I read Feynman's QED back in high school and was fascinated by it, but always had a lingering question.
Sure the math works out by summing all possible paths, and then most of them usually cancel each other out, and so you're left with the ones that mostly contribute to the actual probabilities, as complicated as those can still be.
But I always suspected... surely that just has to be one mathematical way of looking at it. That at some point we'd discover that turns out to be mathematically identical to something else that would be a bit more local, a bit more intuitive, a bit more straightforward. That never even had to deal in the first place with the infinity of paths that all wind up cancelling each other out anyways.
So now that I'm on a forum with smart people, the kind I didn't have access to back in high school... has there been any progress towards something like this at all? Are there any directions for this? Or is there a conceptual reason I'm missing why this fundamentally isn't possible, why the sum of all possible paths is the only way to account for certain behaviors?
In the past 10 or so years, a combinatoric approach to particle interactions has shown surprisingly good results. Simple algebraic calculations have been able to give answers that requires hundreds of pages of math through the feynman diagram approach.
This is the amplituhedron, led by Nima Arkani-Hamed, here's an article also by quanta magazine about it
In addition to giving exact simple calculations to particle scattering amplitudes, the most important aspect is that it does so in a way that does not require notions of space or time. Leading to the notion that there are more fundamental aspects to reality and space and time may be emergent phenomena.
This is the third formulation I've come across that sounds like this with space and time emerging from particle interactions. The other two are from Stephen Wolfram and Carlo Rovelli. Are the formulations related do you know?
I've struggled to get beyond their popular science descriptions to the actual theories, I find descriptions like the amplituhedron being 'a multidimensional jewel' distracting, if you have any suggestions where to start for a deeper understanding of this one I'd be interested.
Until we get evidence of such a thing, those ideas have little credibility though they are nice thought experiments.
What they're really seeing is that mathematically something can be expressed with less dimensions or degrees of freedom than what you observe in the real world, and then make the conclusion that therefor the dimensions and properties etc. that we observe are some emergent property and not fundamental.
But you can't make this conclusion from the mathematical model.
For example, if I have a finite sized two dimensional plane where each point is associated with some function f(x,y),you can trivially express it as a one dimensional system where all the rows of the plane are sort of unwound onto a single line.
This trick does not work for infinite 2D spaces but there are other ways to remap infinite sized spaces onto finite ones (e.g. via tan).
Yet there's nothing fundamental about this, it's just a mathematical modelling trick.
this trick works for infinite sets of the same cardinality and in fact works continuously for the case you are describing. there is a surjective continuous function fron R to R^n for any n
The biggest clue we have right now is illustrated by last year's nobel prize. Particles in an entangled state can interact instantaneously, disregarding space and time entirely.
It's a common thing to see in theories that try to unify QM with GR because in GR spacetime is dynamic, while in QM it's static. For QM to produce GR, there needs to be some way of generating space and time instead of taking it as an assumption.
You can find quite a few lectures by Nima Arkani-Hamed on YouTube including some on the amplituhedron. But they are mostly not gentle introductions and require some knowledge of physics to follow along. I think this one [1] provides a lot of background and explains the motivation.
I used to follow his lectures, but in the past 3 or 4 years he's become really quiet. I wonder if people stopped inviting him because he always overruns his time limit, or if he's too busy making progress on his theory.
Theoretical physicists are, on the whole, not a silly bunch and there do exist various "mathematically identical" ways to look at quantum field theory. But by the very definition of "mathematically identical" one can find the path integrals somewhere in these alternative descriptions anyway.
Contrary to your stated desire, I actually think that the path integral formulation is in some sense the most "intuitive" and "local". I apologize for not taking the time to explain that here, but let me instead try to convince you that "locality" is not at all obvious in quantum field theory.
One intuitively non-local process is the following. If I start with a state with one electron at the origin at time 0, what is the probability that I measure an electron one lightyear away one minute later? Naively this probability is zero, because the electron that you started with cannot move faster than light. But the actual (and measured, in a less idealized setting) probability is not zero, since vacuum fluctuations can create an electron exactly one lightyear away and you end up measuring that one. That, a physicist would say, is perfectly cromulent with locality of the underlying theory.
I hope this gives a glimpse of how subtle causality (and hence locality) really is, and why at least some believe that path integrals may actually be the most "intuitive" description available.
That's like looking at distant synchronized clocks and concluding they communicate faster than light. Your wish to make it look nonlocal isn't supported by physics.
> Theoretical physicists are, on the whole, not a silly bunch
... at least until the workday ends.
My own feeling is that it's obviously local if you consider that there really are virtual particles taking the non-critical paths. I know this is a bit interpretation-dependent though. GP may be interested to read e.g. https://en.m.wikipedia.org/wiki/Double-slit_experiment#Inter...
Of course the simple way around this issue is the best: "shut up and calculate".
Yes, this always bothered me as well. It happens to annoy some physicists too, so there has been a resurgence of interest in alternative (less magical?) interpretations of quantum mechanics in the past decade.
There are other very interesting, and intuitive, features of this line of modeling. I think it's better to have a realistic mental model of the microworld even if the overall QM predictions aren't (obviously, yet) affected. Maybe there's more here than immediately obvious to the mainstream physics community.
