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P. 160, Smith, Quentin, 1997, “The Ontological Interpretation of the Wave Function of the Universe”, The Monist, vol. 80, no.1, pp, 160-185.

THE ONTOLOGICAL INTERPRETATION OF THE WAVE FUNCTION OF THE UNIVERSE

By Quentin Smith

1. Quantum Mechanics and the Real World: The View from Quantum Gravity Theories

There are two distinct questions that arise when one asks about "the interpretation of quantum mechanics" or "how can quantum mechanics be reconciled with the 'real' world-the world we experience."

(i) The usual sense of these questions is about (so-called) ordinary quantum mechanics and its relation to the "world we experience." Ordinary quantum mechanics is largely based on the Schrodinger equation and the ontological interpretations of this equation, Since the late 1920s the "orthodox" interpretation was taken to be the Copenhagen interpretation (which has many variants), but by the 1990s other interpretations have gained ascendancy, at least among philosophers of quantum mechanics and quantum physicists on the "cutting edge" of their field. Although the typical graduate textbook on quantum mechanics presents (a version of) the Copenhagen interpretation, some physicists and most philosophers of quantum mechanics are attracted to Everett-type interpretations (the original Everett-DeWitt many-worlds version, or the Albert-Loewer many-minds version, or the Griffths-Omnes consistent histories version, etc.), Bohmian interpretations, modal interpretations (Healey, van Fraassen, Kochen, Dieks, etc.), the Ghirardi-Rimini- Weber interpretation, and many other interpretations (such as Krips's) that avoid the "anti- realism" often associated with the Copenhagen interpretation. Thus, the first sense of the question about "quantum mechanics and the real world" may be represented as the question: What is the ontological interpretation of the Schrodinger equation and how is this related to the world we experience ?

(ii) However, there is a second sense to these questions about interpretation and the "reconciliation of quantum mechanics with the world of

P. 161, Smith, Quentin, 1997, “The Ontological Interpretation of the Wave Function of the Universe”, The Monist, vol. 80, no.1, pp, 160-185.

experience" that has been extensively discussed by physicists since the mid-1980s, but discussed less often by philosophers of quantum mechanics. This is the question of the interpretation of the gravitational Schrodinger equation (the Wheeler-DeWitt equation) or, more exactly, of the relevant solutions to this equation. These solutions are called "wave functions of the universe" and the most well-known ones are the "no boundary" wave function of Hawking and his collaborators,[1] and Vilenkin's and Linde's[2] "tunnelling" wave functions. The topic of these discussions is not ordinary quantum mechanics, but quantum gravity theory. Ordinary quantum mechanics deals with the Schrodinger equation, where gravity is not quantized, but quantum gravity theory deals with the gravitational Schrodinger equation and proposals about how gravity is to be quantized. Thus, the second sense of the question about "quantum mechanics and the real world" may be represented as the question: What is the ontological interpretation of the gravitational Schrodinger equation and how is this related to the world we experience?

The present paper is about this second sense of the question. The most important contribution to the discussions of the ontological interpretation of the wave function of the universe is Craig Callender's and Robert Weingard's classic article "The Bohmian Model of Quantum Cosmology."[3] They explain how solutions to the gravitational Schrodinger equation can be interpreted If we assume a Bohmian version of quantum mechanics. One of the reasons Callender's and Weingard's paper is noteworthy is that it departs from the assumptions of most of the physicists who work in this field (Hawking, Hartle, Vilenkin, Linde, Halliwell, etc.), who assume an Everett-type interpretation of quantum mechanics. However, Hawking, Vilenkin and others have made only sketchy and inadequate (as I shall argue) remarks about how a wave function of the universe should be interpreted if we assume an Everett-type interpretation of quantum mechanics. In this paper, I shall develop at length an interpretation of a wave function of the universe that assumes an Everett-type interpretation as the back- ground theory of quantum mechanics. This will result in a different picture of the world than is given by the Callender-Weingard Bohmian quantum cosmology. My goal is not to argue that an interpretation based on an Everett-type theory is preferable to a Bohmian quantum cosmology; I shall instead confine myself to the preliminary task of presenting a coherent and reasonably detailed Everett-type interpretation.

P. 162, Smith, Quentin, 1997, “The Ontological Interpretation of the Wave Function of the Universe”, The Monist, vol. 80, no.1, pp, 160-185.

An interpretation of the wave function of the universe that assumes a background Everett-type theory results in a novel "quantum mechanical world/world of experience" split that differs from the .'split" that pertains to interpretations of ordinary quantum mechanics. Hawking and other physicists who work in this area typically call "the world of our experience" the part of the manifold that is described by the oscillating component of the wave function of the universe, which can be reduced by the WKB approximation to Friedman solutions of the general relativistic equations. "The world of our experience" becomes approximately our expanding Friedman universe, the familiar universe of big-bang cosmology (minus the big-bang singularity). These physicists use the phrase "the purely quantum mechanical region of reality" to refer to the part of the manifold that is described by the exponentially growing component of the wave function of the universe; this component is interpreted as corresponding to a Euclidean spacetime. This also corresponds approximately to solutions to Einstein's equations, but these solutions describe a timeless and closed four-dimensional space. This timeless hypersphere and the "join region" where this hypersphere is attached to our expanding Friedman universe plays the role (in quantum gravity theories) of the "bizarre quantum mechanical region"; this is the part of the manifold about which we shall ask the question standardly asked in ordinary quantum mechanics: "how could the world possibly be like that?"

