we already have. In particular, since our knowledge is in principle limited to what occurred within our past light cone, having a framework for prediction/postdiction is the same as coming up with a prescription for finding appropriate samples for past light cones that are consistent with our prior knowledge.

  • From the point of view of a single observer, infinities in eternal inflation arise only if one repeats “experiments” an infinite number of times, i.e. only if the observer travels through the multiverse an infinite number of times. This implies that by considering a huge, but finite, number of observers emigrating from a fixed initial region at a very early moment, only a finite number of—indeed, only a very small fraction of—observers see past light cones that are consistent with the conditions we specify as our prior knowledge. In particular, if our knowledge contains exact information about any non-static observable, then the number of past light cones satisfying the prior conditions and encountered by one of the observers is always finite, no matter how large the number of observers we consider.

These observations naturally lead us to introduce the following general framework for making predictions, which consists of two elements:

  • We first phrase a physical question in the form: given a set of conditions imposed on a past light cone, what is the probability of this light cone to have a property or to evolve into another past light cone having a property ? We will argue that any physical questions associated with prediction/postdiction can always be formulated in this manner.

  • We then find an ensemble of past light cones satisfying conditions by “scanning” what observers actually see. Specifically, we consider a set of geodesics emanating from randomly distributed spacetime points on a fixed, “initial” (space-like or null) hypersurface at a very early moment in the history of spacetime. We then keep track of past light cones along each of these geodesics, and if we find a light cone that satisfies , then we “record” it as an element of the ensemble. Answers to any physical questions can then be obtained by simply counting the numbers of relevant light cones:

    (1)

    where , , and are the numbers of the recorded light cones that satisfy the specified conditions. Note that these numbers are countable and finite if conditions involve a specification of any non-static observable, e.g. the value of a parameter that can play the role of time. The predictions in Eq. (1) can be made arbitrary precise by making the number of the geodesics arbitrarily large.

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Figure 1: A schematic picture for obtaining samples of past light cones that satisfy specified conditions . The light cones to be selected are depicted by shaded triangles (with the figure showing ). Note that a single geodesic may encounter relevant light cones multiple times in its history.

A schematic depiction of the procedure of sampling past light cones is given in Fig. 1. Note that this procedure corresponds precisely to simulating the entire multiverse many times as viewed from a single observer (geodesic). Also, in this method, any time coordinate becomes simply a spurious parameter, exactly like a variable used in a parametric representation of a curve on a plane, . It plays no role in defining probabilities, and time evolution of a physical quantity is nothing more than a correlation between and a quantity that can play the role of time, such as the average temperature of the cosmic microwave background (CMB) in our universe.

It is illuminating to highlight a close similarity between the prescription here and the formulation of usual quantum mechanics. Consider a quantum mechanical system in a state with . If we perform an experiment many times on this system, we would find that anything that can happen (represented by ) will happen (with a nonzero probability ). Moreover, by performing it an infinite number of times, all these results occur an infinite number of times. In our prescription, corresponds to the multiverse, while to a collection of past light cones along a geodesic in this spacetime. Performing an experiment then corresponds to sending an observer (geodesic) from an initial hypersurface located at the “beginning” of spacetime.

The second aim of this paper is to formulate the above framework, described using the semi-classical picture, in the context of quantum mechanics, where we will see that the similarity just illustrated is more than an analogy. The necessity of a quantum mechanical treatment may sound obvious since we live in a quantum mechanical world, but it is much more fundamental than one might naively think. Consider a set of geodesics that scan past light cones dense enough at late times, i.e., when conditions can be satisfied. Because of the rapid exponential expansion of spacetime, these geodesics must have been much closer at early times—in fact, there are generically a huge number of geodesics emanating from an early space-like hypersurface of the Planck size. On the other hand, theories of quantum gravity suggest that the Planck size region can contain only (or smaller) information. How can such a state evolve into many different universes in the future? Is it reasonable to expect that the procedure (ii) above, based on the classical spacetime picture, can select the appropriate ensemble of past light cones, despite the fact that it involves scales shorter than the Planck length at early times?

We find that quantum mechanics plays a crucial role in answering these questions—in fact, the ultimate resolution of the measure problem requires quantum mechanical interpretation of the multiverse. Our basic assumption here is that the laws of quantum mechanics—deterministic, unitary evolution of quantum states and the superposition principle—are not violated when physics is described from an observer’s point of view. As we will argue, this implies that the complete description of the multiverse can be obtained purely from the viewpoint of a single observer traveling the multiverse. All the information on the multiverse is contained in the (stretched) apparent horizon and spacetime therein, as seen from that observer. This situation is similar to describing a black hole from the viewpoint of a distant observer using “complementarity” [7]—indeed the situation for a black hole arises as a special case of our general description. Our construction is strongly motivated by such complementarity view of spacetime, as well as the holographic principle [8].

To exemplify this picture further, let us consider that spacetime was initially in a highly symmetric configuration, e.g. four dimensional de Sitter spacetime, with the fields sitting in a local minimum of the potential. From the semi-classical analysis, we know that even a tiny region of this initial configuration evolves into infinitely many different universes. On the other hand, the holographic principle, or de Sitter entropy, says that such a small region can have only finite degrees of freedom. This implies that in the quantum picture, the origin of various semi-classical universes cannot be attributed to the difference of initial conditions—they must arise as different “outcomes” of a quantum state , which is uniquely determined once an initial condition is given. This clearly answers one question raised above: how can an initial state having only information evolve into different universes? In fact, the initial state evolves into the unique future state, which however is a probabilistic mixture of different (semi-classical) universes. The meaning of the sampling procedure in (ii) also becomes clear—sending a geodesic corresponds to making a “measurement” on , and the sampling corresponds to defining probabilities through “repeated measurements.” In particular, we need not take too seriously the sub-Planckian distances appearing in the procedure—the semi-classical picture of the multiverse is simply a pictorial way of representing probabilistic processes, e.g. bubble nucleation processes, occurring in the quantum universe.111While completing this paper, I learned that Raphael Bousso also arrived at a similar picture in the context of geometric cutoff measures [9].

The remaining task to define a complete, quantum mechanical framework for prediction is to come up with the explicit formalism of implementing procedure (ii), given originally within the semi-classical picture. To do so, we first define more carefully. For simplicity, we take it to be a pure state (although an extension to the mixed state case is straightforward). A crucial point is that we describe the system from a single observer’s point of view. This allows us to consider the state , where the time parameter is taken as a proper time along the observer, which can be assigned a simple, invariant meaning. (We take the Schrödinger picture throughout.) We consider that is defined on observer’s past light cones bounded by the (stretched) apparent horizons, as viewed from the observer. This restriction on spacetime regions comes from consistency of quantum descriptions for systems with gravity. If we provide an initial condition on a space-like hypersurface , then the state is given on past light cones and at early times (as in Fig. 2), and later, on past light cones inside and at the horizons (see Figs. 4, 5 and Eq. (21) in Section 4.4).

\Photon(0,20)(300,20)240 \Text(10,12)[t]initial hypersurface \Line(150,20)(150,180) \Line(150,180)(146,170) \Line(150,180)(154,170) \Text(160,180)[lt]observer (geodesic) \Line(19,130)(19,172) \Line(21,130)(21,172) \Line(20,175)(16,165) \Line(20,175)(24,165) \Text(10,165)[rt]\Text(20,125)[t]unitary evolution \DashLine(110,20)(150,60)2 \DashLine(190,20)(150,60)2 \DashLine(0,20)(110,20)2 \DashLine(190,20)(300,20)2 \Line(70,20)(150,100) \Line(230,20)(150,100) \Line(0,20)(70,20) \Line(230,20)(300,20) \Line(30,20)(150,140) \Line(31,20)(150,139) \Line(29,20)(150,141) \Line(270,20)(150,140) \Line(269,20)(150,139) \Line(271,20)(150,141) \Line(0,20)(30,20) \Line(0,21)(30,20.5) \Line(0,19)(30,19.5) \Line(270,20)(300,20) \Line(270,21)(300,20.5) \Line(270,19)(300,19.5) \LongArrow(40,45)(40,60) \LongArrow(260,45)(260,60) \LongArrow(70,75)(70,90) \LongArrow(230,75)(230,90) \LongArrow(100,105)(100,120) \LongArrow(200,105)(200,120) \LongArrow(130,135)(130,150) \LongArrow(170,135)(170,150) \Text(320,20)[l] on \Photon(375,20)(415,20)25.33 \Text(320,50)[l] on \DashLine(375,50)(415,50)2 \Text(320,80)[l] on \Line(375,80)(415,80) \Text(320,110)[l] on \Line(375,110)(415,110) \Line(375,110.5)(415,110.5) \Line(375,109.5)(415,109.5) \Vertex(370,134)1 \Vertex(370,142)1 \Vertex(370,150)1
Figure 2: The entire multiverse can be described purely from the viewpoint of a single “observer” in terms of a quantum state , which is defined on the observer’s past light cones (and on the initial hypersurface at early times; later, the light cones are bounded by the apparent horizons). Once the initial condition is given, the state is uniquely determined according to unitary, quantum mechanical evolution.

In general, the state can be expanded as

(2)

where span a complete set of orthonormal states in the landscape (i.e. all possible past light cones within the horizons), and the index may take continuous values (in which case the sum over should be interpreted as an integral). Here, we have taken the basis states to be independent of , so that all the dependence of come from those of the coefficients . Once the initial condition is given, the state is determined uniquely. (Our framework does not provide the initial condition; rather, it is modular with respect to initial conditions.)

We next introduce an operator which projects onto the states consistent with conditions A imposed on the past light cone:

(3)

where represent the complete set of states that satisfy conditions . The conditions can be imposed either within or on the past light cone, since they can be transformed to each other using (supposedly known) deterministic, quantum evolution. Note that, as long as conditions are physical (i.e. do not depend on arbitrary parametrization ), the operator does not depend on , so that it can be written in the form of Eq. (3).

The probability that a past light cone satisfying also has a property is then given by

(4)

which is the quantum version of the probability in Eq. (1). Here, we have put the subscript to remind us that the probability depends on initial conditions. The rightmost expression arises when conditions specify an exact “time,” e.g. by requiring a particular value for a parameter that has smooth dependence; in such a case, the conditions select definite (discrete) times . Note that specifying physical “time” is different from specifying time parameter ; in fact, the same “time” can occur multiple times in the history at ().

The probability defined in Eq. (4) inherits the following crucial properties from the semi-classical definition:

  • well-defined — While there are cosmic histories which encounter -satisfying past light cones an infinite number of times, they compose only a measure zero set. In particular, histories encountering such light cones occur only with probabilities with , because, starting from any generic state, the probability of encountering -satisfying light cones is exponentially small. This makes the sums (time integrals) in Eq. (4) converge, and so makes well-defined.

