Structural insight

Tempora mutantur, nos et mutamur in illis.
(Times change, and we change with them.)

— Latin Adage, 16t‍h-century Germany.

I want to understand how a certain kind of mathematical system can act as a foundation for a certain kind of physical cosmos. The ultimate goal of course would be to find a physical cosmos that matches the one we're in; but as a first step I'd like to show it's possible to produce certain kinds of basic features that seem prerequisite to any cosmos similar to the one we're in. A demonstration of that much ought, hopefully, to provide a starting point to explore how features of the mathematical system shape features of the emergent cosmos.

The particular kind of system I've been incrementally designing, over a by-now-lengthy series of posts (most recently yonder), is a rewriting system —think λ-calculus— where a "term" (really more of a graph) is a state of the whole spacetime continuum, a vast structure which is rewritten according to some local rewrite rules until it reaches some sort of "stable" state. The primitive elements of this state have two kinds of connections between them, geometry and network; and by some tricky geometry/network interplay I've been struggling with, gravity and the other fundamental forces are supposed to arise, while the laws of quantum physics emerge as an approximation good for subsystems sufficiently tiny compared to the cosmos as a whole. That's what's supposed to happen for physics of the real world, anyway.

To demonstrate the basic viability of the approach, I really need to make two things emerge from the system. The obvious puzzle in all this has been, from the start, how to coax quantum mechanics out of a classically deterministic rewriting system; inability to extract quantum mechanics from classical determinism has been the great stumbling block in devising alternatives to quantum mechanics for about as long as quantum mechanics has been around (harking back to von Neumann's 1932 no-go theorem). I established in a relatively recent post (thar) that the quintessential mathematical feature of quantum mechanics, to be derived, is some sort of wave equation involving signed magnitudes that add (providing a framework in which waves can cancel, so producing interference and other quantum weirdness). The geometry/network decomposition is key for my efforts to do that; not something one would be trying to achieve, evidently, if not for the particular sort of rewriting-based alternative mathematical model I'm trying to apply to the problem; but, contemplating this alternative cosmic structure in the abstract, starting from a welter of interconnected elements, one first has to ask where the geometry — and the network — and the distinction between the two — come from.

Time after time in these posts I set forth, for a given topic, all the background that seems relevant at the moment, sift through it, glean some new ideas, and then set it all aside and move on to another topic, till the earlier topic, developing quietly while the spotlight is elsewhere, becomes fresh again and offers enough to warrant revisiting. It's not a strategy for the impatient, but there is progress, as I notice looking back at some of my posts from a few years ago. The feasibility of the approach hinges on recognizing that its value is not contingent on coming up with some earth-shattering new development (like, say, a fully operational Theory of Everything). One is, of course, always looking for some earth-shattering new development; looking for it is what gives the whole enterprise shape, and one also doesn't want to become one of those historical footnotes who after years of searching brushed past some precious insight and failed to recognize it, so that it had to wait for some other researcher to discover it later. But, as I noted early in this series, the simple act of pointing out alternatives to a prevailing paradigm in (say) physics is beneficial to the whole subject, like tilling soil to aerate it. Science works best with alternatives to choose between; and scientists work best when their thoughts and minds are well-limbered by stretching exercises. For these purposes, in fact, the more alternatives the merrier, so that as a given post is less successful in reaching a focused conclusion it's more likely to compensate in variety of alternatives.

In this series of physics posts, I keep hoping to get down to mathematical brass tacks; but very few posts in the series actually do so (with a recent exception in June of last year). Alas, though the current post does turn its attention more toward mathematical structure, it doesn't actually achieve concrete specifics. Getting to the brass tacks requires first working out where they ought to be put.

Contents
Dramatis personae
Connections
Termination
Duality
Scale

Dramatis personae

A rewriting calculus is defined by its syntax and rewriting rules; for a given computation, one also needs to know the start term. In this case, we'll put off for the moment worrying about the starting configuration for our system.

The syntax defines the shapes of the pieces each state (aka term, graph, configuration) is made of, and how the pieces can fit together. For a λ-like calculus, the pieces of a term would be syntax-tree nodes; the parent/child connections in the tree would be the geometry, and the variable binding/instance connections would be the network. My best guess, thus far, has been that the elementary pieces of the cosmos would be events in spacetime. Connections between events would, according to the general scheme I've been conjecturing, be separated into local connections, defining spacetime, and non-local connections, providing a source of seeming-randomness if the network connections are sufficiently widely distributed over a cosmos sufficiently vast compared to the subsystem under consideration.

I'm guessing that, to really make this seeming-randomness trick work, the cosmos ought to be made up of some truly vast number of events; say, 10⁶⁰, or 10⁸⁰, or on up from there. If the network connections are really more-or-less-uniformly distributed over the whole cosmos, irrespective of the geometry, then there's no obvious reason not to count events that occur, say, within the event horizon of a black hole, and from anywhere/anywhen in spacetime, which could add up to much more than the currently estimated number of particles in the universe. Speculatively (which is the mode all of this is in, of course), if the galaxy-sized phenomena that motivate the dark-matter hypothesis are too big, relative to the cosmos as a whole, for the quantum approximation to work properly —so one would expect these phenomena to sit oddly with our lesser-scale physics— that would seem to suggest that the total size of the cosmos is finite (since in an infinite cosmos, the ratio of the size of a galaxy to the size of the universe would be exactly zero, no different than the ratio for an electron). Although, as an alternative, one might suppose such an effect could derive, in an infinite cosmos, from network connections that aren't distributed altogether uniformly across the cosmos (so that connections with the infinite bulk of things get damped out).

With the sort of size presumed necessary to the properties of interest, I won't be able to get away with the sort of size-based simplifying trick I've gotten away with before, as with a toy cosmos that has only four possible states. We can't expect to run a simulation with program states comparable in size to the cosmos; Moore's law won't stretch that far. For this sort of research I'd expect to have to learn, if not invent, some tools well outside my familiar haunts.

The form of cosmic rewrite rules seems very much up-for-grabs, and I've been modeling guesses on λ-like calculi while trying to stay open to pretty much any outre possibility that might suggest itself. In λ-like rewriting, each rewriting rule has a redex pattern, which is a local geometric shape that must be matched; it occurs, generally, only in the geometry, with no constraints on the network. The redex-pattern may call for the existence of a tangential network connection —the β-rule of λ-calculus does this, calling for a variable binding as part of the pattern— and the tangential connection may be rearranged when applying the rule, just as the local geometry specified by the redex-pattern may be rearranged. Classical λ-calculus, however, obeys hygiene and co-hygiene conditions: hygiene prohibits the rewrite rule from corrupting any part of the network that isn't tangent to the redex-pattern, while co-hygiene prohibits the rewrite rule from corrupting any part of the geometry that isn't within the redex-pattern. Impure variants of λ-calculus violate co-hygiene, but still obey hygiene. The guess I've been exploring is that the rewriting rules of physics are hygienic (and Church-Rosser), and gravity is co-hygienic while the other fundamental forces are non-co-hygienic.

I've lately had in mind that, to produce the right sort of probability distributions, the fluctuations of cosmic rewriting ought to, in essence, compare the different possible behaviors of the subsystem-under-consideration. Akin to numerical solution of a problem in the calculus of variations.

Realizing that the shape of spacetime is going to have to emerge from all this, the question arises —again— of why some connections to an event should be "geometry" while others are "network". The geometry is relatively regular and, one supposes, stable, while the network should be irregular and highly volatile, in fact the seeming-randomness depends on it being irregular and volatile. Conceivably, the redex-patterns are geometric (or mostly geometric) because the engagement of those connections within the redex-patterns cause those connections to be geometric in character (regular, stable), relative to the evolution of the cosmic state.

The overall character of the network is another emergent feature likely worth attention. Network connections in λ-calculus are grouped into variables, sub-nets defined by a binding and its bound instances, in terms of which hygiene is understood. Variables, as an example of network structure, seem built-in rather than emergent; the β-rule of λ-calculus is apparently too wholesale a rewriting to readily foster ubiquitous emergent network structure. Physics, though, seems likely to engage less wholesale rewriting, from which there should also be emergent structure, some sort of lumpiness —macrostructures— such that (at a guess) incremental scrambling of network connections would tend to circulate those connections only within a particular lump/macrostructure. The apparent alternative to such lumpiness would be a degree of uniform distribution that feels, to my intuition anyway, unnatural. One supposes the lumpiness would come into play in the nature of stable states that the system eventually settles into, and perhaps the size and character of the macrostructures would determine at what scale the quantum approximation ceases to hold.

Connections

Clearly, how the connections between nodes —the edges in the graph— are set up is the first thing we need to know, without which we can't imagine anything else concrete about the calculus. Peripheral to that is whether the nodes (or, for that matter, the edges) are decorated, that is, labeled with additional information.

In λ-calculus, the geometric connections are of just three forms, corresponding to the three syntactic forms in the calculus: a variable instance has one parent and no children; a combination node has one parent and two children, operator and operand; and a λ-expression has one parent and one child, the body of the function. For network connections, ordinary λ-calculus has one-to-many connections from each binding to its bound instances. These λ network structures —variables— are correlated with the geometry; the instances of a variable can be arbitrarily scattered through the term, but the binding of the variable, of which there is exactly one, is the sole asymmetry of the variable and gives it an effective singular location in the syntax tree, required to be an ancestor in the tree of all the locations of the instances. Interestingly, in the vau-calculus generalization of λ-calculus, the side-effectful bindings are somewhat less uniquely tied to a fixed location in the syntax tree, but are still one-per-variable and required to be located above all instances.