The "no boundary" theory of Hawking and Company and the "tunnelling" theory of Vilenkin and others are highly simplified models (most of the gravitational degrees of freedom are not quantized in their models) and they bracket the major problem facing quantum cosmology, viz., that there is no complete and consistent unification of general relativity with quantum mechanics. But the physicists working in this area believe that progress can be made despite the fact that the problem of the renorrnalizability of general relativity and related problems have not been solved. They believe we can make reasonable conjectures about some of the elements that a complete quantum gravity theory will involve; if this is true, the complete ontology corresponding to a complete quantum gravity theory may be approximated (in some degree) by an ontological interpretation of the simplified "toy models" developed by Hawking, Vilenkin and others.

2. Imaginary Time and Real Time

In the standard general relativist model of the universe, the universe may be regarded as beginning at a singularity at the earliest instant t₀ or

P. 163, Smith, Quentin, 1997, “The Ontological Interpretation of the Wave Function of the Universe”, The Monist, vol. 80, no.1, pp, 160-185.

else as having a half-open earliest temporal interval with the singular instant t₀ being regarded as fictional. However, on either interpretation there remains the problem commonly noted by physicists, viz., that during the first 10^-43 second of the universe's history, the laws of general relativity are not expected to apply. This time period is called the Planck era, during which the radius of the universe is less than 10^-33 centimeters. The Planck era is a focus of study in Hawking's and Vilenkin's cosmologies and the key prediction is that the universe exists in imaginary time during the Planck era.

Imaginary time is the temporal dimension of the Euclidean spacetime. A Euclidean spacetime has a metric with positive definite signature (++++), which involves treating the "temporal dimension" on a par with the three spatial dimensions. This is reflected in the fact that the numbers for the time coordinate of a Euclidean spacetime are imaginary and for this reason the time is called an "imaginary time." The physical meaning of using imaginary numbers for the time coordinate is that time loses its distinction from space; time becomes a fourth spatial dimension.

The Euclidean spacetime that exists at the Planck era is a hyper- sphere; it is a closed, four dimensional space. This Euclidean spacetime is connected to our expanding Lorentzian universe, which exists in real time. A Lorentzian spacetime has a metric with a positive indefinite signature (+++-); this is our familiar spacetime with three spatial dimensions and one (real) temporal dimension.

The central problem of quantum gravity ontology should now be apparent: how is the imaginary time of the Euclidean spacetime connected to the real time of our Lorentzian spacetime?

Stephen Hawking, Arthur Vilenkin, Jonathan Halliwell, Christopher Isham, Paul Davies, and other physicists describe this connection in temporal terms. In an article in Scientific American on Hawking's "no boundary" model, Halliwell writes that

the universe began from a perfectly smooth kind of cap [the four dimensional space] rather than a point. The "smoothing off' occurs in imaginary time, so it does not contradict singularity theorems, which refer to real time. Shortly after quantum creation, the universe evolved classically in real physical time.[4]

Christopher Isham summarizes his account of Vilenkin's wave function model, which is relevantly similar to Hawking's model, by writing, "Thus the final picture of the origination of the universe is of an imaginary-time

P. 164, Smith, Quentin, 1997, “The Ontological Interpretation of the Wave Function of the Universe”, The Monist, vol. 80, no.1, pp, 160-185.

spacetime (which is totally non-classical) from which the real-time (and classical) universe emerges with some finite radius."[5]

Paul Davies describes the similar transition in the Hartle-Hawking model as follows:

It may happen, as a result of these quantum effects, that the most probable structure of space-time under some circumstances is actually four-dimensional space... That is, if we imagine going backward in time toward the big bang, then, when we reach about one Planck time after what we thought was the initial singularity, something peculiar starts to happen. Time begins to "turn into" space... Notice that the transition from time to space is gradual... time emerges gradually from space.[6]

Hawking himself says that the four dimensional space existed "during the first minute fraction of a second,"[7] i.e., that the four dimensional space existed during the first interval of 10^-43 second. This four dimensional space, at a later time, smoothly joins onto a three dimensional space that exists in real time. "The geometry of the four-dimensional space. ..has to somehow smoothly join onto the more familiar spacetime once the quantum smearing effects subside."[8]

It seems to me, however, that the description of real time as coming into existence "shortly after" the imaginary time phase is at best a metaphorically accurate description-of the relation between the two times. The temporal description offered by Hawking and others, if taken literally, suggests some logical paradoxes, and if taken metaphorically, we are left by these authors without a literal explanation of what the temporal terms symbolize. Consider these propositions, where the temporal expressions are taken literally:

(1) The first interval of one second at which the universe exists is partly composed of an imaginary time interval of 10^-43 second and partly composed of a longer real time interval.

(2) Real times are later than imaginary times.

(3) A four dimensional space, without a real time coordinate, existed earlier in time than a three dimensional space with a real time co- ordinate.

(4) Time evolves in gradual and successive stages out of space.

P. 165, Smith, Quentin, 1997, “The Ontological Interpretation of the Wave Function of the Universe”, The Monist, vol. 80, no.1, pp, 160-185.

(5) Space existed for a time, "during the first minute fraction of a second," that was earlier than the time at which time began to exist.

Proposition (5) is an obvious contradiction, so we shall examine the other propositions, where the contradiction is less explicit. Consider proposition (2). If a real time t is later than an imaginary time it, then is t later than it in real time or imaginary time? It cannot be later in real time than it, since if it were, then it  would be a position in real time, namely, a position earlier than t, and it would then not be an imaginary time position of the universe but a real time position, contradicting the original supposition. Nor can t be later than it in imaginary time, for then t would be a position in imaginary time, namely, a time later than i, and thus t would not be a real time position but an imaginary time position, contradicting the original supposition. It seems, then, that proposition (2) is an implicit, logical contradiction.

Analogous considerations would show that (I) and (3) are implicit logical contradictions. (4) is an implicit logical contradiction since it !c asserts that time evolved gradually, in successive stages. Since, however, , a succession of stages requires successive times, this statement requires that there exist a succession of times during which time gradually came into existence: This is a contradiction, since a logical condition of time successively coming into existence is that it already exist as the successive positions occupied by the states of time's gradual coming into existence.