  • “gauge invariant” — The probability does not depend on parametrization of time , as is clear from the expressions in Eq. (4). The role of time in making predictions can then be played by a physical quantity that has smooth dependence on  [10], which need not be defined globally in the entire spacetime. In fact, it can be arbitrary information on any physical system, as long as it is not about a static property of the system.

Since quantum evolution of the state is deterministic, the probability for a future event to happen, , is also calculated once the values of are known for all ’s that could affect the likelihood for to occur (assuming, of course, that the evolution law is known).

In principle, the definition of Eq. (4) can be used to answer any physical questions associated with prediction and postdiction in the multiverse. However, since involves degrees of freedom on horizons, as well as those in the bulk of spacetime, its complete evolution can be obtained only with the knowledge of quantum gravity. This difficulty is avoided if we focus on the bulk degrees of freedom, by considering a bulk density matrix

(5)

where represents the partial trace over the horizon degrees of freedom. This description allows us to make predictions/postdictions without knowing quantum gravity, since the evolution of can be determined by semi-classical calculations in low energy quantum field theories. The cost is that (apparent) unitarity violation is induced in processes involving horizons, such as black hole evaporation.

Our framework has a number of implications. Major ones are:

  • The measure problem in eternal inflation is solved. The principle behind the solution is quantum mechanics, where quantum states are defined in the spacetime region bounded by (stretched) apparent horizons as viewed from a single “observer” (geodesic). No ad hoc cutoff needs to be introduced, and problems/paradoxes plaguing other measures are absent. Transformations between descriptions based on different observers are possible, but they are in general highly nonlocal when the observers are not in causal contact at late times.

  • The multiverse and many worlds in quantum mechanics are the same. Our framework provides a unified treatment of quantum measurement processes and the eternally inflating multiverse, usually associated with vastly different scales—smaller than atomic and much larger than the universe. The probabilities in microscopic quantum processes and in the multiverse are both understood as “branching” in the entire multiverse state (or equivalently, as a “misalignment” in Hilbert space between and the basis of local observables).222The conjecture that the multiverse and many worlds may be the same thing appeared earlier in Ref. [11]. (I thank Leonard Susskind for bringing this to my attention.) Here we show that they are, in fact, identical, following the same formula for the probabilities, Eqs. (28) and (29). This provides complete unification of the two concepts.

  • Global spacetime can be viewed as a derived concept. By rearranging various terms in the multiverse state , the global spacetime picture can be “reconstructed.” In particular, this allows us to connect the quantum probabilities with the semi-classical probabilities, defined using geodesics traveling through global spacetime of the multiverse.

  • The multiverse is a transient phenomenon while relaxing into a supersymmetric Minkowski state. Our framework suggests that the ultimate fate of the multiverse is a supersymmetric Minkowski state, which is free from further decay. The other components in , e.g. ones hitting big crunch or black hole singularities, simply “disappear.” This leads to the picture that our entire multiverse arises as a transient phenomenon, during the process of some initial state (either determined by quantum gravity or generated as a “fluctuation” in a larger structure) relaxing into a stable, supersymmetric final state.

Other implications are also discussed throughout the paper.

We emphasize that the multiverse state is literally “everything” in terms of making predictions—even we ourselves appear as a part of at some time(s) . Once the state is given, any physical predictions can be obtained using Eq. (4), which is nothing but the standard Born rule. There is no need to introduce wavefunction collapse, environmental decoherence, or anything like those—indeed, there is no “external observer” that performs measurements on , and there is no “environment” with which interacts. From the viewpoint of a single observer (geodesic), probabilities keep being “diluted” because of continuous branching of the state into different semi-classical possibilities, caused by the fact that the evolution of is not along an axis in Hilbert space determined by operators local in spacetime. Indeed, the probabilities are constantly being dissipated into (fractal) Minkowski space, which acts as an infinite reservoir of (coarse-grained) entropy—this is the ultimate reason behind the well-definedness of the probabilities. It is quite remarkable that this simple and satisfactory picture comes with a drastic change of our view on spacetime: the entire world can be described within the spacetime region inside the causal patch of a single geodesic, bounded by its (stretched) apparent horizons. All the information about the world is contained in a single quantum state defined in this region—we do not even need to consider an ensemble of quantum states. In this sense, we may even say that spacetime exists only to the extent that it can be accessed directly by the single “observer” (geodesic).

The organization of this paper is the following. In Section 2, we provide detailed discussions on what prediction and postdiction mean in physical theories, especially in the context of the multiverse. We present precise definitions of prediction/postdiction, as well as their practical implementations in realistic contexts. In Section 3, we present our explicit framework for prediction, using the semi-classical picture; but we will also see that the framework cries for a quantum mechanical treatment of the multiverse. Based on these results, in Section 4 we introduce a fully, quantum mechanical framework for prediction. We carefully consider quantum states, and argue that they must be defined in the region within (stretched) apparent horizons, viewed from a single “observer” traveling through the multiverse. We also introduce an approximation scheme in which horizon degrees of freedom are integrated out, so that the probabilities can be calculated without knowing quantum gravity.

The issue of initial conditions is discussed in Section 5, both from the perspective of making predictions in eternally inflating universe and addressing fundamental questions regarding spacetime. In Section 6, we present a unified picture of the eternally inflating multiverse and quantum measurement processes—in fact, this unification is an automatic consequence of our treatment of the multiverse. We also discuss a connection of our picture, based on a single “observer,” with the global spacetime picture. In Section 7, we show that the framework does not suffer from problems plaguing other measures proposed so far [5] based on geometric cutoffs; in particular, it avoids a peculiar conclusion that time should “end” [12]. Finally, in Section 8 we summarize what we learned about eternal inflation, and discuss further issues such as the beginning/fate of the multiverse, the possibility of a fully holographic description, and a structure even larger than the entire multiverse (“mega-multiverse”) which is suggested by extrapolating our framework to the extreme.

Appendix A discusses an interpretation of the framework in terms of “fuzzy” time cutoff. Appendix B provides sample calculations of probabilities in toy landscapes, both in the semi-classical and quantum pictures. Appendix C presents an analysis of a gedanken experiment which provides a nontrivial consistency check of our framework.

2 What Questions Should We Ask?

In this section, we discuss in detail what are “legitimate” physical questions we would ask in the context of the eternally inflating multiverse. For example, we may want to ask what are the results of future measurements, what is the probability of certain Lagrangian describing our universe, or if the observations we have already made are consistent with the assumption of typicality in the multiverse. We will see that answering these questions is boiled down to coming up with a well-defined prescription for selecting appropriate samples of past light cones (either finite or infinite numbers) that are consistent with our “prior knowledge” about our past light cone. The actual prescription will be given in the next section, where we will see that it naturally calls for a quantum mechanical treatment of the multiverse.

2.1 Predictions in a strict sense

Suppose you want to “predict” what physics you will see at the LHC experiments, assuming you have a perfect knowledge about the underlying theory governing evolution of systems, including the multi-dimensional scalar potential in the string landscape (but, of course, not about which vacuum you live). Specifically, you want to know what is the probability for you to find that weak scale supersymmetry, technicolor, just the standard model, or something else, is the Lagrangian describing your local universe, i.e. physics within your Hubble horizon.333If the LHC has answers to these questions by the time you read this paper, then you should appropriately replace statements below with those regarding some other experiment whose results are not yet available to you.

In the multiverse context, what this “really” means is the following. First, you specify all you know about your past light cone.444Here, I consider “you” as a point in spacetime, for simplicity. To be precise, “you” must (at least) be some pattern of electromagnetic activities (neural signals) corresponding to a brain in your “current” moment, which occur inside but around the tip of the light cone. In fact, if you are not a Boltzmann brain, which we assume throughout this section, your entire body (indeed, the entire you from the birth to the present) must exist inside the light cone. For more discussions on these and related issues, see Section 7.1. This includes the existence of you, the room around you (if you are reading this in a room), the earth, and (the observable part of) the universe. You then ask what is the probability of this “initial situation”—i.e. the particular configuration within your past light cone—to evolve into a certain “final situation”—a past light cone lying in your future in which you will somehow learn that physics at the TeV scale is, e.g., weak scale supersymmetry (either from a friend, paper, …). In fact, possible past light cones in your future must include ones in which the LHC fails, e.g. due to some accidents, so that no relevant measurement will be made. After including all these possible (mutually exclusive) futures, the probabilities should add up to one.

Of course, you practically do not know everything you need to specify uniquely your past light cone. (You even do not know the location of a coffee cup next to you with infinite precision.) This implies that, even at the classical level, the “initial situation” you prepare must be an ensemble of “situations”—i.e. a collection of past light cones—that are consistent with the knowledge you have about your past light cone.555In fact, in a classical world, if you have a complete knowledge about your past light cone, you would not even ask about future probabilities, since the physics is fully deterministic (assuming the initial conditions at the earliest moment are also known). For example, you would be able to “calculate” if the LHC fails or not, and you would know physics at very high energies through tiny effects encoded in higher dimension operators in the effective theory. This apparently does not sound to be the case in a quantum world, since even if you completely specify your past light cone, outcome of future experiments can only be determined probabilistically. However, for a given initial quantum state, its future sates are uniquely determined. In fact, the concept of making “predictions” in the sense considered here is relevant only if our knowledge about the current state of the system is incomplete. Therefore, a specification of the “initial situation” requires some method that assigns relative weights among these “initial past light cones”—namely, you need to find samples of past light cones that “faithfully” represent incompleteness of your knowledge. In fact, in the global spacetime picture of the multiverse, the number of past light cones that satisfy any input knowledge will be infinite, so some regularization prescription will be needed to define the samples. (This is the measure problem in the eternally inflating multiverse.) On the other hand, once such samples are given, it is conceptually a straightforward matter to work out predictions, following the procedure described above.

2.2 Predictions in practice

While the definition of predictions described in the previous subsection is “rigorous,” it is not very practical. In practice, when we ask questions, e.g., about physics at the TeV scale, we only want to keep track of global quantities which are (reasonably) uniform in our horizon, such as the CMB temperature, the size of its statistical fluctuation, masses of “elementary” particles, the gauge group, and so on. In this context, how can we make predictions, e.g., about the probability of our local universe being described by a certain Lagrangian? We emphasize that this issue is logically separate from that of defining probabilities; rather, the issue here is to come up with reasonable approximation schemes which we may hope to be tractable.

Suppose you only specify that your past light cone—more precisely the intersection of your past light cone and your own bubble universe—is described by the standard models of particle physics and cosmology at energies below e.g. a TeV, and that the CMB temperature at the tip of the cone is . Here, the latter condition is imposed to specify a physical “time,” which in the previous treatment was done by giving a particular configuration in the past light cone (e.g. the “current” state of your brain; see footnote 4). Then the ensemble you need to prepare as an “initial situation” includes past light cones in which physics at the TeV scale is the minimal standard model, weak scale supersymmetry, technicolor, and so on (as long as these theories arise in consistent vacua in the landscape).