Physics doesn't obviously lend itself to a tree structure; there's no apparent way for a binding to be "above" its instances, nor apparent support for an asymmetric network structure. Symmetric structures would seem indicated. A conceivable alternative strategy might use time as the "vertical" dimension of a tree-like geometry, though this would seem rather contrary to the loss of absolute time in relativity.

A major spectrum of design choice is the arity of network structures, starting with whether network structures should have fixed arity, or unfixed as in λ-like calculi. Unfixed arity would raise the question of what size the structures would tend to have in a stable state. Macrostructures, "lumps" of structures, are a consideration even with fixed arity.

Termination

In exploring these realms of possible theory, I often look for ways to defer aspects of the theory till later, as a sort of Gordian-knot-cutting (reducing how many intractable questions I have to tackle all at once). I've routinely left unspecified, in such deferral, just what it should mean for the cosmic rewriting system to "settle into a stable state". However, at this point we really have no choice but to confront the question, because our explicit main concern is with mathematical properties of the probability distribution of stable states of the system, and so we can do nothing concrete without pinning down what we mean by stable state.

In physics, one tends to think of stability in terms of asymptotic behavior in a metric space; afaics, exponential stability for linear systems, Lyapunov stability for nonlinear. In rewriting calculi, on the other hand, one generally looks for an irreducible form, a final state from which no further rewriting is possible. One could also imagine some sort of cycle of states that repeat forever, though making that work would require answers to some logistical questions. Stability (cyclic or otherwise) might have to do with constancy of which macrostructure each of an element's network connections associates to.

If rewriting effectively explores the curvature of the action function (per the calculus of variations as mentioned earlier), it isn't immediately obvious how that would then lead to asymptotic stability. At any rate, different notions of stability lead to wildly different mathematical developments of the probability distribution, hence this is a major point to resolve. The choice of stability criterion may depend on recognizing what criterion can be used in some technique that arrives at the right sort of probability distribution.

There's an offbeat idea lately proposed by Tim Palmer called the invariant set postulate. Palmer, so I gather, is a mathematical physicist deeply involved in weather prediction, from which he's drawn some ideas to apply back to fundamental physics. A familiar pattern in nonlinear systems, apparently, is a fractal subset of state space which, under the dynamics of the system, the system tends to converge upon and, if the system state actually comes within the set, remains invariant within. In my rewriting approach these would be the stable states of the cosmos. The invariant set should be itself a metric space of lower dimension than the state space as a whole and (if I'm tracking him) uncomputable. Palmer proposes to postulate the existence of some such invariant subset of the quantum state space of the universe, to which the actual state of the universe is required to belong; and requiring the state of the universe to belong to this invariant set amounts to requiring non-independence between elements of the universe, providing an "out" to cope with no-go theorems such as Bell's theorem or the Kochen–Specker theorem. Palmer notes that while, in the sort of nonlinear systems this idea comes from, the invariant set arises as a consequence of the underlying dynamics of the system, for quantum mechanics he's postulating the invariant set with no underlying dynamics generating it. This seems to be where my approach differs fundamentally from his: I suppose an underlying dynamics, produced by my cosmic rewriting operation, from which one would expect to generate such an invariant set.

Re Bell and, especially, Kochen-Specker, those no-go theorems rule out certain kinds of mutual independence between separate observations under quantum mechanics; but the theorems can be satisfied —"coped with"— by imposing some quite subtle constraints. Such as Palmer's invariant set postulate. It seems possible that Church-Rosser-ness, which tampers with independence constraints between alternative rewrite sequences, may also suffice for the theorems.

Duality

What if we treated the lumpy macrostructures of the universe as if they were primitive elements; would it be possible to then describe the primitive elements of the universe as macrostructures? Some caution is due here for whether this micro/macro duality would belong to the fundamental structure of the cosmos or to an approximation. (Of course, this whole speculative side trip could be a wild goose chase; but, as usual, on one hand it might not be a wild goose chase, and on the other hand wild-goose-chasing can be good exercise.)

Perhaps one could have two coupled sets of elements, each serving as the macrostructures for the other. The coupling between them would be network (i.e., non-geometric), through which presumably each of the two systems would provide the other with quantum-like character. In general the two would have different sorts of primitive elements and different interacting forces (that is, different syntax and rewrite-rules). Though it seems likely the duals would be quite different in general, one might wonder whether in a special case they could sometimes have the same character, in which case one might even ask whether the two could settle into identity, a single system acting as its own macro-dual.

For such dualities to make sense at all, one would first have to work out how the geometry of each of the two systems affects the dynamics of the other system — presumably, manifesting through the network as some sort of probabilistic property. Constructing any simple system of this sort, showing that it can exhibit the sort of quantum-like properties we're looking for, could be a worthwhile proof-of-concept, providing a buoy marker for subsequent explorations.

On the face of it, a basic structural difficulty with this idea is that primitive elements of a cosmic system, if they resemble individual syntax nodes of a λ-calculus term, have a relatively small fixed upper bound on how many macrostructures they can be attached to, whereas a macrostructure may be attached to a vast number of such primitive elements. However, there may be a way around this.

Scale

I've discussed before the phenomenon of quasiparticles, group behaviors in a quantum-mechanical system that appear (up to a point) as if they were elementary units; such eldritch creatures as phonons and holes. Quantum mechanics is fairly tolerant of inventing such beasts; they are overtly approximations of vastly complicated underlying systems. Conventionally "elementary" particles can't readily be analyzed in the same way —as approximations of vastly complicated systems at an even smaller scale— because quantum mechanics is inclined to stop at Planck scale; but I suggested one might achieve a similar effect by importing the complexity through network connections from the very-large-scale cosmos, as if the scale of the universe were wrapping around from the very small to the very large.

We're now suggesting that network connections provide the quantum-like probability distributions, at whatever scale affords these distributions. Moreover, we have this puzzle of imbalance between, ostensibly, small bounded network arity of primitive elements (analogous to nodes in a syntax tree) and large, possibly unbounded, network arity of macrostructures. The prospect arises that perhaps the conventionally "elementary" particles —quarks and their ilk— could be already very large structures, assemblages of very many primitive elements. In the analogy to λ-calculus, a quark would correspond to a subterm, with a great deal of internal structure, rather than to a parse-tree-node with strictly bounded structure. The quark could then have a very large network arity, after all. Quantum behavior would presumably arise from a favorable interaction between the influence of network connections to macrostructures at a very large cosmic scale, and the influence of geometric connections to microstructures at a very small scale. The structural interactions involved ought to be fascinating. It seems likely, on the face of it, that the macrostructures, exhibiting altogether different patterns of network connections than the corresponding microstructures, would also have different sorts of probability distributions, not so much quantum as co-quantum — whatever, exactly, that would turn out to mean.

If quantum mechanics is, then, an approximation arising from an interaction of influences from geometric connections to the very small and network connections to the very large, we would expect the approximation to hold, not at the small end of the range of scales, but only at a subrange of intermediate scales — not too large and at the same time not too small. In studying the dynamics of model rewriting systems, our attention should then be directed to the way these two sorts of connections can interact to reach a balance from which the quantum approximation can emerge.

At a wild, rhyming guess, I'll suggest that the larger a quantum "particle" (i.e., the larger the number of primitive elements within it), the smaller each corresponding macrostructure. Thus, as the quanta get larger, the macrostructures get smaller, heading toward a meeting somewhere in the mid scale — notionally, around the square root of the number of primitive elements in the cosmos — with the quantum approximation breaking down somewhere along the way. Presumably, the approximation also requires that the macrostructures not be too large, hence that the quanta not be too small. Spinning out the speculation, on a logarithmic scale, one might imagine the quantum approximation working tolerably well for, say, about the middle third of the lower half of the scale, with the corresponding macrostructures occupying the middle third of the upper half of the scale. This would put the quantum realm at a scale from the number of cosmic elements raised to the 1/3 power, down to the number of cosmic elements raised to the 1/6 power. For example, if the number of cosmic elements were 10¹²⁰, quantum scale would be from 10⁴⁰ down to 10²⁰ elements. The takeaway lesson here is that, even if those guesses are off by quite a lot, the number of primitive elements in a minimal quantum could still be rather humongous.

Study of the emergence of quasiparticles seems indicated.

Thus quantum mechanics occupies a very unusual place among physical theories: it contains classical mechanics as a limiting case, yet at the same time it requires this limiting case for its own formulation.

— Lev Landau and Evgeny Lifshitz, Quantum Mechanics: Non-relativistic Theory (3rd edition, 1977, afaik).

Gradually, across a series of posts exploring alternative structures for a basic theory of physics, I've been trying to tease together a strategy wherein quantum mechanics is, rather than a nondeterministic foundation of reality, an approximation valid for sufficiently small systems. This post considers how one might devise a concrete mathematical demonstration that the strategy can actually work.

