Born in an infnite universe a cosmological interpretation of quantum mechanics by anthony aguirre and max tegnark

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We study the quantum measurement problem in the context of an in nite, statistically uniform
space, as could be generated by eternal in
ation. It has recently been argued that when identical
copies of a quantum measurement system exist, the standard projection operators and Born rule
method for calculating probabilities must be supplemented by estimates of relative frequencies of
observers. We argue that an in nite space actually renders the Born rule redundant, by physically
realizing all outcomes of a quantum measurement in di erent regions, with relative frequencies given
by the square of the wave function amplitudes. Our formal argument hinges on properties of what
we term the quantum confusion operator, which projects onto the Hilbert subspace where the Born
rule fails, and we comment on its relation to the oft-discussed quantum frequency operator. This
analysis uni es the classical and quantum levels of parallel universes that have been discussed in
the literature, and has implications for several issues in quantum measurement theory. Replacing
the standard hypothetical ensemble of measurements repeated ad in nitum by a concrete decohered
spatial collection of experiments carried out in di erent distant regions of space provides a natural
context for a statistical interpretation of quantum mechanics. It also shows how, even for a sin-
gle measurement, probabilities may be interpreted as relative frequencies in unitary (Everettian)
quantum mechanics. We also argue that after discarding a zero-norm part of the wavefunction,
the remainder consists of a superposition of indistinguishable terms, so that arguably \collapse" of
the wavefunction is irrelevant, and the \many worlds" of Everett's interpretation are uni ed into
one. Finally, the analysis suggests a \cosmological interpretation" of quantum theory in which the
wave function describes the actual spatial collection of identical quantum systems, and quantum
uncertainty is attributable to the observer's inability to self-locate in this collection.

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Born in an infnite universe a cosmological interpretation of quantum mechanics by anthony aguirre and max tegnark

  1. 1. Born in an Infinite Universe: a Cosmological Interpretation of Quantum Mechanics Anthony Aguirre Dept. of Physics & SCIPP, University of California at Santa Cruz, Santa Cruz, CA 95064; aguirre@scipp.ucsc.edu Max Tegmark Dept. of Physics & MIT Kavli Institute, Massachusetts Institute of Technology, Cambridge, MA 02139; tegmark@mit.edu (Dated: June 13, 2012. To be submitted to Phys. Rev. D) We study the quantum measurement problem in the context of an infinite, statistically uniform space, as could be generated by eternal inflation. It has recently been argued that when identical copies of a quantum measurement system exist, the standard projection operators and Born rule method for calculating probabilities must be supplemented by estimates of relative frequencies of observers. We argue that an infinite space actually renders the Born rule redundant, by physically realizing all outcomes of a quantum measurement in different regions, with relative frequencies givenarXiv:1008.1066v2 [quant-ph] 12 Jun 2012 by the square of the wave function amplitudes. Our formal argument hinges on properties of what we term the quantum confusion operator, which projects onto the Hilbert subspace where the Born rule fails, and we comment on its relation to the oft-discussed quantum frequency operator. This analysis unifies the classical and quantum levels of parallel universes that have been discussed in the literature, and has implications for several issues in quantum measurement theory. Replacing the standard hypothetical ensemble of measurements repeated ad infinitum by a concrete decohered spatial collection of experiments carried out in different distant regions of space provides a natural context for a statistical interpretation of quantum mechanics. It also shows how, even for a sin- gle measurement, probabilities may be interpreted as relative frequencies in unitary (Everettian) quantum mechanics. We also argue that after discarding a zero-norm part of the wavefunction, the remainder consists of a superposition of indistinguishable terms, so that arguably “collapse” of the wavefunction is irrelevant, and the “many worlds” of Everett’s interpretation are unified into one. Finally, the analysis suggests a “cosmological interpretation” of quantum theory in which the wave function describes the actual spatial collection of identical quantum systems, and quantum uncertainty is attributable to the observer’s inability to self-locate in this collection. PACS numbers: 03.65.-w,98.80.Qc I. INTRODUCTION or, in some interpretations consciousness – existed? How can we split the universe into a system and measuring Although quantum mechanics is arguably the most apparatus if the system is the entire universe itself? And successful physical theory ever invented, the century-old cosmology may even be intertwined with quantum mea- debate about how it fits into a coherent picture of the surements here-and-now. In an infinite universe, our ob- physical world shows no sign of abating. Proposed re- servable universe (the spherical region from which light sponses to the so-called measurement problem (e.g., [1]) has had time to reach us during the past 14 billion years) include the Ensemble [2, 3], Copenhagen [4, 5], Instru- may be just one of many similar regions, and even one mental [6–9], Hydrodynamic [10], Consciousness [12–15], of many exact copies – and in fact many versions of the Bohm [16], Quantum Logic [17], Many-Worlds [18, 19], widely-accepted standard inflationary cosmology provide Stochastic Mechanics [20], Many-Minds [21, 22], Con- just such a context [34], as we shall discuss. Don Page has sistent Histories [23], Objective Collapse [24], Transac- recently argued that pure quantum theory and the text- tional [25], Modal [26], Existential [27], Relational [28], book Born rule cannot produce outcome probabilities for and Montevideo [29] interpretations. Moreover, different multiple replicas of a quantum experiment [35–37], and proponents of a particular interpretation often disagree must therefore be augmented in a cosmological scenario about its detailed definition. Indeed, there is not even in which such replicas exist. consensus on which ones should be called interpretations. It is therefore timely to further investigate how such a While it is tempting to relegate the measurement prob- cosmology impacts the quantum measurement problem, lem to “philosophical” irrelevancy, the debate has both and this is the aim of the present paper. We will argue informed and been affected by other developments in that cosmology is not merely part of the problem, but physics, such as the understanding of the importance [30] also part of the solution. Page essentially showed that to and mechanisms [21, 31–33] of decoherence, experimen- calculate the probability that a measurement in a collec- tal probes of quantum phenomena on ever larger scales, tion of N identical experiments has a given outcome, one and the development of cosmology. must supplement the standard Born rule for assigning Putting quantum mechanics in the context of cosmol- probabilities to measurement outcomes, based on projec- ogy creates new issues. How can we apply the measure- tion operators, by a rule that assigns probabilities to each ment postulate at times before measuring apparatuses – of the events “the experiment was performed by the kth
  2. 2. 2observer, k = 1...N ”. We will go further, and argue that Appendix B. In Section V we describe how to formallyif one identifies probabilities with the relative frequen- describe measurement in this context, then discuss pos-cies of experimental outcomes in three-dimensional space, sible interpretation of our mathematical results in Sec-the measurement postulate (Born’s rule) becomes super- tion VI. We discuss some open issues in Section VII, andfluous, as it emerges directly from the quantum Hilbert summarize our conclusions in Section VIII.space formalism and the equivalence of all members of aninfinite collection of exact replicas. We will see that thisis intimately linked to classic frequency operator results II. THE COSMOLOGICAL CONTEXT[38–42], except that a fictitious infinite sequence of iden-tical measurements is replaced by an actually existing When first applying General Relativity to our universe,spatial collection. It is also closely related to arguments1 Einstein assumed the Cosmological Principle (CP): ourby [43, 45]. universe admits a description in which its large-scale This potential cosmological connection between prob- properties do not select a preferred position or direc-abilities in quantum mechanics and the relative fre- tion. This principle has served cosmology well, sup-quencies of actual observers is relevant to most of the plying the basis for the open, flat, and closed universeabove-mentioned quantum interpretations. It is partic- metrics that underly the highly successful Friedmann-ularly interesting for Everett’s Many Worlds Interpreta- Lemaˆ ıtre-Robertson-Walker (FLRW) Big-Bang cosmol-tion (MWI), where we will argue that it eliminates the ogy. We shall argue that this principle and the interpre-perplexing feature that, loosely speaking, some observers tation of QM may be closely intertwined, with the theoryare more equal than others. To wit, suppose a spin mea- of cosmological inflation as a central player. In particu-surement that should yield “up” with probability p = 0.5 lar, we will discuss how eternal inflation naturally leadsis repeated N = 10 times. According to the MWI, the fi- to a universe obeying a strong version of the CP [43, 44],nal wavefunction has 2N = 1024 terms, each correspond- in which space is infinite and has statistically uniform2ing to an equally real