Now, in selecting these light cones, you did not impose any condition about yourself. However, as long as the chance that you are born does not depend much on TeV-scale physics, you can expect

(6)

where and (and similarly for others) are, respectively, the numbers of past light cones selected with and without the condition that you are in the light cones (around the tips). In this case, you can use , , and , instead of , , and , to compute relative probabilities for each TeV-scale physics describing your own universe.666To compute absolute probabilities, these numbers must be divided by the number of past light cones that are selected only with the condition that they are consistent with your current knowledge about the physical laws in your own universe, e.g., constraints from collider and rare decay experiments. Of course, in the context of predicting results at the LHC, “SUSY” should mean the existence of superpartners within the reach of the LHC.

Note that when selecting samples for , , , …, the condition was imposed that physics below a TeV scale is described by the standard models of particle physics and cosmology, with various parameters—such as the electron mass, fine structure constant, dark matter abundance, and baryon-to-photon ratio—taking the observed values (within experimental errors). This is important especially because your own existence may be strongly affected by the values of these parameters. Since some of the parameters, e.g. cosmological ones, will depend highly on physics at high energies, the ratios between , , , …, are in general not the same as the corresponding ratios obtained without conditioning the values of low energy parameters. From the practical point of view, this makes the problem hard(er); but again, once appropriate samples of past light cones are obtained, it is straightforward to make probabilistic predictions for physics at the TeV scale.

2.3 Predictions and postdictions

At first sight, it might seem completely straightforward to apply the method described so far to “postdict” some of the physical parameters we already know—we simply need to select samples of past light cones without imposing any condition on those parameters. This, however, needs to be done with care, because our own existence may be affected by the values of these parameters. In the context of an approximation scheme discussed in the previous subsection, appropriate postdictions are obtained after specifying an “anthropic factor” —the probability of ourselves developing in the universe with parameters taking values . With this factor, the probability density for us to observe is given by

(7)

where is the number of past light cones in which the parameters take values between and . Note that, when selecting samples for , we still need to specify the “time” at which postdictions are made (e.g. through ). In fact, the level of the validity of Eq. (7) is determined by how well and the specification of “time” can mimic ourselves.777A simple approximation, appropriate in some cases, is obtained by assuming that for habitable regions while for regions hostile for life. See e.g. Ref. [13] for more details about this approximation. Postdictions obtained in this way can then be used to test the multiverse hypothesis, by comparing them with the observed data.

A treatment of postdictions analogous to that of predictions in Section 2.1 would require a subtle choice for the condition of selecting light cone samples. Suppose we want to postdict the electron mass, , and compare it with the experimental value, . We would then need to prepare the ensemble of past light cones that have some “consciousnesses” around the tip perceiving similar worlds as we see now except that the electron mass may not take the value . However, since worlds with different electron masses will not be identical, we will not be able to require these consciousnesses to have identical thoughts/memories. In this sense, the concept of postdiction will have intrinsic ambiguities associated with the specification of reference observers, and its treatment can only be “approximate,” e.g., in the sense of Eq. (7).

3 A Framework for Prediction—Selecting the Ensemble of Past Light Cones

We have argued that, assuming the underlying theory is known, answering any physical questions in the multiverse is boiled down to finding appropriate samples for past light cones that satisfy your input information. This is equivalent to specifying relative weights for various “microscopic” possibilities that are consistent with your input. In this section we provide an explicit prescription for obtaining an appropriate ensemble of past light cone samples. We will focus on the implementation of the idea in a semi-classical world, deferring the complete, quantum mechanical formulation to Section 4. We will, however, see that our prescription naturally demands a quantum mechanical treatment of the multiverse.

3.1 Semi-classical picture

A simple prescription for finding the required ensemble is to define it as the result of “repeated experiments.” In the context of the eternally inflating multiverse, this means that we need to “simulate” many times the entire multiverse as viewed from a single “observer.” Defining this way, the result will depend on the initial setup for the “experiments,” i.e. the initial condition for the evolution of the multiverse. We will discuss this issue in more detail in Section 5, but for now we simply assume that the multiverse starts from some initial state that is highly symmetric. In particular, we assume that we can choose a time variable in such a way that, at sufficiently early times, constant-time space-like hypersurfaces are homogeneous and isotropic.

We now focus on a generic, finite region on an initial space-like hypersurface which contains at least one (measure zero) eternally inflating point. Because of the homogeneity of the hypersurface, the choice of the region can be arbitrary. (If the volume of the initial hypersurface is finite, as in the case where the spacetime is born as a closed universe, then may be chosen to be the entire hypersurface.) We then consider a set of randomly distributed spacetime points on , and a future geodesic emanating from each . Here we choose all the ’s to be normal to ; with this choice, the symmetry of the spacetime is recovered in the limit of a large number of , .888The condition that the symmetry recovers in the large limit does not uniquely fix the initial condition; in particular, we can have an arbitrary distribution for the magnitude of the velocities, , as long as their direction is random. While decays exponentially with time during inflation, this can still affect predictions [14] albeit to a small extent [15]. Here we have chosen simply to illustrate our method in a “minimal” setup. The issue of initial conditions in eternally inflating universe will be discussed in Section 5.1. Ultimately, when we apply our framework to the entire multiverse, the initial condition needs to be specified by the “theory of the beginning.” Candidates for such a theory will be discussed in Sections 5.2 and 8.4.

Despite the fact that the spacetime is eternally inflating, any of the geodesics (except for a measure zero subset) experiences only a finite number of cosmic phase transitions and ends up with one of the terminal vacua—i.e. vacua with absolutely zero or negative cosmological constants—or a black hole singularity.999We assume that the landscape has (at least one) terminal vacua, as suggested by string theory. This assumption is also needed to avoid the Boltzmann brain problem; see Section 7.1. This implies that if we follow the “history” of one of these ’s by keeping track of the evolution of its past light cone, the possibility of finding a light cone that is consistent with any input conditions is extremely small. However, since the input conditions necessarily have some “range,” as discussed in Section 2.1, the probability of finding such a light cone is nonzero, .101010If the landscape is decomposed into several disconnected pieces, some choice for the initial state may lead to . This implies that our vacuum does not belong to the same landscape component as the initial state, and we need to (re)choose another initial state to make predictions/postdictions. In order for the entire framework to make sense (i.e. our own existence to be compatible with the multiverse picture), the correct choice for the initial state must lead to . Therefore, if we prepare a large number of ’s:

(8)

then we can find many light cone samples that satisfy the conditions provided. A schematic depiction of this procedure was given in Fig. 1 in Section 1. Note that, if the input involves a specification of an exact physical “time,” e.g. through the precise state of our own brains (Section 2.1) or through (Section 2.2), then the number of light cones selected is countable and finite for each geodesic.111111The specification of an exact time is not a necessity. Indeed, in a real world we never have an exact knowledge about the current time, so that the “initial situation” light cones have some spread in the time direction, requiring us to count the passing of through tips of these light cones as a “single event.” In many cases, however, our knowledge about the current time is sufficiently precise for the present purposes, so that the error arising from treating it as an exact time is negligible in the limit where is large.

Our proposal is to use the ensemble of past light cones obtained in this way to make predictions/postdictions according to the procedures described in Section 2. Suppose, for example, that we obtain past light cone samples satisfying prior conditions , and that among these samples past light cones also satisfy conditions . Then, the conditional probability of under , , is given by

(9)

Since the distribution of ’s is not correlated with the histories afterwards, the resulting probability does not depend on this distribution in the limit of very large . Also, since and are both finite at arbitrarily large , the probability is well-defined.

Similarly, in order to predict the future in the sense of Section 2.1, we can count the number of past light cones in the sample that evolve into a particular future situation : . The probability that occurs under conditions is then given by

(10)

If we select a set of future situations which are exhaustive and mutually exclusive, then we have

(11)

In the example of predicting the result of the LHC experiments (as in Sections 2.1 and 2.2), we may choose, e.g., the LHC will {find just the Higgs boson, find supersymmetry, find technicolor, find any other theory or the result will be inconclusive, fail}.121212In fact, the two definitions of Eqs. (9) and (10) are not independent, since we can calculate the latter, , from the former, , once we know for all ’s that could affect ’s.

We note that the validity of our method is not limited to vacua with four spacetime dimensions. Our prior conditions can be formulated in any numbers of non-compact and compact spatial dimensions. While it is appropriate in most circumstances to limit our discussions to -dimensional spacetime by integrating out all the effects of extra compact dimensions, there is nothing wrong with applying our procedure to the full, higher dimensional context. Such a treatment may be useful if we want to postdict the number of our large spacetime dimensions to be four (unless the anthropic factor is trivially zero except for four large dimensions). For the universe we live today, our knowledge about compact dimensions must come from bounds from experimental searches, as well as the structure of the observed physical laws if relations between four-dimensional physics and extra dimensional geometries are known in the landscape. The initial space-like hypersurface may also have arbitrary numbers of non-compact and compact spatial dimensions.

In summary, the probabilities are defined through the sampling of past light cones that satisfy our prior conditions, by following histories of many geodesics emanating from an initial hypersurface . Note that this procedure corresponds precisely to simulating the entire multiverse many times as viewed from a single “observer.”131313The procedure may also be viewed as a sort of “fuzzy” time cutoff in expanding universes; see Appendix A. It is quite satisfactory that this operational way of defining probabilities also gives a well-defined prescription for calculating probabilities we are interested in: Eqs. (9) and (10). Some sample calculations of the probabilities are given for toy landscapes in Appendix B.

3.2 Necessity of a quantum mechanical treatment

So far, we have been describing our framework using the language of classical spacetime. We know, however, that our world is quantum mechanical. This raises an important question: does the prescription really make sense in the quantum mechanical world? In particular, can we literally follow the histories of ’s classically, emanating from a small initial region ?

Suppose we want to “scan” our current universe finely enough so that we can find a past light cone that satisfies our input conditions. For example, let us imagine that we need one of the ’s for every at the current moment in order to follow the procedure described in Section 2.1. In this case, the average separation between ’s at the time when our observable inflation, i.e. the last -fold of inflation, began is roughly

(12)

where and are the current and reheating temperatures, respectively. Here, we have ignored any focusing of geodesics, e.g. due to structure formation, for simplicity, and used to obtain the final number. We find that the distance obtained is much shorter than the Planck length, ; in other words, we have a large number of ’s passing a Planck size region at the time when the observable inflation started. The number of ’s per a Planck size region increases even more if we extrapolate the history further back. This implies that we need to consider a huge number of ’s in a Planck size region on , from which ’s emanate.

