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Tim Maudlin NYU Physics & Philosophy of Time

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Physics is supposed to be an empirical theory, i.e. a theory that can be tested against empirical data. Fundamental physics is supposed to provide a complete physical ontology for the world. Therefore, either the data against which a fundamental physical theory is tested is not itself physical, or else it is somehow to be understood as contained in the ontology of the theory itself.

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Saying that a theory is tested against empirical data suggests that the data is somehow “empirical”. “Empirical” comes from the Greek “ ἐ μπειρία”, which is usually translated as “experience”. ἐ μπειρία One might then conclude that either experience is itself a physical thing, and hence part of the fundamental physical ontology, or else some extra principle is required to connect claims about the physical ontology to the experiential data.

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Both options here seem unpalatable. One requires giving a physical analysis of experience itself, and hence solving the mind-body problem. The other requires some sort of “bridge principle” from physical claims to experiential claims, and raises the question of both the metaphysical status and justification of the bridge principle.

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In the early 20 th century, the school of logical empiricists tried to dress up the old empiricist account of meaning found in Hume and Locke with the modern garb of predicate calculus. Many (but not all) followed the principle that all meaningful claims must either be analytic (and hence “empty”) or reducible to claims about experience, such as a particular sort of visual experience (“red- spot-here-for-me-now”).

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Since claims about conscious experience do not follow directly from claims about, e.g., the behavior of atoms, some logical empiricists who wanted to connect physical claims to data needed bridge principles.

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Consider the following characterization of a “theory” due to Ernst Nagel, in The Structure of Science: “For the purposes of analysis, it will be useful to distinguish three components in a theory: (1) an abstract calculus that is the logical skeleton of the explanatory system, and that ‘implicitly defines’ the basic notions of the system; (2) a set of rules that in effect assign an empirical content to the abstract calculus by relating it to the concrete materials of observation and experiment; and (3) an interpretation or model for the abstract calculus, which supplies some flesh for the skeletal structure in terms of more or less visualizable materials.” (p. 90)

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Nagel’s presentation avoids any direct discussion of the ontology of a theory. The “skeleton” of the theory is not an ontological postulate about what exists, but an “abstract calculus”, presumably a piece of the mathematics (otherwise, why “abstract”?). But such an “abstract calculus” cannot “implicitly define” any physical notion: it can only implicitly define other abstract, mathematical notions. So the “skeleton”, on this picture, contains no physics at all: it is just mathematics.

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In Nagel’s picture, the abstract calculus comes to have empirical content, and hence comes to be empirically testable, only via the second component, the “set of rules” that “assign empirical content to the abstract calculus”. These are sometimes called “bridge laws”, which are supposed to connect the “unobservable” or “theoretical” parts of the theory to empirical phenomena. With this sort of picture in place, it is easy to make connections to, e.g., textbook quantum theory.

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The standard presentation of the quantum- mechanical calculus in terms of probabilities for measurement outcomes Nagel’s schema, but it completely fails to provide any recognizable physical account of anything. There is no explanation of why any matrix at all, much less some particular one, should be associated with a certain laboratory procedure. There is no explanation of why another operator should be associated with another experimental condition. There is no explanation of how the electron interacts with the experimental apparatus to produce any particular outcome. There even fails to be any clear notion of how to apply the abstract calculus to anything except rather constrained experimental conditions.

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Let’s consider a completely different approach, one that does not fit Nagel’s schema well: Newtonian gravitational theory. Newton does not begin with any abstract calculus. He rather begins by describing the structure of Absolute Space, Time, and Motion and postulating two sorts of physical magnitude: the mass of an object and the notion of a force. With these in hand, Newton can specify his laws of motion, which we would render as F net = mA.

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With these postulates in place, Newton can now derive what the motions of bodies will be in certain circumstances. For example, having postulated the gravitational force law F g = GmM/r 2, Newton can derive the following claim about the motion of bodies: if two bodies of different mass are released together at the same time above the surface of the earth, with the only non-negligible force being that of gravity, then they should hit the ground at the same time.

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If one understands “empirical content” as having to do with the content of experience, then Newton’s theory has not yet been shown to have any empirical content at all. Nothing has been derived about anyone’s experience of anything: all that has been derived is how, according to Newton’s theory, bodies in certain physical circumstances will move. Indeed, it is obvious that no claims about anyone’s experience of anything follow from the whole of Newtonian mechanics, supplemented by whatever force laws one cares to postulate.

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Given appropriate initial conditions, specifying the masses, positions and velocities of the planets, together with the assumption that only gravity produces a non-negligible force, Newtonian theory predicts the orbits of the planets. Adding the laws of geometrical optics (how light moves through empty space), the theory predicts the exact timing of eclipses (given initial data). But again, none of this makes any direct claim about anyone’s experience of anything. So how is Newtonian mechanics an empirical theory?