I came into all this with a gnawing sense that modern physics had taken a conceptual wrong turn somewhere, that it had made some —unidentified— incautious structural assumption that ought not have been made and was leading it further and further astray. (I explored the philosophy of this at some depth in an earlier post in the series, several years ago by now.) The larger agenda here is to shake up our thinking on basic physics, accumulating different ways to structure theories so that our structural choices are made with eyes open, rather than just because we can't imagine an alternative. The particular notion I'm stalking atm —woven around the concept of co-hygiene, to be explained below— is, in its essence, that quantum mechanics might be an approximation, just as Newtonian mechanics is, and that the quantum approximation may be a consequence of the systems-of-interest being almost infinitesimally small compared to the cosmos as a whole. Quantum mechanics suggests that all the elementary parts of the cosmos are connected to all the other elementary parts, which is clearly not conducive to practical calculations. In the model I'm pursuing, each element is connected to just a comparatively few others, and the whole jostles about, with each adjustment to an element shuffling its remote connections so that over many adjustments the element gets exposed to many other elements. Conjecturally, if a sufficiently small system interacts in this way with a sufficiently vast cosmos, the resulting behavior of the small system could look a lot like nondeterminism.

The question is, could it look like quantum mechanics?

As I've remarked before, my usual approach to these sorts of posts is to lift down off my metaphorical shelf the assorted fragments I've got on the topic of interest; lay out the pieces on the table, adding at the same time any new bits I've lately collected; inspect them all severally and collectively, rearranging them and looking for new patterns as I see them all afresh; and record my trail of thought as I do so. Sometimes I find that since the last time I visited things, my whole perception of them has shifted (I was, for example, struck in a recent post by how profoundly my perception of Church's λ-calculus has changed just in the past several years). Hopefully I glean a few new insights from the fresh inspection, some of which find their way into the new groupings destined to go back up on the shelf to await the next time, while some other, more speculative branches of reasoning that don't make it into my main stream of thought are preserved in my record for possible later pursuit.

Moreover, each iteration achieves focus by developing some particular theme within its line of speculation; some details of previous iterations are winnowed away to allow an uncluttered view of the current theme; and once the new iteration reaches its more-or-less-coherent insights, such as they are, a reset is then wanted, to unclutter the next iteration. Most of the posts in this series —with a couple of exceptions (1, 2)— have focused on the broad structure of the cosmos, touching only lightly on concrete mathematics of modern physics that, after all, I've suspected from the start of favoring incautious structural assumptions. This incremental shifting between posts is why, within my larger series on physics, the current post has a transitional focus: reviewing the chosen cosmological structure in order to apply it to the abstract structure of the mathematics, preparing from abstract ground to launch an assault on the concrete.

Though I'll reach a few conclusions here —oriented especially toward guidance for the next installment in the series— much of this is going to dwell on reasons why the problem is difficult, which if one isn't careful could create a certain pessimism toward the whole prospect. I'm moderately optimistic that the problem can be pried open, over a sufficient number of patient iterations of study. The formidable appearance of a mountain in-the-large oughtn't prevent us from looking for a way to climb it.

Contents
Co-hygiene
Primitive wave functions
Probability distributions
Quantum/classical interface
Genericity
The universe says 'hi'
The upper box

Co-hygiene

The schematic mathematical model I'm considering takes the cosmos to be a vast system of parts with two kinds of connections between them: local (geometry), and non-local (network). The system evolves by discrete transformational steps, which I conjecture may be selected based entirely on local criteria but, once selected, may draw information from both local and non-local connections and may have both local and non-local effects. The local part of all this would likely resemble classical physics.

When a transformation step is applied, its local effect must be handled in a way that doesn't corrupt the non-local network; that's called hygiene. If the non-local effect of a step doesn't perturb pre-existing local geometry, I call that co-hygiene. Transformation steps are not required in general to be co-hygienic; but if they are, then local geometry is only affected by local transformation steps, giving the steps a close apparent affinity with the local geometry, and I conjectured this could explain why gravity seems more integrated with spacetime than do the other fundamental forces. (Indeed, wondering why gravity would differ from the other fundamental forces was what led me into the whole avenue of exploration in the first place.)

Along the way, though, I also wondered if the non-local network could explain why the system deviated from "classical" behavior. Here I hit on an idea that offered a specific reason why quantum mechanics might be an approximation that works for very small systems. My inspiration for this sort of mathematical model was a class of variant λ-calculi (in fact, λ-calculus is co-hygienic, while in my dissertation I studied variant calculi that introduce non-co-hygienic operations to handle side-effects); and in those variant calculi, the non-local network topology is highly volatile. That is, each time a small subsystem interacts non-locally with the rest of the system, it may end up with different network neighbors than it had before. This means that if you're looking at a subsystem that is smaller than the whole system by a cosmically vast amount — say, if the system as a whole is larger than the subsystem by a factor of 10⁷⁰ or 10⁸⁰ — you might perform a very large number of non-local interactions and never interact with the same network-neighbor twice. It would be, approximately, as if there were an endless supply of other parts of the system for you to interact non-locally with. Making the non-local interactions look rather random.

Without the network-scrambling, non-locality alone would not cause this sort of seeming-randomness. The subsystem of interest could "learn" about its network neighbors through repeated interaction with them, and they would become effectively just part of its internal state. Thus, the network-scrambling, together with the assumption that the system is vastly larger than the subsystem, would seem to allow the introduction of an element of effective nondeterminism into the model.

But, is it actually useful to introduce an element of effective nondeterminism into the model? Notwithstanding Einstein's remark about whether or not God plays dice, if you start with a classical system and naively introduce a random classical element into it, you don't end up with a quantum wave function. (There is a vein of research, broadly called stochastic electrodynamics, that seeks to derive quantum effects from classical electrodynamics with random zero-point radiation on the order of Planck's constant, but apparently they're having trouble accounting for some quantum effects, such as quantum interference.) To turn this seeming-nondeterminism to the purpose would require some more nuanced tactic.

There is, btw, an interesting element of flexibility in the sort of effective-nondeterminism introduced: The sort of mathematical model I'm conjecturing has deterministic rules, so conceivably there could be some sort of invariant properties across successive rearrangements of the network topology. Thus, some kinds of non-local influences could be seemingly-random while others might, at least under some particular kinds of transformation (such as, under a particular fundamental force), be constant. The subsystem of interest could "learn" these invariants through repeated interactions, even though other factors would remain unlearnable. In effect, these invariants would be part of the state of the subsystem, information that one would include in a description of the subsystem but that, in the underlying mathematical model, would be distributed across the network.

Primitive wave functions

Suppose we're considering some very small physical system, say a single electron in a potential field.

A potential field, as I suggested in a previous post, is a simple summation of combined influences of the rest of the cosmos on the system of interest, in this case our single electron. Classically —and under Relativity— the potential field would tell us nothing about non-local influences on the electron. In this sort of simple quantum-mechanical exercise, the potential field used is, apparently, classical.

The mathematical model in conventional quantum mechanics posits, as its underlying reality, a wave function — a complex- (or quaternion-, or whatever-) valued field over the state space of the system, obeying some wave equation such as Schrödinger's,

iℏ ∂ Ψ

∂t

= Ĥ Ψ .

This posited underlying reality has no electron in the classical sense of something that has a precise position and momentum at each given time; the wave function is what's "really" there, and any observation we would understand as measuring the position or momentum of the electron is actually drawing on the information contained in the wave function.

While the wave function evolves deterministically, the mathematical model as a whole presents a nondeterministic theory. This nondeterminism is not a necessary feature of the theory. An alternative mathematical model exists, giving exactly the same predictions, in which there is an electron there in the classical sense, with precise position and momentum at each given time. Of course its position and momentum can't be simultaneously known by an observer (which would violate the Heisenberg uncertainty principle); but in the underlying model the electron does have those unobservable attributes. David Bohm published this model in 1952. However Bohm's model doesn't seem to have offered anything except a demonstration that quantum theory does not prohibit the existence of an unobservable deterministic classical electron. In Bohm's model, the electron had a definite position and momentum, yes, but it was acted on by a "pilot wave" that, in essence, obeyed Schrödinger's equation. And Schrödinger's equation is non-local, in the sense that not only does it allow information (unobservable information) to propagate faster than light, it allows it to "propagate" infinitely fast; the hidden information in the wave function does not really propagate "through" space, it just shows up wherever the equation says it should. Some years later, Bell's Theorem would show that this sort of non-locality is a necessary feature of any theory that always gives the same predictions as quantum mechanics (given some other assumptions, one of which I'm violating; I'll get back to that below); but my main point atm is that Bohm's model doesn't offer any new way of looking at the wave function itself. You still have to just accept the wave function as a primitive; Bohm merely adds an extra stage of reasoning in understanding how the wave function applies to real situations. If there's any practical, as opposed to philosophical, advantage to using Bohm's model, it must be a subtle one. Nevertheless, it does reassure us that there is no prohibition against a model in which the electron is a definite, deterministic thing in the classical sense.

The sort of model I'm looking for would have two important differences from Bohm's.

