Quantum Mechanics, Spacetime Locality, and Gravity
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Abstract
Quantum mechanics introduces the concept of probability at the fundamental level, yielding the measurement problem. On the other hand, recent progress in cosmology has led to the “multiverse” picture, in which our observed universe is only one of the many, bringing an apparent arbitrariness in defining probabilities, called the measure problem. In this paper, we discuss how these two problems are related with each other, developing a picture for quantum measurement and cosmological histories in the quantum mechanical universe. In order to describe the cosmological dynamics correctly within the full quantum mechanical context, we need to identify the structure of the Hilbert space for a system with gravity. We argue that in order to keep spacetime locality, the Hilbert space for dynamical spacetime must be defined only in restricted spacetime regions: in and on the (stretched) apparent horizon as viewed from a fixed reference frame. This requirement arises from eliminating all the redundancies and overcountings in a general relativistic, global spacetime description of nature. It is responsible for horizon complementarity as well as the “observer dependence” of horizons/spacetime—these phenomena arise to represent changes of the reference frame in the relevant Hilbert space. This can be viewed as an extension of the Poincaré transformation in the quantum gravitational context. Given an initial condition, the evolution of the multiverse state obeys the laws of quantum mechanics—it evolves deterministically and unitarily. The beginning of the multiverse, however, is still an open issue.
Keywords
Quantum cosmology Quantum gravity Multiverse1 Introduction—The Basic Picture
This paper discusses two subjects: quantum mechanics and gravity, especially in the context of cosmology. Quantum mechanics introduced the concept of probability to physics at the fundamental level. This has led to the issue of the quantumtoclassical transition, in particular the measurement problem. Despite much progress, a complete and satisfactory picture, particularly the one including the entire universe, still seems missing.
Recent progress in cosmology has led to the “multiverse” picture—our observed universe may be one of the many in which low energy physical laws take different forms. This view is suggested by both observation and theory: it provides a successful understanding of the order of magnitude of the observed dark energy [1], and arises naturally as a result of eternal inflation [2, 3, 4, 5] and the string landscape [6, 7, 8, 9]. This elegant picture, however, suffers from the issue of predictivity—in the multiverse, any event that can happen will happen in infinitely many times, making any definition of probabilities extremely subtle [10, 11, 12, 13]. Many proposals have been put forward to regulate these infinities, but they seem to be arbitrary, without relying on a solid fundamental principle. This arbitrariness of defining probabilities in the multiverse is called the measure problem.^{1}
In Ref. [14], it was argued that the state Ψ(t)〉 must be defined only in restricted spacetime regions—in and on the (stretched) apparent horizons—consistently with what we learned about quantum gravity in the past two decades: the holographic principle [15, 16, 17] and black hole complementarity [18, 19, 20]. In the cosmological context, however, the locations of horizons are “observer dependent.” What does this really mean? Moreover, Ref. [14] also discussed the meaning of spacetime singularities from the low energy viewpoint, and argued that it implies that the multiverse evolves asymptotically into a supersymmetric Minkowski world. Do these results have any implications for the problem of quantum measurement?
We argue that, as quantum mechanics has helped the measure problem in eternal inflation, the multiverse can help the measurement problem in quantum mechanics. In particular, the fact that we observe an ordered, classical world can be explained by a combination of spacetime locality and the fact that the multiverse ultimately evolves into a Minkowski (or singularity) world, which has an infinitedimensional Hilbert space. This results in irreversibility of quantum measurement, despite the fact that the evolution of the multiverse state is unitary.
We also elucidate the meaning of the Hilbert space structure in Eq. (2). It is well known that to do Hamiltonian quantum mechanics, all the gauge redundancies must be fixed—and a theory of gravity has huge redundancies. Defining a state in Eq. (2) provides a simple way to fix these redundancies and to extract causal relations among events, which are physical (coordinate reparameterization invariant). In other words, we need to fix a reference frame when we describe a system with gravity quantum mechanically—this is the real meaning of the phrase: “physics must be described from the viewpoint of a single observer” in Ref. [14]. In particular, the location of a physical object/observer (with respect to “the origin of the coordinates” p) has physical meaning, so it needs to be included as a part of specification in conditions A and B when applying Eq. (1).
Since the Hilbert space Open image in new window in Eq. (2) is defined on restricted spacetime regions, changes of the reference frame represented in Open image in new window in general mix elements of different Open image in new window as well as the degrees of freedom associated with Open image in new window and Open image in new window . (More generally, changing the reference frame can also mix elements of Eq. (2) with intrinsically quantum mechanical states associated with singularities.) This is the origin of horizon complementarity (mixture between different Open image in new window ) and of the “observer dependence” of cosmic horizons (mixture between the bulk and horizon degrees of freedom)! This general transformation can be viewed as an extension of the Lorentz/Poincaré transformation in the quantum gravitational context. It introduces more “relativeness” in physical descriptions—it makes even the concept of spacetime relative, as it mixes the bulk and horizon degrees of freedom in general.
The above picture provides a complete account for the process of quantum measurement in the eternally inflating multiverse. While not all the aspects of the dynamics described above are fully proven, the basic picture is strongly supported by recent progress on understanding the quantumtoclassical transition (e.g. [21, 22, 23, 24, 27]). What are the implications of this in calculating physical probabilities in Eq. (1)? Physical information we can handle is only the “robust” kind, i.e. the one that can appear multiple times in physical systems. It therefore only makes sense to ask questions about information that is amplified in some component of the state. This corresponds to choosing projection operators Open image in new window and Open image in new window to extract only such information; in particular, it corresponds to projecting onto classically welldefined configurations when we ask questions about macroscopic systems.