observer, most of whom have mea- properties. In this context, any given finite region issured a random-looking sequence of ups and downs. This replicated throughout the infinite space, which in turnsuggests that quantum probabilities can be given a simple requires a re-appraisal of quantum probabilities.frequentist interpretation. However, for an unequal prob-ability case such as p = 0.001, the final wavefunction stillhas 2N terms corresponding to real observers, but now A. The Cosmological Principle and infinite spacesmost of them have measured approximately 50% spin upand concluded that the Born rule is incorrect. (We are In a finite space, the CP has a curious status: withsupposed to believe that everything is still somehow con- a single realization of a finite space, there is no mean-sistent because the observers with a smaller wave func- ingful way for the statistical properties to be uniform.tion amplitude are somehow “less real”.) We will show There would, for example, always be a unique point ofthat in an infinite inflationary space, probabilities can be highest density. We could compare our realization to agiven a frequentist interpretation even in this case. hypothetical ensemble of universes generated assuming a The rest of this paper is organized as follows. In Sec- set of uniform statistical properties, but we could nevertion II, we describe the cosmological context in which recover these putative statistical properties beyond a cer-quantum mechanics has found itself. We then investigate tain degree of precision. In this sense the CP in a finitehow this yields a forced marriage between quantum prob- space is really nothing more than an assumption (as byabilities and relative frequencies, in both a finite space Einstein) that space and its contents are “more or less”(Section III) and an infinite space (Section IV). Rather homogeneous on large scales; a precise description wouldthan launching into an intimidating mathematical for- require the specification an enormous amount of infor-malism for handling the most general case, we begin with mation.a very simple explicit example, then return to the ratherunilluminating issue of how to generalize the result in 2 The matter distribution in space is customarily modeled as evolved from some random initial conditions, so properties in1 In brief, these works argue that a complete description of the any given region must be described statistically, e.g., by the universe does not single out a particular place. So instead of de- mean density, 2-point and higher correlation functions, etc. By scribing what happens “here” it describes an ensemble (in Gibbs’ statistically uniform, we mean that all such statistics are transla- sense) of identical experiments uniformly scattered throughout tionally invariant. (For example, the homogeneous and isotropic an infinite (expanding) space (a cosmological ensemble). It fol- Gaussian random fields generated by inflation – and anything lows from a description of the measuring process in which the evolved from such initial conditions – are statistically uniform in measuring apparatus is assigned a definite macrostate but not this sense.) Another way to look at this is that the probability a definite microstate that the measurement outcomes and their distribution for different realizations in a given region of space relative frequencies in the cosmological ensemble coincide with is the same as the distribution across different spatial regions in those given by the measurement postulate. (A similar idea was a given realization. (See [45] for an extended discussion of this independently put forward in schematic form by [46].) point and its implications.)
  3. 3. 3 An infinite space is quite different: by examining arbi- but in this case expansion will tend to win forever.3trarily large scales, its statistical properties can in prin- The result is that eternal inflation does provide post-ciple be assessed to arbitrarily high accuracy about any inflationary regions with the requisite properties, but aspoint, so there is a precise sense in which the properties part of an ultimately infinite spacetime.can be uniform. Moreover, if (as the holographic prin- It might seem that a given post-inflationary region isciple suggests) a region of some finite size and energy necessarily finite, because no matter how long inflationcan only take on a fixed finite set of possible configura- goes on, it can only expand a given finite initial regiontions, then the full specification of a statistically-uniform into a much larger yet still finite space. But this is not theinfinite space would require only those statistics. This case. General Relativity forbids any fundamental choiceimplies [43, 44] that in contrast to a finite system, there of time variable, but there is a physically preferred choice,would be only one possible realization of such a system, which is to equate equal-time surfaces with surfaces ofas any two systems with the same statistical properties constant inflaton field value (and hence constant energywould be indistinguishable. density), so that the end of inflation occurs at a single The CP might be taken as postulated symmetry prop- time. In eternal inflation, this choice leads to multipleerties of space and its contents, consistent with the near disconnected surfaces on which inflation ends, each oneuniformity of our observed universe. In an infinite (open generally being both infinite and statistically uniform.or flat) FLRW universe, these symmetries can be exact in Likewise, in each region and in these coordinates, thethe above sense, and such a postulated cosmology would ensuing cosmic evolution occurs homogeneously.4support the arguments of this paper beginning in Sec- This occurs in all three basic types of eternal infla-tion II D, or those of [45]. tion: “open” inflation (involving quantum tunneling, and driven by an inflaton potential with multiple minima), in Alternatively, we might search for some physical expla- “topological” inflation (driven be an inflaton field stucknation for the near-uniformity of our observable universe. around a maximum in its potential), and “stochastic”This was a prime motivation for cosmological inflation. inflation (in which upward quantum fluctuations of theYet inflation can do far more than create a large uniform field can overwhelm the classical evolution of the fieldregion: in generic models inflation does, in fact, create toward smaller potential values). These three particu-an infinite uniform space. lar scenarios are discussed in more detail in Appendix A, where we also provide heuristic arguments as to why in- finite, statistically uniform spaces are a generic product of eternal inflation, by its very nature.5 B. Infinite spaces produced by eternal inflation Thus eternal inflation, if it occurs, provides a causal mechanism for creating a space (or set of spaces) obeying Inflation was devised ([47]; see [48] for some history) a form of the CP, in the sense that each space is infiniteas a way to grow a finite-size region into an extremelylarge one with nearly uniform properties, and if inflationis realized in some region, it does this effectively: the 3 This is not to say that every inflation model has eternal behavior:exponential expansion that inflates the volume also di- it is not hard to devise non-eternal versions; but the need to do solutes or stretches into near homogeneity any particles or deliberately in most cases suggests that eternal behavior is morefields within the original region. The post-inflationary generic. (An exception is hybrid inflation, which is genericallyproperties are then primarily determined not by cosmic non-eternal [51]; such models however tend to predict a scalarinitial conditions, but by the dynamics of inflation, which spectral index n > 1 [52], which is in some conflict with currentare uniform across the region; although particular initial constraints [53].) In scenarios where inflation might take place in parallel in different parts of a complicated potential energyconditions are required for such inflation to arise, once it “landscape”, regions of the landscape with eternal inflation willdoes, information about the initial conditions is largely naturally outcompete those with non-eternal inflation, predictinginflated away. by almost any measure that the region of space we inhabit was generated by eternal inflation. On the other hand, it has been It was soon discovered, however, that in generic mod- argued that inflation eternal inflation ([55, 56]) and perhaps evenels, inflation is eternal: although inflation eventually ends inflation (e.g. [54]) may be difficult to realize in a landscape thatwith probability unity at any given location, the expo- is generated as a low-energy effective potential from a true high-nential expansion ensures that the total inflating volume energy quantum gravity theory. 4 Moreover, a given point on the spatial surface at which infla-always increases exponentially (see [48–50] for recent re- tion ends will occur an enormously or infinitely long durationviews.) In many cases, one may think of this as a compe- after any putative initial conditions for inflation. Thus, inso-tition between the exponential expansion exp(Ht), and far as inflation makes these initial conditions irrelevant, they arethe “decay” from inflation to non-inflation with charac- arguably completely irrelevant in eternal inflation. 5 In a cosmology with a fundamental positive cosmological con-teristic time tdecay . This means that an initial inflat-ing volume V has, at some later time, inflating volume stant, this issue becomes more subtle, as some arguments sug- gest such a cosmology should be considered as having a finite∼ V exp(3Ht) exp(−t/tdecay ) = V exp[(3H −t−1 )t]; for decay total number of degrees of freedom (see, e.g. [57]). How thisinflation to work at all requires the expansion to win for can be understood consistently with the semi-classical spacetimea number of e-foldings, implying a positive exponent; structure of eternal inflation is an open issue.