On the other hand, we expect that at the length scale gravity becomes strong, so that quantum effects become important even for spacetime.141414If the theory has a large number of species , the scale where quantum gravity becomes important is actually (or the string length [16]. This, however, does not affect any of our discussions, including the one here. In the rest of the paper, we will set (or ) for simplicity, but it is straightforward to recover the dependence on , if needed. In particular, the holographic principle (or de Sitter entropy) suggests that a Planck size region in de Sitter space can contain at most of bits of information. Then, how can we make sense out of histories of many ’s distributed inside a Planck size region? This naturally suggests the following interpretation of our procedure. Starting from a small (even a Planck size) region in the early universe, we follow its evolution quantum mechanically. Because of the quantum nature, this probabilistically leads to many cosmic histories; and it is these possible histories that correspond to sets of past light cones associated with various ’s in the semi-classical picture.

In the next section, we will see that this interpretation is, in fact, forced by the laws of quantum mechanics. Interestingly, quantum mechanics cannot be viewed as “small corrections” to the classical picture—quantum mechanics is essential to (correctly) interpret the multiverse.

4 Multiverse as a Quantum Mechanical Universe

In this section, we show that the quantum mechanical interpretation of the multiverse is unavoidable if we assume that the laws of quantum mechanics are not violated when physics is described from an observer’s viewpoint. In particular, we argue that the entire multiverse can be described from the point of view of a “single observer.” We develop an explicit quantum mechanical formalism for making predictions/postdictions, which corresponds to the framework described in Section 3 in the semi-classical picture. We first consider the case in which the multiverse is in a pure quantum state, and then generalize it to the mixed state case. We also discuss how the quantum-to-classical transition is incorporated in our framework.

4.1 Quantum mechanics and the “multiverse”

Recall the laws of quantum mechanics—deterministic, unitary evolution of the states and the superposition principle. (We adopt the Schrödinger picture throughout.) These laws say: (i) given a pure quantum state , its history is uniquely determined by solving quantum evolution equation, where is a unitary operator; and (ii) if and are both solutions of the evolution equation, then is also a solution for arbitrary, complex numbers and .

While these properties are manifest in the usual formulation of non-relativistic quantum mechanics, they are sometimes obscured in quantum field theory—can’t an initial “state” evolve into , or some other “states”? Such an evolution, however, does not contradict the fact that quantum evolution is deterministic. What is happening is simply that the state that is initially , , evolves into a unique final state that has nontrivial overlaps with the states approaching in the future, . Expanding in terms of the states in free field theories (which is possible at ), we can write

(13)

where , , are uniquely determined.151515Here we have suppressed the momenta and spins of the particles, but including them is straightforward. In fact, the evolution of states in quantum field theory is unitary, and satisfies the superposition principle.

These properties of quantum mechanics essentially force us to take the quantum mechanical view of the multiverse, outlined at the end of Section 3.2. Consider a small initial region at an early universe, , which contains an eternally inflating point. If we follow the evolution of this initial (presumably highly symmetric) state quantum mechanically, we should find that future states are uniquely determined. On the other hand, the semi-classical picture of eternal inflation says that various observers emigrating from will see different semi-classical histories, or universes. Note that, assuming the energy density on is (much) smaller than the Planck density, we do not have any reason to doubt this semi-classical picture. The only way to reconcile these two facts is to consider that various semi-classical histories correspond to possible outcomes obtained from a unique quantum state . Schematically,

(14)

Below, we present more precise arguments on this point and give an explicit formalism which makes well-defined predictions/postdictions possible in the multiverse.

4.2 Lessons from black holes

The two expressions in Eqs. (13) and (14) look very similar, but there is one significant difference. While the usual “in-out” formalism of quantum field theory, exemplified in Eq. (13), deals only with the states at infinite past or future (which is indeed quite sufficient for the purpose of describing experiments involving scattering or decay), the cosmological setup of Eq. (14) requires states defined throughout the whole history. As a relativistic version of quantum mechanics, quantum field theory must be able to describe such a state, . However, a naive implementation of this under the existence of gravity could cause a problem—the laws of quantum mechanics may be violated.

To see what might happen, let us consider a traveler falling into a black hole, carrying some information. For this traveler, the information is always with him/her until he/she hits the singularity; in particular, it will be inside the black hole at late times. On the other hand, for a distant observer, the information appears to be absorbed into the (stretched) horizon, and then sent back in Hawking radiation. This apparently indicates that the same information exists in two different locations, contradicting the no-cloning theorem of quantum mechanics, which says that a faithful duplication of quantum information is not possible (see Fig. 3).161616The no-cloning theorem can be proved using the laws of quantum mechanics stated in Section 4.1.

\Line(0,0)(0,160) \Text(-5,80)[r]\Line(0,0)(150,150) \Line(150,150)(70,230) \Line(70,160)(70,230) \Line(0,160)(70,160) \Photon(0,160)(70,160)28 \Text(35,168)[b]\DashLine(0,90)(70,160)3 \CArc(-116,40)(170,12,40) \CArc(-115.5,40)(170,12,40) \Vertex(50.8,74)2 \Line(25,132.5)(13,148.5) \Line(27.5,135)(15.5,151) \Line(13,152)(13,142) \Line(13,152)(22.6,149.2) \Photon(40,120)(70,150)24 \DashLine(74,156)(88.5,170.5)2 \DashLine(76,153.5)(91,168.5)2 \DashLine(92,172)(88.8,162.5)2 \DashLine(92,172)(82.5,168.8)2 \DashLine(0,145)(51.7,145)1 \DashCArc(52.7,160.0)(15,270,315)1 \DashLine(64.0,150.2)(66.1,152.3)1 \DashCArc(95,124.8)(40,77.6,135)1 \DashCArc(70,-33.3)(200,66.4,80)1
Figure 3: A Penrose diagram representing a traveler who falls into an evaporating black hole (solid curve) carrying some information. For the traveler, information appears to be always with him/her (solid arrow), while from a distant observer, the information appears to be sent back from the black hole in Hawking radiation (dashed arrow) An example of “wrong” constant time hypersurfaces is depicted with the dotted line.

This paradox, however, can be resolved by the so-called black hole complementarity [7]. Since one cannot be both a distant observer and the falling traveler at the same time, it is perfectly consistent that the two descriptions disagree about something that cannot in principle be compared; in particular, they may not agree about where the information exists after the traveler crossed the horizon. When physics is described consistently by a distant observer, or by the falling traveler, the violation of quantum mechanical principles does not occur.

One can consider this phenomenon to arise as a result of noncommutativity of local measurements performed by the two experimenters: the traveler and distant observer . Suppose that and have their own complete sets of operators and , and that we take a Hilbert space basis in which the states that appear local to can be written schematically in the form

(15)

where the symbol means that is given by the direct product of all the elements in the vector. Here, each element corresponds to a different spatial point viewed from , and and represent excited and unexcited states at each point. The local operators for then take the (block-)diagonal form:

(16)

(ignoring any exponentially damping tails). In the same Hilbert space basis, however, operators local for are not necessarily diagonal, e.g.

(17)

The states in Eq. (15) may then look highly delocalized for the distant observer . (This happens at a time after the traveler passed the horizon). In fact, we expect that (local) information inside the black hole are stored in long-range correlations in Hawking radiation (or in horizon degrees of freedom) from an outside observer point of view [7].

Now, if we want to discuss physics as viewed from distant observer , it is absurd to take the basis in Eq. (17)—we should go to the basis where local operators for are diagonalized. In the new basis, Eq. (17) becomes

(18)

and the states in Eq. (15) look like

(19)

Two equations (15) and (19) clearly show that when we describe physics from an experimenter’s point of view, the two experimenters and describe the same system very differently—one localized and the other delocalized. (Of course, in the Hilbert space, the ’s in the two equations are the same states: they simply correspond to different coordinate representations.) Below, when we talk about an observer, we always assume that we take the basis such as Eqs. (16) and (18), i.e. the basis where local operators are “diagonalized,” which is the usual basis for local quantum fields.171717Later we will quantize the system on null hypersurfaces, in which case Schrödinger picture operators at different points do not all commute. This subtlety, however, does not affect the basic picture; see Section 4.5. This means, in particular, that when we change the viewpoint, we must in general perform the associated basis change in the Hilbert space.

The black hole example discussed here also tells us that the choice of constant time slices—a set of hypersurfaces on which quantum states are defined—is extremely important in quantum theories with gravity. Suppose we want to describe the system from a distant observer’s point of view, using the local operator basis as described above. How should we define the states? From Fig. 3, it is evident that if we choose a “wrong” time slice (often called a “nice” slice), we would have a state in which quantum information is duplicated, contradicting the no-cloning theorem. In fact, choosing this kind of slices, nonlocal effects from quantum gravity do not decouple at low energies, and we cannot simultaneously “diagonalize” operators localized in different spatial points [17]. Therefore, if we want to maintain a local description of physics, such choices of time slices should be avoided. The usual in-out formalism of quantum field theory bypasses this issue by considering only states at in asymptotically Minkowski (or anti de Sitter) spacetime, but it is in general important in the context of the cosmology of the multiverse.

4.3 Quantum observer principle—relativistic quantum mechanics for cosmology

We now present our quantum mechanical framework for describing the multiverse. Our central postulate, which we may call the quantum observer principle, is the following:

  • Physics obeys the laws of quantum mechanics when described from the viewpoint of an “observer” (geodesic) traveling the multiverse, although this need not be the case if described in other ways, e.g., using the global spacetime picture with synchronous time slicing. The description involves only spacetime regions inside the (stretched) apparent horizons, as well as the degrees of freedom associated with these horizons.

As we discussed in the previous subsection, we choose a Hilbert space basis which “diagonalizes” local operators as viewed from the observer; in particular, we take the same operator basis for all observers. (We can always do this because, being state independent, complete operator sets for observers are all isomorphic.) To realize the framework explicitly, we still need to define quantum states carefully, which will be discussed in the next subsection. However, without such a realization, the single principle stated above already leads, together with the usual statistical interpretation of horizon entropies, to many nontrivial, general consequences for the description of the multiverse.