The framework described here provides a solid theoretical ground for asking any physical questions in the quantum universe. However, to make actual predictions in the context of the multiverse, e.g. of the value of a physical parameter we observe, we still need to know the explicit form of the time evolution operator as well as the initial condition for the multiverse state (except for a few special cases, including that for calculating the distribution of the cosmological constant [28]). In particular, knowing the complete evolution of the state requires understanding of the dynamics of the horizon degrees of freedom as well as the full string landscape. The former can be bypassed if we adopt the semiclassical approximation based on the “bulk density matrix” [14], ρ _{bulk}(t)=Tr_{horizon}Ψ(t)〉〈Ψ(t), while the latter needs further progress in string theory. The initial condition for the multiverse state must be given by some external theory. Some (speculative) possibilities are presented in Ref. [14], but here we leave this issue aside and simply assume that an appropriate initial state is provided by some theory of initial conditions. (For a recent proposal to address the issue of initial conditions within the framework presented here, see Ref. [29].)
The organization of this paper is as follows. In the first half of the paper, Sects. 2, 3 and 4, we discuss quantum measurement without taking into account the effect of gravity. We discuss carefully how spacetime locality selects a basis in the Hilbert space, and we analyze quantum measurement in the context of applying quantum mechanics to the whole universe. We argue that the preferred basis for measurement is determined purely by the dynamics; in particular, the ultimate openness of the system is not required.
In the second half of the paper, Sects. 5, 6, 7 and 8, we discuss quantum mechanics in a system with gravity. Some of the results presented there were obtained in Ref. [14]; there are, however, some important refinements, e.g. on the precise definition of Open image in new window in Eq. (2), the treatment of spacetime singularities, and a useful probability formula applying in many practical cases. We argue that complementarity as well as the observer dependence of horizons can be understood in a unified manner from the fact that changes of the reference frame are represented in the Hilbert space defined in restricted spacetime regions. We restate that the eternally inflating multiverse and many worlds in quantum mechanics are the same [14].
Relations between quantum mechanics and the multiverse have been discussed in other work as well. Reference [30] considered the issue of basis selection in the context of the multiverse, although the resulting picture is crucially different from the one here, especially about unitarity of quantum mechanics. Earlier considerations of quantum mechanics in the multiverse/universe can be found in Refs. [31, 32]. The picture of the multiverse from a local viewpoint, which arises here as a consequence of quantum mechanics, has been promoted in the context of geometric cutoff measures; see Refs. [33, 34, 35] for example.
2 Probabilistic Interpretation of Quantum Mechanics
In this and the next two sections, we discuss how the probabilistic interpretation of quantum mechanics arises in a complete quantum mechanical system that includes a physical observer. We emphasize that the discussion does not require that the system is open as is often assumed in the treatment of quantum measurements. Having a precise understanding of this is crucial to apply quantum mechanics to the entire universe (or the eternally inflating multiverse).
Quantum Measurement a la von Neumann
Let us consider a simple nonrelativistic state in two dimensional Hilbert space. For definiteness, we take this to be a spin1/2 system: Ψ _{sys}〉=c _{↑}↑〉+c _{↓}↓〉, where c _{↑}^{2}+c _{↓}^{2}=1. The conventional Copenhagen interpretation says that if we measure the spin of this system at some time t=t _{m}, then we find it up or down with the probabilities P _{↑}=c _{↑}^{2} and P _{↓}=c _{↓}^{2}, respectively. We may write this as Open image in new window (α=↑,↓), where Open image in new window is the operator that projects onto the state with a definite spin α. After we measure a definite outcome, e.g. spin up, the wavefunction Ψ _{sys}〉 of the system “collapses,” \(\varPsi_{\mathrm{sys}}\rangle \stackrel{t=t_{\mathrm{m}}}{\longrightarrow} \uparrow \rangle\).
The Problem of the Preferred Basis
A standard answer to this question is environmental decoherence [37, 38, 39], whose implementation in the present context goes as follows. We first regard the apparatus and spin as open quantum systems, interacting with some “environment” E _{0}〉. We can then define the preferred states for the combined apparatusspin system as the states that are least sensitive to the interaction with the environment, i.e. those that are least entangled with the environment by dynamical evolution. For instance, if the interaction between the apparatus and environment is such that Open image in new window and Open image in new window with 〈E _{1}E _{2}〉→0, then the preferred states are the two terms in the righthand side of Eq. (7) because each of them will not get entangled with the environment. The measurement is then claimed to be performed in this preferred state basis.
In this picture of environmentinduced basis selection, the openness of quantum systems plays a crucial role in understanding measurement processes. In fact, such a picture is appropriate for the purpose of discussing consequences of quantum measurement performed in terrestrial experiments, which are indeed open. At the fundamental level, however, this raises the following question: what if we include the environment in the description of our quantum state? One might say that there is always some environment for any system in practice, but here we are talking about the fundamental issue. This question becomes particularly acute if we try to apply quantum mechanics to describe the entire universe, since then it is not even clear what one can take as an environment for the entire universe.
A line of reasoning like this has recently led the authors of Ref. [30] to claim that quantum mechanics is operationally well defined only under the existence of intrinsically inaccessible degrees of freedom, which they took to be those escaping a cosmic horizon in the eternally inflating multiverse. In this picture, quantum mechanical evolution is intrinsically irreversible—to obtain probabilistic interpretation of quantum mechanics, degrees of freedom outside the horizon must be traced out. Here we will argue differently—we need not introduce such irreversibility at the fundamental level. We argue that, as discussed in Ref. [14], the principles of quantum mechanics, including deterministic unitary evolution of the states, are fully respected if one describes physics as viewed from a single reference frame. The ambiguity of the basis is fixed by a feature in the dynamics, specifically spacetime locality as encoded in the algebra of (low energy) operators. We now see in detail what the implications of spacetime locality are in our context.