Some philosophers and physicists, when confronted with the aforementioned charges of contradiction, respond that the criticism is based on an adherence to classical (general relativistic) concepts of space and time and that when one enters the quantum realm of the Planck era, these concepts need to be abandoned in favor of a quantum framework for describing the universe. For example, Christopher Isham argues that there is no absurdity involved in the various statements about the imaginary time/real time relations I quoted and responds that "what is absurd is the desire to stick to classical concepts when discussing all domains of reality!"[9]

But a response of this sort misses the point of the criticism. The point is not that the description of the Planck era is inconsistent with classical

P. 166, Smith, Quentin, 1997, “The Ontological Interpretation of the Wave Function of the Universe”, The Monist, vol. 80, no.1, pp, 160-185.

(general relativistic) concepts. The point is that the temporal description of the relation of the Planck era to the classical era is inconsistent with itself. It is an implicit self-contradiction to assert that a four dimensional space existed earlier than the earliest time.

At this point, defenders of the coherency of the, above quotations may suggest that the temporal terms are used metaphorically and therefore there is no contradiction. Isham adopts this view in some of his writings. In his interpretation of Vilenkin's "tunnelling" model, he defends the coherency of the transition from the imaginary time phase to the real time phase by suggesting the transition is not a temporal one. After saying that the real-time universe "emerges" from the imaginary-time universe, he writes: "Of course, the words 'emerge' and 'process' must be understood in a symbolic sense since their usual temporal implications are not appropriate in the present situation: time as we normally understand the word is applicable only in the region… that is well away from the originating four-sphere."[10]

Isham should be commended for pointing out that the temporal phrases used in his description of the relation between imaginary and real time should not be taken literally. However, Isham's remark that his temporal terms should not be used in their normal temporal sense raises several questions. In what sense is the four-sphere an "originating" sphere if the real-time universe does not temporally succeed it? And in what sense is the real-time region "well away from" the four-sphere if "well- away from" does not mean "temporally distant from"? Furthermore, if the words "emerge" and "process" are used symbolically or metaphorically, what do they symbolize? Unfortunately, Isham offers no clue in his paper as to how these questions are to be answered; he does not specify in literal terms any physical relation that could satisfy these predicates.

Perhaps, however, Isham is not attempting to offer a physical or ontological interpretation of the relation between imaginary and real time. Isham elsewhere writes that" ...if one tries to track backwards in time from the WKB semi-classical region [i.e., the real-time region] one comes eventually to a regime where the classical category of time becomes harder and harder to sustain and 'then' to a region where it fades out all together. But the 'then' refers to a hypothetical movement along a mathematical path in superspace: it is not a temporal statement "[11] These remarks suggest that Isham is talking only about the relation between the

P. 167, Smith, Quentin, 1997, “The Ontological Interpretation of the Wave Function of the Universe”, The Monist, vol. 80, no.1, pp, 160-185.

Euclidean region and the Lorentzian region in the mathematical space on which the wave function equation is defined. Is this reading of Isham correct?

In his well-known articles "Creation of the Universe as a Quantum Tunneling Process" and "Quantum Theories of the Creation of the Universe"[12] (which most philosophers rely upon for their understanding of quantum gravity cosmologies) Isham describes both the real-time region and the imaginary-time region in language that (if taken literally) implies that these regions are physically existent and physically interrelated. However, Isham, when questioned (by the author) about the paradoxical consequences of treating these regions as temporally related, has asserted that he does not intend to offer any physical interpretation of the wave function equations. "Unfortunately, I cannot give you a realist (or, for that matter, any other physical) interpretation of the mathematical model because this is part of the basic interpretation problem of quantum theory which, as I have emphasized, is still unresolved."[13] Isham suggests that it , is best to interpret his physicalist language in a purely symbolic sense that 11 refers to properties of the mathematical model, not to what physically exists. Writing about the statement that "the universe at first is a four dimensional space and then gradually evolves. ...", Isham notes; "My preferred interpretation is that the existential statement refers to what is in the mathematical model, and the process statement refers to how various mathematical ingredients in the model change as one looks at various parts of the model."[14] Given this denial that his remarks are to be read as suggesting an ontological interpretation of the wave function of the universe, we may allow that Isham cannot properly be charged with a contradiction or lack of specificity in an ontological interpretation of the equation. However, this way of avoiding the problems comes at a price, for Isham's two highly influential articles on the Hawking and Vilenkin wave function models ("Creation of the Universe as a Quantum Tunneling Process" and "Quantum Theories of the Creation of the Universe") can no longer be read as contributions to the most pressing problem in contemporary quantum cosmology, namely, providing an ontological interpretation of the wave functions of the universe. Furthermore, the position taken by Isham implies that his two articles should not be understood as they are commonly understood by philosophers, namely, as providing an ontological interpretation of these models.

P. 168, Smith, Quentin, 1997, “The Ontological Interpretation of the Wave Function of the Universe”, The Monist, vol. 80, no.1, pp, 160-185.

Isham makes some helpful remarks that are directly pertinent to our present concerns. Isham advocates that "philosophers… should get entangled with the great ontological question in quantum theory: the theory in which imaginary time makes its appearance."[15] Isham remarks, "1 should emphasize how much the theoretical physicists who work in quantum gravity tend to disagree among themselves about what is going on, and so attempts to describe this situation in non-technical terms are particularly difficult."[16] He suggests that "the 'ontological question' of precisely how the mathematics relates to reality" is of exceptional difficulty to physicists and "throws up a real challenge to metaphysically- inclined philosophers."[17] Isham's point is well-taken, for if we take the ontological remarks made by Hawking, Vilenkin, Halliwell, Davies and others in literal terms, we end up with contradictions, but if we take these remarks as metaphors or symbolic terms, we are either left with the unanswered question "what sort of ontological items do these terms symbolically denote?" or we are left with Isham's non-ontological position that these terms merely symbolize properties of the mathematical model. In the following sections, I will outline an ontology that I believe corresponds to the mathematical model.