  4. 4. 4and has uniform properties determined on average by the of the current matter distribution (such as what you ateclassical evolution of the inflaton, with statistical varia- for breakfast) were determined by these same inflation-tions provided by the quantum fluctuations of the field ary initial conditions, augmented by subsequent quantumduring inflation. fluctuations amplified by chaotic dynamics, etc. Because such small-scale processes (and any microscopic “initial” conditions connected with them) are decoupled from the C. Infinite statistically uniform space, and super-horizon large-scale dynamics giving rise to the infi- probabilities, from inflation nite space, the overall space should again be statistically uniform, here in the sense that the probability distribu- The fact that post-inflationary spacetime is infinite in tion of microstates in each finite region depends not oneternal inflation leads to some rather vexing problems, in- its location in space, but only on its macroscopic proper-cluding the “measure problem” of how to count relative ties, which are themselves drawn randomly from a region-numbers of objects so that statistical predictions for the independent statistical distribution.cosmic properties surrounding those objects can be made In short, inflation creates an infinite set of cosmic re-(see, e.g., [58] for a recent review.) This paper is not an gions, each with “initial conditions” and subsequently-attempt to solve that problem. In particular, we do not evolving properties that are characterized (and only char-address the comparison of observer numbers across re- acterized) by a statistical distribution that is independentgions with a different inflationary history and hence dif- of the choice of region.ferent gross properties. Rather, we will ask about whathappens when we apply the formalism of quantum the-ory to a system in the context of a single infinite space D. Quantum mechanics and replicaswith uniform and randomly-determined statistical prop-erties.6 Let us now make our link to everyday quantum me- One of the greatest successes of cosmological inflation chanics. For a simple example that we shall followis that small-scale quantum fluctuations required by the throughout this paper, consider a spin 1/2 particleHeisenberg uncertainty relation get stretched with the and a Stern-Gerlach experiment for measuring the z-expanding space, then amplified via gravitational insta- component of its spin, which has been prepared in thebility into cosmological large-scale structure just like that state ψ = α| ↓ + β| ↑ . Here α and β are com-we observe in, e.g., the galaxy distribution and in the plex numbers satisfying the usual normalization condi-cosmic microwave background [53, 59]. In a given finite tion |α|2 + |β|2 = 1. If we assume that a finite volume re-cosmic region, this process creates pattern of density fluc- gion with a roughly flat background metric has a finite settuations representing a single realization of a statistical of possible microscopic configurations7 (as suggested by,process with a probability distribution governed by the e.g., the holographic principle and other ideas in quantumdynamics of inflation and the behavior of quantum fields gravity), and that our system plus experimenter config-within the inflating space. uration evolved from one of finitely many possible sets Eternal inflation also creates infinitely many other of initial conditions drawn from the distribution govern-nearly-homogeneous regions with density fluctuations ing the statistically uniform space at some early time,drawn from the same distribution (because the dynam- then it follows that this configuration must be replicatedics are just the same) that evolve independently of each elsewhere.8 That is, there are infinitely many places inother (because the regions are outside of causal contact this space where an indistinguishable experimenter hasif they are sufficiently widely separated, where “widely” prepared the same experiment using a classically indis-means being farther apart than the horizon scale dur- tinguishable procedure, and therefore uses the same αing inflation, say 10−24 m). The resulting space thenhas statistically uniform properties, and the probabilitydistribution governing the fluctuations in any single re-gion is recapitulated as the relative frequencies of these 7 Meaning a finite number of meaningfully distinct ways in whichfluctuation patterns across the actually-existing spatial the state can be specified. Note that although the real numbercollection of regions. coefficient α would seem to allow an uncountably infinite set of specifications, this is misleading: the maximum von Neumann Now, these inflationary fluctuations constitute the entropy S = −Trρ log ρ for our system is just log 2, and at mostclassical cosmological “initial” conditions that determine two classical bits of information can be communicated using athe large-scale variation of material density and thus, single qubit. We should note, however, that while our assump-e.g., the distribution of galaxies. Smaller-scale details tion is quite standard, the precise way in which the continuous α would over-specify the state is a subtle question that we do not address here. 8 This does assume some additional subtleties. For example it is argued in [43] that “statistical predictions do not prescribe all6 While it is our assumption for present purposes, it is not a given the properties of infinite collections. ...Any outcome that occurs that these questions are inseparable, as some “global” measures a finite number of times has zero probability.” In particular, an would also “induce” a measure over the otherwise uniform sub- outcome that is consistent with physical laws could in principle spaces we are considering. occur in only one observable universe.