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The answer to the last question is obvious. Galileo, for example, reported the result of some experiments he did in exactly the terms that Newton’s theory predicts: two objects of different mass, let fall off a tall tower at the same time, hit the ground (nearly) together. Or again, one can compare the predictions of Newtonian gravitation to the observed periods of planetary orbits, times of eclipses, etc. The language of observational data is the same as that found in the theory: it reports the (relative) motions and positions of bodies. Note: no “rules” connecting an abstract calculus to observation are needed. Rather, the theory postulates laws for items moving in space-time, and the predictions of the theory are checked against data reporting the same.

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A complete physical theory (as opposed to an effective practical tool for making predictions) must contain within it a physical account of the facts that are taken to constitute the data against which the theory is tested. One need not treat an effective device for calculating predictions as a serious proposal for a physical theory in this sense. Indeed, an effective tool for making predictions may not be of the form of a physical theory with a clear ontology.

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One approach to forging a connection between the mathematical quantum formalism of wavefunctions, operators, etc., and data was pioneered by Bohr, and is the foundation of one understanding of the Copenhagen interpretation. On the data side, Bohr insisted that the data of the theory, typically laboratory operations, be described using “classical language”.

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“For this purpose, it is decisive to recognize that, however far the phenomena transcend the scope of classical physical explanation, the account of all evidence must be expressed in classical terms. The argument is simply that by the word “experiment” we refer to a situation where we can tell others what we have done and what we have learned, and that, therefore, the account of the experimental arrangement and of the results of the observations must be expressed in unambiguous language with suitable application of the terminology of classical physics.”

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“As stressed in the [1927 Como] lecture, an adequate tool for a complementary way of description is offered precisely by the quantum- mechanical formalism which represents a purely symbolic scheme permitting only predictions, on lines of the correspondence principle, as to results obtainable under conditions specified by means of classical concepts. It must be remembered that even in the indeterminacy relation [uncertainty relation] we are dealing with an implication of the formalism which defies unambiguous expression in words suited to describe classical physical pictures.”

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“Thus, a sentence like ‘we cannot know both the momentum and the position of an atomic object’ raises at once questions as to the physical reality of two such attributes of the object, which can be answered only by referring to the conditions for the unambiguous use of space-time concepts, on the one hand, and dynamical conservation laws, on the other. While the combination of these concepts into a single picture of a causal chain of events is the essence of classical mechanics, room for regularities beyond the grasp of such a descriptions is afforded by the circumstance that the study of the complementary phenomena demands mutually exclusive experimental arrangements.”

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Bohr’s insistence that classical concepts such as space-time trajectory or energy-and- causation can only be applied to quantum systems in certain classically described circumstances implies that the classically described circumstances themselves can’t be the consequences of a quantum-mechanical description. It is this that yields the “measurement problem” in its most virulent form.

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To work our way out of Bohr’s trap we need for the physics to attribute some features to systems that obtain independently of their environment. These features constitute the “experimental arrangements” that in turn may play a role in defining some of the properties of the “quantum system”.

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It is important to distinguish the two senses of “classical” that Bohr uses. One sense is tied to classical dynamics, and describes a system that has both “space-time” and a “energy- momentum” properties that obey classical equations. Bohr insists that “quantum mechanical” systems are not “classical” in this sense. The other sense is that in which the experimental conditions must be described in “classical” terms. This does not and cannot require that the experimental apparatus obeys classical dynamics.

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Bohr describes the “classical terms” in which the phenomena (data) are reported as communicable in unambiguous language. This, of course, does not demand that the phenomena are even approximately “classical” in terms of dynamics. It is also hard to see what the issue of “communicability” is: is there a fear that the phenomena cannot even be described in natural language?

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Obviously, the language in which the data is described must be comprehensible natural language. That is a very weak constraint. The more difficult and important requirement on the language of the data is that it be unambiguously connected to the description given by the theory. It is only in this way that the data can be derived from the theory in a principled way.

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The most obvious way that theory can connect to data in a principled way has been clearly articulated by John Bell. Bell notes that any complete physical theory must postulate some beables, i.e. physical elements that are simply there. “In particular we will exclude the notion of ‘observable’ in favor of that of ‘beable’. The beables of the theory are those elements which might correspond to elements of reality, to things which exist. Their existence does not depend on ‘observation’. Indeed observation and observers must be made out of beables.”

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Beables are needed to end Bohr’s regress. If some “properties” that a quantum system has depend on the environmental conditions, then at least some environmental conditions must hold of their own accord. These will be beables in Bell’s sense. And if the environment is describable in the language of quantum theory, then there must be some beables in quantum systems.

