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Remarks on Flat-Footed Ontology Shellyfest! Oct. 9, 2007 Tim Maudlin
Whence “Flat-Footed”? • Time: July, 2007 • Place: Oxford, England • Participants: David Albert, T. M. • Albert: “Oh, I get it. I’ve been trying to be flat-footed about all of the ontology, and you only want to be flat-footed about part of the ontology.” • Maudlin: “Yeah, and I want it to be the right part.”
Comprehesibility Part 1:Mathematical Transparency • The objects used to represent physical reality in physics are mathematical objects, even though, contra Pythagoras, the reality is not mathematical. • So the question arises of exactly how a mathematical object can appropriately represent a physical structure. • In some cases, this representation relation is clear and comprehensible.
Example: Euclidean Space • There is, on the one hand, the physical space (or space-time) that we live in. It is possible (or was reasonably thought to be possible) that this space be Euclidean. • The Elements studies the structure of this space. It is not a numerical or algebraic object. It does have an intrinsic metrical structure, affine structure, etc. • Although there are ratios of magnitudes intrinsic to Euclidean space, the space does not consist of numbers:it consists of points.
Euclidean Space Con’t • We can, of course, use purely numerical objects to represent Euclidean space, and thereby allow geometrical questions to be translated into algebraic ones. • Thus, under obvious conventions, we can use R3 to represent Euclidean space. • But it is simply false and misleading to say that R3is Euclidean space. Euclidean space, for example, has no origin. And there is no fact about whether 2 points in Euclidean space “have the same x value”.
Transparency of R3 • Although Euclidean space is not R3, it is easily comprehensible why R3 is a good numerical object to use to represent Euclidean space, and also why there should be gauge transformations: there are different ways of using R3 to represent the same Euclidean space. The assignment of a particular triple of real numbers to a point in space is significantly a product of conventions. It is only relative to those conventions that two points “have the same x value”.
Comprehensibilty • The relation between Euclidean space, as a (possible) physical object, and R3 as a mathematical representation is an example of a sort of comprehensibility of a mathematical representation. Note that • 1) The relation is not isomorphism. R3 has more structure than Euclidean space has. • 2) This extra structure means that the relation between represented and representation is one-many: i.e. there are gauge freedoms. • 3) The gauge freedoms reflect arbitrary conventions (e.g. which point is the “origin”). We understand these freedoms, and so expect them.
Comprehensibility Is Not Visualizability • Although much of the early discussion of quantum theory (influenced by Kant) concerned whether the microscopic reality was anschaulich (visualizable), that is not the issue here. • If one postulates a space that is 11-dimensional, with all dimensions open, then is is equally obvious why R11 is a good representation, and what the gauge freedoms will be. Still, one can’t visualilze any of this.
Contrast With Brian Greene • Although it is transparent why there must be different co-ordinatizations of an 11-dimensional space, it is not obvious that dual theories are just different ways of representing the same physical situation, despite the fact that they may be provably empircally equivalent. Understanding a mathematical transformation as a change of gauge requires more than proving empirical equivalence.
Contrast with Tumulka et al. • Despite the fact that theories that yield the same possible distributions of Primitive Ontology are even more than empirically equivalent (physically equivalent), it is not obvious that any further mathematical differences in the theories must be like gauge transformations: just different ways of representing the same physical reality. This would require an argument.
Flat-footedness About the Wavefunction • David Albert, in “Elementary Quantum Metaphysics” proceeds in two steps. • The first is to assert that, in order to avoid the anti-realism of, e.g., Copenhagen, one must “learn to think of wave functions as physical objects in and of themselves”. • The second is to conclude that the way to do this is, as it were, to work backwards from the mathematical representation and postulate a physical object that the mathematics would transparently represent.
Verbatim • “And of course the space these sorts of objects live in, and (therefore) the space we live in, the space in which any realistic account of quantum mechanics is necessarily going to depict the world as playing itself out…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 flatly illusory.”
John Bell on • “No one can understand this theory [Bohm’s theory] until he is willing to think of as a real objective field rather than just a ‘probability amplitude’. Even though is propagates not in 3-space but in 3N-space.” (Q.M. for Cosmologists) • “The QRW type theories have nothing in their kinematics but wavefunction. It gives the density (in a multidimensional configuration space!) of stuff.” (Against Measurement)
What kind of stuff? • “The sorts of 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 states one specifies by specifying the values of some set of numbers at every point in the space where they live…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.” Albert
Amplitude and Phase • “The values of the amplitude and phase are thought of (as with all fields) as intrinsic properties of the points in the configuration space with which they are associated. And so (for example) the fact that the integral over the entirety of the configuration [space] of the square of the amplitude…is invariably equal to one is going to have to be thought of not as following analytically from the sorts of physical objects wave function are (which is certainly can not), but as a physical law, or perhaps an initial condition.”