3. A Spaceless and Timeless Physical Region

I will begin by discussing the physical region of the universe at which there is no space or time. This region is the "join" region that Hawking alludes to in this sentence; "The question then arises as to the geometry of the four-dimensional space which has to somehow smoothly join onto the more familiar spacetime once the quantum smearing effects subside."[18] Although the "once" has to be taken metaphorically in order for this sentence to avoid logical difficulties, the italicized portion of the sentence mentions the "join" region that I shall discuss. I shall argue that the "smooth join" consists in topological and other nonmetrical relations between the four-dimensional space and our familiar spacetime, relations that obtain on a part of the manifold where there is no space or time.

The "join" region is a metrically amorphous, four-dimensional and differentiable region of the manifold. This unfamiliar idea is based on a familiar idea, that the metric is not the most fundamental structure of the manifold. It is a commonplace that there are affine, differentiable and topological structures of the manifold that underlie the manifold's metric. But

P. 169, Smith, Quentin, 1997, “The Ontological Interpretation of the Wave Function of the Universe”, The Monist, vol. 80, no.1, pp, 160-185.

it is not a commonplace that differentiable and topological relations can obtain on parts of a physical manifold where no metrical relations obtain; indeed, some philosophers and physicists believe this is impossible, as we shall see below.

To understand the conception of this metrically undefined "join" region, let us review some of the basic concepts of a manifold.[19]

The most basic structure of a manifold is topological. The topological structure involves the number of a manifold's dimensions (e.g., whether it is three or twelve dimensional), whether it has a boundary or is unbounded, whether it has holes in it (like a doughnut) and the like. Our manifold has the general structure of any topological space (X, T), which consists of a set X together with a collection T of subsets of X that meet three conditions: (1) the union of an arbitrary collection of subsets, each of which is in T, is also in T. (2) The intersection of a finite number of subsets in T is in T. (3) The complete set X and the empty set are in T. r is the topology on X.

Our manifold will be coordinatizable by R­⁴, where R⁴ is the set of quadruples of real numbers. Given any point p on the manifold, there is ~ neighborhood of p (a set of all points close to p), such that there is a one-to-one map from the neighborhood into R⁴ that is sufficiently continuous (nearby points in the neighborhood of p are mapped into nearby points in R⁴, and vice versa). Our manifold will also have further topological features, e.g., it will be connected, Hausdorff and paracompact.

Furthermore, our manifold has differentiability. There are maps M, such that M maps nearby points in a neighborhood of some point p onto nearby points in R4. The real-valued function f defined on the neighbor- hood is differentiable if and only if f o M^-1 (the result of applying f to the result of applying the inverse of M) is differentiable for every map M. Thus, the manifold (at least locally) is a four-dimensional differential manifold.

The differentiable structure of the manifold allows us to define a curve and the tangent field of that curve. A curve on the manifold is a continuous and differentiable map s from an interval of the real line into the manifold. The tangent field is a set of tangent vectors, which indicate direction from a certain point on the curve s.

At the "join" region where the metric is undefined, there is a topologically four-dimensional, coordinatizable and differentiable region of

P. 170, Smith, Quentin, 1997, “The Ontological Interpretation of the Wave Function of the Universe”, The Monist, vol. 80, no.1, pp, 160-185.

the manifold. This region joins the timeless four-dimensional space and our familiar spacetime at least in the sense that there is some curve s in the Euclidean spacetime that continues into the Lorentzian spacetime by means of passing through the metrically amorphous region of the manifold. In the spatially four-dimensional region, this curve has a metric with a positive definite signature ( ++++ ); in the .'join" region, the curve is metrically undefined; and in our familiar spacetime, this curve has a metric with positive indefinite signature ( +++- ).

It may be doubted if this is ontologically possible. Some philosophers have objected that if there is no temporal relation between the two parts of the manifold, there is no relation that is merely topological. Others, such as William Craig, Robert Deltete and Reed Guy,[20] have argued that it is ontologically impossible for there to exist a metrically amorphous region of a manifold. For example, Deltete argues that the topological points in the “join” region can be separated from one another only if they are spatially or temporally separated, and since the metric tensor is undefined in this region, there are no spatial or temporal separations. And Shannon Maudlin[21] has argued that the topological structure is a conceptual abstraction from a spacetime that has a well defined metric and that this structure can no more exist on its own than a shape can exist without a body of which it is a shape. He argues that a topology without a metrically well-defined spacetime is not a physical entity but a mathematical one.

These critics may be called “anti-realists” about the “join region.” In the following, I will present an argument for the realist position. My argument concentrates on showing that a manifold without a metric but with a topology is possible.

I shall first present an argument that a metrically undefined topology is a physical possibility, and then shall present a simplified example that helps to make this notion more intuitively clear.

My argument is about physical particulars; one way to frame the issue is to ask if a topological point is an abstract particular, such as a set, or a concrete and physical particular, such as a spacetime point. (We must avoid the false assumption, sometimes made, that all abstract objects are universals and all particulars are concrete objects.) The anti-realist position about the "join region" can then be phrased as the position that topological points that lack metrical properties are abstract objects that belong only to the mathematical formulae of quantum cosmology, and the realist position about the "join" region is that such points are concrete

P. 171, Smith, Quentin, 1997, “The Ontological Interpretation of the Wave Function of the Universe”, The Monist, vol. 80, no.1, pp, 160-185.

physical particulars. My argument (A) is an argument for the realist position.

