Sunday, June 28, 2015

Thinking outside the quantum box

Doctor:  Don't tell me you're lost too.
Shardovan:  No, but as you guessed, Doctor, we people of Castrovalva are too much part of this thing you call the occlusion.
Doctor:  But you do see it, the spatial anomaly.
Shardovan:  With my eyes, no — but, in my philosophy.
— Doctor Who, Castrovalva.

I've made no particular secret, on this blog, that I'm looking (in an adventuresome sort of way) for alternatives to quantum theory.  So far, though, I've mostly gone about it rather indirectly, fishing around the edges of the theory for possible angles of attack without ever engaging the theory on its home turf.  In this post I'm going to shave things just a bit closer — fishing still, but doing so within line-of-sight of the NO FISHING sign.  I'm also going to explain why I'm being so indirect, which bears on what sort of fish I think most likely here.

To remind, in previous posts I've mentioned two reasons for looking for an alternative to quantum theory.  Both reasons are indirect, considering quantum theory in the larger context of other theories of physics.  First, I reasoned that when a succession of theories are getting successively more complicated, this suggests some wrong assumption may be shared by all of them (here).  Later I observed that quantum physics and relativity are philosophically disparate from each other (here), a disparity that has been an important motivator for TOE (Theory of Everything) physicists for decades.

The earlier post looked at a few very minor bits of math, just enough to derive Bell's Inequality, but my goal was only to point out that a certain broad strategy could, in a sense, sidestep the nondeterminism and nonlocality of quantum theory.  I made no pretense of assembling a full-blown replacement for standard quantum theory based on the strategy (though some researchers are attempting to do so, I believe, under the banner of the transactional interpretation).  In the later post I was even less concrete, with no equations at all.

The quantum meme
How to fish
Why to fish
Hygiene again
The structure of quantum math
The structure of reality
The quantum meme

Why fish for alternatives away from the heart of the quantum math?  Aside, that is, from the fact that any answers to be found in the heart of the math already have, presumably, plenty of eyeballs looking there for them.  If the answer is to be found there after all, there's no lasting harm to the field in someone looking elsewhere; indeed, those who looked elsewhere can cheerfully write off their investment knowing they played their part in covering the bases — if it was at least reasonable to cover those bases.  But going into that investigation, one wants to choose an elsewhere that's a plausible place to look.

Supposing quantum theory can be successfully challenged, I suggest it's quite plausible the successful challenge might not be found by direct assault (even though eventual confrontation would presumably occur, if it were really successful).  Consider Thomas Kuhn's account of how science progresses.  In normal science, researchers work within a paradigm, focusing their energies on problems within the paradigm's framework and thereby making, hopefully, rapid progress on those problems because they're not distracting themselves with broader questions.  Eventually, he says, this focused investigation within the paradigm highlights shortcomings of the paradigm so they become impossible to ignore, researchers have a crisis of confidence in the paradigm, and after a period of distress to those within the field, a new paradigm emerges, through the process he calls a scientific revolution.  I've advocated a biological interpretation of this, in which sciences are a variety of memetic organisms, and scientific revolution is the organisms' reproductive process.  But if this is so, then scientific paradigms are being selected by Darwinian evolution.  What are they being selected for?

Well, the success of science hinges on paradigms being selected for how effectively they allow us to understand reality.  Science is a force to be reckoned with because our paradigms have evolved to be very good at helping us understand reality.  That's why the scientific species has evolved mechanisms that promote empirical testing:  in the long run, if you promote empirical testing and pass that trait on to your descendants, your descendants will be more effective, and therefore thrive.  So far so good.

In theory, one could imagine that eventually a paradigm would come along so consistent with physical reality, and with such explanatory power, that it would never break down and need replacing.  In theory.  However, there's another scenario where a paradigm could get very difficult to break down.  Suppose a paradigm offers the only available way to reason about a class of situations; and within that class of situations are some "chinks in the armor", that is, some considerations whose study could lead to a breakdown of the paradigm; but the only way to apply the paradigm is to frame things in a way that prevents the practitioner from thinking of the chinks-in-the-armor.  The paradigm would thus protect itself from empirical attack, not by being more explanatory, but by selectively preventing empirical questions from being asked.