  5. 5. 5and β to describe the initial wavefunction of her particle. to consider these N experiments as a single quantum sub-The rather conservative estimate in [63] suggests that system of the universe, this is problematic because in thisthe nearest indistinguishable copy of our entire observ- situation there is no set of projection operators that can 115able universe (“Hubble volume”) is no more than 1010 assign outcome probabilities purely via the Born rule.10meters away, and the nearest subjectively indistinguish- Thus it seems that the quantum formalism by itself isable experimenter is likely to be much closer.9 We will insufficient, and must be in some way supplemented bynow argue that the quantum description of this infinite additional ingredients.set of systems sheds light on the origin of probabilities in While this conflicts with the idea that quantum theoryquantum mechanics. alone should suffice when applied to the whole universe, we can recover the usual Born rule results for the sin- gle system in a fairly straightforward way if we augment III. PROBABILITIES FOR MEASUREMENT the Born rule as applied to the product state for the OUTCOMES IN A FINITE REGION N systems with probabilities assigned according to rela- tive frequencies among the N systems.11 In particular, A. The problem since the 2N terms are orthogonal (being a basis for the tensor-product state space) we might in principle imag- In a statistically uniform space, consider a finite region ine measuring the whole system, and attribute a quantumthat is large enough to contain N identical copies of our probability to each term given by its squared amplitude.Stern-Gerlach experiment prepared in the simple above- Yet even if just one of these terms is “realized”,12 therementioned state. is still uncertainty as to which spin is measured, because The state of this combined N -particle system is simply there is complete symmetry between the N indistinguish-a tensor product with N terms. For example, N = 3 gives able measuring apparatuses. You should thus accord athe state probability for ↑ given by the relative frequencies of ↑ and ↓. |ψ = (α|↓ + β|↑ ) ⊗ (α|↓ + β|↑ ) ⊗ (α|↓ + β|↑ ) = The total probability P↑ of measuring ↑ would then = α3 |↓↓↓ + α2 β|↓↓↑ + ... + β 3 |↑↑↑ . (1) come from a combination of quantum probabilities and frequentist estimates of probability, using P (A) = If we order the 2N basis vectors of this 2N -dimensional i P (Bi )P (A|Bi ) where P (A|Bi ) is the conditionalHilbert space by increasing number of “up” vectors, the probability of A given Bi . Thusvector of wavefunction coefficients takes the simple form N N n  ↓↓↓ |ψ   3  α P↑ = (β ∗ β)n (α∗ α)(N −n) . (3) n=0 n N  ↓↓↑ |ψ   α2 β   ↓↑↓ |ψ   α2 β        2  Each term in this sum is just the binomial coefficient  ↑↓↓ |ψ   α β   = (2) f (n; N, p) with p = β ∗ β (the quantum probability from  ↑↑↑ |ψ   αβ 2     ↑↑↓ |ψ   αβ 2  equation (2) of getting n ↑-factors) times n/N (the prob-     ↑↓↑ |ψ   αβ 2   ability that among the N identical observers, you are one of the n who observed ↑). Mathematically, this sum ↑↑↑ |ψ β3 simply computes 1/N times the mean of the binomial distribution, which is N p, giving P↑ = p. In this way, Nfor our N = 3 example. For general N , there are nterms with n spins up, each with coefficient αN −n β n . 10 ˆ ˆ In particular, define Pi,↑ and Pi,↓ to be operators that project onto |↑ and |↓ for the ith observer (leaving the N − 1 other B. Probabilities in the finite region components of the product vector unchanged.) Then Page [37] shows that for N = 2, there is no state-independent projection Suppose we would like to ask the core question: “given ˆ ˆ operator P↑ that gives Born-rule probabilities P (↑) = P↑ forthat I have prepared the quantum system as described, measuring ↑ (absent information about which particular observer one is) that are a weighted combination (with positive weights)what is the probability that I will measure ↑?” This is of the probabilities Pi,↑ for measuring |↑ for each given knownmore subtle than it would appear, because “I” might be observer i.part of any one of the N indistinguishable experimental 11 This is essentially ‘T5’ suggested in [35] as one possible way tosetups assumed. As argued by Page [35–37], if one wants restore probabilities. 12 Readers preferring the Everettian perspective can accord a prob- ability to each of these terms as branches of the wavefunction, and make a similar argument. Note, however, that it is some- what less satisfying because if we add up the relative frequency9 The frequency of such repetitions depends on very poorly under- of observers across all branches, it is by symmetry 50% for ↑ stood questions such as the probability for certain types of life and 50% for ↓; this is the uncomfortable issue of some observers to evolve, etc. being more real than others noted in the introduction.
  6. 6. 6the standard Born rule probability β ∗ β to measure ↑ is we find that [39]recovered, using a combination of the Born rule appliedto the 2N -state superposition, and the relative frequen- (F − p)|ψ 2 = ψ|(F − p)2 |ψ =cies of |↑ and |↓ contained in each (product) state in N N n 2that superposition. = (1 − p)n pN −n −p = n=0 n N p(1 − p) C. Frequency and confusion operators = . (7) N If the reality of indistinguishable systems has forced Note that as in equation (3), α and β enter only in theus to augment quantum probabilities with probabilities combinations α∗ α = (1 − p) and β ∗ β = p (because Fbased on observer frequencies, as above, it is very inter- is diagonal), and just as equation (3) is the mean of aesting to examine more carefully how these two notions of binomial distribution (divided by N ), here the second lineprobability connect. To do so, let us now define two Her- is simply the variance of a binomial distribution (dividedmitean operators on this Hilbert space, both of which are by N 2 ).diagonal in this basis. The first is the frequency operator As for the confusion operator, using equation (2) andF introduced by [38, 39, 41, 42], which multiplies each equation (5) we obtainbasis vector by the fraction of the arrows in its symbolthat point up; for our N = 3 example, ○ |ψ ∼ ¨ 2 = ψ| ○ |ψ = ∼ ¨   0 0 0 0 0 0 0 0 N N n 0 1 0 0 0 0 0 0 = (1 − p)n pN −n θ −p − n N 0 0 1 0 0 0 0 0   n=0 1 0 0 0 1 0 0 0 0  N F =  (4) (1 − p)n pN −n  = 3 0 0 0 0 2 0 0 0  n 0 0 0 0 0 2 0 0 |n−N p|>N   2 ≤ 2e−2 0 0 0 0 0 0 2 0 N , (8) 0 0 0 0 0 0 0 3 where θ again denotes the Heaviside step function, andThe second is the confusion operator ○ , which projects ∼ ¨ we have used Hoeffding’s inequality13 in the last step.onto those basis vectors where the spin-up fraction differs Thus ○ |ψ is exponentially small if N ∼¨ −2 ; thisby more than a small predetermined value from the mathematical result will prove to be important below.Born rule prediction p = |β 2 |; for our N = 3 example For large enough N , our rescaled Binomial distribu-with any < 1/3, tion approaches a Gaussian with mean p and standard   deviation p(1 − p)/N , so 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1/2 0  0 0 0 0 0 0 0  ○ |ψ 2 N ∼ ¨ ≈ erfc . (9) ○=  0  0 0 0 0 0 0 0  2p(1 − p) ∼  ¨ (5) 0 0 0 0 1 0 0 0  0 0 0 0 0 1 0 0  0  (erfc denotes the complementary error function, i.e., the 0 0 0 0 0 1 0 area in the Gaussian tails). 0 0 0 0 0 0 0 1 In summary, for finite N , the product state of our N copies represents a sum of many terms. In some of them,More generally, the two operators are clearly related by the relative frequencies of up and down states in indi- vidual members of the spatial collection closely approxi- ○ = θ |F − p| − ∼ ¨ , (6) mate the corresponding probabilities given by Borns rule. In other terms the frequencies differ greatly from thewhere θ denotes the Heaviside step function, i.e., θ(x) = corresponding Born-rule probabilities. However, as N1 for x ≥ 0, vanishing otherwise. increases, these confusing states contribute smaller and smaller amplitude, as measured by the quickly diminish- Since both F and ○ are Hermitean operators, one ∼ ¨would conventionally interpret them as observables, with ing expectation value of ○ ∼. ¨F measuring the fraction of spins that are up, and ○ ∼¨measuring 1 if this fraction differs by more than fromp, 0 otherwise. 13 Specifically, apply theorem 2 of [11], representing the binomial Examining the norm of the state multiplied by (F −p), distribution as the sum of N independent Bernoulli distributions.