(A)

(1) The concept of a metric is defined in part in terms of the concept of a topology, but the concept of a topology is not defined in terms of a metric; the concept of a metric analytically entails the concept of a topology, but the concept of a topology does not analytically entail the concept of a metric.

Therefore,

(2) A metrical property is exemplified by something x only if some topological property is exemplified by x, but something x can exemplify a topological property without exemplifying a metrical property. (To deny that #1 entails #2, the anti-realist must resort to the claim that there is a synthetically necessary relation between a topology and a metric, but there is no logical justification for such a claim and it is not self-evident.)

(3) If a particular exemplifies a physical property, then it is a physical particular, at least in respect of the physical property it exemplifies. (This is the innocuous point illustrated by the case of humans: since a human exemplifies the property of having a brain, a human is a physical particular at least in respect of having a brain.)

(4) A spacetime point is a physical particular that has zero spatial extension and zero temporal extension, but is endowed with a metric g. (1 take #4 to be an allowable assumption since the denial of physicality to a spacetime point involves separate issues, e.g., scientific anti-realism, a non-modal relationalism about spacetime, or a finitist rejection of the divisibility of spacetime intervals into a continuum of points.)

(5) The topological properties of spacetime points, such as being part of a continuum, being part of a four-dimensional manifold, being part of a closed manifold, being part of a connected manifold, being part of a Hausdorff manifold, are physical properties. (The denial of #5 requires a sort of phenomenalism or subjective idealism that rejects the thesis that we exist in a physically four-dimensional universe.)

Therefore, [from #3 and #5]

(6) A sufficient condition of a spacetime point being a physical particular is that it exemplify topological properties.

Therefore, [from #2],

P. 172, Smith, Quentin, 1997, “The Ontological Interpretation of the Wave Function of the Universe”, The Monist, vol. 80, no.1, pp, 160-185.

(7) A particular that has zero spatial and temporal dimensions can exemplify topological properties without exemplifying metrical properties. Therefore, [from #6 and #7]

(8) If we abstract or remove the metrical properties from a spatiotemporally unextended particular x, we are not thereby abstracting from x all the sufficient conditions for that particular being a physical particular.

Therefore, [from #8] (9) If we remove all the metrical properties from the points in a certain region R of a manifold that admits of a metric in other regions, we are not removing from the points of R their property of being physical particulars.

Therefore, [from #9] (10) There can be a metrically undefined and physical "join" region of a manifold that topologically connects a Euclidean region to a Lorentzian region.

This seems to me to be a plausible argument for the realist position. But this argument may not be enough to persuade; part of what motivates the anti-realist position about the "join" region is that a physical region that has no spatial or temporal points seems unimaginable and inconceivable. I grant that it is impossible to imagine such a region, but I believe the con- junction of the following examples' will show that such a region can be clearly and coherently conceived.

First, conceive a spatiotemporal point, i.e, a physical particular that has no spatial or temporal extension but has a metric g. Think of this point without thinking of its spatial and temporal relations to other points. What is now before your mind is a physical topological point without a metric g .

But the principal issue is whether we can clearly conceive a metrically undefined region of topological points. The following example shows how this can be done in a simplified way. Consider lines L_l and L₂:

L_l and L₂ have the same topology: L₁ can be transformed into L₂ by a one- one continuous transformation. Both L₁ and L₂ share the topological properties of being one dimensional, continuous, and having a beginning and end. However they have different metrics. If we suppose O₁, O₂ and

P. 173, Smith, Quentin, 1997, “The Ontological Interpretation of the Wave Function of the Universe”, The Monist, vol. 80, no.1, pp, 160-185.

O₃ are temporal occurrences, we can say that the metric of L₁ assigns an equal duration to each of these three occurrences, but the metric of assigns greater measures of duration to earlier temporal occurrences.[22]

If we relate this example to our "join" region, the metrical properties that distinguish L₁ from L₂ will be absent; we conceive a line that has only the properties shared in common by L₁ and L_­2. our metrically amorphous line contains O₁, O₂ and O₃ ordered by a relation R; R is not a temporal relation but a topological relation, such that R merely places O₂ in between O₁ and 0₃ in a static structure. O₁, O₂ and O₃ are stripped of their status as temporal extended occurrences and become physical complexes with a topological structure.

It seems to me that this example suggests that a physical and metrically undefined "join" region, although unimaginable and unlike what we perceive and unlike anything in the familiar world of common sense, is not as difficult to conceive as the anti-realists about the "join" region wish to make out. I think the burden of proof is on the anti-realists to show that this "join" region is "physically unintelligible," to borrow a phrase from William Craig.

4. The Vague Beginning of Real Time :

In, section 3, only one-half of the quantum ontology was presented. The other half is the ontology of vagueness. We recall that Paul Davies said that "time emerges gradually from space."[23] And Isham says in one of his articles that as we go backwards towards the origination phase "the harder it becomes to sustain an interpretation of anything 'evolving' in time."[24] What sort of literal ontological interpretation should we give to the word "gradually" in the expression "time emerges gradually from space"?

I believe the ontology of the "join" region requires a logic of vagueness. According to this logic, there are three truth values: true, false and indefinite (neither true nor false). The truth tables for the logical connectives accommodate these three values. For example, a disjunction is true if either disjunct is true, and is false if both disjuncts are false; otherwise it is indefinite. A conjunction is true if both conjuncts are true, and false if either conjunct is false; otherwise it is indefinite. The "gradual emergence of time" will be taken to imply that in part of the "join" region it is neither true nor false that there is time; time is vague in part of this region.