First, the wave function would not be primitive at all, but instead would be a consequence of the way the local-geometric aspect of the cosmos is distorted by the new machinery I'm introducing. The Schrödinger equation, above, seems to have just this sort of structure, with Ĥ embodying the classical behavior of the system while the rest of the equation is the shape of the distorting lens through which the classical behavior passes to produce its quantum behavior. The trick is to imagine any sensible way of understanding this distorting lens as a consequence of some deeper representation (keeping in mind that the local-geometric aspect of the cosmos needn't be classical physics as such, though this would be one's first guess).

A model with different primitives is very likely to lead to different questions; to conjure a quote from Richard Feynman, "by putting the theory in a certain kind of framework you get an idea of what to change". Hence a theory in which the wave function is not primitive could offer valuable fresh perspective even if it isn't in itself experimentally distinguishable from quantum mechanics. There's also the matter of equivalent mathematical models that are easier or harder to apply to particular problems — conventional quantum mechanics is frankly hard to apply to almost any problem, so it's not hard to imagine an equivalent theory with different primitives could make some problems more tractable.

Second, the model I'm looking for wouldn't, at least not necessarily, always produce the same predictions as quantum mechanics. I'm supposing it would produce the same predictions for systems practically infinitesimal compared to the size of the cosmos. Whether or not the model would make experimentally distinguishable predictions from quantum mechanics at a cosmological scale, would seem to depend on how much, or little, we could work out about the non-local-network part of the model; perhaps we'd end up with an incomplete model where the network part of it is just unknown, and we'd be none the wiser (but for increased skepticism about some quantum predictions), or perhaps we'd find enough structural clues to conjecture a more specific model. Just possibly, we'd end up with some cosmological questions to distinguish possible network structures, which (as usual with questions) could be highly fruitful regardless of whether the speculations that led to the questions were to go down in flames, or, less spectacularly, were to produce all the same predictions as quantum mechanics after all.

Probability distributions

Wave functions have always made me think of probability distributions, as if there ought to be some deterministic thing underneath whose distribution of possible states is generating the wave function. What's missing is any explanation of how to generate a wave-function-like thing from a classical probability distribution. (Not to lose track of the terminology, this is classical in the sense of classical probability, which in turn is based on classical logic, rather than classical physics as such. Though they all come down to us from the late nineteenth century, and complement each other.)

A classical probability distribution, as such, is fairly distinctive. You have an observable with a range of possible values, and you have a range of possible worlds each of which induces an observable value. Each possible world has a non-negative-real likelihood. The (unnormalized) probability distribution for the observable is a curve over the range of observable values, summing for each observable value the likelihoods of all possible worlds that yield that observable value. The probability of the observable falling in a certain interval is the area under the curve over that interval, divided by the area under the curve over the entire range of observable values. If you add together two mutually disjoint sets of possibilities, the areas under their curves simply add, since for each observable value the set of possible worlds yielding it is just the ones in the first set and the ones in the second set.

The trouble is, that distinctive pattern of a classical probability distribution is not how wave functions work. When you add together two wave functions, the two curves get added all right, but the values aren't unsigned reals; they can cancel each other, producing an interference pattern as in classic electron diffraction. (I demonstrated the essential role of cancellation, and a very few other structural elements, in quantum mechanical behavior in a recent post.) As an additional plot twist, the wave function values add, but the probability isn't their sum but (traditionally) the square of the magnitude of their sum.

One solution is to reject classical logic, since classical logic gives rise to the addition rule for deterministic probability distributions. Just say the classical notion of logical disjunction (and conjunction, etc.) is wrong, and quantum logic is the way reality works. While you're at it, invoke the idea that the world doesn't have to make sense to us (I've remarked before on my dim view of the things beyond mortal comprehension trope). Whatever its philosophical merits or demerits, this approach doesn't fit the current context for two reasons: it treats the wave function as primitive whereas we're interested in alternative primitives, so it doesn't appear to get us anywhere new/useful; and, even if it did get us somewhere useful (which it apparently doesn't), it's not the class of mathematical model I'm exploring here. I'm pursuing a mathematical model spiritually descended from λ-calculus, which is very much in the classical deterministic tradition.

So, we're looking for a way to derive a wave function from a classical probability distribution. One has to be very canny about approaching something like this. It's not plausible this would be untrodden territory; the strategy would naturally suggest itself, and lots of very smart, highly trained physicists with strong motive to consider it have had nearly a century in which to do so. Yet, frankly, if anyone had succeeded it ought to be well-known in alternative-QM circles, and I'd hope to have at least heard of it. So going into the thing one should apply a sort of lamppost principle, and ask what one is bringing to the table that could possibly allow one to succeed where they did not. (A typical version of the lamppost principle would say, if you've lost your keys at night somewhere on a dark street with a single lamppost, you should look for them near the lamppost since your chances of finding them if they're somewhere else are negligible. Here, to mix the metaphors, the something-new you bring to the table is the location of your lamppost.)

I'm still boggled by how close the frontier of human knowledge is. In high school I chose computer science for a college major partly (though only partly) because it seemed to me like there was so much mathematics you could spend a lifetime on it without reaching the frontier — and yet, by my sophomore year in college I was exploring extracurricularly some odd corner of mathematics (I forget what, now) that had clearly never been explored before. And now I'm recently disembarked from a partly-mathematical dissertation; a doctoral dissertation being, rather by definition, stuff nobody has ever done before. The idea that the math I was doing in my dissertation was something nobody had ever done before, is just freaky. At any rate, I'm bringing to this puzzle in physics a mathematical perspective that's not only unusual for physics, but unique even in the branch of mathematics I brought it from.

The particular mathematical tools I'm mainly trying to apply are:

• "metatime" (or whatever else one wants to call it), over which the cosmos evolves by discrete transformation steps. This is the thing I'm doing that breaks the conditions for Bell's Theorem; but all I've shown it works for is reshaping a uniform probability distribution into one that violates Bell's Inequality (here), whereas now we're not just reshaping a particular distribution but trying to mess with the rules by which distributions combine.

  My earlier post on metatime was explicitly concerned with the fact that quantum-mechanical predictions, while non-local with respect to time, could still be local with respect to some orthogonal dimension ("metatime"). Atm I'm not centrally interested in strict locality with respect to metatime; but metatime still interests me as a potentially useful tactic for a mathematical model, offering a smooth way to convert a classical probability distribution into time-non-locality.

• transformation steps that aggressively scramble non-local network topology. This seems capable of supplying classical nondeterminism (apparently, on a small scale); but the apparent nondeterminism we're after isn't classical.

• a broad notion that the math will stop looking like a wave function whenever the network scrambling ceases to sufficiently approximate classical nondeterminism (which ought to happen at large scales). But this only suggests that the nondeterminism would be a necessary ingredient in extracting a wave function, without giving any hint of what would replace the wave function when the approximation fails.

These are some prominent new things I'm bringing to the table. At least the second and third are new. Metatime is a hot topic atm, under a different name (pseudo-time, I think), as a device of the transactional interpretation of QM (TI). Advocates recommend TI as eliminating the conceptual anomalies and problems of other interpretations — EPR paradox, Schrödinger's cat, etc. — which bodes well for the utility of metatime here. I don't figure TI bears directly on the current purpose though because, as best I can tell, TI retains the primitive wave function. (TI does make another cameo appearance, below.)

On the problem of deriving the wave function, I don't know of any previous work to draw on. There certainly could be something out there I've simply not happened to cross paths with, but I'm not sanguine of finding such; for the most part, the subject suffers from a common problem of extra-paradigm scientific explorations: researchers comparing the current paradigm to its predecessor are very likely to come to the subject with intense bias. Researchers within the paradigm take pains to show that the old paradigm is wrong; researchers outside the paradigm are few and idiosyncratic, likely to be stuck on either the old paradigm or some other peculiar idea.

The bias by researchers within the paradigm, btw, is an important survival adaptation of the scientific species. The great effectiveness of paradigm science — which benefits its evolutionary success — is in enabling researchers to focus sharply on problems within the paradigm by eliminating distracting questions about the merits of the paradigm; and therefore those distracting questions have to be crushed decisively whenever they arise. It's hard to say whether this bias is stronger in the first generation of scientists under a paradigm, who have to get it moving against resistance from its predecessor, or amongst their successors trained within the zealous framework inherited from the first generation; either way, the bias tends to produce a dearth of past research that would aid my current purpose.

A particularly active, and biased, area of extra-paradigm science is no-go theorems, theorems proving that certain alternatives to the prevailing paradigm cannot be made to work (cf. old post yonder). Researchers within the paradigm want no-go theorems to crush extra-paradigm alternatives once and for all, and proponents of that sort of crushing agenda are likely, in their enthusiasm, to overlook cases not covered by the formal no-go-result. Extra-paradigm researchers, in contrast, are likely to ferret out cases not covered by the result and concentrate on those cases, treating the no-go theorems as helpful hints on how to build alternative ideas rather than discouragement from doing so. The paradigm researchers are likely to respond poorly to this, and accuse the alternative-seekers of being more concerned with rejecting the paradigm than with any particular alternative. The whole exchange is likely to generate much more heat than light.

Quantum/classical interface

A classical probability distribution is made up of possibilities. One of them is, and the others are not; we merely don't know which one is. This is important because it means there's no way these possibilities could ever interact with each other; the one that is has nothing to interact with because in fact there are no other possibilities. That is, the other possibilities aren't; they exist only in our minds. This non-interaction is what makes the probability distribution classical. Therefore, in considering ways to derive our wave function from classical probability distributions, any two things in the wave function that interact with each other do not correspond to different classical possibilities.