  7. 7. 7 tor F case, since the spectrum of the former is constant (zero) near the limit point.) Third, let us define the complement of the confusion operator, which projects onto the states where the up fraction is within of the Born rule prediction p: 1 ○ ≡ I − ○, ¨ ∼ ¨ (11) 0.8 where I is the identity operator. Both ○ and ○ are ∼ ¨ ¨ projection operators, and they by definition satisfy the 2 2 0.6 relations ○ = ○, ○ = ○ , and ○ ○ = ○ ○ = 0. ∼ ¨ ∼ ¨ ¨ ¨ ∼ ¨ ¨ ¨ ∼¨ Now let us decompose our original state into two or- thogonal components: |ψ = ○ |ψ + (I − ○ )|ψ = ○ |ψ + ○ |ψ . 0.4 ∼ ¨ ∼ ¨ ∼ ¨ ¨ (12) Substituting equation (10) now implies the important re- 0.2 sult that |ψ → ○ |ψ ¨ as N → ∞, (13) 0 0 0.2 0.4 0.6 0.8 1 Spin up frequency n/N with convergence in the sense that the correction term approaches zero Hilbert space norm as N → ∞. But the right-hand-side ○ |ψ is a state which by definition is a ¨ superposition only of states where the relative frequencies of |↑ and |↓ are in precise accord with Born-rule prob- abilities, with an up-fraction as close to p = |β|2 as weFIG. 1: As a function of the spin up fraction n/N , the figure chose to require with our -parameter. If we now assumeshows the scaled binomial distribution for (n, p) = (500, 1/3) that the outcome of a single quantum measurement is one(shaded curve) together with the spectra of the frequency of the measured observables eigenvalues, almost all com-operator (diagonal line) and the = 0.1 confusion operator ponents of the superposition of pre-measurement states(shaded, equal to 1 for |n/N − p| > ). weve been discussing represent post-measurement states in which the relative frequencies of eigenstates are equal to the corresponding Born probabilities for the possible IV. PROBABILITES IN AN INFINITE SPACE outcomes of a single measurement. (Measurement and decoherence are discussed in the next section.) As N → ∞, four key things happen. Finally, as N → ∞, |ψ approaches (again in the sense First, the binomial distribution governing the spread that the correction term approaches zero Hilbert spacein values of the frequency operator in Eq. 7 approaches norm) a state where every element in the grand super-a δ-function centered about p, so that (F − p)|ψ 2 = position becomes statistically indistinguishable from thep(1 − p)/N → 0. others, in the following sense. Suppose, as above, that Second, the norm of the state projected by ○ vanishes, ∼ ¨ each term in the grand superposition is taken to repre- sent a collection of identical apparatuses that have each ○ |ψ ∼ ¨ 2 → 0, (10) registered a definite outcome corresponding to either |↑ or |↓ . Consider a volume of enormous but finite radius Rwith exponentially rapid convergence as per equation (6). that contains some large number M of our identical ex-This is illustrated in Fig. 1: in the equation (8) sum, the periments, order them any way you like and write downθ-term vanishes outside the shaded area, whereas the re- their readings as a sequence such as ... ↑↑↓↓↓↑↓ ..., andmaining binomial distribution factor (the plotted curve) let n↓ and n↑ denote the number of readings of ↓ andgets ever narrower as N → ∞, eventually ending up al- ↑, respectively (n↓ = M − n↑ ). In our infinite space,most entirely inside the region where θ = 0. (That |ψ is just as there are infinitely many realizations of a singlean eigenvector of ○ also follows directly from |ψ being ∼ ¨ experiment, there will be infinitely many such spherical regions, in each of which one of the 2M possible outcomesan eigenvector of F and the fact that ○ is a function of F ∼ ¨ is realized. Across this collection, these outcomes will beas per equation (6). However, the figure also illustrates realized with frequencies that are within of the Bornthat convergence is much faster (indeed exponential) for rule predictions (1 − p)M −n↑ pn↑ , except for a correctionthe confusion operator ○ than for the frequency opera- ∼ ¨ term with zero Hilbert space norm. This follows from
  8. 8. 8the exact same argument given above, generalized to the likecase of multiple outcomes as we do below in Appendix B.Since we must have (1 − p)M −n↑ pn↑ > 0 for any sequence |ψ = (α|↓ + β|↑ )|ar ⊗ (α|↓ + β|↑ )|ar ⊗ ...(15)that actually occurs in some sphere in some branch ofthe wavefunction, a corollary is that this exact same se- Now, after the interaction between the system and appa-quence will occur in every branch of the wavefunction ratus described by equation (14), and following the ex-(after neglecting the correction term with zero Hilbert act same reasoning as Section III and Section IV, whenspace norm), regardless of how vast this sphere is. More- N → ∞, our product state becomes an infinite super-over, the two-outcome result implies that the relative fre- position of terms, all of which (except for a set of totalquencies in a randomly selected volume in a given branch Hilbert space norm zero) look likegive no information whatsoever about which branch it isin, because they all have the same average frequencies. ...|↑ |a↑ ⊗ |↑ |a↑ ⊗ |↓ |a↓ ⊗ |↑ |a↑ ⊗ |↓ |a↓ ..., (16)Finally, the multiple outcomes result tells us that even if where the relative frequencies of the terms | ↑ |a↑ andwe compute any finite amount of statistical data from a |↓ |a↓ are given by |α|2 and |β|2 , respectively.truly infinite volume (by computing what fraction of the In this way, the interaction between system and ap-time various combinations of outcomes occur in the infi- paratus has evolved |ψ from a superposition containingnite volume), the different branches remain statistically infinitely many identical apparatuses into one of statisti-indistinguishable. In just the sense of Sec. II A, the dif- cally indistinguishable terms, where each term describesferent branches are equivalent to different realizations of two different sets of apparatuses: one in which each ap-an infinite universe with the same statistical properties, paratus reads ‘up’, and one in which each reads ‘down’,and therefore cannot be told apart. with relative frequencies |α|2 and |β|2 . Now, each system described by equation (14) will fur- ther interact with the degrees of freedom making up its V. MEASUREMENT AND DECOHERENCE environment, which we assume have a random charac- ter. This causes the local superposition to undergo deco- In our discussion above we have mentioned “outcomes” herence [21, 31, 33]. This decoherence is typically quiteof experiments. It is generally agreed that in a measure- rapid, with timescales of order 10−20 seconds being com-ment process, the post-measurement state of the mea- mon [32, 61, 62]. Let us consider the effect of this de-surement system is encoded in the degrees of freedom of coherence on the density matrix ρ of the full N -particlea macroscopic device (say the readout of a Stern-Gerlach system. Initially, ρ = |ψ ψ|, where |ψ is the pure N -apparatus, the position of a macroscopic pointer, or the particle state given by equation (15). Since each appara-brain state of an observer). We can sketch how