Let us now discuss implications of each element of the laws of quantum mechanics in turn:

  • Deterministic evolution — Consider a small, eternally inflating space-like region at some early time. According to the semi-classical picture, observers in this region can see infinitely many different universes in the future (i.e. the entire multiverse), even if they are “equivalent” (i.e. related by an element of the kinematic subgroup of the de Sitter group). On the other hand, the quantum observer principle says that quantum evolution of states is deterministic; and the holographic principle (or de Sitter entropy) says that the amount of information can carry, i.e. the number of different initial conditions one can prepare on , is finite. This implies that in the quantum picture, the origin of various semi-classical universes cannot be attributed to the difference of initial conditions on .181818Mathematically, the de Sitter group cannot be exact at the quantum level, because of the discreteness of energy levels [18]. However, since typical level spacing is of order the inverse Poincaré recurrence time, it is plausible that the effect is physically irrelevant, and semi-classically equivalent observers see equivalent quantum systems. (See also [19].) Indeed, string theory suggests that lifetimes of de Sitter vacua are (much) shorter than the recurrence times [20], so that no energy measurement can physically resolve the discreteness of the levels. The only possible interpretation, then, is that these different universes arise as possible outcomes obtained from a quantum state , which is uniquely determined once an initial condition is given at some early moment.191919Note that, taking the same operator basis for all observers, the two descriptions in Fig. 3, i.e. by the traveler and distant observer , correspond to having different (though equivalent) states and , subjected to different initial conditions: one falling and the other not falling into the black hole. On the other hand, if the two experimenters and arise in the future of an eternally inflating region , then the two descriptions become parts of a single multiverse state, , in the complete description of the multiverse.

  • Unitarity — The quantum state of the multiverse is meaningful if (and, presumably, only if) it is described from the point of view of an observer. In particular, this implies that the state is not defined outside horizons (of de Sitter, black hole, or any other kind). On the other hand, in the semi-classical picture, there are certainly processes which carry information to outside these horizons. How can unitary evolution of be ensured then? The answer is that, as in the black hole case, this information is stored in the stretched horizons when seen by the observer. (Indeed, there is evidence that any cosmological horizons virtually act as black hole horizons; see e.g. [21, 22, 23].) Namely, assuming a pure initial state, subsequent evolution of the multiverse is described by a pure quantum state if we include the microscopic description of the stretched horizons. For such a state, the usual, thermal description of a horizon emerges only after considering a suitable statistical ensemble in the multiverse, by “coarse-graining” the horizon degrees of freedom.202020For a generic state , the entanglement entropy between the bulk and stretched-horizon degrees of freedom is smaller than the full horizon entropy, which takes into account all possible microstates associated with the horizon. A nontrivial consistency check of the statement made here is provided in Appendix C.

  • Superposition principle — Suppose we want to compute the probability of an initial situation at time to develop into a future situation at later time . We would then evolve the state representing by , and take the overlap with the state(s) corresponding to : , where is the Hamiltonian describing the dynamics of the system. On the other hand, in the description of the entire multiverse, the situation occurs multiple times at () as one of the possible universes: , where in general. Because of the superposition principle, however, the evolution of the term in is exactly the same as the state (if we use the complete multiverse evolution operator for , which is most accurate though often overkill). This implies that we may compute the probability using the entire multiverse state , instead of ; in particular, we need not consider that collapses to when we compute .

An important consequence of the framework described here is that the quantum state , defined from a single observer’s viewpoint, describes the entire multiverse. This is indicated by the fact that eternal inflation populates the entire landscape in the semi-classical picture, even starting from a measure zero point on an initial hypersurface.212121Here we have assumed that the multiverse is irreducible, i.e. any two points in the landscape are connected by some physical processes. The case with a reducible landscape will be mentioned in Section 5 (in footnote 26). Namely, the state (which may in general be a pure or mixed state) provides a complete description of the multiverse.

4.4 The entire multiverse from the viewpoint of a single observer

How can the multiverse state be defined explicitly? Let us suppose, for simplicity, that the multiverse is described by a pure state . (An extension to the mixed state case is straightforward and will be discussed later.) From an observer’s point of view, the subsequent evolution of this state is uniquely determined according to the laws of quantum mechanics. A question is: what constant time slices should we choose to describe the system “as viewed from an observer”? In order to maintain locality of the description, such slices should not extend to the region that cannot be accessed by the observer.

Let us consider how we can define time slices in an operationally well-defined manner. In the multiverse, one obviously cannot carry a physical clock through (some of) the cosmic phase transitions occurring in the landscape—even low energy degrees of freedom may change across bubble walls associated with the transitions. This implies that we cannot use any physical quantities, e.g. average CMB temperature, to define equal time hypersurfaces throughout the entire multiverse. Of course, we can adopt any time parametrization along the geodesic (observer)—a natural choice is the proper time, which can be assigned a simple, invariant meaning. But how can we extend it to the rest of spacetime without referring to any physical quantities?

We use the causal structure to determine the equal time hypersurfaces, i.e. the hypersurfaces on which quantum states are defined. While this is not an absolute necessity (see discussion in Section 4.5), it provides the conceptually simplest formulation of our framework. Specifically, consider a spacetime point along the geodesic (observer) corresponding to a proper time . We then define a quantum state on the past light cone whose tip is at . We assume that is defined only within the stretched horizon, which is located roughly proper distance in front of the real, mathematical horizon in the static coordinates. The rationale behind this is the following. In the “coarse-grained picture,” which considers an ensemble of spacetime regions having the same macroscopic properties, the local temperature on the stretched horizon is of , due to the blueshift effect. Since this thermal entropy already saturates the horizon entropy, the number of physical degrees of freedom behind the stretched horizon should be at most of , where is the horizon area (see, e.g., [24]). This implies that we can describe any physics in that region using degrees of freedom “on” the stretched horizon, which we include in .

Suppose that an initial condition for the multiverse is given on a space-like hypersurface (which need not be the case, as we will see shortly). In this case, the multiverse state is defined at early times on the past light cone within and on outside , where is the intersection between the light cone and (see Fig. 2 in Section 1). This situation, however, does not last long. After a brief initial period, specifically for after the initial moment, where is the Hubble parameter, the state is given entirely on the past light cones bounded by the stretched horizon, as depicted in Fig. 4.

Figure 4: The state is defined on the past light cones (thin solid lines) bounded either by the initial hypersurface (wavy line) or the stretched horizon (thick solid). The tips of these light cones are on the geodesic corresponding to the “observer,” and the time parameter is chosen such that it agrees with the proper time associated with the observer. If the initial condition is given on a past light cone and the stretched horizon bounding it, then the introduction of the space-like hypersurface is not necessary.

One may wonder what happens if the observer enters into a Minkowski or anti de Sitter bubble at a late stage in the multiverse evolution. In this case, the (stretched) event horizon disappears, which would be bounding the region where is defined. Does this mean that we need to know the entire (future) cosmic history to determine the region where is defined? Or should we suddenly change the defining region of to the entire past light cone all the way back to , when the observer undergoes a transition from a de Sitter to a Minkowski/anti de Sitter phase? Both of these would be highly unreasonable. Considerations along these lines lead us to the following picture: the defining region for is determined not by the event horizon but by the apparent horizon, a surface on which the (local) expansion of the cross sectional area of the past light cone turns from positive to negative.222222For a use of apparent horizons in the context of a geometric cutoff, see [25]. With this definition, the state lives on a smaller portion of the light cone, as sketched in Fig. 5 for nucleations of Minkowski and anti de Sitter bubbles.

\Line(0,0)(20,0) \Line(0,0.5)(20,0.5) \Line(20,0)(100,80) \Line(20,0.7)(100,80.7) \Line(100,80)(180,0) \Line(100,80.7)(180,0.7) \Line(180,0)(200,0) \Line(180,0.5)(200,0.5) \DashLine(20,0)(86.5,-68.5)2 \DashCArc(77,-80)(14.7,0,42)2 \DashLine(180,0)(113.5,-68.5)2 \DashCArc(123,-80)(14.7,138,180)2 \CArc(297,-277)(300,135,160) \CArc(297.3,-277.3)(300,135,159.95) \CArc(74.3,-54.3)(15,-45,45) \CArc(74.8,-54.3)(15,-47,47) \Line(84.9,-43.7)(71.6,-30.4) \Line(85.2,-43.4)(71.9,-30.1) \CArc(-13.3,-115.4)(120.1,45,74) \CArc(-13.0,-115.1)(120.1,45,74.3) \CArc(-97,-277)(300,20,45) \CArc(-97.3,-277.3)(300,20.05,45) \CArc(125.7,-54.3)(15,135,225) \CArc(125.2,-54.3)(15,133,227) \Line(115.1,-43.7)(128.4,-30.4) \Line(114.8,-43.4)(128.1,-30.1) \CArc(213.3,-115.4)(120.1,106,135) \CArc(213.0,-115.1)(120.1,105.7,135) \Line(100,75)(24,-1) \Line(100,75)(176,-1) \Line(100,65)(31.2,-3.8) \Line(100,65)(168.8,-3.8) \Line(100,50)(41.5,-8.5) \Line(100,50)(158.5,-8.5) \Line(100,25)(57,-18) \Line(100,25)(143,-18) \Line(100,0)(71,-29) \Line(100,0)(129,-29) \Line(100,-60)(45,-115) \Line(100,-60)(155,-115) \Line(100,-85)(18,-167) \Line(100,-85)(182,-167) \Line(99,-50)(99,-23) \Line(101,-50)(101,-23) \Line(100,-20)(96,-30) \Line(100,-20)(104,-30) \Text(90,-23)[l]\Text(100,-155)[t]de Sitter \LongArrow(30,50)(60,30) \Text(30,57)[b]Minkowski \Line(270,0)(290,0) \Line(270,0.5)(290,0.5) \PhotonArc(370,-40)(89.7,26.5,153.5)222 \PhotonArc(370,-40)(90.3,26.5,153.5)222 \Line(450,0)(470,0) \Line(450,0.5)(470,0.5) \DashLine(290,0)(356.5,-68.5)2 \DashCArc(347,-80)(14.7,0,42)2 \DashLine(450,0)(383.5,-68.5)2 \DashCArc(393,-80)(14.7,138,180)2 \CArc(567,-277)(300,135,160) \CArc(567.3,-277.3)(300,135,159.95) \CArc(344.3,-54.3)(15,-45,45) \CArc(344.8,-54.3)(15,-47,47) \Line(354.9,-43.7)(341.6,-30.4) \Line(355.2,-43.4)(341.9,-30.1) \CArc(256.7,-115.4)(120.1,45,74) \CArc(257.0,-115.1)(120.1,45,74.3) \CArc(173,-277)(300,20,45) \CArc(172.7,-277.3)(300,20.05,45) \CArc(395.7,-54.3)(15,135,225) \CArc(395.2,-54.3)(15,133,227) \Line(385.1,-43.7)(398.4,-30.4) \Line(384.8,-43.4)(398.1,-30.1) \CArc(483.3,-115.4)(120.1,106,135) \CArc(483.0,-115.1)(120.1,105.7,135) \Line(370,50)(311.5,-8.5) \Line(370,50)(428.5,-8.5) \Line(370,45)(315,-10) \Line(370,45)(425,-10) \Line(370,35)(321,-14) \Line(370,35)(419,-14) \Line(370,20)(330,-20) \Line(370,20)(410,-20) \Line(370,-5)(343,-32) \Line(370,-5)(397,-32) \Line(370,-60)(315,-115) \Line(370,-60)(425,-115) \Line(370,-85)(288,-167) \Line(370,-85)(452,-167) \Line(369,-50)(369,-28) \Line(371,-50)(371,-28) \Line(370,-25)(366,-35) \Line(370,-25)(374,-35) \Text(360,-28)[l]\Text(370,-155)[t]de Sitter \LongArrow(290,40)(320,20) \Text(285,47)[b]Anti de Sitter
Figure 5: A quantum state for the multiverse, , is defined on past light cones ( lines) bounded by apparent horizons (thick solid lines). The left (right) diagram represents a nucleation of a Minkowski (anti de Sitter) bubble in a meta-stable de Sitter vacuum. The bubble walls are depicted by dashed lines.