3 Physical Predictions and Spacetime Locality
Physical Information is in Matrix Elements
The simple analysis above features two points. First, expressed in the form of conditional probabilities as in Eq. (11), physical predictions do not depend on how the state is written, including in what basis it is expanded. Second, our (more fundamental) formula of Eq. (11) reproduces Eq. (10) even if the state Ψ(t)〉 contains additional terms that are not selected by the projection operator Open image in new window . In fact, given an initial condition, the state at late times might contain a term representing a possibility that is not listed in Eq. (9); for example, the apparatus might break before the observer reads it, or the observer might change his/her mind and never look at the apparatus. Because of the way we asked the question, however, our answer always satisfies ∑_{ α } P(αobs)=1. Namely, the possible additional terms in the state Ψ(t)〉 are irrelevant for the question we are asking.
As discussed in detail in Ref. [14], any physical question can be phrased in the form of a conditional probability; in the simplest setup, we can phrase it as: “Given what we know about our past light cone, A, what is the probability of that light cone to have properties B as well?” This eliminates the question of what the right basis is to expand the state. The answer is that it doesn’t matter. Once the question is phrased using the appropriate projection operators Open image in new window and Open image in new window (e.g. Open image in new window and Open image in new window in the above example), the desired probability P(BA) is defined unambiguously.
Note that this, however, still leaves the question of “what is the right question to ask?” For example, if one asks the question in the form “what value of cosmological parameter X will I measure?,” then the ambiguity (of the question, not of the formalism) lies in the definition of “I” (or “we” unless the complete brain state is specified). This comprises a part of the measure problem in eternal inflation. The present framework offers the possibility of formulating it explicitly in terms of the projection operators used in the quantum mechanical probability formula, hence relating the measure problem with the issue of defining quantum mechanical probability.
Spacetime Locality and the Basis in Hilbert Space
Is there any ambiguity in writing operators Open image in new window and Open image in new window , and if so, wouldn’t that just be trading the basis ambiguity of states for that of operators? First of all, we note that it is appropriate to discuss the issue of basis in terms of operators, rather than states, as we will do here. This is because Hilbert space by itself does not carry any physical information other than its dimensionality—any (complex) Hilbert spaces having the same dimension are identical with each other. Of course, being operators acting on a vector space, quantum operators may also be written in an arbitrary basis. However, we now have dynamical structures that may distinguish some basis over the others. In particular, there can be a special basis in which algebraic relations among operators look particularly simple.
We should emphasize that the choice of the Hilbert space basis discussed here does not by itself address the issue of basis selection for quantum measurement described in Sect. 2, although the former is needed for the discussion of the latter. Indeed, the choice described here is, in some sense, “a matter of convenience,” in that we can also describe physics using the ϕ(z) basis in principle (because the matrix elements, appearing in the probability formula, do not depend on the basis). In this basis, however, the time evolution operator has an extremely complicated form, which completely obscures the fact that the dynamics respects spacetime locality. Therefore, in practice one always needs to choose a Hilbert space basis associated with locality: either κ _{ m }〉 in Eq. (14) or a basis that has a simple relation to it (such as the momentum basis).
What Physical Questions May One Ask?
Let us choose the “locality basis” κ _{ m }〉, given in Eq. (14). Then there is no ambiguity in expanding states as in Eq. (15). The question, however, still remains: how can we choose the “correct form” for projection operators Open image in new window and Open image in new window appearing in the probability formula? Experience says that all the information we can explicitly handle (in the sense that it can be duplicated in physical systems) is given in the form of, e.g., Eq. (11)—i.e. by operators projecting onto states that have welldefined macroscopic configurations in the phase space (up to some uncertainties). Why is that?
In general, states having welldefined macroscopic configurations are obtained as superpositions of κ _{ m }〉 that have “similar” spatial configurations. For each macroscopic configuration i, we have a set of n _{ i } corresponding microstates: \(\psi^{\mathrm{(i)}}_{a}\rangle = \sum_{m} f_{a,m}^{(i)}  \kappa_{m} \rangle\) (a=1,…,n _{ i }), which we collectively call α _{ i }〉. Here, \(f_{a,m}^{(i)}\) for each (i,a) play the role of a smearing function in position space, ensuring that the configuration has a welldefined momentum at the macroscopic level. The projection operator onto macroscopic configuration i can then be defined as \( \alpha_{i} \rangle\langle\alpha _{i} \equiv \sum_{a=1}^{n_{i}} \psi^{\rm(i)}_{a}\rangle\langle\psi^{\mathrm{(i)}}_{a}\), where we have taken \(\langle\psi^{\rm(i)}_{a}  \psi^{\rm(i)}_{b} \rangle = \delta_{ab}\). Since \(\langle\psi^{\rm(i)}_{a}  \psi^{\rm(j)}_{b} \rangle \approx0\) for different macroscopic configurations i≠j, these projection operators satisfy P _{ i } P _{ j }≈P _{ j } P _{ i }≈0 for i≠j, where P _{ i }=α _{ i }〉〈α _{ i }.
4 Classical Reality in the Quantum Mechanical Universe
In this section, we discuss the origin of the following basic observational facts: (i) probabilistic processes in quantum mechanics are well described by density matrices that are diagonal in the “classical state basis” α _{ i }〉 at least for macroscopic systems; (ii) a measurement selects an outcome; namely, we can ignore other possible outcomes after a measurement is performed. In the standard treatment of these problems, the openness of a system is emphasized [21, 22, 23, 24, 27]. Here we ask if the openness is really necessary at the fundamental level to account for these features. We will argue that the answer is no—the quantumtoclassical transition may occur consistently with observation even in a closed, finite dimensional quantum mechanical system.