What characteristics might we expect such a paradigm to have, and would they be heritable?  Advanced math that appears unavoidable would seem a likely part of such a complex.  If learning the subject requires indoctrination in the advanced math, then whatever that math is doing to limit your thinking will be reliably done to everyone in the field; and if any replacement paradigm can only be developed by someone who's undergone the indoctrination, that will favor passing on the trait to descendant paradigms.  General relativity and quantum theory both seem to exhibit some degree of this characteristic.  But while advanced math may be an enabler, it might not be enough in itself.  A more directly effective measure, likely to be enabled by a suitable base of advanced math, might be to make it explicitly impossible to ask any question without first framing the question in the form prescribed by the paradigm — as quantum theory does.

This suggests to me that the mathematical details of quantum theory may be a sort of tarpit, that pulls you in and prevents you from leaving.  I'm therefore trying to look at things from lots of different perspectives in the general area without ever getting quite so close as to be pulled in.  Eventually I'll have to move further and further in; but the more outside ideas I've tied lines to before then, the better I'll be able to pull myself out again.

How to fish

What I'm hoping to get out of this fishing expedition is new ideas, new ways of thinking about the problem.  That's ideas, plural.  It's not likely the first new idea one comes up with will be the key to unlocking all the mysteries of the universe.  It's not even likely that just one new idea would ever do it.  One might need a lot of new ideas, many of which wouldn't actually be part of a solution — but the whole collection of them, including all the ones not finally used, helps to get a sense of the overall landscape of possibilities, which may help in turning up yet more new ideas inspired from earlier ones, and indeed may make it easier to recognize when one actually does strike on some combination of ideas that produce a useful theory.

Hence my remark, in an aside in an earlier post, that I'm okay with absurd as long as it's different and shakes up my thinking.

Case in point.  In the early 1500s, there was this highly arrogant and abrasive iconoclastic fellow who styled himself Philippus Aureolus Theophrastus Bombastus von Hohenheim; ostensibly our word "bombastic" comes from his name.  He rejected the prevailing medical paradigm of his day, which was based on ancient texts, and asserted his superiority to the then-highly-respected ancient physician Celsus by calling himself "Paracelsus", which is the name you've probably heard of him under.  He also shook up alchemical theory; but I mention him here for his medical ideas.  Having rejected the prevailing paradigm, he was rather in the market for alternatives.  He advocated observing nature, an idea that really began to take off after he shook things up.  He advocated keeping wounds clean instead of applying cow dung to them, which seems a good idea.  He proposed that disease is caused by some external agent getting into the body, rather than by an imbalance of humours, which sounds rather clever of him.  But I'm particularly interested that he also, grasping for alternatives to the prevailing paradigm, borrowed from folk medicine the principle of like affects like.  Admittedly, you couldn't do much worse than some of the prevailing practices of the day.  But I'm fascinated by his latching on to like-effects-like, because it demonstrates how bits of replicative material may be pulled in from almost anywhere when trying to form a new paradigm.  Having seen that, it figured later into my ideas on memetic organisms.

It also, along the way, flags out the existence of a really radically different way of picturing the structure of reality.  Like-affects-like is a wildly different way of thinking, and therefore ought to be a great limbering-up exercise.

In fact, like-affects-like is, I gather, the principle underlying the anthropological phenomenon of magic — sympathetic magic, it's called.  I somewhat recall an anthropologist expounding at length (alas, I wish I could remember where) that anthropologically this can be understood as the principle underlying all magic.  So I got to thinking, what sort of mathematical framework might one use for this sort of thing?  I haven't resolved a specific answer for the math framework, yet; but I've tried to at least set my thoughts in order.

What I'm interested in here is the mathematical and thus scientific utility of the like-affects-like principle, not its manifestation in the anthropological phenomenon of magic (as Richard Cavendish observed, "The religious impulse is to worship, the scientific to explain, the magical to dominate and command").  Yet the term "like affects like" is both awkward and vague; so I use the term sympathy for discussing it from a mathematical or scientific perspective.