Physics is a model of the way the universe works. There's no reason to be sure that it does work the way that we model it. Practically speaking though it doesn't really matter as long as the model gives us accurate predictions. At that point you are entering the realm of philosophy.
A model can become a cage. Although the QM "infinite trajectories" interpretation is fine at explaining what happens in a quantum system from the perspective of an observer. But it doesn't provide any intuition about how the systems really behave. There may be ways to get access to more realistic interpretations. If we had them they might help us work with quantum systems.
That's a fascinating philosophical presupposition you have there: That there is such a thing as a difference between the observable* behavior of a system and its real* behavior. I won't agree or disagree with you. Just pointing it out because it's really interesting to me.
*You used essentially those words ('observer', 'really'). I won't try to pin you down into a particular definition of them, but I'm guessing we at least approximately agree on what they mean.
You could make a similar but less controversial statement by substituting "low-fidelity observer" & "high-fidelity observer" for "observer" & "reality". I wonder if that's what you mean, or if you meant what you said?
Hm... I suppose this could be mathematically/logically equivalent, but the thing that bugs me in GP's answer (and the way I hear QM talked about in general) is the very concept of an observer. I've never heard a good explanation of why an observer would be a thing - I'm not sure if there could be, without creating some kind of magical realm for it to live in, detached from the rest of physics.
What "an observer" really means, in the context of QM, is a large, approximately classical system interacting weakly with the system being "observed" (such that the interaction can be modeled perturbatively). Humans "observe" photons, but so does a pail of water.
It's a way of framing certain classes of problems, not a distinct sort of entity.
We, humans, are limited in our understanding of the worlds by what we can perceive and conceive. What does it means to understand how a system really behave when we are so limited? That’s an interesting metaphysical question but we have left the realm of sciences by asking it. Sciences care about what can be experienced.
The aim of sciences is to accurately predict the results of experiments. A model doesn’t have to be simple or intuitive, just accurate.
Obviously everyone would like a simpler model predicting the same things but not because it would be more real, just because simpler is nicer.
Feynman's QED talks pretty extensively about this as well. An intuitive mental model that humans can picture in their heads is a nice to have, not a requirement. Understanding how a system "really works" is also not really a goal of physics - the science is about predicting physical phenomena - and we generally choose whatever is the simplest equation we have currently available that does that. If you want to define the bar for explaining how a "system really works" as being able to give you an intuitive human-centric mental model you can fully understand and capture in your head - well, you are moving the goal posts over what physics really is and quantum physics has already left you far behind.
> Understanding how a system "really works" is also not really a goal of physics - the science is about predicting physical phenomena
I have to point out the above is a rather strong and debatable claim. I think most people who study science do want to understand how things really work.
Even if we accept that physicists only want to predict phenomena as opposed to understanding them, the same can't be said of other sciences (let's say astronomy, or molecular genetics). Mathematicians seek almost exclusively to understand things rather than predict them. So the claim attributes a certain lack of curiosity specifically to physicists, that doesn't apply to biologists, mathematicians, etc. To me, that makes it even more peculiar. So I'm skeptical.
I put "really works" in quotes because the definition of how to get to the bottom of how something "really works" is up for debate itself. The original context for claiming something like path integrals can't be how the system "really works" because it is some combination of being too complex, lacking an intuitive mental model for how to conceptualize it, and invoking things that seem extraneous to the person making the claim (possible paths that aren't necessarily realized).
I would agree that physicists have the same, or perhaps even more, curiosity regarding discovery as other fields (partly why they are drawn to understanding the nature of the existence/universe itself) - but there is certainly an error with this narrow definition of "this can't be how it really works because X" is simply wrong - we choose the simplest explanation which meets the scientific criteria for the time, and up until something new is discovered - that is de facto how the system "really works".
Ultimately people have this assumption from classical mechanics that everything can neatly fit into intuitive thought experiments to understand the nature of reality, but quantum mechanics has not followed the same human-centric intuitive modeling. Does it mean we haven't figured out how the system "really works"? Ultimately, it's a philosophical question, not question directly related to physics which is showing the way through predictable repeatable science.
The path integral formulation doesn't sound that counterintuitive to me, though I don't know anything about it in terms of actually computing with it. Aren't there similar integrals in statistical thermodynamics, where what is "really going on" is random motion of molecules and the math has to consider every possible path, though in Euclidean space? I'd like to understand this stuff better someday. The "really going on" of quantum mechanics is perhaps more mysterious, but that doesn't mean physicists don't want to find it.
You mean you don't understand when you don't know something? If you had no problem reasoning about quantum physics, you wouldn't say it's a philosophical question.
My post is not a statement about animal sentience. Until we can have a discussion about empiricism with another kind of sentience, I think beginning sentences with "we, humans" will be fine. I would appreciate if you could go farm PC points somewhere else than in reply to my comments and kept condescending winks out of it.
Unless you are making a pun about AI bots and I’m just having a bad day, in which case, I will have to ask you to be kind enough to bear with my rudeness.