It follows that quantum states — those things that can be superposed, interfere with each other, and partly cancel each other out — are not separated by a boundary between different classical possibilities. This does not, on the face of it, prohibit superposable elements from being prior or orthogonal to such boundaries, so that the mathematical model superposes entities of some sort and then applies them to a classical probability distribution (or applies the distribution to them). Also keep in mind, though we're striving for a model in which the wave function isn't primitive, we haven't pinned down yet what is primitive.

Now, the wave function isn't a thing. It isn't observable, and we introduce it into the mathematics only because it's useful. So if it also isn't primitive, one has to wonder whether it's even needed in the mathematics, or whether perhaps we're simply to replace it by something else. To get a handle on this, we need to look at how the wave function is actually used in applying quantum mechanics to physical systems; after all, one can't very well fashion a replacement for one part of a machine unless one understands how that part interacts with the rest of the machine.

The entire subject of quantum mechanics appears imho to be filled with over-interpretation; to the extent any progress has been made in understanding quantum mechanics over the past nearly-a-century, it's consisted largely in learning to prune unnecessary metaphysical underbrush so one has a somewhat better view of the theory.

The earliest, conventional "interpretation" of QM, the "Copenhagen interpretation", says properties of the physical system don't exist until observed. This, to be brutally honest, looks to me like a metaphysical statement without practical meaning. There is a related, but more practical, concept called contextuality; and an associated — though unfortunately technically messy — no-go theorem called the Kochen–Specker theorem, a.k.a. the Bell–Kochen–Specker theorem. This all relates to the Heisenberg uncertainty principle, which says that you can't know the exact position and momentum of a particle at the same time; the more you know about its position, the less you can know about its momentum, and vice versa. One might think this would be because the only way to measure the particle's position or momentum is to interact with it, which alters the particle because, well, because to every action there is an equal and opposite reaction. However, in the practical application of the wave function to a quantum-mechanical system, there doesn't appear to be any experimental apparatus within the quantum system for the equal-and-opposite-reaction to apply to. Instead, there's simply a wave function and then it collapses. Depending on what you choose to observe (say, the position or the momentum), it collapses differently, so that the unobservable internal state of the system actually remembers which you chose to observe. This property, that the (unobservable) internal state of the system changes as a result of what you choose to measure about it, is contextuality; and the Kochen–Specker theorem says a classical hidden-variable theory, consistent with QM, must be contextual (much as Bell's Theorem says it must be non-local). Remember Bohm's hidden-variable theory, in which the particle does have an unobservable exact position and momentum? Yeah. Besides being rampantly non-local, Bohm's model is also contextual: the particle's (unobservable, exact) position and momentum are guided by the wave function, and the wave-function is perturbed by the choice of measurement, therefore the particle's (unobservable, exact) position and momentum are also perturbed by the choice of measurement.

Bell, being of a later generation than Bohr and Einstein (and thus, perhaps, less invested in pre-quantum metaphysical ideas), managed not to be distracted by questions of what is or isn't "really there". His take on the situation was that the difficulty was in how to handle the interface between quantum reality and classical reality — not philosophically, but practically. To see this, consider the basic elements of an exercise in traditional QM (non-relativistic, driven by Schrödinger's equation):

• A set of parameters define the classical state of the system; these become inputs to the wave equation. [typo fixed]

• A Hamiltonian operator Ĥ embodies the classical dynamics of the system.

• Schrödinger's equation provides quantum distortion of the classical system.

• A Hermitian operator called an "observable" embodies the experimental apparatus used to observe the system.  The wave function collapses to an eigenstate of the observable.

The observable is the interface between the quantum system and the classical world of the physicist; and Bell ascribes the difficulty to this interface. Consider a standard double-slit experiment in which an electron gun fires electrons one at a time through the double slit at a CRT screen where each electron causes a scintillation. As long as you don't observe which slit the electron passes through, you get an interference pattern from the wave function passing through the two slits, and that is quantum behavior; but there's nothing in the wave function to suggest the discreteness of the resulting scintillation. That discreteness results from the wave function collapse due to the observable, the interface with classical physics — and that discreteness is an essential part of the described physical reality. Scan that again: in order to fully account for physical reality, the quantum system has to encompass only a part of reality, because the discrete aspect of reality is only provided by the interface between the quantum system and surrounding classical physics. It seems that we couldn't describe the entire universe using QM even if we wanted to because, without a classical observable to collapse the wave function, the discrete aspect of physical reality would be missing. (Notice, this account of the difficulty is essentially structural, with only the arbitrary use of the term observable for the Hermitian operator as a vestige of the history of philosophical angst over the "role of the observer". It's not that there isn't a problem, but that presenting the problem as if it were philosophical only gets in the way of resolving it.)

The many-worlds interpretation of QM (MWI) says that the wave function does not, in fact, collapse, but instead the entire universe branches into multiples for the different possibilities described by the wave function. Bell criticized that while this is commonly presented as supposing that the wave function is "all there is", in fact it arbitrarily adds the missing discreteness:

the extended wave does not simply fail to specify one of the possibilities as actual...it fails to list the possibilities. When the M‍WI postulates the existence of many worlds in each of which the photographic plate is blackened at particular position, it adds, surreptitiously, the missing classification of possibilities. And it does so in an imprecise way, for the notion of the position of a black spot (it is not a mathematical point) [...] [or] reading of any macro‍scope instrument, is not mathematically sharp. One is given no idea of how far down towards the atomic scale the splitting of the world into branch worlds penetrates.

— J.S. Bell, "Six possible worlds of quantum mechanics", Speakable and unspeakable in quantum mechanics (anthology), 1993.

I'm inclined to agree: whatever philosophical comfort the M‍WI might provide to its adherents, it doesn't clarify the practical situation, and adds a great deal of conceptual machinery in the process of not doing so.

The transactional "interpretation" of QM is, afaik, somewhat lower-to-the-ground metaphysically. To my understanding, TI keeps everything in quantum form, and posits that spacetime events interact through a "quantum handshake": a wave propagates forward in time from an emission event, while another propagates backward in time from the corresponding absorption event, and they form a standing wave between the two while backward waves cancel out before the emission and forward waves cancel after the absorption. Proponents of the TI report that it causes the various paradoxes and conceptual anomalies of QM to disappear (cf. striking natural structure), and this makes sense to me because the "observable" Hermitian operator should be thus neatly accounted for as representing half of a quantum handshake, in which the "observer" half of the handshake is not part of the particular system under study. Wherever we choose to put the boundary of the system under study, the interface to our experimental apparatus would naturally have this half-a-handshake shape.

The practical lesson from the transactional interpretation seems to be that, for purposes of modeling QM, we don't have to worry about the wave function collapsing. If we can replicate the wave function, we're in. Likewise, if we can replicate the classical probability distributions that the wave function generates; so long as this includes all the probability distributions that result from weird quantum correlations (spooky action-at-a-distance). That the latter suffices, should be obvious since generating those probability distributions is the whole point of quantum theory; that the latter is possible is demonstrated by Bohm's hidden-variable theory (sometimes called the "Bohm Interpretation" by those focusing on its philosophy).

Genericity

There is something odd about the above list of basic elements of a QM exercise, when compared to the rewriting-calculus-inspired model we're trying to apply to it. When one thinks of a calculus term, it's a very concrete thing, with a specific representation (in fact over-specific, so that maintaining it may require α-renaming to prevent specific name choices from disrupting hygiene); and even classical physics seems to present a rather concrete representation. But the quantum distortion of the wave equation apparently applies to whatever description of a physical system we choose; to any choice of parameters and Ĥ, regardless of whether it bears any resemblance to classical physics. It certainly isn't specific to the representation of any single elementary unit, since it doesn't even blink (metaphorically) at shifting application from a one-electron to a two-electron system.

This suggests, to me anyway, two things. On the negative/cautionary side, it suggests a lack of information from which to choose a concrete representation for the "local" part of a physical system, which one might have thought would be the most straightforward and stable part of a cosmological "term". Perhaps more to the point, though, on the positive, insight-aiding side it suggests that if the quantum distortion is caused by some sort of non-local network playing out through rewrites in a dimension orthogonal to spacetime, we should consider trying to construct machinery for it that doesn't depend, much, on the particular shape of the local representation. If our distortion machinery does place some sort of constraints on local representation, they'd better be constraints that say something true about physics. Not forgetting, we expect our machinery to notice the difference between gravity and the other fundamental forces.

My most immediate goal, though, lest we forget, is to reckon whether it's at all possible any such machinery can produce the right sort of quantum distortion: a sanity check . Clues to the sort of thing one ought to look for are extremely valuable; but, having assimilated those clues, I don't atm require a full-blown theory, just a sense of what sort of thing is possible. Anything that can be left out of the demonstration probably should be. We're not even working with the best wave equation available; the Schrödinger equation is only an approximation covering the non-relativistic case. In fact, the transactional-interpretation folks tell us their equations require the relativistic treatment, so it's even conceivable the sanity check could run into difficulties because of the non-relativistic wave equation (though one might reasonably hope the sanity check wouldn't require anything so esoteric). But all this talk about relativistic and non-relativistic points out that there is, after all, something subtle about local geometry built into the form of the wave equation even though it's not directly visible in the local representation. In which case, the wave equation may still contain the essence of that co-hygienic difference between gravity and the other fundamental forces (although... for gravity even the usual special-relativistic Dirac equation might not be enough, and we'd be on to the Dirac equation for curved spacetime; let's hope we don't need that just yet).