How might a rigorous model of this work, structurally?  Taking a stab at it, one might have objects, each capable of taking on characteristics with a potentially complex structure, and patterns which can arise in the characteristics of the objects.  Interactions between the objects occur when the objects share a pattern.  The characteristics of objects might be dispensed with entirely, retaining only the patterns, provided one specifies the structure of the range of possible patterns (perhaps a lattice of patterns?).  There may be a notion of degrees of similarity of patterns, giving rise to varying degrees of interaction.  This raises the question of whether one ought to treat similar patterns as sharing some sort of higher-level pattern and themselves interacting sympathetically.  More radically, one might ask whether an object is merely an intersection of patterns, in which case one might aspire to — in some sense — dispense with the objects entirely, and have only a sort of web of patterns.  Evidently, the whole thing hinges on figuring out what patterns are and how they relate to each other, then setting up interactions on that basis.

I distinguish between three types of sympathy:

  • Pseudo-sympathy (type 0).  The phenomenon can be understood without recourse to the sympathetic principle, but it may be convenient to use sympathy as a way of modeling it.

  • Weak sympathy (type 1).  The phenomenon may in theory arise from a non-sympathetic reality, but in practice there's no way to understand it without recourse to sympathy.

  • Strong sympathy (type 2).  The phenomenon cannot, even in theory, arise from a non-sympathetic reality.

All of which gives, at least, a lower bound on how far outside the box one might think.  One doesn't have to apply the sympathetic principle in a theory, in order to benefit from the reminder to keep one's thinking limber.

(It is, btw, entirely possible to imagine a metric space of patterns, in which degree of similarity between patterns becomes distance between patterns, and one slides back into a geometrical model after all.  To whatever extent the merit of the sympathetic model is in its different way of thinking, to that extent one ought to avoid setting up a metric space of patterns, as such.)

Why to fish

Asking questions is, broadly speaking, good.  A line comes to mind from James Gleick's biography of Feynman (quoted favorably by Freeman Dyson):  "He believed in the primacy of doubt, not as a blemish upon our ability to know but as the essence of knowing."  Nevertheless, one does have to pick and choose which questions are worth spending most effort on; as mentioned above, the narrow focus of normal scientific research enables its often-rapid progress.  I've been grounding my questions about quantum mechanics in observations about the character of the theory in relation to other theories of physics.

By contrast, one could choose to ground one's questions in reasoning about what sort of features reality can plausibly have.  Einstein did this when maintaining that the quantum theory was an incomplete theory of the physical world — that it was missing some piece of reality.  An example he cited is the Schrödinger's cat thought-experiment:  Until observed, a quantum system can exist in a superposition of states.  So, set up an experiment in which a quantum event is magnified into a macroscopic event — through a detector, the outcome of the quantum event causes a device to either kill or not kill a cat.  Put the whole experimental apparatus, including the cat, in a box and close it so the outcome cannot be observed.  Until you open the box, the cat is in a superposition of states, both alive and dead.  Einstein reasoned that since the quantum theory alone would lead to this conclusion, there must be something more to reality that would disallow this superposition of cat.

The trouble with using this sort of reasoning to justify a line of research is, all it takes to undermine the justification is to say there's no reason reality can't be that strange.

Hence my preference for motivations based on the character of the theory, rather than the plausibility of the reality it depicts.  My reasoning is still subjective — which is fine, since I'm motivating asking a question, not accepting an answer — but at least the reasoning is then not based on intuition about the nature of reality.  Intuition specifically about physical reality could be right, of course, but has gotten a bad reputation — as part of the necessary process by which the quantum paradigm has secured its ecological niche — so it's better in this case to base intuition on some other criterion.

Hygiene again

To make sure I'm fully girded for battle — this is rough stuff, one can't be too well armed for it — I want to revisit some ideas I collected in earlier blog posts, and squeeze just a bit more out of them than I did before.

My previous thought relating explicitly to Theories of Everything was that, drawing an analogy with vau-calculi, spacetime geometry should perhaps be viewed not as a playing field on which all action occurs, but rather as a hygiene condition on the interactions that make up the universe.  This analogy can be refined further.  The role of variables in vau-calculi is coordinating causal connections between distant parts of the term.  There are four kinds of variables, but unboundedly many actual variables of each kind; and α-renaming keeps these actual variables from bleeding into each other.  A particular variable, though we may think of it as a very simple thing — a syntactic atom, in fact — is perhaps better understood as a distributed, complex-structured entity woven throughout the fabric of a branch of the term's syntax tree, humming with the dynamically maintained hygiene condition that keeps it separate from other variables.  It may impinge on a large part of the α-renaming infrastructure, but most of its complex distributed structure is separate from the hygiene condition.  The information content of the term is largely made up of these complex, distributed entities, with various local syntactic details decorating the syntax tree and regulating the rewriting actions that shape the evolution of the term.  Various rewriting actions cause propagation across one (or perhaps more than one) of these distributed entities — and it doesn't actually matter how many rewriting steps are involved in this propagation, as for example even the substitution operations could be handled by gradually distributing information across a branch of the syntax tree via some sort of "sinking" structure, mirror to the binding structures that "rise" through the tree.