P. 174, Smith, Quentin, 1997, “The Ontological Interpretation of the Wave Function of the Universe”, The Monist, vol. 80, no.1, pp, 160-185.

Consider a curve s in the "join" region that is gradually picking up a Lorentzian metric and, specifically, a real temporal dimension. The curve s has a direction that is "headed towards" the Lorentzian region where the metric is well-defined, and as the curve s comes closer to the metrically well-defined region, it gradually acquires a temporal dimension. Suppose that at a part of the "join" region that is topologically distant from the Lorentz region, there is a topological point P₁ on the curve s, such that P₁ definitely has no temporal dimension, and that at a point P₅ on this curve, the curve definitely has a temporal dimension. In the journey between P₁ and P₅, the curve gradually acquires a temporal dimension. This implies that at some of the points between P₁ and P₅, the temporal dimensionality of the curve is indefinite. This means that at point PJ' it is false that the curve has a temporal dimension; at some further point, P₃, it is neither true nor false that the curve has a temporal dimension; and at P₅, it is true that the curve has a temporal dimension. Our theory of temporal indefiniteness implies that the indefiniteness is in physical reality itself, not merely in our description of reality: something x is temporally indefinite if and only if there is no physical fact of the matter as to whether x has a temporal dimension or lacks a temporal dimension. There being no physical fact of the matter is the ontological ground of the semantic property of having a truth-value gap (being neither true nor false). Thus, we adopt a realist theory of vagueness, as do David Sanford and Michael Tye[25] and not an anti-realist theory of vagueness, as does Timothy Williamson.[26]

The vague beginning of time also requires the notion of increasing similarity to time. The fact that the temporal dimension is gradually acquired as the point P₅ on the curve s is approached implies that at some points on 0", the curve is more similar to a curve with a temporal dimension than it is at other points. The physical property F of being temporally indefinite is a degreed property and its exemplification includes the exemplification of being similar to a temporal dimension to a certain degree.

It may be objected to this account of the vague beginning of time that it uses temporal phrases to describe an allegedly non-temporal region, and thus that my own description of the "join" region succumbs to the very objection I made against Hawking's, Halliwell's and Davies's descriptions of this region.[27]

I would respond that I am using expressions such as 'gradually acquires', 'journey', 'is approached', 'distant from' in non-temporal senses

P. 175, Smith, Quentin, 1997, “The Ontological Interpretation of the Wave Function of the Universe”, The Monist, vol. 80, no.1, pp, 160-185.

that are definable entirely in coordinate and topological terms. For the purposes of the present discussion, I have chosen a coordinate system in which the numbers assigned to the points on the curve 0- increase in the direction of the Lorentz region. The clause "the temporal dimension is gradually acquired as the point P5 on the curve s is approached" means just this: at the point P₁ s is nontemporal; at P₂, s is temporally indefinite; at P₃, s is also temporally indefinite but more similar to a curve with a temporal dimension than it is at P₂; the same for P₄; and at P₅, s is temporally definite. Ontologically, we have a static structure; the Euclidean region, the "join" region, and the Lorentz region, are merely different regions of the manifold. We may say there is temporal evolution inside the Lorentz region, and that curves "gradually acquire" properties in a temporal sense in the Lorentz region, but we must distinguish this temporal sense of "gradually acquire" from the sense that is definable in terms of an arbitrary coordinate system.

The ontology of a vaguely beginning time requires several more notions in addition to our account of the degreed property F of being temporally indefinite. One notion is that there is no fact of the matter  as to whether there is a first temporal interval of each actual length (a first hour, a first second, etc.). Consider intervals of some very short length; the “no boundary” and “tunneling” models predict the various stages in the very brief inflationary era and allow that there are temporal intervals at least as short as 10^-35 second. Regarding intervals of this length, we can say that before any such interval there are not infinitely many intervals of that length; but there is also no interval of this length that is definitely not preceded by an interval of this length. Although there are some intervals of this length that are definitely preceded by another interval of that length (for example, all the intervals in the year we call "1996"), near the vague beginning of time there are some intervals about which there is no fact of the matter as to whether they are preceded by another interval of that length. But this property G (which we may call the property of being in- definitely preceded by an interval of the same length) cannot be possessed by only one interval of length 10^-35 second. If there were only one such interval, call it ti' then the immediately succeeding interval t₂ would have the property that it is the first interval of length 10^-35 second that definitely has at least one interval of this length earlier than it. This would imply that the earlier interval t_l is the earliest interval of 10^-35 second. In this case, time would not have a vague beginning; it would begin at a precisely

P. 176, Smith, Quentin, 1997, “The Ontological Interpretation of the Wave Function of the Universe”, The Monist, vol. 80, no.1, pp, 160-185.

specified interval, viz., t₁. The case is rather that there is no fact of the matter as to how many intervals of length 10^-35 second near the beginning of time have the property G and there is no fact of the matter as to precisely when intervals of this length cease to have this property and acquire the property of being definitely preceded.

This implies that for some very large finite number n, fewer than n intervals of 10-35 second have elapsed before the present one, but there is no finite number n " such that exactly n' intervals of length 10-35 second have elapsed before the present one.

Certain metrical implications follow about dating systems. There is no first year, and so there is no second year or ten-thousandth year or fifteenth-billionth year-and thus the conclusion may seem to follow that there is no present year. But what really follows is that all dating systems must have their origin point sufficiently far away from the beginning of time. For example, we date the present year 0, the previous year -1, the next earlier year -2, and so on. But as we approach the beginning of time, our dating system breaks down; for example, there will be no fact of the matter as to whether there is a year -15 billion (assuming the big bang occurred about 15 billion years ago).