Consequences of this Flat-Footedness • Note that in trying to specify the nature of the physical object by flat-footed back-formation from the mathematical representation of , Albert arrives at a non-standard account of gauge freedom: it is not the case that C and represent the same physical state. This would have come out differently if one took the mathematical representation to be a ray, but then the status of the “space”, and the idea of as a “field” would be obscure.
“Configuration Space” • Both Albert and Bell help themselves to the phrase “configuration space” in contexts where it is clear that one is not postulating the existence of a multiplicity of localized objects in a common space, i.e. in a situation where there are no configurations. So the connotations of “configuration space” must be ignored. In the flat-footed ontology there is simply a high-dimensional space, filled with “stuff” or a “field”. One arrives at this ontology via a flat-footed interpretation of the (mathematical) wavefunction.
Albert’s GRW and Albert’s Bohm • Having started with the wavefunction, Albert then specifies additional ontology to suit. In the case of GRW, as Bell suggests, there is no additional ontology: just a field evolving in a specified way in a very high-dimensional space. That is what the world is made of. In the case of Bohm, David adds only a single particle, the “world particle” or “marvelous point”, which moves in the space of the wavefunction.
Claims It Would be Hard to Approach Flat-footedly • Observable properties are, or are represented by, or correspond to, Hermetian matrices, or PVMs, or POVMs. • The fundamental ontology of this theory is matrices. • The history of the world is a series of projections. • An event is a set of “histories” (no single one of which actually occurs).
Comprehensibility Part 2:transparency of the lived world • Albert’s GRW and Albert’s Bohm present a mathematical apparatus and a physical ontology such that the relation between the two is transparent: it is obvious why these mathematical objects would be good representations of such a world. • But it is not transparent, in either case, how the physical ontology could be that of the world we live in. That is, it is not obvious how we could locate ourselves in such a world.
The World Particle • Consider, for example, Schrödinger’s cat. In any Bohmian theory, since the wavefuction does not collapse, the outcome of the experiment must be sought in the additional variables. • But in Albert’s Bohm, the only additional piece of ontology is a single particle, wiggling one way or another in a very, very high dimensional space. If this theory is right, one way of such wiggling is a universe like ours with a live cat in it, while another way of wiggling is a universe like ours with a dead cat.
Being Vs. Representation • It is trivial, of course, that a single mathematical point moving in a high-dimensional mathematical space can represent one or the other outcome: if there are many physical particles in a common low dimensional space moving around, then there is an evolving configuration of particles, and this can be represented (under obvious conventions) by a single point in a high-dimensional space. This abstract (non-physical) space is configuration space.
But… • The fact that is is trivial to represent an evolving configuration of many particles by a single point (using obvious conventions) does not imply that it is comprehensible how something we thought to be an evolving configuration of many particles (such as a cat) could really be just a single particle! Note that in this case there is no room for any conventions.
Configuration Space vs. High-Dimensional Physical Space • It is easy to miss this if one calls the high-dimensional physical space “configuration space”. • The individual points of a physical space are all intrinsically the same: a single point does not correspond to a complexly structured state of affairs. • In contrast, every point in an abstract configuration space does correspond to a complexly structured state of affairs, viz, a multiplicity of particles being configured in some way.
Simple example • 12 particles in a 1-D space, 6 in a 2-D space, 3 in a 4-D space etc. all are associated with a 12 dimensional configuration space. But these are different with respect to what they represent. If there are 4 particles in a 3-space, then some points, but not others, correspond to the particles being the vertices of a regular tetrahedron. If there are 6 particles in a 2-D space, no points correspond to this. So the points in a configuration space correspond to structured states of affairs.
Cheating • If one posits a high-dimensional physical space, and calls it a “configuration space”, one can be tricked into thinking that the individual points of the space, automatically and transparently, correspond to, or represent, or are complexly structured physical states of affairs. But they are not. So it is obscure how something happening at a point (such as a particle occupying a point or a field being concentrated near a point) could be a complexly structured physical state of affairs.
The Lived World • But the world we live in- the world whose job it is for physics to explain- appears to be filled with complexly structured physical objects that inhabit a common low-dimensional physical space. So although it is easy to understand the physical structures of Albert’s Bohmian world or Albert’s GRW world (worlds in which the fundamental physical space is high-dimensional), it is not easy to understand how those physical structures could constitute cats, or chairs, or people.
Albert’s Challenge • So by a particular choice of physical ontology, David buys complete transparency of the relation between the mathematical representation and the physical ontology, but at the expense of a transparent relation between the physical ontology and the lived world. He is left with the problem of understanding how, e.g., our world could just be a single particle moving in a high-dimensional space.