The universe says 'hi'

Let's just pause here, take a breather and see where we are. The destination I've had my eye on, from the start of this post, was to demonstrate that a rewriting system, of the sort described, could produce some sort of quantum-like wave function. I've been lining up support, section by section, for an assault on the technical specifics of how to set up rewriting systems — and we're not ready for that yet. As noted just above, we need more information from which to choose a concrete representation. If we try to tangle with that stuff before we have enough clues from... somewhere... to guide us through it, we'll just tie ourselves in knots. This kind of exploration has to be approached softly, shifting artfully from one path to another from time to time so as not to rush into hazard on any one angle of attack. So, with spider-sense tingling —or perhaps thumbs pricking— I'll shift now to consider, instead of pieces of the cosmos, pieces of the theory.

In conventional quantum mechanics, as noted a couple of sections above, we've got basically three elements that we bring together: the parameters of our particular system of study, our classical laws of physics, and our wave equation. Well, yeah, we also have the Hermitian operator, but, as remarked earlier, we can set that aside since it's to do with interfacing to the system, which was our focus in that section but isn't what we're after now. The parameters of the particular system are what they are. The classical laws of physics are, we suppose, derived from the transformation rules of our cosmic rewriting system, with particular emphasis on the character of the primitive elements of the cosmos (whatever they are) and the geometry, and some degree of involvement of the network topology. The wave equation is also derived from the transformation rules, especially from how they interact with the network topology.

This analysis is already deviating from the traditional quantum scenario, because in the traditional scenario the classical laws of physics are strictly separate from the wave equation. We've had hints of something deep going on with the choice of wave equation; Transactional Interpretation researchers reporting that they couldn't use the non-relativistic wave equation; and then there was the odd intimation, in my recent post deriving quantum-like effects from a drastically simplified system that lacked a wave equation, that the lack of a wave equation was somehow crippling something to do with systemic coherence buried deep in the character of the mathematics. Though it does seem plausible that the wave equation would be derived more from the network topology, and perhaps the geometry, whereas the physical laws would be derived more from the character of the elementary physical components, it is perhaps only to be expected that these two components of the theory, laws and wave equation, would be coupled through their deep origins in the interaction of a single cosmological rewriting calculus.

Here is how I see the situation. We have a sort of black box, with a hand crank and input and output chutes, and the box is labeled physical laws + wave equation. We can feed into it the parameters of the particular physical system we're studying (such as a single electron in a potential field), carefully turn the crank (because we know it's a somewhat cantankerous device so that a bit of artistry is needed to keep it working smoothly), and out comes a wave function, or something akin, describing, in a predictive sense, the observable world. What's curious about this box is that we've looked inside, and even though the input and output are in terms of a classical world, inside the box it appears that there is no classical world. Odd though that is, we've gotten tolerably good at turning the crank and getting the box to work right. However, somewhere above that box, we are trying to assemble another box, with its own hand crank and input/output chutes. To this box, we mean to feed in our cosmic geometry, network topology, and transformation rules, and possibly some sort of initial classical probability distribution, and if we can get the ornery thing to work at all, we mean to turn the crank and get out of it — the physical laws plus wave equation.

Having arrived at this vision of an upper box, I was reading the other day a truthfully rather prosaic account of the party line on quantum mechanics (a 2004 book, not at all without merit as a big-picture description of mainstream thought, called Symmetry and the Beautiful Universe ), and encountered a familiar rhetorical question of such treatments: when considering a quantum mechanical wave function, "‍[...] what is doing the waving?" And unlike previous times I'd encountered that question (years or decades before), this time the answer seemed obvious. The value of the wave function is not a property of any particular particle in the system being studied, nor is it even a property of the system-of-interest as a whole; it's not part of the input we feed into the lower box at all, rather it's a property of the state of the system and so part of the output. The wave equation describes what happens when the system-of-interest is placed into the context of a vastly, vastly larger cosmos (we're supposing it has to be staggeringly vaster than the system-of-interest in order for the trick to work right), and the whole is set to jostling about till it settles into a stable state. Evidently, the shape that the lower box gives to its output is the footprint of the surrounding cosmos. So this time when the question was asked, it seemed to me that what is waving is the universe.

The upper box

All we have to work with here are our broad guesses about the sort of rewriting system that feeds into the upper box, and the output of the lower box for some inputs. Can we deduce anything, from these clues, about the workings of the upper box?

As noted, the wave function that comes out of the lower box assigns a weight to each state of the entire system-of-interest, rather than to each part of the system. Refining that point, each weight is assigned to a complete state of the system-of-interest rather than to a separable state of a part of the system-of-interest. This suggests the weight (or, a weight) is associated with each particular possibility in the classical probability distribution that we're supposing is behind the wave equation generated by the upper box. Keep in mind, these possibilities are not possible states of the system-of-interest at a given time; they're possible states of the whole of spacetime; the shift between those two perspectives is a slippery spot to step carefully across.

A puzzler is that the weights on these different possibilities are not independent of each other; they form a coherent pattern dictated by the wave equation. Whatever classical scenario spacetime settles into, it apparently has to incorporate effective knowledge of other possible classical scenarios that it didn't settle into. Moreover, different classical scenarios for the cosmos must —eventually, when things stabilize— settle down to a weight that depends only on the state of our system-of-interest. Under the sort of structural discipline we're supposing, that correlation between scenarios is generated by any given possible spacetime jostling around between classical scenarios, and thus roaming over various possible scenarios to sample them. Evidently, the key to all of this must be the transitions between cosmic scenarios: these transitions determine how the weight changes between scenarios (whatever that weight actually is, in the underlying structure), how the approach to a stable state works (whatever exactly a stable state is), and, of course, how the classical probabilities eventually correlate with the weights. That's a lot of unknowns, but the positive insight here is that the key lever for all of it is the transitions between cosmic scenarios.

And now, perhaps, we are ready (though we weren't a couple of sections above) to consider the specifics of how to set up rewriting systems. Not, I think, at this moment; I'm saturated, which does tend to happen by the end of one of these posts; but as the next step, after these materials have gone back on the shelf for a while and had a chance to become new again. I envision practical experiments with how to assemble a rewriting system that, fed into the upper box, would cause the lower box to produce simple quantum-like systems. The technique is philosophically akin to my recent construction of a toy cosmos with just the barest skeleton of quantum-like structure, demonstrating that the most basic unclassical properties of quantum physics require almost none of the particular structure of quantum mechanics. That treatment particularly noted that the lack of a wave equation seemed especially problematic; the next step I envision would seek to understand how something like a wave equation could be induced from a rewriting system. Speculatively, from there one might study how variations of rewriting system produce different sorts of classical/quantum cosmos, and reason on toward what sort of rewriting system might produce real-world physics; a speculative goal perhaps quite different from where the investigation will lead in practice, but for the moment offering a plausible destination to make sail for.

"Why is a raven like a writing desk?"
...
"No, I give it up," Alice replied. "What's the answer?"
"I haven't the slightest idea," said the Hatter.

— Alice's Adventures in Wonderland, Chapter 7, Lewis Carroll.

I'm always in the market for new models of how a system can be structured. A wider range of models helps keep your thinking limber; the more kinds of structure you know, the more material you have to draw inspiration from when looking for alternatives to a given theory.

Several years ago, in developing vau-calculi, I noticed a superficial structural similarity between the different kinds of variable substitution I'd introduced in my calculi, and the fundamental forces of nature in physics. (I mentioned this in an earlier blog post.) Such observed similarities can, of course, be white noise; but it's also possible that both seemingly unrelated systems could share some deep pattern that gives rise to the observed similarity. In the case of vau-calculi and physics, the two systems are so laughably disparate that for several years I didn't look past being bemused by it. But just recently I was revisiting my interest in physics TOEs (that's Theory of Everything, the current preferred name, last I heard, for what colloquially used to be called Grand Unified Theory, and before that, Unified Field Theory), and I got to thinking.

This will take a bit of set-up, and the payoff may be anticlimactic; but given the apparent extreme difficulty of making progress in this area at all, I'll take what I can get.

Contents
Substitution in vau calculi
Theories of Everything
Hygienic physics

Substitution in vau calculi

Traditionally in λ-calculi, all variables are bound by a single construct, λ, and manipulated by a single operation, called substitution. Substitution is used in two ways.

The major rearrangements of calculus terms take place in an action called β-rewriting, where a variable is completely eliminated by discarding its binding λ and replacing all references to it in the body of the old λ with some given argument. The part about eliminating the old λ is just a local adjustment in the structure of the term; but the replacement of references is done by β-substitution, which is not localized at a particular point in the term structure but instead broadcast across the body (an entire branch of the term's syntax tree). When you do this big β-substitution operation, you have to be careful. A naive rule for substituting argument A for variable x in body B would be "replace every reference to x in B with A". If you naively do that you'll get into trouble, because B might contain λs that bind either x, or some other variable that's referred to in A. Then, by following that naive rule you would lose track of which variable reference is really meant to refer to which λ. This sort of losing track is called bad hygiene.