If it is even possible that our quantum gravity theories correctly describe time, then most standard definitions of time's beginning are false. According to most definitions, time begins if and only if there is a first instant of time or there is a first and half-open interval of each actual length. But there is one standard definition that is consistent with a vague beginning, namely, that time begins if and only if for some interval t of length L, it is not the case that there are infinitely many intervals of length L prior to t.

5. The Mathematical Justification for a Vague Beginning

What justifies this thesis that there is a "join" region where time vaguely begins? In a response to an earlier presentation of this thesis, Storrs McCall argued that we can avoid logical contradictions about temporal relations between imaginary and real time without supposing that there is a "join region" where real time gradually emerges; intervals of imaginary time and real time can be immediately connected. McCall provides several models in which "space-time as a whole is the topologically connected union of two regions I [an imaginary time region] and R [a real time region]. A curve s can be drawn linking the two, with part of

P. 177, Smith, Quentin, 1997, “The Ontological Interpretation of the Wave Function of the Universe”, The Monist, vol. 80, no.1, pp, 160-185.

s in I and part in R. In every case, let I be topologically closed at the intersection between I and R, and let R be open at the intersection."[28]

I agree with McCall that there is no obvious philosophical reason for believing that the connection between imaginary and real time requires a metrically amorphous "join" region where the logic of vagueness needs to be applied to the beginning of real time. But such a region is required by the mathematical formulae of Hawking's "no boundary" model and Vilenkin's "tunnelling" model. The mathematical reasons for introducing a logic of vagueness, however, are not any of the familiar reasons offered in discussions of quantum mechanics. Specifically, the vagueness at the "join" region is not due to a superposition of probability amplitude functions (y= y_E +y _L) for the Euclidean and Lorentzian metrics in the mathematical description of this region; there is no such superposition. More generally, the vagueness is not due to any of the reasons offered in the relevant versions of the Copenhagen Interpretation bf quantum mechanics. For example, sometimes it is said that before a dynamical attribute F is measured, then for every admissible value v of F, sentences of the form "the value of F is v" are neither true nor false. Some thinkers may extend this line of reasoning and suggest that if spacetime is quantized and thereby becomes a dynamical attribute F, the same sort of ,vagueness applies to it. But this suggestion is off the mark. Hartle, Hawking, Vilenkin and other physicists who work on the wave function of the universe reject the Copenhagen Interpretation of quantum mechanics and use an Everett-type Interpretation of quantum mechanics. Hartle, Halliwell, Isham and others favor a "consistent histories" Everett- type Interpretation, where there are not "many universes" (the original Everett-DeWitt interpretation) or "many minds" (the Albert-Loewer interpretation) but instead many possible histories, only some of which are actual.[29] Consequently, if all the arguments for vagueness that are based on the Copenhagen Interpretation are unsound, that would not affect the argument for vagueness based on the Everett-type "no boundary" or "tunnelling" models.

Furthermore, this vagueness is not based on any simplistic argument for temporal vagueness involving Heisenberg's uncertainty relation between time and energy,

P. 178, Smith, Quentin, 1997, “The Ontological Interpretation of the Wave Function of the Universe”, The Monist, vol. 80, no.1, pp, 160-185.

which implies in part that the smaller the interval delta-E in which the values for energy are located, the larger the interval delta-t in which the values of time are located. Sometimes this is interpreted as meaning that (in a relevant situation) for any value v in the interval delta-t, any sentence of the form "the value of t is v" is neither true nor false. But the unsoundness of any such argument for temporal vagueness is consistent with the soundness of the quantum gravity argument for vagueness, if only for the reason that the "t" in the Heisenberg uncertainty relation does not denote an observable represented by an operator in Hilbert space, but is a parameter describing evolution (e.g., it is the external time parameter t in the time-dependent Schrodinger equation . There is no external time parameter t in the "no boundary" and "tunnelling" models, which are based on the gravitational Schrodinger equation (the Wheeler- DeWitt equation). There is no external time parameter in the Wheeler-De- Witt equation and time is defined internally in terms of an observable for spatial curvature or a matter field and is represented by a corresponding operator in Hilbert space.

Moreover, by means of the WKB semi-classical approximation used in these theories, "time" disappears into the "spacetime" of general relativity and the vagueness of time becomes a part of the vagueness of spacetime. The WKB semi-classical approximation is a reduction of the solutions of the gravitational Schrodinger equation to approximate solutions of equations in general relativity. We are no longer talking about the "time" of ordinary quantum mechanics, but about the "temporal dimension" of spacetime that results from quantizing general relativity. Talk of "the beginning of real time" being vague is, in effect, talk of the beginning of the temporal dimension of Lorentzian spacetime being vague.

The introduction of the logic of vagueness is needed because the WKB semi-classical approximation to the Wheeler-DeWitt equation becomes a more and more distant approximation to the Wheeler-DeWitt equation in the part of the mathematical space that corresponds to the "join" region. In quantum cosmology, the main mathematical formula that is given an ontological interpretation is the WKB semi-classical approximation. There are two WKB forms for writing the wave function, y= e^is (which are oscillatory) and y= e^-I (which are exponential). Regarding y= e^is, S is the classical Einstein action and is a rapidly varying function

P. 179, Smith, Quentin, 1997, “The Ontological Interpretation of the Wave Function of the Universe”, The Monist, vol. 80, no.1, pp, 160-185.

of a radius a and scalar field f. Regarding y = e^-I, I  is the Euclidean Einstein , action, obtained from the classical Einstein action by rotating t to ( -it) and adjusting the sign so it is positive. Wave functions of the form y = e^is correspond to Lorentzian ( +++- ) space-times and wave functions of the form y = e^-I correspond to Euclidean ( ++++ ) space-times. If we insert y = e^is in the Wheeler-DeWitt equation, S will obey to leading order the Harnilton-Jacobi equation

where U is the potential relevant to the radius a and scalar field f. This enables us to obtain Lorentzian solutions to Einstein's field equations. If we insert y= e^-I in the Wheeler-DeWitt equation, we will be able to obtain Euclidean solutions to Einstein's field equations.