Note that since the apparent horizon agrees with the Hubble horizon in the limit of small curvature, the picture of Fig. 4 is essentially unchanged. In particular, the horizon should still be stretched.

In fact, there are several reasons to choose the apparent horizon as the boundary of the defining region for the multiverse state:

  • The concept of apparent horizon does not depend on low energy physics, such as the gauge group or matter content, so that the definition can be extended to the entire multiverse.

  • While the apparent horizon does not exist for all spacetimes, it does exists in any Friedmann-Robertson-Walker (FRW) universe that starts from a big-bang or bubble nucleation in a higher energy meta-stable vacuum, i.e., spacetime relevant for cosmology in the multiverse.

  • The location of an apparent horizon is determined “locally” within the bubble universe. In particular, unlike a particle or event horizon, it does not depend on what happens in the infinite past or future. This is very comfortable. In particular, it allows us to describe physics without knowing the future.

  • The apparent horizon plays a role of the preferred screen in the holographic principle [26], i.e., its area bounds the amount of entropy on a light sheet associated with it. Here, a light sheet is defined as a congruence of light rays orthogonal to the screen whose cross sectional areas are non-increasing in the direction away from it.

The last point is especially interesting, since our constant time slice is given by a past light cone, which is precisely one of the light sheets associated with the horizon. This allows us to state the holographic principle in a very simple form: at a given time , the maximal number of degrees of freedom in the bulk is bounded by the area of the boundary in Planck units. Namely,

(20)

where is the bulk quantum state, and the area of the horizon bounding the “space” (i.e. the light cone). Another important point is that the mandatory existence of the apparent horizon (the second item above) allows us to specify the initial condition on a stretched apparent horizon and the past light cone bounded by it. This has an advantage that the introduction of a space-like hypersurface is not necessary, which considerably simplifies the formalism. In fact, by specifying initial conditions this way, the state is defined consistently on the past light cone bounded by the stretched apparent horizon.

We now have the multiverse state which is consistent with the principles of quantum mechanics. In general, consists of a superposition of different semi-classical spacetime configurations, and the above analysis applies to each of these components. The Hilbert space for the multiverse state is, therefore, given by

(21)

where and represent Hilbert spaces for the degrees of freedom on the past light cones inside and on the stretched apparent horizon, respectively, for a fixed semi-classical configuration . The representation of in Eq. (21) is analogous to the Fock space in usual quantum field theories.

How does the multiverse state evolve? The quantum observer principle states that the evolution is deterministic and unitary:

(22)

where is a unitary operator acting on the multiverse state . How can we determine the form of ? Since the definition of the state involves horizon degrees of freedom, an explicit expression of is obtained only with knowledge of quantum gravity.232323In particular, quantum gravity must provide a picture of how unitarity is ensured despite the fact that some of the semi-classical histories represented in end up with big crunch or black hole singularities; a conjecture on this will be given in Section 8.1. Also, since the apparent horizon can have space-like world volumes in some regions, its “time evolution” should be into space-like directions in those regions. This is not a problem. Since the holographic principle allows the apparent horizon to encode all the information on the portion of the observer’s past light cone that lies in the past of the horizon, such a “space-like evolution” can be equivalent to the standard time evolution of the system outside the horizon. Therefore, while the description in terms of is essential for the understanding of the conceptual issues, it is not very “practical” in making predictions for low energy physics (with current theoretical technology). This difficulty, however, can be circumvented if we adopt the following “approximation.” Suppose we are maximally ignorant about the state of the horizon degrees of freedom, so the multiverse is described by a bulk density matrix

(23)

where means the partial trace over the horizon degrees of freedom. This corresponds to the usual statistical description of the horizons, so that the evolution of can be determined by semi-classical calculations in low energy quantum field theories (for example, through thermal treatments for future horizons [27, 21] and stochastic approaches for past horizons [28]). Note that the unitarity of the evolution is not preserved in this description, so that information appears to be lost in some processes, as was found in the classic black hole analysis by Hawking [29]. The unitarity, however, is recovered once we include the horizon degrees of freedom, as in Eq. (22).

We finally mention the possibility that the multiverse itself is in a mixed state, rather than a pure state (which occurs, e.g., if the theory of initial conditions requires it or if our knowledge of the system is incomplete). In this case, the multiverse is described by a density matrix

(24)

whose evolution is given by

(25)

where we take following the convention. Again, quantum gravity is needed to obtain the complete evolution. The description corresponding to Eq. (23) can be obtained by considering a reduced density matrix

(26)

whose evolution can be determined using semi-classical calculations.

4.5 Probabilities in the quantum universe

Having defined states, we now consider operators. As discussed in Section 4.2, we take the Hilbert space basis in which local operators are “diagonalized.” Strictly speaking, with quantization on past light cones, operators at different points do not all commute, as in the case of usual light front quantization [30]. For example, in an ordinary Minkowski space, operators in the same angular direction do not necessarily commute due to possible causal connections. This subtlety, however, is not essential. In fact, we can avoid it if we adopt quantization on space-like hypersurfaces, instead of null hypersurfaces. The only crucial thing for our framework is to restrict the spacetime region to inside the causal patch bounded by apparent horizons—with this restriction, the problem discussed in Section 4.2 does not arise, and our previous formulae all persist. Here, however, we keep using past light cones as our “equal time” hypersurfaces. This has an advantage that observational conditions are easier to impose, since our “direct” knowledge (without using the evolution equation) is intrinsically limited to the spacetime region inside our past light cone.

The probabilities are defined through operators projecting Hilbert space onto subspaces which satisfy specified (observational) conditions. Let be a set of orthonormal states that satisfy condition . Then the corresponding projection operator is given by

(27)

The probability that a past light cone satisfying also has a property is then given by

(28)

where we have assumed that the multiverse is in a pure state . This probability corresponds to the semi-classical probability given in Eq. (9). We can similarly define the probability of a past light cone to evolve into a particular future situation :

(29)

where is the operator projecting onto states that satisfy and evolve into . This corresponds to Eq. (10) in the semi-classical picture. Note that the comment in footnote 12 still applies for the probabilities defined here, since the quantum evolution of is deterministic.

The probabilities in Eqs. (28) and (29) are well-defined. In particular, the numerators and denominators in these expressions are separately finite.242424Strictly speaking, they are finite only if is normalized. If not, they can be infinite, but these infinities cancel between the numerators and denominators, giving well-defined, finite probabilities. This finiteness can be understood as follows. Let us expand as

(30)

where () corresponds to a component which encounters -satisfying past light cones  times in the cosmic history. The quantity in the denominators in Eqs. (28) and (29) is then given by

(31)

Now, starting from any generic initial conditions, the probability of finding a past light cone that satisfies is exponentially small (since it requires the geodesic to tunnel into a particular vacuum, or vacua, in which can be satisfied). This implies that scales as

(32)

so that the sum in Eq. (31) converges. The expression in Eq. (31) is thus finite, as long as the multiverse state is appropriately normalized, e.g. . The finiteness of other quantities can also be understood in a similar way.

The probabilities defined here are also “gauge invariant,” i.e. they do not depend on time parametrization. This is fairly obvious from the expressions in Eqs. (28) and (29), but it can also be understood along the lines above. Basically, our probabilities count the number of times each history, , encounters a particular past light cone(s). Since the coefficients do not depend on time parametrization, the resulting probabilities do not depend on it either.

As discussed in Section 4.4, the complete evolution of requires the knowledge of quantum gravity, so that the probabilities in Eqs. (28) and (29) can only serve the role of defining the framework. To do a “practical” calculation, we need to focus on bulk physics, i.e.  introduced in Eq. (23) (at the cost of unitarity in processes involving horizons). The observational conditions, such as , should then be imposed only on the bulk part of the state. This is not a strong restriction, since our observational data are always on bulk physics in practice. The relevant projection operator is given by

(33)

where is an orthonormal set of the bulk part of the states satisfying condition ; see Eq. (21). The probability is then given by

(34)

where the trace is over the bulk part of Hilbert space. The probability is defined similarly, with the replacement .

The definition of Eq. (34) allows us to calculate probabilities without knowing quantum gravity. In principle it allows us to answer any questions, except for the ones regarding information stored on a horizon at some time in the history. In Appendix B.2, we give sample calculations in toy landscapes, where it is shown that the results agree with those obtained using the semi-classical definition. Of course, to have definite numbers, an initial condition for needs to be specified. The issue of initial conditions will be discussed in Sections 5 and 8.

We finally mention the case where the multiverse is in a mixed state . In this case, the probability is given by

(35)

where the trace and projection operators act on the entire Hilbert space. The probability can be defined similarly.

4.6 Quantum-to-classical transition

As we have seen, our framework is (necessarily) quantum mechanical. On the other hand, our daily experience is certainly (almost) classical. How does this dichotomy arise? Why do we not observe, e.g., a state which is a superposition of different macroscopic configurations?

To be specific, let us suppose that the multiverse was in some highly symmetric state at an early moment. In particular, we consider that respects rotational symmetry. This leads to a question: why do we not observe a chair next to us in a rotationally invariant, -wave state, i.e., a superposition of chairs with all different orientations? One might think that various physical processes, such as bubble nucleations, spontaneously break rotational symmetry. This is, however, too naive. The resulting state will still be a linear combination of states with various bubbles nucleating in all different locations such that rotational invariance is respected. Note that this problem is particularly acute in our context. With being the quantum state for the entire multiverse, there is no “environment” with which interacts to feel violation of rotational invariance.