Bohm’s Bohm • The more usual understanding of Bohm’s theory is that it starts by postulating a multiplicity of particles in a common 3-dimensional space. If we suppose the space to be Euclidean then • 1) it is transparent how to represent the space and the configuration of the particles in the space. There will be a true abstract configuration space. (This will be different for identical vs. distinguishable particles.) • 2) It is transparent how to identify at least the gross aspects of the lived world. A live cat and a dead cat will be different configurations of particles.
The Price • What one loses, of course, is a transparent relation between the wavefunction as a physical item and the mathematical representation of the wavefunction. Since there is no physical high-dimensional space, one can’t understand the wavefunction as a physical field on such a space.
How Steep a Price? • Since it is not supposed that cats are made of wavefunction, but rather that cats are made of particles, the obscurity of the physical nature of the wavefunction does not threaten the transparency of the lived world. There is no price here. In a monistic theory, in which there is only wavefunction, there would be a price.
How Steep a Price? Con’t • The appropriateness of representing the wavefunction as a mathematical function on configuration space is easily understood even though configuration space is abstract. For our epistemic access to the wavefunction goes through the particles, it is only due to its influence on the particles that we are aware of it. So what we know of it is reflected in how it makes the configuration evolve. This is represented by a velocity field on configuration space, so it is not surprising that the wavfefunction might be represented by an object on configuration space too.
More Bell • “Absurdly, such theories are known as “hidden variables” theories. Absurdly, for there it is not in the wavefunction that one finds an image of the visible world, but in the complementary ‘hidden’(!) variables…In any case, the most hidden of all variables, in the pilot wave picture, is the wavefunction, which manifests itself to us only by its influence on the complementary variables.” (Are There Quantum Jumps?)
Advantages for a “hidden” wavefunction • If our only access to the wavefunction is via its effect on the particles, and if the connection to the lived world is primarily through the particles, then we are not constrained about the physical nature of the wavefunction. • In particular, although we no longer expect certain degrees of freedom in the mathematical representation to be gauge, we also can postulate without penalty that they are gauge. • We can, for example, say that different states of the wavefunction correspond 1-to-1 with rays in Hilbert space rather than vectors, so the choice of a normalized vector to represent the wavefunction is understood as a convention, rather than a reflection of a physicalfact about wavefunctions.
Pushed to the extreme • Even more extremely, since the lived world changes in time, the part of the physical ontology that makes connection with the lived world must change in time. But if that connection is not made via the wavefunction, then it is possible for the universal wavefunction to be static. This opens new possibilities for understanding what it is.
Problems for pure wavefunction ontology • Suppose, as in Albert’s GRW, the physical theory postulates only the wavefunction: no low-dimensional physical space, no localized objects, and a fortiori no configuration of anything. • The relation between the mathematical representation and the physical ontology can then be made transparent (flat-footed). • But the relation between the physical ontology and the lived world is far from transparent- this is so whether the wavefunction evolves linearly or non-linearly.
Bell’s GRW: flash ontology • One way of understanding Bell’s presentation of GRW is a dualistic ontology: wavefunction and point-events in a low dimensional spacetime. • The connection to the lived world can be made reasonably transparent. • The use of a “configuration space” to represent the wavefunction is somewhat curious: although there are flashes, there is no evolving configuration of flashes. At most times, there are no flashes at all.
Ghirardi’s GRW: mass density • Giancarlo Ghirardi has proposed supplementing the wavefunction with a continuous mass density. Such a mass density could transparently connect to the lived world- even more transparently than the flashes. • But now, the relevant configuration is the configuration of the mass distribution. So it is again curious that the wavefunction should be defined on an abstract space that looks life the configuration space of particles.
Morals • I take it to be a virtue that a physical theory have both sorts of transparency: from the mathematical representation to the physical ontology and from the physical ontology to the lived world. • I take Bohm’s Bohm to be an example of transparency of the (the gross features) of the lived world to (part of) the ontology, and from that part of the ontology to its mathematical representation.
Morals • The Bohmian ontology also has the wavefunction, and there is no transparent connection between the wavefuction and the lived world (the wavefunction is hidden). The relation between the mathematical representation of the wavefunction and its physical nature is also not transparent, but there are reasons to expect some features of the mathematical representation.
Grounds for Optimism • There is no logical guarantee that the true physical theory of the world display any kind of transparency at all. Perhaps the desire for either sort of comprehensibility pushes us in the wrong direction. • But it is notable that transparency is provably possible: Bohm’s theory provides a clear example.
Shelly’s Wisdom • As Shelly has often said: given the common claim that adding additional variables to the quantum state is impossible, it is remarkable that if one tries the simplest possible way to do it, it works. • Similarly, given the old Copenhagen claim that having a comprehensible account of microscopic reality is impossible, it is remarkable that the most obvious thing works, at least for non-relativistic QM. Perhaps there is no way to retain comprehensibility while extending the theory to cover gravity. • But we shouldn’t abandon hope without very good reason.