Projecting some of this, cautiously, through the analogy to physics, we find ourselves envisioning a structure of reality in which spacetime is a hygiene condition on interwoven, sprawling complex entities that impinge on spacetime but are not "inside" it; whose distinctness from each other is maintained by the hygiene condition; and whose evolution we expect to describe by actions in a dimension orthogonal to spacetime.  The last part of which is interestingly suggestive of my other previous post on physics, where I noted, with mathematical details sufficient to make the point, that while quantum physics is evidently nondeterministic and nonlocal as judged relative to the time dimension, one can recover determinism and locality relative to an orthogonal dimension of "meta-time" across which spacetime evolves.

One might well ask why this hygiene condition in physics should take the form of a spacetime geometry that, at least at an intermediate scale, approximates a Euclidean geometry of three space and one time dimension.  I have a thought on this, drawing from another of my irons in the fire; enough, perhaps, to move thinking forward on the question.  This 3+1 dimension structure is apparently that of quaternions.  And quaternions are, so at least I suspect (I've been working on a blog post exploring this point), the essence of rotation.  So perhaps we should think of our hygiene condition as some sort of rotational constraint, and the structure of spacetime follows from that.

I also touched on Theories of Everything in a recent post while exploring the notion that nature is neither discrete nor continuous but something between (here).  If there is a balance going on between discrete and continuous facets of physical worldview, apparently the introduction of discrete elementary particles is not, in itself, enough discreteness to counterbalance the continuous feature provided by the wave functions of these particles, and the additional feature of wave-function collapse or the like is needed to even things out.  One might ask whether the additional discreteness associated with wave-function collapse could be obviated by backing off somewhat on the continuous side.  The uncertainty principle already suggests that the classical view of particles in continuous spacetime — which underlies the continuous wave function (more about that below) — is an over-specification; the need for additional balancing discreteness might be another consequence of the same over-specification.

Interestingly, variables in λ-like calculi are also over-specified:  that's why there's a need for α-renaming in the first place, because the particular name chosen for a variable is arbitrary as long as it maintains its identity relative to other variables in the term.  And α-renaming is the hygiene device analogized to geometry in physics.  Raising the prospect that to eliminate this over-specification might also eliminate the analogy, or make it much harder to pin down.  There is, of course, Curry's combinatorial calculus which has no variables at all; personally I find Church's variable-using approach easier to read.  Tracing that through the analogy, one might conjecture the possibility of constructing a Theory of Everything that didn't need the awkward additional discreteness, by eliminating the distributed entities whose separateness from each other is maintained by the geometrical hygiene condition, thus eliminating the geometry itself in the process.  Following the analogy, one would expect this alternative description of physical reality to be harder to understand than conventional physics.  Frankly I have no trouble believing that a physics without geometry would be harder to understand.

The idea that quantum theory as a model of reality might suffer from having had too much put into it, does offer a curious counterpoint to Einstein's suggestion that quantum theory is missing some essential piece of reality.

The structure of quantum math

The structure of the math of quantum theory is actually pretty simple... if you stand back far enough.  Start with a physical system.  This is a small piece of reality that we are choosing to study.  Classically, it's a finite set of elementary things described by a set of parameters.  Hamilton (yes, that's the same guy who discovered quaternions) proposed to describe the whole behavior of such a system by a single function, since called a Hamiltonian function, which acts on the parameters describing the instantaneous state of the system together with parameters describing the abstract momentum of each state parameter (essentially, how the parameters change with respect to time).  So the Hamiltonian is basically an embodiment of the whole classical dynamics of the system, treated as a lump rather than being broken into separate descriptions of the individual parts of the system.  Since quantum theory doesn't "do" separate parts, instead expecting everything to affect everything else, it figures the Hamiltonian approach would be particularly compatible with the quantum worldview.  Nevertheless, in the classical case it's still possible to consider the parts separately.  For a system with a bunch of parts, the number of parameters to the Hamiltonian will be quite large (typically, at least six times the number of parts — three coordinates for position and three for momentum of each part).