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Among the beables, Bell further distinguishes the local beables. “These are the mathematical counterparts in the theory to real events at definite places and times in the real world (as distinct from the many purely mathematical constructions that occur in the working out of physical theories, as distinct from things that may be real but not localized, and as distinct from the ‘observables’ of other formulations of quantum mechanics, for which we have no use here.”

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“The concept of ‘observable’ lends itself to very precise mathematics when identified with ‘self-adjoint operator’. But physically, it is a rather wooly concept. It is not easy to identify precisely which physical processes are to be given the status of ‘observations’ and which are to be relegated to the limbo between one observation and another. So it could be hoped that some increase in precision might be possible by concentration on the beables, which can be described in ‘classical terms’, because they are there. The beables must include the settings of switches and knobs on experimental equipment, the currents in coils, and the readings of instruments. ‘Observables’ must be made, somehow, out of beables. The theory of local beables should contain, and give precise physical meaning to, the algebra of local observables.”

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Note Bell’s use of “classical” in the last quote. It has nothing to do with classical dynamics. Rather, the “classical” terms allow one to describe the experimental conditions at macroscopic scale in terms of the disposition of objects in space and time.

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Bell’s point is that the data and description of experimental conditions is always given in terms of local beables at macroscopic scale (“the pointer moved to the right”). If this is to be connected in a principled way to the language of the theory, the theory must have some local beables from which the macroscopic objects are composed.

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Bell further notes that there is a simple, transparent, comprehensible way for macroscopic objects to have locations in space-time at macroscopic scale: postulate that they are composed of local beables at microscopic scale. The bizarre position defended by Bohr seems to be that there must be macroscopic localized objects in a familiar space-time but they can’t be composed of microscopic localized parts described by quantum theory.

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Bell explicates Bohr’s theory in terms of a dual ontology: macroscopic “classical” objects that define the experimental condition and a “quantum world” the is associated only with a symbolic wavefunction. The significance of the wavefunction is exhausted by its role in calculating predictions for the behavior of the classical objects. The “quantum system” never can be described by both a space-time and a “causal” vocabulary, but each individually may be appropriate given experimental conditions.

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“The kinematics of the world, in this orthodox picture, is given by a wavefunction (maybe more than one?) for the quantum part, and classical variables—variables which have values—for the classical part: (Ψ(t,q…),X(t)…). The Xs are somehow macroscopic. This is not spelled out very explicitly. The dynamics is not very precisely formulated either. It includes a Schrödinger equation for the quantum part, and some sort of classical dynamics for the classical part, and ‘collapse’ recipes for their interaction.”

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“It seems to me that the only hope of precision with the dual (Ψ,x) kinematics is to omit completely the shifty split [between classical and quantum], and let both Ψ and x refer to the world as a whole. Then the xs must not be confined to some vague macroscopic scale, but must extend to all scales. In the picture of de Broglie and Bohm, every particle is attributed a position x(t). Then instrument pointers—assemblies of particles—have positions, and experiments have results.”

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“The dynamics is given by the world Schrödinger equation plus precise ‘guiding’ equations prescribing how the x(t)s move under the influence of Ψ. Particles are not attributed angular momenta, energies, etc., but only positions as functions of time. Peculiar ‘measurement’ results for angular momenta, energies, and so on, emerge as pointer positions in appropriate experimental setups.”

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If all of this works, then one gets from the description of the physical world offered by the theory to the description of the data provided by experiment without any mention of observers, observables, systems, apparatuses, measurements, or anything like that. The work is done by local beables that are postulated to constitute the macroscopic objects, the dynamics (including other beables) provided by the physical theory, and coarse-graining.

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The methodological role of the local beables in this picture can be fulfilled by many different sorts of things at microscopic scale, including: Particles Continuous matter density Flashes Fermion Number Density Strings

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Bell’s local beables are local in space-time, but he does not specify exactly what sort of space- time structure he has in mind. We can get a clear constraint on the space-time structure by reflecting on the methodology: if we want to derive the language of the data from a description in the language of the theory by coarse-graining, then the theoretical description of the local beables and the space-time should, in the right circumstances, coarse-grain to a familiar, everyday description at macroscopic scale.

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1: Newtonian Absolute Space and Time 2: Neo-Newtonian Space-Time 3: Newton-Cartan Space-Time 4: Minkowski Space-Time 5: Globally Hyperbolic Solution to the Einstein Field Equations on a 4-D Manifold

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6: Solutions to the Field Equations with Closed Timelike Curves, or other non- globally hyperbolic solutions 7: 10- or 11-Dimensional Space-Time with 6 or 7 Compactified Dimensions Forming a Calabi-Yau Space

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8: The Holographic Principle, together with the claim that space-time is “really” just the “boundary space” and that the “mass” is an illusion (parallel to how an actual hologram is really only 2-dimensional but nonetheless gives the appearance of being 3-dimensional)

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9: The fundamental physical space (for non- relativistic quantum mechanics) is geometrically isomorphic to a classical configuration space 10: “We live in Hilbert space” (Sean Carroll)

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The last two suggestions are qualitatively completely unlike the first 7. (Let’s leave aside the Holographic Principle.) According to them, the fundamental “space” is extremely high-dimensional, with no four dimensions picked out at macroscopic scale. In the first 7, the macroscopic structure of space-time is four-dimensional. So the weird suggestions cannot be used for our purposes. Why should weird structures be suggested?