this pro- tus is rapidly entangled with its own local environment,cess plays out in the simple example and cosmological all degrees of freedom of the full N -particle system de-context of this paper by considering, along with the set cohere on the same rapid timescale. This means thatof replica quantum systems (each a single-spin system when we compute the resulting N -particle reduced den-represented by α|↑ + β|↓ ), a corresponding set of in- sity matrix ρ by partial-tracing the global density matrixdistinguishable measuring devices in a ‘ready’ state just over the other (environment) degrees of freedom, it be-prior to measurement. Following the scheme of Von Neu- comes effectively diagonal in our basis (given by Equa-mann [60], if the apparatus is in a particular ‘ready’ state tion 2). The vanishing off-diagonal matrix elements are|ar , independent of the system’s state, then interaction those that connect different pointer states in the grandbetween the system and the apparatus causes the com- superposition, and also therefore in the individual sys-bined system to evolve into an entangled state: tems; thus any quantum interference between different states becomes unobservable. (α|↑ + β|↓ )|ar −→ α|↑ |a↑ + β|↓ |a↓ , (14) In other words, decoherence provides its usual two ser- vices: it makes quantum superpositions for all practicalwhere |a↑ and |a↓ are states of the apparatus in which purposes unobservable in the “pointer basis” of the mea-it records an ‘up’ or ‘down’ measurement. surement [31], and it dynamically determines which basis Let us consider a set of N perfect replicas of this this is (in our case, the one with basis vectors like the ex-setup14 that exist in an infinite statistically uniform ample in Equation (16)).space for just the same reason that there are copies ofα| ↑ + β| ↓ . Following the same reasoning as in Sec-tion III, we can consider the product state, which looks VI. INTERPRETATION Application of quantum theory to the actually existing14 infinite collection of identical quantum systems that is In appendix C we generalize this discussion to the arguably more relevant case where the measuring devices are macroscopically in- present in an infinite statistically uniform space (such as distinguishable, and in particular, described by the same density provided by eternal inflation) leads to a very interesting matrix. quantum state. As long as we are willing to neglect a part
  9. 9. 9of the wavefunction with vanishing Hilbert-space norm, cosmic wavefunction, each of the “many worlds” are thethen we end up with a superposition of a huge number of same world, where “world” here refers to the state of andifferent states, each describing outcomes of an infinite infinite space. In the terminology of [63, 64], the Level Inumber of widely separated identical measurements in Multiverse is the same as the Level III Multiverse (andour infinite space. In all of them, a fraction p = |α|2 of if inflation instantiates more than one solution to a morethe observers will have measured spin up.15 fundamental theory of physics, then the Level II Multi- In this way, the quantum probabilities and frequentist verse is the same as the Level III Multiverse).observer-counting that coexisted in the finite-N case have All this suggests that we take a different and radi-merged. Born’s rule for the relative probabilities of ↑ and cally more expansive view of the statevector for a finite↓ emerges directly from the relative frequencies of actual system: this quantum state describes not a particularobservers within an unbounded spatial volume; Born’s system “here”, but rather the spatial collection of iden-rule as applied to the grand superposition is superflu- tically prepared systems that already exist. This pro-ous since all give the same predictions for these relative vides a real collection rather than fictitious ensemble forfrequencies. In particular, the “quantum probabilities” a statistical interpretation of quantum mechanics. It alsoassumed in Eq. 3 to be given by α∗ α and β ∗ β are replaced allows quantum mechanics to be unitary in a very sat-by the assumption that two vectors in Hilbert space are isfactory way. Rather than the world “splitting” intothe same if they differ by a vector of zero norm. a decohered superposition of two outcomes as seen by One of the most contentious quantum questions is an experimenter, infinitely many observers already existwhether the wavefunction ultimately collapses or not in different parts of space, a fraction |α|2 of which willwhen an observation is made. Our result makes the measure one outcome, and a fraction |β|2 of which willanswer to this question anti-climactic: insofar as the measure the other. The uncertainty represented by thewavefunction is a means of predicting the outcome of superposition corresponds to the uncertainty before theexperiments, it doesn’t matter, since all the elements measurement of which of the infinitely many otherwise-in the grand superposition are observationally indistin- identical experimenters the observer happens to be; afterguishable. In fact, since each term in the superposi- the measurement the observer has reduced this uncer-tion is indistinguishable from all the others, it is unclear tainty. Moreover, the “partially real” observers in the Ev-whether it makes sense to even call this a superposition erett picture have vanished: observers either exist equallyanymore.16 Since each state in the superposition has if they are part of the wavefunction with unity, or don’tformally zero norm, there is no choice but to consider exist if they are part of the zero norm branch that hasclasses of them, and if we class those states with indistin- been discarded.guishable predictions together, then this group has totalHilbert space norm of unity, while the class of “all otherstates” has total norm zero. In term of prediction, then, VII. DISCUSSIONthe infinite superposition of states is completely indistin-guishable from one quantum state (which could be taken The above-mentioned results raise interesting issuesto be any one of the terms in the superposition) with that deserve further work, and we comment on a cou-unity norm. In this sense, a hypothetical “collapse” of ple below.the wavefunction would be the observationally irrelevantreplacement of one statevector with another functionallyidentical one. A. Levels of indistinguishability In more Everettian terms we might say that in the Considering our finite or infinite sequences, in which ways are two such sequences distinguishable, and what15 does this mean physically? For finite N , if each system is For readers who are concerned whether infinity should be ac- cepted as a meaningful quantity in physics, it is interesting to labeled, then each term in the superposition is different. also consider the implications of a very large but finite N . In However, if we consider just statistical information such this case, the Hilbert space norm of the wavefunction component as the relative frequencies of |↑ and |↓ , then many se- 2 where the Born rule appears invalid is bounded by 2e−2 N , so quences will be statistically indistinguishable. Moreover, although it is not strictly zero, it is exponentially small as long if we do not label the terms, so that sequences can be re- as N −2 . For example, if N = 101000 (a relatively modest ordered when they are compared, then only the relative number in many inflation contexts), then the Born rule proba- bility predictions are correct to 100 decimal places except in a frequencies are relevant. wavefunction component of norm around 10−10 . 