Now, the quantum state of the system is described by a vector over a complex Hilbert space of, typically, infinite dimension.  Wait, what?  Yes, that's an infinite number of complex numbers.  In fact, it might be an uncountably infinite number of complex numbers.  Before you completely freak out over this, it's only fair to point out that if you have a real-valued field over three-dimensional space, that's an uncountably infinite number of real numbers (the number of locations in three-space being uncountably infinite).  Still, the very fact that you're putting this thing in a Hilbert space, which is to say you're not asking for any particular kind of simple structure relating the different quantities, such as a three-dimensional Euclidean continuum, is kind of alarming.  Rather than a smooth geometric structure, this is a deliberately disorganized mess, and honestly I don't think it's unfair to wish there were some more coherent reality "underneath" that gives rise to this infinite structure.  Indeed, one might suspect this is a major motive for wanting a hidden variable theory — not wishing for determinism, or wishing for locality, but just wishing for a simpler model of what's going on.  David Bohm's hidden variable theory, although it did show one could recover determinism with actual classical particles "underneath", did so without simplifying the mathematics — the mathematical structure of the quantum state was still there, just given a makeover as a potential field.  In my earlier account of this bit of history, I noted that Einstein, seeing Bohm's theory, remarked, "This is not at all what I had in mind."  I implied that Einstein didn't like Bohm's theory because it was nonlocal; but one might also object that Bohm's theory doesn't offer a simpler underlying reality, rather a more complicated one.

The elements of the vector over Hilbert space are observable classical states of the system; so this vector is indexed by, essentially, the sets of possible inputs to the Hamiltonian.  One can see how, step by step, we've ended up with a staggering level of complexity in our description, which we cope with by (ironically) not looking at it.  By which I mean, we represent this vast amorphous expanse of information by a single letter (such as ψ), to be manipulated as if it were a single entity using operations that perform some regimented, impersonal operation on all its components that doesn't in general require it to have any overall shape.  I don't by any means deride such treatments, which recover some order out of the chaos; but it's certainly not reassuring to realize how much lack of structure is hidden beneath such neat-looking formulae as the Schrödinger equation.  And the amorphism beneath the elegant equations also makes it hard to imagine an alternative when looking at the specifics of the math (as suspected based on biological assessment of the evolution of physics).

The quantum situation gets its structure, and its dynamics, from the Hamiltonian, that single creature embodying the whole of the rules of classical behavior for the system.  The Schrödinger equation (or whatever alternative plays its role) governs the evolution of the quantum state vector over time, and contains within it a differential operator based on the classical Hamiltonian function.

iℏ Ψ
 =   Ĥ Ψ .
One really wants to stop and admire this equation.  It's a linear partial differential equation, which is wonderful; nonlinearity is what gives rise to chaos in the technical sense, and one would certainly rather deal with a linear system.  Unfortunately, the equation only describes the evolution of the system so long as it remains a purely quantum system; the moment you open the box to see whether the cat is dead, this wave function collapses into observation of one of the classical states indexing the quantum state vector, with (to paint in broad strokes) the amplitudes of the complex numbers in the vector determining the probability distribution of observed classical states.

It also satisfies James Clerk Maxwell's General Maxim of Physical Science, which says (as recounted by Ludwik Silberstein) that when we take the derivatives of our system with respect to time, we should end up with expressions that do not themselves explicitly involve time.  When this is so, the system is "complete", or, "undisturbed".  (The idea here is that if the rules governing the system change over time, it's because the system is being affected by some other factor that is varying over time.)

The equation is, indeed, highly seductive.  Although I'm frankly on guard against it, yet here I am, being drawn into making remarks on its properties.  Back to the question of structure.  This equation effectively segregates the mathematical description of the system into a classical part that drives the dynamics (the Hamiltonian), and a quantum part that smears everything together (the quantum state vector).  The wave function Ψ, described by the equation, is the adapter used to plug these two disparate elements together.  The moment you start contemplating the equation, this manner of segregating the description starts to seem inevitable.  So, having observed these basic elements of the quantum math, let us step back again before we get stuck.