On the other hand, we will also argue that cosmology based on a finite dimensional system with a generic initial condition fails to explain another basic observational fact: (iii) we observe an ordered world, i.e., we perceive a world that obeys consistent laws of physics. This argument will force us to consider that the Hilbert space for the entire universe (multiverse) is infinitely large: Open image in new window , unless we abandon unitarity of quantum mechanical evolution or genericity of the quantum state.
A DoubleSlit Experiment in a Large System
We begin by a standard analysis of the doubleslit experiment, which sets the stage for later discussions. For the moment, we can be agnostic about whether the entire system is open or closed. The setup of the experiment is such that an electron, initially prepared at slits as \( \psi_{e,{\mathrm{init}}} \rangle= \frac{1}{\sqrt{2}} ( 1 \rangle+  2 \rangle)\) evolves according to 1〉→∫dx ψ _{1}(x)x〉 and 2〉→∫dx ψ _{2}(x)x〉. Here, 1〉 and 2〉 represent the electron localized at slits 1 and 2, respectively, while x〉 represents the electron localized at position x on the screen.
The question, again, is why the result of the experiment is described by Eq. (18). In particular, why do we perceive the world in such a way that a macroscopic quantum system decoheres in the classical state basis, in this case the location of the pointer of the apparatus? In the standard explanation due to environmentinduced basis selection, the openness of a system plays a crucial role. Below, we will obtain (essentially) the same result without invoking an openness of the whole system, which elucidates the real origin of the basis selection in quantum measurement.
Dynamical Selection of a Measurement Basis—Spacetime Locality
Let us keep following the state of the system after the doubleslit experiment was performed. Denoting the state after the doubleslit measurement by ψ _{ x }(t)〉, where Open image in new window , the state of the entire system is given by Ψ(t)〉=∫dx c _{ x }ψ _{ x }(t)〉, where c _{ x }≡(ψ _{1}(x)+ψ _{2}(x))/(∫dy ψ _{1}(y)+ψ _{2}(y)^{2})^{1/2}.
The condition used above to obtain Eq. (21), 〈r _{ x }(t _{exp})r _{ y }(t _{exp})〉≪1 for x≠y, is exactly what we expect. Since the detector states Open image in new window for different x have different macroscopic configurations and since the Hamiltonian of the system is local, the future states corresponding to different Open image in new window are almost orthogonal: 〈ψ _{ x }(t)ψ _{ y }(t)〉∼δ(x−y).^{5} This implies that 〈r _{ x }(t)r _{ y }(t)〉∼δ(x−y) because ϕ _{ x }(t)〉 is only a very small subset of the entire degrees of freedom in ψ _{ x }(t)〉, leading to Eq. (21).
The argument presented here is the heart of the basis selection in describing any experimental result in a way that our classical intuition is manifest. It also provides a real rationale behind Eq. (16), which was chosen to be diagonal in the classical state basis α _{ i }〉. The information about classical configuration α _{ i }〉 is that which can be amplified by the dynamical evolution—this is not very surprising given that α _{ i }〉’s are deeply related to the locality basis states, which are determined by the form of the time evolution operator. Questions we ask are about information that can be objectively accessed by multiple physical processes, hence the form of Eq. (16). Note that by integrating out R〉 as well as all the histories after t=0, this reproduces the usual einselection criterion in the decoherence paradigm. The argument here, however, makes it clear that the basis selection has nothing to do with the ultimate openness of the system—indeed, the present argument still applies even if the entire system is finite dimensional, being subject to thermalization and recurrences at later times. The origin of the basis selection lies entirely in the dynamics—specifically, the fact that the time evolution operator takes a special form of Eq. (13) in the locality basis.
Ordered Observations Require Infinitely Large Hilbert Space for a Generic State
Does the preceding argument ensure that the two features listed at the beginning of this section, (i) and (ii), are valid in any quantum system described by a local theory? In other words, can we always consider that a sufficiently macroscopic measurement collapses the wavefunction to one of the possible states having a welldefined classical configuration?
The answer is yes, but not trivially if the system is finite dimensional and we are interested in arbitrarily long timescales. In this case the entire system thermalizes at ∼t _{th}, after which it occasionally experiences rare fluctuations producing low entropy regions, and eventually comes back to a state arbitrarily close to the original state at timescale \(t_{*} \sim e^{S_{\mathrm{th}}}\), where S _{th} is the thermal entropy of the system. This picture applies regardless of the details of the system, as long as quantum mechanical evolution is unitary and the initial condition is generic, which we assume here. Since the process of producing low entropy fluctuations generically involves interferences between macroscopically different terms, one might think that replacing Ψ _{fin}〉 by a collapsed state Ψ _{col}〉 gives an obviously wrong answer. This conclusion, however, is too naive.
Hence a description of the universe consistent with our observation, i.e. item (iii) listed at the beginning of this section, is obtained only if the thermalization timescale of the system, t _{th}, is (much) larger than the timescale of interest t. Since \(t_{*} \sim e^{S_{\mathrm{th}}} \gg t_{\mathrm{th}}\) and Open image in new window , where Open image in new window is the Hilbert space dimension of the entire system, this implies Open image in new window . In particular, this implies that if we want to describe the entire history of the universe (t→∞), which we must do if quantum mechanical evolution is fundamentally unitary, then we need to take Open image in new window , i.e. the Hilbert space describing the quantum universe must be infinitely large.^{6}
As we will see in Sect. 7, this condition is satisfied in the eternally inflating multiverse because the multiverse evolves asymptotically to a supersymmetric Minkowski (or singularity) world, which contains an infinite number of states. This completes an ultimate picture for quantum measurement—a quantum measurement is a process in which a coherence existing in a (microscopic) system is dissipated into larger systems, ultimately into states in a supersymmetric Minkowski (or singularity) world. Recoherence of the resulting, macroscopically different worlds does not occur because of an infinitely large coarsegrained entropy of Minkowski (and singularity) space.