To maintain hygiene during β-substitution, we apply α-renaming, which simply means that we replace the variable of a λ, and all the references to it, with some other variable that isn't being used for other purposes and so won't lead to confusion. This is a special case of the same sort of operation as β-substitution, in which all references to a variable are replaced with something else; it just happens that the something else is another variable. These two cases, β-substitution and α-renaming, are not perceived as separate functions, just separate uses of the same function — substitution.

It's possible to extend λ-calculus to encompass side-effectful behaviors — say, continuations and mutable storage — but to do so with well-behaved (technically, compatible ) rewriting rules, you need some sort of bounding construct to define the scope of the side-effect. In my construction of vau-calculus (a variant λ-calculus), I developed a general solution for bounding side-effects with a variable-binding construct that isn't λ, and operating the side-effects using a variable-substitution function different from β-substitution. (Discussed here.)

I ended up with four different kinds of variables, each with its own substitution operation — or operations. All four kinds of variables need α-renaming to maintain hygiene, though, and for the three kinds of side-effect variables, α-renaming is not a special case of operational substitution. If you count each variable type's α-renaming as a separate kind of substitution, there are a total of nine substitution functions (one both α and operational, three purely α, and five purely operational). λ-variables emerge as a peculiarly symmetric case, since they're the only type of variable whose substitutions (α and β) are commensurate.

This idea of multiple kinds of variables was not, btw, an unmixed blessing. One kind of variable — environment variables — turned out to be a lot more complicated to define than the others. Two kinds of variables (including those) each needed a second non-α substitution, falling into a sort of gray area, definitely not α-renaming but semi-related to hygiene and not altogether a full-fledged operational substitution. The most awkward part of my dissertation was the chapter in which I developed a general theory of rewriting systems with multiple kinds of variables and substitution functions — and the need to accommodate environment variables was at the heart of the awkwardness.

Theories of Everything

Theoretical physics can be incredibly complicated; but when looking for possible strategies to tackle the subject, imho the only practical way to think about it is to step back from the details and look at the Big Picture. So here's my take on it.

There are, conventionally, four fundamental forces: gravity, electromagnetism, the weak nuclear force, and the strong nuclear force. Gravity was the first of these we got any sort of handle on, about three and a half centuries ago with Isaac Newton's law of universal gravitation. Our understanding of electromagnetism dates to James Clerk Maxwell, about one and a half centuries ago. We've been aware of the weak nuclear force for less than a century, and the strong nuclear force for less than half a century.

Now, a bit more than a century ago, physics was based on a fairly simple, uniform model (due, afaics, to a guy about two and a half centuries ago, Roger Joseph Boscovich). Space had three Euclidean dimensions, changing with respect to a fourth Euclidean dimension of time; and this three-dimensional world was populated by point particles and space-filling fields. But then in the early twentieth century, physics kind of split in two. Two major theories arose, each of them with tremendous new explanatory power... but not really compatible with each other: general relativity, and quantum mechanics.

In general relativity, the geometry of space-time is curved by gravity — and gravity more-or-less is the curvature of space-time. The other forces propagate through space-time but, unlike gravity, remain separate from it. In quantum mechanics, waves of probability propagate through space, until observation (or something) causes the waves to collapse nondeterministically into an actuality (or something like an actuality); and various observable quantities are quantized, taking on only a discrete set of possible values. These two theories don't obviously have anything to do with each other, and leave gravity being treated in a qualitatively different way than the other forces.

Once gravity has become integrated with the geometry of space-time — through which all the forces, including gravity, propagate — it's rather hard to imagine achieving a more coherent view of reality by undoing the integration already achieved in order to treat gravity more like the other forces. As a straightforward alternative, various efforts have been taken to modify the geometry so as to integrate the other forces into it as well. This is made more challenging by the various discrete-valued quantities of quantum mechanics, as the geometry in general relativity is continuous. The phenomena for which these two theories were created are at opposite scales, and the two theories are therefore perceived as applying primarily at those scales: general relativity to the very large, and quantum mechanics to the very small; so in attempting to integrate the other forces into the geometry, modification of the geometry tends to be at the smallest scales. The two most recently-popular approaches to this are, to my knowledge, string theory and loop quantum gravity.

I've remarked in an earlier blog post, though, that the sequence of increasingly complex theories in physics seems to me likely symptomatic of a wrong assumption held in common by all the theories in the sequence (here). Consequently, I'm in the market for radically different ways one might structure a TOE. In that earlier post, I considered an alternative structure for physics, but I wasn't really looking at the TOE problem head-on; just observing that a certain alternative structure could, in a sense, eliminate one of the more perplexing features of quantum mechanics.

Hygienic physics

So here we have physics, with four fundamental forces, one of which (gravity) is somehow "special", more integrated with the fabric of things than the others are. And we have vau-calculus, with four kinds of variables, one of which (λ-variables) is somehow "special", more integrated with the fabric of things than the others are. Amusing, perhaps. Not, in itself, suggestive of a way to think about physics (not even an absurd one; I'm okay with absurd, if it's different and shakes up my thinking).

Take the analogy a bit further, though. All four forces propagate through space-time, but only gravity is integrated with it. All four operational substitutions entail α-renaming, but only β-substitution is commensurate with it. That's a more structural sort of analogy. Is there a TOE strategy here?

Well, each of the operational substitutions is capable of substantially altering the calculus term, but they're all mediated by α-renaming in order to maintain hygiene. There's really quite a lot more to term structure than the simple facets of it affected by α-renaming, with the quite a lot more being what the rewriting actions, with their operational substitutions, engage. There is, nonetheless, for most purposes only one α-renaming operation, which has to deal with all the different kinds of variables at once, because although each operational substitution directly engages only one kind of variable, doing it naively could compromise any of the four kinds of variables.

Projecting that through the structural analogy, we envision a TOE in which the geometry serves as a sort of "hygiene" condition on the forces, but is really only a tangential facet of the reality that the forces operate on — impinging on all the forces but only, after all, a hygiene condition rather than a venue. Gravity acts on the larger structure of reality in a way that's especially commensurate with the structure of the hygiene condition.

Suggestively, quantum mechanics, bearing on the three non-gravitational forces, is notoriously non-local in space-time; while the three kinds of non-λ variables mediate computational side-effects — which is to say, computational effects that are potentially non-local in the calculus term.

The status of gravity in the analogy suggests a weakness in the speculations of my earlier post on "metaclassical physics": my technique for addressing determinism and locality seems to divorce all forces equally from the geometrical structure of reality, not offering any immediately obvious opportunity for gravity to be any more commensurate with the geometry than any other force. I did mention above, that post wasn't specifically looking at TOEs; but still, I'm inclined to skepticism about an approach to fundamental physics that seeks to mitigate one outstanding problem and fails to suggest mitigation for others — that's kind of how we ended up with this bifurcated mess of relativity and quantum mechanics in the first place. As I also remarked in that post when discussing why I suspect something awry in theoretical physics, while you can tell you're using an unnatural structure by the progression of increasingly complicated descriptions, you can also tell you've hit on the natural structure when subsidiary problems just seem to melt away, and the description practically writes itself. Perhaps there's a way to add a hygiene condition to the metaclassical model, but I'd want to see some subsidiary problems melting.

Supposing one wants to try to construct a TOE, metaclassical or not, based on this strategy, the question that needs answering is, what is the primary structure of reality, to which the geometry serves a sort of tangential hygiene-enforcement role? For this, I note that the vau-calculus term structure is just a syntactic representation of the information needed to support the rewriting actions, mostly (though not exclusively) to support the substitutions. So, the analogous structure of reality in the TOE would be a representation of the information needed to support... mainly, the forces, and the particles associated with them. What we know about this information is, thus, mainly encapsulated in the table of elementary particles. Which one hopes would give us the wherewithal to encompass gravity — the analog to β-substitution — since it includes a mediating particle for mass: the Higgs boson.

[Note: I've further explored the rewriting/physics analogy in a later post, here.]

Nature shows us only the tail of the lion. But there is no doubt in my mind that the lion belongs with it even if he cannot reveal himself to the eye all at once because of his huge dimension.

— Albert Einstein

What theoretical physics needs, I've long believed, is to violate some assumption that is shared in common by both classical physics and quantum mechanics. Everyone nowadays seems to understand that something different is needed, but I suspect the "radical" new theories I've heard of aren't radical enough.

So in this post I'm going to suggest a class of physical theory that seems to me to be radical enough (or if, you prefer, weird enough) to shake things up.

Indirect evidence of a wrong assumption

Suppose you're trying to find the best way to structure your description of something. (Examples: choosing the structure for a computer program to perform some task; or choosing the structure for a theory of physics.)

What you hope to find is the natural structure of what you're describing — a structure that affords a really beautiful, simple description. When you strike the natural structure, a sort of resonance occurs, in which various subsidiary problems you may have had with your description just melt away, and the description practically writes itself.