The key structural feature of the quantum description, in contrast to classical physics, is that the parts can't be considered separately.  This classical separability produced the sense of simplicity that, I speculated above, could be an ulterior motive for hidden variable theories.  The term for this is superposition of states, i.e., a quantum state that could collapse into any of multiple classical states, and therefore must contain all of those classical states in its description.

A different view of this is offered by so-called quantum logic.  The idea here (notably embraced by physicist David Finkelstein, who I've mentioned in an earlier post because he was lead author of some papers in the 1960s on quaternion quantum theory) is that quantum theory is a logic of propositions about the physical world, differing fundamentally from classical propositional logic because of the existence of superposition as a propositional principle.  There's a counterargument that this isn't really a "logic", because it doesn't describe reasoning as such, just the behavior of classical observations when applied as a filter to quantum systems; and indeed one can see that something of the sort is happening in the Schrödinger equation, above — but that would be pulling us back into the detailed math.  Quantum logic, whatever it doesn't apply to, does apply to observational propositions under the regime of quantum mechanics, while remaining gratifyingly abstracted from the detailed quantum math.

Formally, in classical logic we have the distributive law

P and (Q or R)  =  (P and Q) or (P and R) ;
but in quantum logic, (Q or R) is superpositional in nature, saying that we can eliminate options that are neither, yet allowing more than the union of situations where one holds and situations where the other holds; and this causes the distributive law to fail.  If we know P, and we know that either Q or R (but we may be fundamentally unable to determine which), this is not the same as knowing that either both P and Q, or both P and R.  We aren't allowed to refactor our proposition so as to treat Q separately from R, without changing the nature of our knowledge.

[note: I've fixed the distributive law, above, which I botched and didn't even notice till, thankfully, a reader pointed it out to me.  Doh!]

One can see in this broadly the reason why, when we shift from classical physics to quantum physics, we lose our ability to consider the underlying system as made up of elementary things.  In considering each classical elementary thing, we summed up the influences on that thing from each of the other elementary things, and this sum was a small tidy set of parameters describing that one thing alone.  The essence of quantum logic is that we can no longer refactor the system in order to take this sum; the one elementary thing we want to consider now has a unique relationship with each of the other elementary things in the system.

Put that way, it seems that the one elementary thing we want to consider would actually have a close personal relationship with each other elementary thing in the universe.  A very large Rolodex indeed.  One might object that most of those elementary things in the universe are not part of the system we are considering — but what if that's what we're doing wrong?  Sometimes, a whole can be naturally decomposable into parts in one way, but when you try to decompose it into parts in a different way you end up with a complicated mess because all of your "parts" are interacting with each other.  I suggested, back in my first blog post on physics, that there might be some wrong assumption shared by both classical and quantum physics; well, the idea that the universe is made up of elementary particles (or quanta, whatever you prefer to call them) is something shared by both theories.  The quantum math (Schrödinger equation again, above) has this classical decomposition built into its structure, pushing us to perceive the subsequent quantum weirdness as intrinsic to reality, or perhaps intrinsic to our observation of reality — but what if it's rather intrinsic to that particular way of slicing off a piece of the universe for consideration?

The quantum folks have been insisting for years that quantum reality seems strange only because we're imposing our intuitions from the macroscopic world onto the quantum-scale world where it doesn't apply.  Okay...  Our notion that the universe is made up of individual things is certainly based on our macroscopic experience.  What if it breaks down sooner than we thought — what if, instead of pushing the idea of individual things down to a smaller and smaller scale until they sizzle apart into a complex Hilbert space, we should instead have concluded that individual things are something of an illusion even at macroscopic scales?

The structure of reality

One likely objection is that no matter how you split up reality, you'd still have to observe it classically and the usual machinery of quantum mechanics would apply just the same.  There are at least a couple of ways — two come to mind atm — for some differently shaped 'slice' of reality to elude the quantum machinery.

  • The alternative slice might not be something directly observable.

