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.
Primitive wave functions
The universe says 'hi'
The upper box
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,
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.
iℏ ∂ Ψ ∂t = Ĥ Ψ .
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.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.— J.S. Bell, "Six possible worlds of quantum mechanics", Speakable and unspeakable in quantum mechanics (anthology), 1993.
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.