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The inspiration of the last two suggestions is evidently the quantum state. In non- relativistic quantum mechanics, the wavefunction, which represents the quantum state, is a complex function on a classical configuration space. And since the wavefunction must be square-integrable, it lives in (i.e. is an element of) a high- dimensional Hilbert space.

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These last two, radical, positions begin with the wavefunction (the mathematical representation of the quantum state) and then retrofit an account of the fundamental structure of physical space or space-time that is inspired by the wavefunction. I want to argue that this strategy, which tries to derive a physical ontology by focusing on the mathematical structure of the wavefunction, is a bad hangover from the historical avoidance of the “measurement problem”.

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The most straightforward example of this sort of inference is in “Elementary Quantum Metaphysics”: “..it has been essential (that is) to the project of quantum-mechanical realism…to learn to think of wave functions as physical objects in and of themselves.”

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“And of course the space those sorts of objects live in, the space which any realistic understanding of quantum mechanics is necessarily going to depict the history of the world as playing itself out (if space is the right name for it—of which more later) is configuration-space. And whatever impression we have to the contrary (whatever impression we have, say, of living in a three- dimensional space, or a four-dimensional space-time) is somehow flatly illusory.”

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“The sorts of physical objects that wave functions are, on this way of thinking, are (plainly) fields— which is to say that they are the sorts of objects whose state one specifies by specifying the values of some set of numbers at every point in the space where they live, the sorts of objects whose states one specifies (in this case) by specifying the values of two numbers (one of which is usually referred to as an amplitude, and the other as a phase) at every point in the universe’s so-called configuration space”.

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If we accept Bell’s approach, which completely solves the ‘measurement problem’, we arrive at an entirely different understanding of the quantum state (the thing represented by the wavefunction) and its role in the theory, and therefore an entirely different understanding from Albert’s of the significance of the mathematical form of the wavefunction.

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If the data of the theory are a function of the distribution of local beables (via simple coarse-graining to get macroscopic variables), then the quantum state has empirical consequences only insofar as it helps determine the behavior of the local beables. That is, the quantum state manifests itself in the data exactly to the extent that it influences the configuration of local beables.

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The same conclusion would hold, according to this analysis, for any non-local beable, because the non-local beables are not directly given in the data. The mathematical form of the representative of the quantum state must therefore be of a kind to generate (by some equation) a specification of how configurations of local beables (will or might) change. An obvious mathematical object that can play this role is a function on configuration space.

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It is essential, in this picture, that the mathematical space represent the configuration of local beables in a space-time of the form 1-7, at least in the present state of the universe. (The space-time, and hence configuration space, could be quite different at different times and places in the history of the universe.) Note that Albert’s naïve interpretation of the wavefunction as indicating the fundamentality of “configuration space” has entirely evaporated.

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If we follow Bell’s suggestions, a clear, complete physical theory that makes transparent connections to data will have three main components: 1) a space-time structure that (now) coarse- grains to a familiar 4-dimensional form 2) a set of postulated physical entities, or beables, divided into two types: local and non-local 3) a dynamics for the beables and (ultimately) the space-time structure

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The beables divide into local and non-local because, on the one hand, familiar macroscopic objects, as reported in the data, should be made of microscopic local beables and, on the other hand, we want to implement the non-locality without superluminal local entities (tachyons).

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The division of beables suggests a parallel division of dynamics: one for the non-local beable (quantum state) and another for the local beables (ultimately including space-time structure). In Bohm’s theory, this yields, e.g., Schrödinger’s equation for the quantum state and the guidance equation for the local beables. In a GRW-type theory, it yields a GRW dynamics for the quantum state and a second equation specifying the distribution of local beables in terms of the quantum state.

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GRW-type theories therefore have only one fundamental dynamical law: the dynamics of the local beables piggy-backs on that of the quantum state via the relation between quantum state and local beables. Bohm-type theories have the potential of having only one fundamental dynamical law if the universal quantum state turns out to be static. The observable changes that constitute the data are then generated by the guidance equation.

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I have not argued that it is impossible to create a clear, complete physical theory with a transparent connection between physics and data in some other way. But Bell’s basic approach leave a wide scope for variation in terms of local beables, non- local beables, space-time structure, and dynamical law. Both Bohmian mechanics and GRW fit easily in the scheme. Any other sort of approach ought to be able to explain its basic architecture as precisely.

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