But here's a problem that often occurs: You've got a structure that affords a pleasing approximate description. But as you try to tweak the description for greater precision, instead of the description getting simpler, as it should if you were really fine-tuning near the natural structure, instead the description gets more and more complicated. What has happened, I suggest, is that you've got a local optimum in solution space, instead of the global optimum of the natural structure: small changes in the structure won't work as well as the earlier approximation, and may not work at all, so fundamental improvement would require a large change to the structure.

I suggest that physics over the past century-and-change has experienced just such a phenomenon. Classical physics was pleasing, but an approximation. Our attempts to come closer to reality gave us quantum mechanics, which has hideously ugly mathematics. And our attempts to improve QM... sigh.

You'll find advocates of these modern theories, and before them advocates of QM, gushing about how beautiful the math is, but frankly I find this wishful thinking. They focus on part of the math that's beautiful, and try to pretend the ugliness out of existence. Ignoring the elephant in the room. In the case of QM, the elephant is commonly called "observation", and in more formal social situations, "wave function collapse".

But it would be a mistake to focus too much on the messiness of QM math. If physics is stuck in a local optimum, we need to look foremost at big things that classical and quantum have in common, rather than getting too focused on details by which they contrast.

The saga of determinism and locality

Two really big ideas that have been much considered, in the contrast between classical and quantum physics, are determinism and locality.

In the classical view of reality, there are three dimensions of space and one dimension of time. In space, there are point particles, and space-filling fields. The particles move continuously through space over time, each tracing out a one-dimensional curve in four-space. The fields propagate in waves over time. The model is deterministic because, in principle, the state of everything at one moment in time completely determines the state of everything at all later moments in time. The model is local because neither particles nor waves travel faster than a certain maximum speed (the speed of light).

QM depicts nature as being fundamentally nondeterministic. Einstein didn't approve of that (it apparently offended his sense of the rhyming scheme of nature, as I've remarked before). God does not play dice, he said.

It's important to realize that Einstein was personally responsible, through his theory of special relativity, for making classical physics more local. Prior to relativity, classical physics did not prohibit things from moving arbitrarily fast; consequently, in considering what would happen to a given volume of space in a given interval of time, there was always the chance that by the end of the interval, some really fast particle might come zooming through the volume that had been on the other side of the universe at the beginning of the interval.

This relation between Einstein and locality helps us appreciate why Einstein, in attempting to demonstrate that quantum mechanics is flawed, constructed with two of his colleagues the EPR paradox showing that QM requires information to propagate across space faster than light. That is, in an attempt to discredit nondeterminism he reasoned that quantum nondeterminism implies non-locality, and since non-locality is obviously absurd, quantum nondeterminism must be wrong.

Perhaps you can see where this is going. Instead of discrediting nondeterminism, he ultimately contributed to discrediting locality.

Okay, let's back up a few years, to 1932. As an alternative to quantum nondeterminism, Einstein was interested in hidden variable theories. A hidden variable theory says that the state of reality is described by some sort of variables that evolve deterministically over time, but these underlying variables are fundamentally unobservable, so that the nondeterministic quantum world is merely our statistical knowledge about the hidden deterministic reality. In 1932, John von Neumann proved, formally, that no hidden variable theory can produce all exactly the same predictions as QM. (All hidden variable theories are experimentally distinguishable from QM.)

This is an example of a no-go theorem, a formal proof that a certain kind of theory cannot work. Often, the most interesting thing about a (correct) no-go theorem is its premise — precisely what it shows to be impossible. Because twenty years later, in 1952, David Bohm published a hidden variable theory experimentally indistinguishable from QM. The hidden variable theory was correct. The no-go theorem was correct. We can therefore deduce that what Bohm did must be different from what von Neumann proved impossible.

And so it was. On careful examination, von Neumann's proof assumes the hidden variable theory is local. Bohm's hidden variable theory has a quantum potential field that can propagate arbitrarily fast, so Bohm's theory is non-local. Einstein remarked, "This is not at all what I had in mind."

Now we come to Bell's Theorem. Published in 1964 (nine years after Einstein's death), this was another no-go theorem, based on a refinement of the EPR experiment. Bell showed that the particular probability distribution predicted by QM, in that experiment, could not possibly be produced by a deterministic local hidden variable theory.

Okay. Here again, maybe you can see already where I'm going with this. I'm preparing to propose a particular way of violating an assumption in the premise of Bell's Theorem. This particular violation may allow construction of a theory that isn't, to my knowledge, what any of the above cast of characters had in mind, but that might nevertheless be plausibly called a deterministic local hidden variable theory.

Changing history

The probability distribution Bell was considering, the one that couldn't be produced by a deterministic local hidden variable theory, has to do with the correlation between observations at two distant detectors, where both observations are based on a generating event that occurred earlier in time, at a point in space in between the detectors.

And one day some years ago, reading about all this, it occurred to me that if you think of these three events —two observations and one generation— as just three points between which signals can be sent back and forth, it's really easy to set up a simple mathematical model in which if you start with a uniform probability distribution, set the system going, and let the signals bounce back and forth until they reach a steady state, the probability distribution of the final state of the system will be exactly what QM predicts. This idea is somewhat reminiscent of a modern development in QM called the transactional interpretation (different, but reminiscent).

The math of this is really easy; it doesn't involve anything more complicated than a dot product of vectors. Wait, propagating back and forth in time? What does that even mean?

There are a lot of really badly thought-out depictions of time travel in modern science fiction. For which I'm sort-of grateful, because over the years I've been annoyed by them, and therefore thought about what was wrong with them, and thereby honed my thinking about time travel.

It seems to me the big problem with the idea of changing history is, what does "change history" mean? In order for something to change, it has to change relative to some other dimension. If a board is badly milled, its thickness may vary (change) along the length of the board, meaning its thickness depends on how far along its length you measure. The apparent magnitude of a star may vary with distance from the star. The position of a moving train varies over time. But if history changes, relative to what other dimension does it change? It isn't changing relative to any of the four dimensions of spacetime.

Let's suppose there is a fifth dimension, relative to which the entire four-dimensional spacetime continuum can change. As a simple name, let's call it "meta-time". This would, of course, raise lots of metaphysical questions; a favorite of mine is, if there's a fifth dimension of meta-time, why not a sixth of meta-meta-time, seventh of meta-meta-meta-time, and so proceed ad infinitum? Though fun to muse on, those sorts of questions aren't needed right now; just suppose for a moment there's meta-time, and let's see where it leads.

While we're at it, let's suppose this five-dimensional model is deterministic in the sense that, in principle, the state of spacetime at one moment in meta-time completely determines the state of spacetime at all later moments in meta-time. And let's also suppose the five-dimensional model is local in the sense that changes to spacetime (whatever they are) propagate, over meta-time, at some maximum rate. (So if you hop in your TARDIS and make a change to history on a particular day in 1963 London, that change to history propagates outward in space at, say, no more than 300,000km per meta-second, and propagates forward and backward in time at no more than one second per meta-second.)

That bit of math I mentioned earlier, in which the QM probability distribution of Bell's Theorem is reproduced? That can be made to use just this kind of model — a five-dimensional system, with three dimensions of space, one of time, and one of meta-time, with determinism and locality relative to meta-time. Granted, it's only a toy: it's nothing like a serious attempt to model reality with any generality at all, just a one-off model describing the particular experimental set-up of Bell's Theorem.

I've got one more suggestion to make, though. And I still won't have a full-blown theory, such as Bohm had (there's stuff Bohm's theory didn't include, but it did have some generality to it), but imho this last point is worth the price of admission. I wouldn't call it "compelling", because atm this is all too outre to be compelling, but I for one found it... remarkable. When I first saw it, it actually made me laugh out loud.

Metaclassical physics

Wondering what a full-blown theory of physics of this sort might look like, I tried to envision what sorts of things would inhabit this five-dimensional model.

In classical physics, as remarked, space contains point particles interacting with fields. And when you add in time, those things that used to look like point particles appear instead as one-dimensional curves, tracing the motion of the particle through spacetime. I was momentarily perplexed when I tried to add in meta-time. Would the three events in Bell's experiment, two observations and one generation, interact through vibrations in these one-dimensional curves traced through spacetime? Modern string theory does make a big thing out of stuff vibrating. Also, though, a one-dimensional curve vibrating, or otherwise evolving, over meta-time traces out a two-dimensional surface in the five-dimensional space-time-metatime continuum. We set out on this journey hoping to simplify things, hoping ideally to strike on the natural structure of physical reality and achieve resonance (the ring of truth?).

But wait. Why have point particles in space? Point particles in classical physics are nice because of the shape of the theory they produce – but points in space don't produce that shape of theory when they're moving through both time and meta-time. And those one-dimensional curves in spacetime don't play nearly such a central role in QM, notwithstanding they make a sort of cameo appearance in Feynman's path integral formulation.

What is really fundamental to QM is the elephant in the room, the thing that makes such a hideous mess out of QM mathematics: observation, known at up-scale parties as wave function collapse. QM views spacetime as a continuum punctuated by zero-dimensional spacetime events — essentially, observations.

And as spacetime evolves over meta-time, a zero-dimensional spacetime event traces out a one-dimensional curve.

So now, apparently, we have a theory in which a continuum is populated by zero-dimensional points and fields, evolving deterministically over a separate dimension with a maximum rate of propagation. Which is so much like classical physics that (as mentioned) when I saw it I laughed out loud.