800 Now for an infinite product state, we have argued16 As well as giving identical predictions, this replacement is irrel- above that once we discard a zero-norm portion, the evant because decoherence has removed any practical possibility remaining states are statistically indistinguishable using of interference. This does not mean that the superposition has any finite amount of statistical information. If the ele- mathematically gone away, however, any more than when de- coherence is applied to a single quantum system. Nor does it ments are unlabeled, this statistical information is simply mean that mathematically the sequences are necessarily equiva- p (the |↑ fraction). We can ask, however, if these states lent; see Section VII A. are mathematically distinguishable. To see that indeed
  10. 10. 10they are, consider two such states such as ...|↑ |↓ |↑ |↑ ..., First, we might from an Everettian perspective con-and ...|↓ |↓ |↑ |↑ ... and represent them as binary strings sider such processes as simply parts of the unitary evo-with 1 = | ↑ and 0 = | ↓ . Now for each state, arbi- lution (or non-evolution, if considering the Wheeler-trarily select one system, and enumerate all systems as DeWitt equation) of a wave-function(al). Processes suchi = 1, 2, ... by increasing spatial distance from this cen- as measurement of quantum systems by apparatuses andtral element. Each state then corresponds to a real num- observers would not be meaningful until later times atber in binary notation such as 0.a1 a2 ... where ai = 0, 1. which a classical approximation of spacetime had alreadyBecause the selected system is arbitrary given the trans- emerged to describe an infinite space, in which such quan-lation symmetry of the space, we can consider our two tum measurement outcomes can be accorded probabili-sequences as indistinguishable if these real number repre- ties via the arguments of this paper. How this view con-sentations match for any choice of the central element of nects with the existing body of work on the emergence ofeach sequence. But there are only countably many such classicality is an interesting avenue for further research.choices, and uncountably many real numbers, so a givensequence is indeed distinguishable from some (indeed al- Second, we might retain the Born rule as a axiomaticmost all) other sequences in this sense. assumption, and employ it to describe such processes; Now, physically, we can ask two key questions. First, then, for quantum measurements that exist as part ofis the difference between (finitely) statistically indistin- a spatial collection, the Born rule would be superfluous,guishable and mathematically indistinguishable impor- as the probabilities for measurement outcomes are bettertant, given that any actual operation will only be able considered as relative frequencies as per the argumentsto gather a finite amount of statistical information? Sec- above.ond, given that these systems are by assumption identi-cal and indistinguishable, and far outside of each others’horizon, is it meaningful to think of them as labeled (even Third, we might consider a classical spacetime descrip-if there were a preferred element in terms of their global tion as logically prior to the quantum one, and in par-distribution)? ticular postulate certain symmetries that would govern If the answer to either question is negative, then it the spacetime in the classical limit. In this perspectivebecomes unclear what purpose is served by distinguish- we could simply assume an FLRW space obeying the CPing the elements in the post-measurement superposition, [45]. Or, we might include inflation and eternal inflation,and one might ask whether in some future formulation but postulate an appropriately generalized inflationaryof quantum cosmology they might be meaningfully iden- version of the CP (or perfect cosmological principle) gov-tified, thereby rendering the issue wavefunction collapse erning the semi-classical universe (see [65, 66] for ideasfully irrelevant. along these lines). Within this context, quantum events such as bubble nucleations could also be considered as part of a spatial collection of identical regions, and the B. The Chicken-and-Egg Problem of Quantum whole set of arguments given herein could be applied to Spacetime accord them probabilities. This would, in the language of [63], unify the “Level II” (inflationary) multiverse with We have argued that an infinite statistically uniform the “Level III” (quantum) multiverse.space can naturally emerge in modern cosmology, andplace the quantum measurement problem in a very differ- Fourth, we might imagine that a full theory of quantument light. Yet quantum processes affect spacetime in any gravity in some way fundamentally changes the quan-theory, and in inflation are responsible for the large-scale tum measurement problem, and that the considerationsdensity fluctuations. Moreover, some versions of eternal herein, based on standard quantum theory, apply onlyinflation themselves rely on quantum processes: stochas- processes in a well-defined background spacetime.tic eternal inflation is eternal solely due to quantum fluc-tuations of the inflaton, and in open eternal inflation, This subtle issue is similar to the – possibly related –inflation ends due to quantum tunneling. If quantum matter of Mach’s principle in General Relativity (GR):probabilities (and rates, etc.) are to be understood by applications of GR almost invariably implicitly assumemaking use of a cosmological backdrop, how do we make a background frame that is more-or-less unacceleratedsense of the quantum processes involved in creating that with respect to the material contents of the application;backdrop?17 There are at least four alternative ways in yet this coincidence between local inertial frames and thewhich we might view this chicken-and-egg problem. large-scale bulk distribution of matter is almost certainly of cosmological origin. Should Mach’s principle simply be assumed as convenient, or explained as emerging in17 a particular limit from dynamics that do not assume it, Similar considerations might apply to other fields – determined by quantum events but stretched into homogeneity by inflation – or does it require specification of cosmological boundary that go into making up the “background” in which the quantum conditions, or must new physics beyond GR be intro- experiment is posed, and to which the CP applies. duced?