Let us focus on a particular macroscopic object, such as a chair, desk, or human. As discussed above, we expect it to be in a rotationally invariant state:

(36)

where we have used a chair as an example and displayed explicitly only two configurations (upward and downward).252525Since the chair is a macroscopic object, () itself can be an arbitrary superposition of microscopic states that macroscopically look like an upward (downward) chair. Now, consider a second object, say a human, next to the chair. Is the quantum state a direct product of the rotationally invariant chair state and the rotationally invariant human state? Namely,

(37)

The answer is no. To see this explicitly, consider the Hamiltonian for the chair

(38)

where we have reduced the system to two states for presentation purposes, and the particular form of the matrix in the right-hand side is dictated by “rotational” symmetry ( in the reduced system). We find that the two eigenstates of the Hamiltonian are with eigenvalues , which are also eigenstates of “rotational” symmetry. However, for a macroscopic object, is exponentially small, since it involves quantum tunneling between the upward and downward configurations. Consequently, the two states are nearly degenerate, and the preferred basis for the chair is determined by any small “rotational”-symmetry violating perturbations [31]. In the above chair-human system, the perturbation is the existence of the human, so that the state of the entire system is not given by Eq. (37), but by

(39)

where we have arbitrarily assumed that the dynamics prefers to align the orientations of the chair and human, rather than anti-align. (Our conclusion does not depend on this particular choice.) The process transforming a (macroscopic) composite system to an entangled form, e.g. as in Eq. (39), is called decoherence, which typically occurs with extremely short timescales [32].

We now see why we do not observe a superposition of different macroscopic configurations in our daily life. Equation (39) says that the chair always has a definite orientation with respect to the human, which we may identify with ourselves. We do not observe a superposition of chairs, which would have been possible if the state contained a term such as as in Eq. (37). Note that quantum interferences between different terms in Eq. (39) are extremely small, since overlaps between macroscopically different configurations, such as and , are suppressed by the huge dimensionality of the corresponding Hilbert space. In fact, for any observables constructed out of local operators, matrix elements between macroscopically distinct states are highly suppressed, e.g. . This, therefore, provides preferred bases for any macroscopic systems.

Of course, the entire state in Eq. (39) is still rotationally invariant. This, however, does not matter. Since we ourselves are a part of the state, we never observe the rotationally invariant state . In fact, rotational invariance of the state implies that for any component in , there are terms whose entire histories are related to via rotational symmetry (see e.g. the first two terms in Eq. (39)). For a macroscopic system, the evolution of these terms are mutually independent for all practical purposes. Thus, we may simply use (or any other term related to it by the symmetry) when making predictions. Of course, we may still use the entire state if we want—the two procedures give identical results.

The consideration here addresses various questions associated with the quantum-to-classical transition, for example, why spontaneous symmetry breaking occurs at all [31], why density perturbation in inflation becomes classical [33], and why we do not observe a superposition of bubble universes. In our context, the multiverse state is indeed a superposition of various macroscopically different configurations. We see (almost) classical physics because we—who are a part of the state—are correlated (entangled) with the rest of the multiverse. Technically, our framework incorporates quantum measurement processes in the form of the imposition of observational conditions, such as , , and in Eqs. (28) and (29). This procedure exactly extracts information encoded in the correlations between ourselves and the rest of the world.

4.7 The possibility of a reduced Hilbert space

We finally mention the possibility that the Hilbert space of the theory is actually smaller than that in Eq. (21).

Consider an arbitrary semi-classical configuration of spacetime . In the multiverse, such a configuration arises as components of at various times (). Let us define an ensemble of all these states:

(40)

where the states are at fixed times , and is the projection operator. Since a component of a pure state is also a pure state, elements of are all pure states, which take the form

(41)

where .

Now, the holographic principle says that (the logarithm of) the number of all possible bulk configurations is bounded by the horizon area in Planck units:

(42)

We expect that this inequality is saturated, since the multiverse realizes all possible states throughout the history. It is also reasonable to assume that the horizon can contain only bits of information per Planck area. Suppose that this number takes a special value of , i.e. . Then we find

(43)

i.e. the numbers of bulk and horizon degrees of freedom are the same for a fixed .

If Eq. (43) is true for arbitrary , then it allows for an interesting possibility that the Hilbert space of the theory is actually the square root of Eq. (21), namely, there are one-to-one correspondences between elements of and for all . If this is the case, then we can provide a complete description of the multiverse in terms of either or .

5 Initial Conditions

The framework developed so far allows us to make predictions/postdictions once initial conditions are given. On the other hand, the framework itself does not provide a unique initial condition. This is, in fact, as it should be. Remember that the measure problem of eternal inflation has a priori nothing to do with the “beginning” of spacetime. Once eternal inflation occurs in the regime where semi-classical analyses are valid, then it already leads to the predictivity crisis—any event that can happen will happen infinitely many times. Our general framework should be able to regulate these infinities, starting from any eternally inflating vacuum. Namely, the framework should be sufficiently modular such that any of such vacua can be used as initial conditions, e.g., on .

On the other hand, the framework also provides a useful tool to probe the real beginning of spacetime, i.e., the initial condition of the multiverse. In this section, we discuss the issue of initial conditions from both these two perspectives. We first consider what initial conditions we need to use to calculate probabilities starting from an arbitrary eternally inflating state. We then briefly discuss the beginning of spacetime, deferring full discussions to Section 8.

5.1 Semi-classical predictions from eternal inflation

Suppose we consider a spacetime region that is eternally inflating, with the fields taking values collectively denoted as . We want to derive predictions for the future, e.g.  and , using our framework. What initial condition should we impose on the multiverse state?

We first note that the quantum state of the universe is not uniquely determined by saying that the universe is in a (approximately) de Sitter state with field values . There are two issues associated with this. The first comes from the fact that the de Sitter space has a nontrivial entropy , which implies that this space, in fact, represents a statistical ensemble of many quantum states with degrees of freedom. Here, is Newton’s constant and is the potential energy density. The only thing we know is that the system is in a thermal state with de Sitter temperature at the level of semi-classical approximation. In our framework, such a thermal picture arises (only) after integrating out horizon degrees of freedom, assuming complete ignorance about these degrees of freedom. This implies that we can (only) use the formalism based on the bulk density matrix, , when we address the questions discussed here.

The second issue is that eternally inflating spacetime is in fact not pure de Sitter—it has a nonzero decay width. This implies that we need to specify an initial hypersurface on which the universe was still in an eternally inflating phase. This specification affects physical predictions at late times, albeit weakly [14, 15]; namely, different choices for the hypersurface correspond to different physical setups. In our framework, a canonical choice for equal time hypersurfaces is to take them to be the observer’s past light cones bounded by the stretched horizon. Therefore, it is natural to define an initial condition on one of these hypersurfaces. Indeed, this choice is physically well motivated, since the spacetime region the observer can actually see is limited to that within his/her past light cone. With this choice, we are specifying that the entire universe was in an eternally inflating phase at some point in the history, as seen by the observer.

The initial condition we need to use, then, is that the universe is in a (approximately) de Sitter state at some initial moment, say at :

(44)

where , which depends on , and is the Hamiltonian operator defined for a theory which mimics the original theory around but has the stable vacuum at (see Fig. 6). (The trace in the denominator is taken in the Hilbert space of this theory.)

Figure 6: The Hamiltonian is that of a theory with the potential having the global minimum at and mimicking the original theory around . The potential of this theory and that of the original theory are drawn by the dotted and solid lines, respectively.

There is, of course, an ambiguity in defining such a theory, and thus , since there is no guideline for choosing a potential of the new theory above the potential barrier of the original theory. This ambiguity, however, leads to only exponentially suppressed corrections in the final results, as long as the de Sitter temperature is much smaller than the barrier height, which is necessary anyway for the minimum to be meta-stable. (And we do not expect semi-classical calculations to correctly reproduce exponentially suppressed contributions anyway.)

With the initial condition of Eq. (44), we can now derive any predictions of eternal inflation using Eq. (34). As discussed before, this can be done without knowing quantum gravity.

5.2 Issues with the initial condition of the multiverse

Let us now consider the beginning of spacetime. This will require a theory of initial conditions beyond what we have developed so far. Here we only discuss several issues associated with it, leaving more complete discussions to Section 8.

One possibility for the beginning is that it is purely determined by semi-classical considerations. Suppose that spacetime starts from some highly symmetric state, with the fields sitting at an extremum of the potential. Let () denote such extrema that have positive energy densities. We may then consider the initial bulk density matrix

(45)

where , , and is the Hamiltonian defined around the extremum at .262626This equation may also apply if the landscape is reducible, i.e., if there are multiple sectors in the potential that are not mutually connected by any physical processes. For

(46)

this corresponds to Hartle-Hawking [34] (the upper sign) and tunneling [35] (the lower sign) proposals.272727The Hartle-Hawking probability can be understood not as a probability of creating universes, but as a thermal distribution of universes in a theory with a de Sitter ground state [36]. In addition, this probability has the serious problem of overwhelming Boltzmann brain observers [37]. Therefore, the tunneling probability seems (relatively) more suitable in the present context. There are some unsatisfactory features in these “semi-classical beginning” pictures (although they may, in some sense, be unavoidable; see discussion in Section 8.4). First, the fact that the initial condition specifies only , and not , means that there is an intrinsic uncertainty which we cannot hope to reduce; in particular, we in principle cannot predict exact states for certain (horizon) degrees of freedom. Second, the expression of Eq. (45) is only approximate, as the definition of has an ambiguity, and we do not see any obvious way to make it exact.

An alternative possibility for the beginning is that the theory of initial conditions determines the multiverse to be in a specific pure state at the earliest moment. According to our hypothesis, this implies that the multiverse is in a pure state for arbitrary time , .

In either of these cases, if we admit the existence of the initial state—which is the case in any scenario along the lines of creation from “nothing” [38]—then the quantum observer principle is necessarily violated there, since we cannot evolve the state further back. While this is possible, in Section 8 we will explore the possibility that the principle is in fact not violated throughout the whole history, which will suggest that our multiverse is a “fluctuation” in some larger structure.

6 Quantum Measurements and Global Spacetime

In this section, we see that our framework allows for a unified treatment of quantum measurement processes and the eternally inflating multiverse. We conclude that the eternally inflating multiverse is the same as many worlds in quantum mechanics. We also discuss the relation of our single observer picture to the conventional, global spacetime picture.282828A similar relation is discussed independently by Raphael Bousso in the context of geometric cutoff measures [9].

6.1 Unification of the multiverse and many worlds in quantum mechanics

So far, we have mainly focused on issues at very large scales, e.g. bubble universes in eternal inflation, in applying our formalism. This, however, need not be the case. In fact, the formalism applies equally to any (even microscopic) quantum processes without modification.

Suppose the multiverse starts from a highly symmetric state . This state evolves into a superposition of states in which various bubble universes nucleate in various spacetime locations. As time passes, a state representing each universe further evolves into a superposition of states representing various possible cosmic histories, including different outcomes of “experiments” performed within that universe. (These “experiments” may, but need not, be scientific experiments—they can be any physical processes.) At late times, the multiverse state will thus contain an enormous number of terms, each of which represents a possible world that may arise from consistently with the laws of physics. A schematic picture of these “branching” processes is given in Fig. 7.