5 Spacetime Locality in Theories with Gravity
We have seen that spacetime locality plays a crucial role in quantum measurement processes. In theories with gravity, however, this property is not automatically guaranteed. In particular, if we take wrong hypersurfaces to quantize the system, theories are not local even at distances much larger than the quantum gravity scale.
In the rest of the paper, we study the structure of the Hilbert space describing the entire quantum universe, starting with the wellknown discussion on quantum mechanics of black holes. Some of the results presented in the following were obtained in Ref. [14]. There are, however, some important refinements, including the treatment of spacetime singularities and a useful probability formula that applies in many practical cases. We also provide a clearer argument leading to the results, and discuss their meaning, especially in the context of physical measurements.
Black Hole Complementarity
Here we review black hole complementarity [18, 19, 20, 43], which we assume gives the correct description of black hole physics.
Now, let us consider the fate of the information originally carried by the traveler. From the distant observer’s viewpoint, elements of Eq. (26) will be mapped, after the back hole evaporates, into those of the Hilbert space associated with spacetime without the black hole: Open image in new window . The information is then first in Open image in new window , and later in Open image in new window . From the falling traveler’s viewpoint, on the other hand, this information is in Open image in new window . A problem arises when we mix these two viewpoints in the global spacetime picture. In this picture, we can draw spacelike hypersurfaces—often called nice slices—on which the information exists both in Hawking radiation and inside spacetime. From a general relativistic point of view, there is nothing wrong with defining states on such hypersurfaces. This, however, leads to contradiction with the laws of quantum mechanics, specifically the nocloning theorem [25].
Black hole complementarity asserts that the problem arises because we have taken the global viewpoint that does not have any operational meaning. Indeed, because of the existence of the horizon, no physical observer can obtain the same information from inside region and Hawking radiation [45, 46]. This implies that if quantum mechanics is defined on equaltime hypersurfaces that pass through both the inside and outside information, then the low energy theory (not just states) must be nonlocal in such a way that these spatially separated degrees of freedom are not independent. Alternatively, if we want to keep locality in our low energy description of nature (which we do), then the Hilbert space should be restricted to the one associated with appropriate spacetime regions, e.g. Eq. (26) or (27)—including both the inside spacetime region and Hawking radiation in a single description is overcounting.
Quantum States are Defined in Restricted Spacetime Regions

The states are defined on the past light cone bounded by the “(stretched) apparent horizon.” Following the refinement in Ref. [47], here we take this horizon as the closer of the “observer horizon” and the surface on which the expansion of a light ray generating the past light cone turns from positive to negative. The former is defined in turn as a surface on which local proper acceleration of a constant affine parameter point along a light ray generating the cone exceeds the cutoff scale, where the affine parameter is normalized such that it agrees with the radial coordinate of a local Lorentz frame erected at the tip of the light cone. (This surface is reduced to the conventionally defined stretched horizon in the static limit; see Ref. [47] for details.) The degrees of freedom exist both inside and on the (stretched) apparent horizon, and the system is described as viewed from a local Lorentz frame at the tip of the light cone.

The definition above provides the simplest way of avoiding the overcounting of the type described above, making the time evolution operator local at distances larger than the quantum gravity scale. The evolution of a quantum state is deterministic and unitary in this Hilbert space (until spacetime singularities are hit; see Sect. 7).
The examples considered above clearly demonstrate that the restriction of spacetime regions is crucial to keep locality of the low energy theory while being consistent with quantum mechanics. Assuming that information absorbed into the de Sitter horizon can be retrieved later (which is necessary for stable de Sitter space to be regarded as a finite dimensional system, as suggested by the holographic principle), quantum states cannot be defined on hypersurfaces that pass through both Hawking radiation and outside spacetime containing the same information. This situation is analogous to that in the black hole case. The only difference is that the de Sitter horizon is “observer dependent”: its location changes depending on from whose point of view the system is described.
In Ref. [14], the definition of quantum states considered above was stated as: the system is described from the viewpoint of a single “observer” (geodesic). Here we phrase the same thing as: physics should be described using a single reference frame, which captures the essential physics better.
6 Importance of Fixing a Reference Frame
Fixing a Gauge—Physics Should be Described in a Single Reference Frame
What are observables in physical theories? They should be “gauge invariant,” i.e. quantities that do not depend on arbitrary parameterizations of the system corresponding to the redundancy of the description. In theories with gravity, the coordinatization of spacetime is precisely one such parameterization, so it might be thought that only observables are certain global quantities, e.g. the ones associated with the topology of spacetime. This is not true—causal relations among events are invariant under general coordinate transformations, and thus are physically observable.
It is well known that to do Hamiltonian quantum mechanics, all the gauge redundancies must be fixed. A theory of gravity has huge redundancies associated with general coordinate transformations. The definition of the states described in the previous section provides a simple way to fix these redundancies and extract causal relations among the events. The origin of a special point p (i.e. the tip of the past light cone used to define states) is now clear—it arises from the fact that the theory is invariant under local spatial translations and that we need to fix the resulting redundancies. By choosing a local Lorentz frame with the origin at p, all the redundancies associated with p are fixed. While this prescription by itself does not completely determine the gauge for general covariance, fixing the residual ones, coming from coordinate transformations on the past light cone, is simple conceptually and gives only minor effects on the overall picture.