  11. 11. 11 VIII. CONCLUSIONS worlds” are all the same; moreover, frequentist statis- tics emerge even for a single quantum measurement19 (rather than a hypothetical infinite sequence of them). In Modern inflationary cosmology suggests that we exist this “cosmological interpretation” of quantum mechan-inside an infinite statistically uniform space. If so, then ics, then, quantum uncertainty ultimately derives fromany given finite system is replicated an infinite number uncertainty as to which of many identical systems theof times throughout this space. This raises serious con- observer actually is.ceptual issues for a prototypical measurement of a quan-tum system by an observer, because the measurer can- In conclusion, the quantum measurement issue isnot know which of the identical copies she is, and must fraught with subtlety and beset by controversy; similarly,therefore ascribe a probability to each one [35–37, 40]. infinite and perhaps diverse cosmological spaces raise aMoreover, as shown by Page [35], this cannot be seam- host of perplexing questions and potential problems. Welessly done using the standard projection operator and suggest here that perhaps combining these problems re-Born rule formalism of quantum mechanics; rather, it im- sults not in a multiplication of the problems, but ratherplies that quantum probabilities must be augmented by an elegant simplification in which quantum probabilitiesprobabilities based on relative frequencies, arising from are unified with spatial observer frequencies, and thea measure placed on the set of observers. We have ad- same infinite, homogeneous space that provides a real,dressed this issue head-on by suggesting that perhaps it is physical collection also provides a natural measure withnot observer-counting that should be avoided, but quan- which to count. Further comprehensive development oftum probabilities that should emerge from the relative this quantum-cosmological unification might raise furtherfrequencies across the infinite set of observers that exist questions, but we hope that it may unravel further theo-in our three-dimensional space. retical knots at the foundations of physics as well. To make this link between quantum measurement andcosmology, we have built on the classic work concerning Acknowledgements: The authors are indebted andfrequencies of outcomes in repeated quantum measure- grateful to David Layzer for providing inspiration and co-ments [38, 39, 41, 42]. Our goal has not been to add pious insight, assistance, and feedback during the prepa-further mathematical rigor (see [67–70] for the current ration of this paper. Andy Albrecht and Don Page pro-state-of-the-art18 ), but instead to develop these ideas in vided very helpful comments. This work was supportedthe new context of the physically real, spatial collection by NASA grants NAG5-11099 and NNG 05G40G, NSFprovided by cosmology. The argument shows that the grants AST-0607597, AST-0708534, AST-0908848, PHY-product state of infinitely many existing copies of a quan- 0855425, and PHY-0757912, a “Foundational Questionstum system can be rewritten as an infinite superposition in Physics and Cosmology” grant from the John Tem-of terms. Because each term has zero norm, these must pleton Foundation, and fellowships from the David andbe grouped in terms of what they predict. Projecting Lucile Packard Foundation and the Research Corpora-these states with a “confusion operator” shows that a tion.grouping with total Hilbert space norm unity consists ofterms all of which are functionally indistinguishable, andcontain relative frequencies of measurement outcomes inprecise accordance with the standard Born rule. The Appendix A: Infinite statistically uniform spacesremaining states, which would yield different relative fre- from eternal inflationquencies, have total Hilbert space norm zero. Predictions of measurement outcomes probabilities in In this appendix, we describe in greater detail howthis situation are, then, provided entirely by relative fre- eternal inflation produces infinite spaces. Open inflationquencies; if conventional quantum probabilities enter at is perhaps the most well-studied case. The Coleman-all, it is only to justify the neglect of the zero-norm por- DeLuccia instanton [72] describes the spacetime and fieldtion of the global wavefunction. Because all terms in the configuration resulting from the nucleation of a singleunity-norm portion are indistinguishable, quantum inter- bubble where the inflaton field has lower energy. Inpretation must be done in a cosmological light. Any “col- single-field models, each constant-field surface is a spacelapse” of the wavefunction is essentially irrelevant, since of constant negative curvature, so that the bubble in-collapse to any of the wavevectors corresponds to ex- terior can be precisely described as an open Friedmannactly the same outcome. In Everettian terms, the “many universe [72]. Bubble collisions (which are inevitable) complicate this picture; see [73] for a detailed review. In18 The core mathematical question is how to deal with the measure- zero set of confused freak observers; but as emphasized in [71], 19 For example, the author ordering for this paper was determined this issue is not unique to quantum mechanics, but occurs also by a single quantum measurement, and the order you yourself in classical statistical mechanics and virtually other theory in- read is shared by exactly half of all otherwise-indistinguishable volving infinite ensembles. worlds spread throughout space.
  12. 12. 12no case, however, do collisions prevent the existence in- occurs when some obstruction prevents the inflaton fieldside the bubble of an infinite, connected, spatial region20 φ from evolving away from some value φ0 during a typical In topological eternal inflation [74, 75], a region of in- Hubble time, so that on average, the physical volumeflating spacetime is maintained by a topological obstruc- containing that field value increases. This implies thattion such as a domain wall or monopole, where the field is there are some (extremely rare) worldlines threading aat a local maximum of its potential. The causal structure horizon volume in which φ ≈ φ0 forever. For such aof these models is fairly well-understood if a bit subtle model to be observationally viable, however, there must(see, e.g., [74, 77]), and also contains infinite spacelike exist a route through field-space that crosses a value φssurfaces after inflation has reheating surfaces. at which slow-roll inflation begins, then a value φe at The precise structure of post-inflation equal-field sur- which inflation ends and matter or radiation-dominationfaces in stochastic eternal inflation is less well-defined, as begins.the spacetimes cannot be reliably calculated with ana- This field evolution plays out in spacetime as well, con-lytic or simple numerical calculations. However, they are necting the eternally inflating spacetime region to theexpected to be infinite (e.g., [34, 78]), with volume domi- post-inflationary region, as sketched in Fig. 2 for a singlenated by regions in which inflation has emerged naturally inflaton field. Whether eternal inflation is open, topo-from slow roll [79], and thus space is relatively flat and logical, or stochastic, this diagram must look essentiallyuniform. the same. The surface of constant field φs , at which eternality fails, is by definition one for which very few worldlines cross back to field values near φ0 ; this surface reheating must therefore be spacelike nearly everywhere. Morever, slow-roll inflation to the future of this surface exponen- i0 Φe tially suppresses field gradients, so that surfaces of con- stant field quickly become uniformly spacelike21 as the slow roll field approaches φe . Now, if we consider a spatial region with φ = φe (soon Φ0 to evolve to the reheating surface), and follow the surface Et of constant field φ = φs in a direction toward the eternal er na Eternal worldline lw region, we see three things. First, the φ = φe surface Φs or ld must be infinite, since the φ = φs surface, for example, lin e’s is infinite, and further inflates into the constant field sur- pa st faces with φs ≤ φ ≤ φe . Second, by the above argument the φ = φe surface is spacelike, so there is no obstacle to continuing the foliation in our original region as far as we like toward the eternal region. Third, since each point on our surface has roughly the same classical field historyFIG. 2: Generic conformal structure of eternal inflation and since φs (variations due to the initial field velocity beingpost-inflationary reheating surface in open, topological, or bounded by the slow-roll condition, and curvature beingstochastic eternal inflation. A region in which inflation is stretched away), the φ = φe surface should be uniformeternal (i.e. contains timelike worldlines that can extend to up to variations induced by quantum fluctuations duringinfinite proper time while remaining in the inflation region)is bounded by an infinite surface of field value φs at which the field’s evolution.eternality fails. Between this value and φe lie some number of Multi-field models are more complex, in that we mighte-folds of slow-roll inflation. This slow-roll inflation ends on a imagine many routes for the vector of field valued φ tospacelike surface that represents a natural equal-time surface take from a given (set of) ‘eternal’ field values φ0 to fieldfor the subsequent evolution including reheating, etc. This values φe at which inflation ends. Yet it seems likelysurface is spatially infinite, with spatial infinity denoted byi0 . (The future infinities following the reheating surface are that in this case the prime difference would be for thenot depicted). φ = φs surface to be replaced by an infinite, spacelike surface Σs on on which eternality failes, and on which φ is inhomogeneous.22 Yet for a given value φs of this That infinite spacelike reheating surfaces are generic vector on this surface, subsequent evolution would bein eternal inflation is not surprising, as per the followingheuristic argument. Roughly speaking, eternal inflation 21 For a more precise specification of this point, see [76, 77]. 22 This is precisely what happens in “Quasi-open” inflation [80].20 This region could be delimited by, e.g. defining some criterion There, the bubble interior cannot be foliated into constant-field by which to identify regions affected by the collision, and exclud- equal-time surfaces. However, there is still an infinite space with ing them. After this removal, the remaining region would have infinitely many finite-sized regions in which the field is constant; uniform properties (as would the statistics of the excisions; see a subset of these with the same field value could be taken as a below.) statistically uniform (though potentially disconnected) space.

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