\Line(0,0)(0,36) \Line(0.5,0)(0.5,36) \Line(-0.5,0)(-0.5,36) \Text(0,-7)[t]The “beginning”\Text(0,-20)[t]of the multiverse\Line(0,36)(-48,84) \Line(-48,84)(-78,102) \Line(-48,84)(-70.8,102) \Line(-48,84)(-60,102) \Vertex(-90,126)1 \Vertex(-100,136)1 \Vertex(-110,146)1 \Vertex(-120,156)1 \Line(0,36)(-24,84) \Text(9.6,72)[]\Line(-24,84)(-60,132) \Line(-60,132)(-84,150) \Line(-60,132)(-78,150) \Line(-60,132)(-68.4,150) \Line(-24,84)(-42,132) \Text(-16.8,120)[]\Line(-24,84)(12,132) \Line(12,132)(20.4,150) \Line(12,132)(25.2,150) \Line(12,132)(36,150) \Text(7,121)[lt]different cosmic\Text(12,112)[lt]histories\Line(-42,132)(-66,180) \Line(-66,180)(-78,198) \Line(-66,180)(-74,198) \Line(-66,180)(-67,198) \Line(-42,132)(-50.4,180) \Text(-34.8,168)[]\Line(-50.4,180)(-58,198) \Line(-50.4,180)(-54,198) \Line(-50.4,180)(-46,198) \Line(-42,132)(-18,180) \Line(-18,180)(-17,198) \Line(-18,180)(-13,198) \Line(-18,180)(-6,198) \Text(-21,169)[lt]different outcomes\Text(-17,162)[lt]for “experiments”\Line(0,36)(48,84) \Line(48,84)(60,102) \Line(48,84)(66,102) \Line(48,84)(78,102) \Vertex(85,131)1 \Vertex(95,141)1 \Vertex(105,151)1 \Text(35,65)[lt]different universes\Line(-168,174)(168,174) \Text(176,174)[l]\Line(-90,15)(90,15) \Text(98,15)[l]
Figure 7: A schematic picture for the evolution of the multiverse state . The state is described from the viewpoint of a single “observer” (geodesic) traveling the multiverse.

The resulting picture is remarkably simple. From the initial state , the multiverse simply evolves deterministically according to the quantum mechanical evolution law. This evolution, however, is not along an axis in Hilbert space that is determined by operators local in spacetime. Therefore, at late times, the multiverse state is misaligned with (an enormous number of) axes determined by the local operators. (These axes are analogues of Fock states in usual quantum field theories.) This makes the multiverse state a superposition of a huge number of terms corresponding to different cosmic histories:

(47)

when expanded in the basis determined by local operators. As discussed before, all these histories are described from the viewpoint of a single “observer” (geodesic).

The picture given above is precisely that of the many-worlds interpretation of quantum mechanics [39]. Therefore, we conclude that the multiverse is the same as (or a specific manifestation of) many worlds in quantum mechanics. In fact, when we ask physical questions, we “inject” these questions to the theory by imposing conditions such as , , and in Eqs. (28) and (29). The questions may be about global properties of the universe, or about outcomes of a specific experiment being performed. Our framework, therefore, provides a fully unified treatment of (even microscopic) experiments and the multiverse. Indeed, it applies to any physical processes from the smallest (the Planck length) to the largest (the apparent horizon) scales.

Incidentally, if we are interested only in the future of a particular macroscopic configuration at time , then we may drop all the terms in Eq. (47) except for ones corresponding to the specified configuration. This is because the superposition principle guarantees that the evolution of the retained terms is independent of the dropped terms, and matrix elements of macroscopic observables between states with different macroscopic configurations are highly suppressed (see Section 4.6) so the dropped terms do not affect future predictions. This operation of truncating the state is precisely what is called wavefunction collapse, which is indeed an extremely good approximation when we ask questions about a system with large degrees of freedom.292929The approximation may even be “exact” if the system is large enough that the effect from the dropped terms is smaller than what is measurable within the limitation of the uncertainty principle. (Note that a system here includes an experimental apparatus, in addition to the object being measured.) It is, however, not an exact procedure in general, nor necessary for making predictions.

6.2 Connection to (“reconstruction” of) the global picture

What is the relation of our picture based on a single quantum observer to the picture based on global spacetime? One (extreme) attitude is to consider that “physical reality” is simply the multiverse quantum state, so that no other picture is needed. This indeed makes sense because all physical questions (regarding predictions/postdictions) can be answered using that state, following the prescription in Section 4. On the other hand, it is useful to understand a connection between the present picture and more conventional, global spacetime picture. Here we study this issue.

One way of developing intuition about the connection is to consider a large (infinite) number of identical multiverse states. As shown in conventional analyses of frequency operators [40], the resulting product state can be reorganized as

(48)

up to a zero-norm part of the state. Along the lines of Ref. [41], we may interpret the rightmost expression to represent each outcome spreading over (global) spacetime. In particular, any experimental results may be viewed as distributed over the multiverse, allowing us to interpret outcomes of a quantum measurement in the frequentist’s sense over spacetime.303030In contrast to Ref. [41], in which the assumption of statistical uniformity played an important role, our argument here requires only Eq. (48), since we do not have the measure problem. In fact, ’s in Eq. (48) are identical copies by construction.

Let us now study the connection between the two pictures in more detail. (The following analysis does not require Eq. (48), which was presented simply to help developing intuition.) Remember that the multiverse state contains all possible worlds that can consistently arise in the single observer’s viewpoint. For example, in a spatially homogeneous region, an event that can occur may occur anywhere in space, so the multiverse state contains a superposition of terms in which the event occurs in all different locations, as schematically illustrated in the upper line of Fig. 8.

\Text(-71,100)[l]\DashLine(61,80)(101,120)1 \DashLine(101,120)(141,80)1 \Text(90,92)[]\Text(150,100)[]\Line(156,80)(196,120) \Line(196,120)(236,80) \Text(197,92)[]\Text(245,100)[]\DashLine(251,80)(293,120)2 \DashLine(291,120)(333,80)2 \Text(304,92)[]\Text(333,100)[l]\Line(65,33)(85,33) \Line(80,36)(90,30) \Line(70,36)(60,30) \Line(65,27)(85,27) \Line(80,24)(90,30) \Line(70,24)(60,30) \Line(62.5,31.5)(60.625,29.625) \Line(66,33)(61.875,28.875) \Line(68,33)(63.125,28.125) \Line(70,33)(64.375,27.375) \Line(72,33)(66,27) \Line(74,33)(68,27) \Line(76,33)(70,27) \Line(78,33)(72,27) \Line(80,33)(74,27) \Line(82,33)(76,27) \Line(84,33)(78,27) \Line(85.7,32.7)(80,27) \Line(87.0,32.0)(82,27) \Line(88.2,31.2)(84,27) \Line(89.4,30.4)(87.6,28.6) \DashLine(150,0)(150,65)2 \DashLine(150,65)(147,60)2 \DashLine(150,65)(153,60)2 \DashLine(100,0)(150,50)2 \DashLine(150,50)(200,0)2 \Line(165,0)(165,65) \Line(165,65)(162,60) \Line(165,65)(168,60) \Line(115,0)(165,50) \Line(165,50)(215,0) \DashLine(180,0)(180,65)1 \DashLine(180,65)(177,60)1 \DashLine(180,65)(183,60)1 \DashLine(130,0)(180,50)1 \DashLine(180,50)(230,0)1 \Text(165.7,15)[]
Figure 8: A schematic depiction of the relation between the single quantum observer picture (the upper line) and the global spacetime picture (the lower line). The location of an event, represented by stars, is discretized for illustrative purposes.

Now, in the global spacetime picture, these terms can be interpreted to correspond to the same spacetime region as viewed from different observers (geodesics), as illustrated in the lower line of the figure.

In general, space is not homogeneous, i.e., the values of the coefficients are not identical for terms that have the event occurring in different locations relative to some other object (since the existence of the object breaks the symmetry). In the global spacetime picture, this means that the “density” of observers (geodesics) are not uniform in such regions. Note that the picture that spacetime is scanned/penetrated by a large number of geodesics is precisely what we had in Section 3. In fact, (semi-classical) global spacetime can always be “reconstructed” from the single quantum observer picture in the manner described here.

7 (No) Problems associated with Geometric Cutoffs

In a conventional treatment of the multiverse, one introduces a geometric cutoff to regularize infinite spacetime. The (relative) probabilities are then defined by counting the number of events within the cutoff, which is eliminated at the end of the calculation either by sending it to infinity or through a certain averaging procedure [5]. This treatment, however, introduces an arbitrariness associated with the choice of the cutoff, and also leads to peculiar conclusions such as the “end” of time [12]. Here we show that our framework does not suffer from these difficulties.

7.1 Youngness paradox, Boltzmann brains, and other problems

Arguably, the earliest, and numerically most severe, problem encountered in the context of eternal inflation is the youngness paradox [42]. Suppose we take FRW time while the universe is still in the eternally inflating phase, and then extend this time coordinate to future keeping a synchronous gauge condition. This defines a global time coordinate over the entire (semi-classical) multiverse. We can now define the relative probability of two classes of events by the ratio of the numbers of events occurring before a fixed time , taking at the end of the calculation [43].

This most naive—and seemingly innocuous—procedure, however, leads to violent contradictions with observations. Since the spatial volume of the eternally inflating region is growing exponentially with time, the rate at which bubble universes form is also increasing exponentially, proportional to the volume. This implies that, at any given time, there are enormously more younger universes than older ones. For instance, the number of universes with is a factor of larger than that with . It is hard to imagine that such an immense bias towards younger universes is compensated by any possible anthropic weight factor.

Our framework does not suffer from this problem. This is because the increase of spatial volume is not rewarded in calculating probabilities. (This is also true in classes of geometric cutoff measures proposed in Refs. [44, 45].) To see this, let us consider the quantum probabilities defined in Section 4. In this picture, the probabilities are calculated from the viewpoint of a single observer, so it is rather evident that the volume increase is not rewarded. For instance, there is obviously no gain in probabilities by staying longer in an eternally inflating phase. The same can also be seen in the semi-classical picture—spacetime expansion increases spatial volume, but only at the cost of diluting observers (geodesics). Since there is no reward for volume increase, any problems associated with volume weighting (such as -catastrophe [46]) do not arise in our framework.

The present framework is also free from ambiguities associated with the question “what is an observer?”;313131So far, we have been using the word “observer” to simply mean a geodesic in the multiverse. In contrast, the observer here means a “real observer” (called an experimenter in Section 4.2), who performs experiments/observations and collects data. I hope there is no confusion. e.g., does an observer mean a civilization, an individual, or some sort of consciousness? (Even what about a dog seeing a tree?) In fact, the answer to it should be specified already in a question we ask, i.e. in prior c