Together with the restriction of spacetime within the (stretched) apparent horizons, this comprises the statement in Ref. [14] that “physics is described from the viewpoint of a single observer.” The choice of the local Lorentz frame at p implies that the tip of the light cone follows a geodesic at the semiclassical level. The overcounting of the type encountered before does not arise, and the time evolution operator is local at large distances.
Hilbert Space for Dynamical Spacetime—Analogy with Fock Space
We now construct the Hilbert space for dynamical spacetime, following the discussion so far. To do so, it is instructive to draw a close analogy with the construction of the Hilbert space in usual (nongravitational, nonconformal) quantum field theory.
The situation in quantum gravity is analogous. We first need to fix the Hilbert space basis to discuss states unambiguously. We assume that, with a fixed local Lorentz frame associated with a fixed reference point p, we have a set of local operators at low energies; specifically, we have a set of quantum fields ϕ _{ i }(x) defined on the past light cone of p. This can provide “meaning” to the states according to the responses to these field operators, and we can now construct states using the language of, e.g., spacetime points.
The general evolution of a state in dynamical spacetime is assumed to be unitary in the full Hilbert space Open image in new window in Eq. (33), but not in each Open image in new window . Unitarity of the evolution is a hypothesis of the framework. Its consistency was discussed in Ref. [14]. In particular, since the full information contained on the past light cone outside the horizon (in the sense of the conventional global spacetime picture) can be encoded on the horizon due to the covariant entropy bound, we can have enough degrees of freedom on the horizon that can determine the future evolution of the state as in conventional null quantization in global spacetime; namely, the evolution inside the horizon can be equivalent to the standard time evolution.
“Reference Frame Dependence” of the Concept of Spacetime
What happens if we change the reference frame, e.g. by a spatial translation or boost? As in any symmetry transformation, this operation must be represented by a unitary transformation in Hilbert space. In particular, if we focus on histories before any component of the state hitting spacetime singularities (the effect of which will be discussed in Sect. 7), then it must be represented entirely in the Hilbert space Open image in new window in Eq. (33), but not necessarily in each component Open image in new window . Namely, the transformation can mix elements in different Open image in new window . Moreover, even if the transformation maps all the elements in Open image in new window onto themselves for some Open image in new window , there is no reason that it should not mix the degrees of freedom associated with Open image in new window and Open image in new window .
The statement in Eq. (39) is nothing but black hole complementarity. Black hole complementarity (or more generally, horizon complementarity) arises because changes of the reference frame are represented in the Hilbert space of Eq. (33), which contains components Open image in new window that are defined only in restricted spacetime regions because of the existence of horizons. In general, these changes transform degrees of freedom associated with spacetime to those with a horizon, or vice versa—the concept of spacetime depends on the reference frame.
The transformation discussed here is supposed to be consistent with the global spacetime picture of general relativity. Under current theoretical technology, we may view this as a sort of “corresponding principle”—understanding the precise mathematical structure of the transformation law would require the (yet unknown) fundamental theory. The transformation can also be viewed as an extension of the Poincaré transformation in the quantum gravitational context. For a given t, the transformation here is specified by (d−1)(d+2)/2 parameters: r _{ i },η _{ i },θ _{[ij]}. In the limit G _{ N }→0, where relative accelerations between all families of geodesics vanish, the transformation is reduced to an element of the (d−1)(d+2)/2 parameter subset of standard Poincaré transformations consisting of spatial translations, rotations and boosts. Time translation also arises from invariance under a shift of the origin of t in the proper time parameterization. The set of these transformations, therefore, is reduced to the standard Poincaré transformations in the limit G _{ N }→0.
This is very much analogous to the fact that the standard Lorentz transformation is reduced to the Galilean transformation in the limit c→∞, where c is the speed of light. In the Galilean transformation a change of the reference frame leads only to a constant shift of all the velocities, while in the Lorentz transformation it also alters temporal and spatial lengths (time dilation and Lorentz contraction) and makes the concept of simultaneity relative. With gravity, a change of the reference frame makes even the concept of spacetime relative. The trend is consistent—as we “turn on” fundamental constants in nature (c=∞→finite and G _{ N }=0→finite), physics becomes more and more “relative,” i.e. the description of the same physical system from different reference frames differ more. The transformations described here (together with time translation) provide the extension of the Galilean group with c, G _{ N }, and ħ all finite.
7 Hilbert Space for Quantum Gravity
Here we discuss the full Hilbert space for quantum gravity. We argue that it contains an infinite number of “intrinsically quantum mechanical” states associated with spacetime singularities, which do not admit any classical interpretation. It is in this full Hilbert space in which the evolution of the multiverse state is unitary and an arbitrary reference frame change is represented as a unitary transformation.
Meaning of Spacetime Singularities

The covariant entropy bound does not count the degrees of freedom that have hit a singularity. Imagine sending a light sheet inwards from a black hole horizon H. The degrees of freedom swiped by the light sheet are then bounded by the area of the horizon Open image in new window . The entropy bound, however, does not limit the amount of information that hits the singularity before being swiped by the light sheet.

SuperPlanckian physics does not have degrees of freedom as suggested by field theory. Consider a black hole (or de Sitter) horizon. Because of blueshift, the local temperature at the mathematical horizon formally diverges. In fact, the physical horizon, from which Hawking radiation arises, is a Planckian distance away from the mathematical horizon: the “stretched” horizon. This suggests that superPlanckian physics does not have degrees of freedom as indicated by field theory.
What does dropping from consideration mean? In Ref. [14], it was postulated that a component that hits a singularity at time t _{ s } is simply eliminated from the state Ψ(t)〉 at that time, t=t _{ s }. This, however, leads to the following paradoxical situations. What if Ψ(t)〉 contains only components that hit singularities in the future? This must violate unitarity, and therefore so does any state containing them. Furthermore, even if we accepted unitarity violation, systems having only de Sitter and anti de Sitter vacua would allow for (a constant fraction of) Ψ(t)〉 to stay in a de Sitter phase at t→∞, since components tunneled into the anti de Sitter vacuum keep disappearing. This is extremely counterintuitive.