    Here an extreme example comes in handy (as hoped).  Recall the sympathetic hypothesis, above.  A pattern would not be subject to direct observation, any more than a Platonic ideal like "table" or "triangle" would be.  (Actually, it seems possible a pattern would be a Platonic ideal.)

    This is also reminiscent of the analogy with vau-calculus.  I noted above that much of the substance of a calculus term is made up of variables, where by a variable I meant the entire dynamically interacting web delineated by a variable binding construct and all its matching variable instances.  A variable in this sense isn't, so to speak, observable; one can observe a particular instance of a variable, but a variable instance is just an atom, and not particularly interesting.

  • The alternative slice might be something quantum math can't practically cope with.  Quantum math is very difficult to apply in practice; some simple systems can be solved, but others are intractable.  (It's fashionable in some circles to assume more powerful computers will solve all math problems.  I'm reminded of a quote attributed to Eugene Wigner, commenting on a large quantum calculation:  "It is nice to know that the computer understands the problem.  But I would like to understand it, too.")  It's not inconceivable that phenomena deviating from quantum predictions are "hiding in plain sight".  My own instinct is that if this were so, they probably wouldn't be just on the edge of what we can cope with mathematically, but well outside that perimeter.

    This raises the possibility that quantum mechanics might be an idealized approximation, holding asymptotically in a degenerate case — in somewhat the same way that Newtonian mechanics holds approximately for macroscopic problems that don't involve very high velocities.

We have several reasons, by this point, to suspect that whatever it is we're contemplating adding to our model of reality, it's nonlocal (that is, nonlocal relative to the time dimension, as is quantum theory).  On one hand, bluntly, classical physics has had its chance and not worked out; we're already conjecturing that insisting on a classical approach is what got us into the hole we're trying to get out of.  On the other hand, under the analogy we're exploring with vau-calculus, we've already noted that most of the term syntax is occupied by distributed variables — which are, in a deep sense, fundamentally nonlocal.  The idea of spacetime as a hygiene condition rather than a base medium seems, on the face of it, to call for some sort of nonlocality; in fact, saying reality has a substantial component that doesn't follow the contours of spacetime is evidently equivalent to saying it's nonlocal.  Put that way, saying that reality can be usefully sliced in a way that defies the division into elementary particles/things is also another way of saying it's nonlocal, since when we speak of dividing reality into elementary "things", we mean, things partitioned away from each other by spacetime.  So what we have here is several different views of the same sort of conjectured property of reality.  Keeping in mind, multiple views of a single structure is a common and fruitful phenomenon in mathematics.

I'm inclined to doubt this nonlocality would be of the sort already present in quantum theory.  Quantum nonlocality might be somehow a degenerate case of a more general principle; but, again bluntly, quantum theory too has had its chance.  Moreover, it seems we may be looking for something that operates on macroscopic scales, and quantum nonlocality (entanglement) tends to break down (decohere) at these scales.  This suggests the prospect of some form of robust nonlocality, in contrast to the more fragile quantum effects.

So, at this point I've got in my toolkit of ideas (not including sympathy, which seems atm quite beyond the pale, limited to the admittedly useful role of devil's advocate):

  • a physical structure substantially not contained within spacetime.
    • space emergent as a hygiene condition, perhaps rotation-related.
    • robust nonlocality, with quantum nonlocality perhaps as an asymptotic degenerate case.
    • some non-spacetime dimension over which one can recover abstract determinism/locality.
  • decomposition of reality into coherent "finite slices" in some way other than into elementary things in spacetime.
    • slices may be either non-observable or out of practical quantum scope.
    • the structural role of the space hygiene condition may be to keep slices distinct from each other.
    • conceivably an alternative decomposition of reality may allow some over-specified elements in classical descriptions to be dropped entirely from the theory, at unknown price to descriptive clarity.
I can't make up my mind if this is appallingly vague, or consolidating nicely.  Perhaps both.  At any rate, the next phase of this operation would seem likely to shift further along the scale toward identifying concrete structures that meet the broad criteria.  In that regard, it is probably worth remarking that current paradigm physics already decomposes reality into nonlocal slices (though not in the sense suggested here):  the types of elementary particles.  The slices aren't in the spirit of the "finite" condition, as there are only (atm) seventeen of them for the whole of reality; and they may, perhaps, be too closely tied to spacetime geometry — but they are, in themselves, certainly nonlocal.