The existence of a metrically amorphous "join" region is due to the fact that the WKB semi-classical approximation breaks down for the region in mathematical space where the solution to the oscillating component of the wave function of the universe is matched to the solution for the exponentially growing component by the WKB matching procedure. The "breakdown" is not due to the presence of ill-defined mathematical terms, but is instead due to the lack of ontological significance of certain well-defined mathematical terms. The WKB solution is a power solution and pertains to the first term in the power series expansion of the Wheeler-DeWitt equation. When ontologically interpreting the WKB solution, we do not go beyond the lowest order semi-classical approximation to the Wheeler-DeWitt equation; the higher-order terms have no ontological significance. The lowest order term in the power series expansion dominates the higher order terms in the Lorentzian and Euclidean regions of mathematical space, which enables the higher-order terms to be suppressed. This gives us a mathematical solution which has ontological significance. However, in the region of mathematical space where the exponentially growing solution is matched (by the WKB matching method) to the oscillating solution of the Wheeler-DeWitt equation, the first term in the power series expansion does not dominate the higher-order terms but is of the same order of magnitude as these higher-order terms. Since these higher-order terms do not provide a good approximation to the Wheeler-DeWitt equation, the WKB solution no

P. 180, Smith, Quentin, 1997, “The Ontological Interpretation of the Wave Function of the Universe”, The Monist, vol. 80, no.1, pp, 160-185.

longer is an approximate solution and fails to give us variables and relations that have ontological meaning. Since our Lorentzian and Euclidean metrics are present only when the WKB solution is an approximation to the Wheeler-DeWitt equation, we do not have a Lorentzian or Euclidean metric in the physical region that corresponds to this "matching" region in mathematical space. But we still have a physical manifold with a topology and a differentiable structure.

The vagueness has its mathematical basis in the gradual nature of this "breakdown." The breakdown of the WKB approximation does not mean there is no WKB solution; there is an analytic continuation of the e^-I solution to the eis solution. The gradual breakdown has a different nature. In Hawking's theory, the physical variable h^½ in the Wheeler-DeWitt equation, representing volume, is the internal time variable; this means that values of the volume variable do double duty as surrogate time values. The wave function enables us to predict correlations between this internal time variable and other variables, such as the density of the matter field f. For example, the probability amplitude y is large when y is big (the density is high) and h^½ small (the volume small). We expect a higher density of matter as the volume of the universe decreases. But where the oscillating and exponential solutions are matched, y  ceases to describe correlations between h^½ and other physical variables. Indeed, since the higher-order terms in the power solution expansion of the Wheeler-DeWitt equation approach the same order of magnitude as the first-order terms, the WKB solution ceases to give us values of h^½ that can be interpreted as volume measurements (and surrogate time values). However, this is not an abrupt transition. As we approach the match region in mathematical space, the values of the variables in e^is remain precise, but become less amenable to an interpretation as corresponding to a volume due to the gradual increase in the order of magnitude of the values in the higher-order terms.

Thus, the ontology of vagueness is not motivated by a superposition of Lorentzian and Euclidean metrics (there is no such superposition), or to variables ceasing to have definite values, but due to a gradual departure of the precise numerical values of h^½ from a range where they have onto- logical significance.[30]

This gradual breakdown of the Lorentzian metric at the "join" region is also a feature of Vilenkin's "tunnelling" model. Let us consider Vilenkin's 1988 model.[31] In this case, the WKB breakdown is due to the increase in order of magnitude of the higher-order terms of the scalar field f. To see

P. 160, Smith, Quentin, 1997, “The Ontological Interpretation of the Wave Function of the Universe”, The Monist, vol. 80, no.1, pp, 160-185.

how this happens in this model, let us write the Wheeler-DeWitt equation as follows:

If we evaluate this equation where the scalar field potential V(f) depends slowly on f, and set V'/V <<I, then as the radius a approaches zero in the very early universe, the coefficient of the second derivative with respect to the scalar field f increases in order of magnitude to a point where the WKB solution fails to be a good approximation. In this "tunnelling" model, a key to the WKB solution being a good approximation to the Wheeler-DeWitt equation is that we can neglect the second derivative with respect to the scalar field f in the Wheeler-DeWitt equation, but we can no longer do this when the coefficient of the second derivative to the scalar field increases in order of magnitude to the point where it can no longer be ignored. With this WKB breakdown, we have a "join" region somewhat analogous to the "join" region in the "no boundary" model.

Thus, we see that the vague beginning of time that is involved in the ontological interpretation of the wave function of the universe has a different sort of mathematical basis than the standard reasons for which vagueness is introduced in ordinary quantum mechanics. The vague beginning of time in a metrically amorphous region of the manifold, along with the Euclidean spacetime that occupies another part of this manifold, are the two most significant aspects of this ontological interpretation of the "no boundary" and "tunnelling" wave functions of the universe. This ontology differs from Callender's and Weingard's Bohmian ontology, which features an external and Newtonian-like time, and no Euclidean spacetime and no "join" region. The presence of the Newtonian-like time in the Callender-Weingard ontology, which involves a rejection of the general relativistic notion of time, is the most interesting aspect of their quantum gravity ontology. Which ontology is preferable will depend in part on whether a Bohmian or Everett-type interpretation of quantum mechanics is more justified. But it also depends on future developments in the quantization of gravity. Our ontology, like the Callender-Weingard ontology, is a proposal whose justification remains hostage to the ongoing search for a complete and consistent theory of quantum gravity.[32]

Quentin Smith

Western Michigan University