We assume that the evolution of the multiverse state Ψ(t)〉 is unitary in Open image in new window , and that any change of the reference frame at an arbitrary time t is represented by a unitary transformation in this Hilbert space. According to the current understanding of string theory, a full quantum gravitational theory possesses many de Sitter, anti de Sitter, and Minkowski vacua. In particular, it possesses exactly supersymmetric Minkowski vacua, which are absolutely stable due to the positive energy theorem [50, 51]. This implies that the dimension of Open image in new window is infinite, Open image in new window , since the Hilbert space dimension of (stable) Minkowski space is infinite. What about Open image in new window ? Consider a set of states that hit singularities at some late time. In the eternally inflating multiverse, these states can be mapped into those that evolve into stable supersymmetric Minkowski states, by appropriate boost transformations. This suggests that Open image in new window , which makes it possible that components that hit singularities do never return to states in Open image in new window , associated with spacetime. Namely, stable Minkowski and anti de Sitter vacua can act as “sinks” in the landscape.
The Heat Death of the Multiverse
What is the ultimate fate of the multiverse state? Starting from a generic eternally inflating state, the coefficients of any components in unstable vacua will eventually decay. In the string landscape, we expect that all the de Sitter as well as nonsupersymmetric Minkowski vacua decay into some lower energy vacua, given that there is a huge number of possible decay channels. The multiverse state, therefore, will asymptotically become a superposition of supersymmetric Minkowski and singularity states (associated with black hole and big crunch singularities). Here, supersymmetric Minkowski states mean configurations in a bubble universe on a supersymmetric Minkowski vacuum, approaching asymptotically the vacuum states as the universe expands. In fact, this is simply a consequence of the second law of thermodynamics, given that the Hilbert space dimensions of Minkowski and singularity worlds are infinite. In other words, the coarsegrained entropy of the multiverse diverges in the asymptotic future: S _{final}≡S _{multiverse}(t→∞)=∞—in this picture, the ultimate future of the multiverse is the “heat death.”
8 Probabilities in the Quantum Multiverse
The (Extended) Born Rule
Having understood how the state Ψ(t)〉 is defined, the probability formula can be obtained following the earlier discussions in Sects. 2–4. An important new point here is that the “time” t in quantum gravity is simply an auxiliary parameter introduced to describe the “evolution” of the state—the physical information is only in correlations between events [52].
Equation (41) is our final formula for the probabilities. The integrations over “time” t are taken from t=t _{0}, where the initial condition for Ψ(t)〉 is specified, to t=∞, which arise because conditions A _{pre} and A _{post} may be satisfied at any values of t (denoted by t _{1} and t _{2} in the equation). This, together with appropriate transformations of H, ensures that P(BA) is invariant under reparameterization of t, as required by general covariance. If conditions A _{pre} and A _{post} are selecting configurations at same t, then Eq. (41) is reduced to the formula of Eq. (1).^{7}
We note that the present framework does not suffer from the problem discussed in Refs. [53, 54, 55]: the ambiguity in the Born rule that arises if an observer can condition only a part of a wavefunction, which is the case in the conventional global description of the quantum universe. Since the quantum states in our framework represent only spacetime regions within the apparent horizons, all the systems included in a state can (at least in principle) be physically probed [14].
Unification of the Eternally Inflating Multiverse and Many Worlds in Quantum Mechanics
Under the usual situation of a terrestrial experiment, the formula of Eq. (41) is reduced to the standard Born rule. This can be seen by isolating the degrees of freedom relevant to the experiment. Suppose \(O_{A_{\mathrm{pre}}}\) acts on these degrees of freedom and selects a particular premeasurement situation A _{pre}, which is realized in components of Ψ(t)〉 at multiple values of \(t = \hat{t}_{i}\) (i=1,2,…): \(O_{A_{\mathrm{pre}}} U(t_{1},0)  \varPsi(0) \rangle = \sum_{i} c_{i}  \phi(\hat{t}_{i}) \rangle\otimes \hat{\varPsi }_{i}(\hat{t}_{i}) \rangle\, \delta(t_{1}\hat{t}_{i})\), where \( \phi(\hat{t}_{i}) \rangle\) represents the degrees of freedom relevant to the experiment, and \( \hat{\varPsi}_{i}(\hat{t}_{i}) \rangle\) the rest. In the limit that A _{pre} selects the initial experimental setup infinitely accurately, which we are considering here for simplicity, the initial state for the experiment is \( \phi(\hat{t}_{1}) \rangle =  \phi(\hat{t}_{2}) \rangle = \cdots\equiv \phi(t_{\mathrm{before}}) \rangle\). Then, assuming that \(O_{A_{\mathrm{post}}}\), which selects a particular postmeasurement situation, acts on the same degrees of freedom as \(O_{A_{\mathrm{pre}}}\), we obtain \(O_{A_{\mathrm{post}}} \hat{U}(t_{2},t_{\mathrm{before}})  \phi (t_{\mathrm{before}}) \rangle \approx c  \phi(t_{\mathrm{after}}) \rangle\delta(t_{2}  t_{\mathrm{after}})\), where \(\hat{U}(t_{1},t_{2})\) is a factor in U(t _{1},t _{2}) acting only on the experimental degrees of freedom, and t _{after} is the smallest value of t _{2} consistent with the projection \(O_{A_{\mathrm{post}}}\). Rewriting Open image in new window and Open image in new window as Open image in new window and Open image in new window , respectively, Eq. (41) gives the probability of obtaining outcome α: Open image in new window . This is nothing but the usual Born rule.
The formula of Eq. (41) can also be used to answer questions regarding global properties of our universe. To predict/postdict physical parameters x, for example, we need to choose A to select the situation of making a measurement of x. We can then use various different values (ranges) of x for B, to obtain the probability distribution P(x). Despite the fact that the t integrals run to ∞, the resulting P(BA) is welldefined, since Ψ(t)〉 is continually “diluted” into supersymmetric Minkowski and singularity states [14]. The procedure to make predictions/postdictions in this way was discussed in Ref. [28], where the probability distribution of the vacuum energy, x=ρ _{ Λ }, was computed.
It is striking that the simple, basic formalism developed here applies to physics at all scales. In particular, the single probability formula Eq. (41) can be used to answer any physical questions, given a state Ψ(t)〉. This, therefore, provides complete unification of the eternally inflating multiverse and many worlds in quantum mechanics. These two are really the same thing—they simply refer to the same phenomenon occurring at (vastly) different scales.
9 Summary
An essential feature of quantum mechanics is that information is fragile. Quantum information cannot be faithfully duplicated—the exact identification of a single state is not possible without having a prior knowledge of the state. Moreover, quantum information transmitted through physical processes will in general become nonlocal, encoded in the entanglement structure of a quantum state.
Despite this intrinsically nonlocal nature of quantum states, however, the dynamics is local. Specifically, the time evolution operator takes a special form such that the concept of locality can be defined in spacetime. This feature allows for a limited set of information (among the full quantum information) to be copiously duplicated, i.e. “amplified,” and it is (only) this information that we can meaningfully store, compare, and handle. Because of the structure of the evolution operator, the relevant information is associated with welldefined classical configurations in phase space, at least for macroscopic systems.
In a system with gravity, the whole picture is more subtle, since if we choose wrong quantization hypersurfaces, then spacetime locality is not manifest even at distances much larger than the quantum gravity scale. We argued, however, that spacetime locality can be preserved if we define quantum states in restricted spacetime regions: in and on (stretched) apparent horizons as viewed from a local Lorentz frame of a fixed spatial point p. This can be viewed as a “unitary gauge for quantum gravity,” on which our intuition should be based. By appropriately limiting the dimensions of the Hilbert subspaces corresponding to a fixed semiclassical geometry, all the redundancies/overcountings associated with a general relativistic, global spacetime description of nature are fixed/eliminated. These include general covariance, global overcounting related to complementarity, and local overcounting implied by the holographic principle.
The need for fixing a reference frame in describing the gravitational system quantum mechanically was emphasized. We identified the transformation associated with changes of the reference frame, which is specified by d(d+1)/2 parameters in d spacetime dimensions. This transformation is the origin of horizon complementarity as well as the “observer dependence” of horizons and spacetime. The transformation is reduced to the standard Poincaré transformation in the limit G _{ N }→0, so it can naturally be regarded as an extension of the Poincaré transformation in the quantum gravitational context. This is much like that the standard Lorentz transformation is regarded as an extension of the Galilean transformation, where the former is reduced to the latter in the limit c→∞.
It is remarkable that the simple framework described in this paper is applicable to physics at all scales, from the smallest (Planck length) to the largest (multiverse). Indeed, it is quite striking that quantum mechanics does not need any modification to be applied to phenomena at such vastly different scales. Does quantum mechanics break down at some point? We don’t know. But perhaps, the beginning of the multiverse might provide one.
Footnotes
 1.
There are several varying, though related, definitions of the measure problem in the literature. In this paper we adopt the one as stated above, which appears, e.g., in Refs. [10, 11, 12, 13]; namely, even if one knows the initial condition and dynamical evolution, the probability is not defined because of the infinity associated with eternally inflating spacetime; and physical predictions depend highly sensitively on how these infinities are regulated, for which there is no fundamental principle to choose one over others.
 2.
This does not mean that we cannot observe a superposition of classically different configurations. It just says that the statement “the system was in a superposition state” is already classical, and it is this information that is actually amplified.
 3.
Here we consider the apparatus (or observer) as a microscopic pure state for illustrative purposes. In practice (or perhaps even in principle), we do not know which microstate a given macroscopic configuration is in, in which case we should instead consider a density matrix in which all the microscopic information is coarsegrained. Our formalism is straightforwardly extended to this case; see e.g. Eq. (42) in Sect. 8.
 4.
It is important to realize that spacetime locality is the property of the operator algebra, and not states. In fact, a state can be easily (and, indeed, is generically) nonlocal, e.g., as the Bell state appearing in the EinsteinPodolskyRosen experiment.
 5.
Note that this would not generally be the case if the relevant object were microscopic, i.e. described by a Hilbert space with small dimensions. In that case, two different states could significantly overlap (recohere) in the future.
 6.
If the multiverse state is not generic, the Hilbert space relevant for it can be (effectively) finite dimensional; see Ref. [29].
 7.
The formula in Eq. (41) treats the state between t _{1} and t _{2} as that evolved from Open image in new window , not Ψ(t _{1})〉. This is justified under the current setup, i.e. a generic initial state Ψ(t _{0})〉 evolving into the heat death, but not in the case where the possibility of recoherence cannot be ignored for macroscopic objects. In such a case, the formula in Eq. (1) must be used, which applies in any quantum mechanical system (and allows for answering any physical questions in principle).
Notes
Acknowledgements
I am grateful for useful discussions with Alan Guth, Grant Larsen, and ISheng Yang. This work was supported in part by the Director, Office of Science, Office of High Energy and Nuclear Physics, of the US Department of Energy under Contract DEAC0205CH11231, and in part by the National Science Foundation under grant PHY0855653.
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