My comment was not motivated by any notion of politics. Notice I didn't say "organism", but something far more general. A galaxy is a subjective agent in the sense that experiences incident upon it are subjective in nature (localized in time and space vs its environment, subject to some notion of "interpretation", broadly defined, on the part of the receiver).
Besides, there is nothing about a public-facing forum that prohibits anyone commenting on your posts for whatever reason. Since you brought politics into this, I will just say we live in a society and you have to accept that if you are to avoid such unhealthy and reactionary outbursts in the future.
There is no politic in my post. You were literally trying to correct my use of language. My previous comment is a polite way to tell you it’s unwelcome.
> But it doesn't provide any intuition about how the systems really behave.
In physics, intuition is much more of a cage than the models.
In fact, I would argue that our intuition itself is such a model. It's just one that vertebrates evolved over 100s of millions of years that enabled their brains to optimize Darwinian fitness.
It has layers upon layers of simplifications and heuristics to reduce complexity to a levels it can use to compute viable behaviour.
Models answer the questions What? and How? They do not answer the question Why?
The simplest model explaining all observed behavior, and even better, making new predictions that are then confirmed, is the model we tend to call "reality." But it's just a model. When you step back to think about it - everything is a model: colors, sounds, the table you're sitting at, the chair you're sitting on. We've just become so accustomed to these models we think of them as "reality" and it's super convenient to do so. I think that's a big point the Buddha was trying to get across.
Such a "truest" formula doesn't really exist, or at least what you determine to be the truest is just a matter of definition and taste.
For any equation it's mathematically trivial to come up with a different set of equations that produce the same result, e.g. for example just via approximating the original function with some infinite series that is guaranteed to give back the original result in the limit.
But even beyond such trite examples, it's not unlikely that there will simply be multiple competing ways to model the same data that the equation takes in and spits out that are very different in form and function, and perhaps even in mathematically incompatible ways (this could happen if e.g. the equations model more than what exists in reality but all of reality is described by some subset of the parameters of these equations, like how gravity works for negative masses but such a thing does not exist from what we know).
You could then decide on some reasonable criteria which of all your models is the truest one, but the criteria themselves will be up for subjective debate.
It isn't the only way. I would argue that the "sum over paths" is more of a convenient language for categorizing QFTs than an actual statement of physical reality. Path integrals have the benefit that they can be easily connected to correlation functions by a Wick rotation, are naturally invariant under relativity, and work (kinda) well with gauge theories (well better than any other way we know).
However, in practice they are not used directly. Instead, the path integral is either treated perturbatively giving you Feynman diagrams, or by working out the Hamiltonian prescription (probably closest to what you are thinking of), or by using Monte Carlo approaches (there are probably other approaches I don't know of too). All of these approaches are easier to calculate with, but are a less natural language for abstractly describing a QFT.
Alright fair enough, I'm not an expert on how those work. I was under the impression that there is quite a bit of work that has to be done to turn a path integral into something amenable to Monte Carlo though.
Well it depends on what you mean by "quite a bit of work"! Wick rotate, focus on a finite volume of spacetime, and discretize the spacetime. That's all. The first step changes a difficult-to-sample distribution exp( i S ) into an easy-to-sample exp( – S ). The other two steps keep you from needing infinite RAM.
The discretization and finite-volume approximations may be removed by doing calculations with different spacings and volumes and extrapolating. The Wick rotation is more subtle, and typically means only some observables are accessible. To study the path integral with real time causes a very difficult sign problem [essentially: you need an exponentially large number of samples in order to get cancellations under control], which is why real-time quantum dynamics is an exciting potential application of quantum computers.
You can take an amplitude-vs-time representation of a signal and via a Fourier Transform instead represent the same information as a sum of weighted complex exponentials in the frequency domain. It works mathematically, sure- but does that mean that- physically, at the core of existence- every RF, acoustic, seismic, or financial data ripple is actually a bunch of sines and cosines which are getting summed together to create the real phenomena?
99% of modern physics was developed for pen & paper mathematics, not numeric simulations. Even when computers became readily available it was seen as somehow "less than" symbolic solutions. More practically, special-case solutions that were tractable symbolically require far too much computer power if approached from a fully general "local" simulation perspective.
QED is essentially a Monte Carlo simulation, just like ray tracing in computer graphics. The difference is that QED will correctly model diffraction (and other more complex physical effects) that the simplified graphics-only model doesn't bother with.
Both QED and raytraced computer graphics rely on a mathematical dualism where a wavefront can be tracked either as a surface OR as a vector perpendicular to the wavefront. The whole "summing up but mostly cancelling" is just accounting for the wavefronts. The use of complex numbers is mostly also just a method for tracking the cyclic nature of wavefronts.
If you skip the "waves" part, it becomes traditional computer graphics. Less accurate, but essentially the same.
Computer graphics could also simulate light transport by using a 3D volume, which is roughly speaking how some diffuse light simulations work in real-time games like the Unreal engine. The problem is that to get decent accuracy, you need a lot of memory to store that 3D volume. For low-resolution scene lighting this is fine.
Monte Carlo enables a solution to be built up incrementally with much less memory required, needing only a 2D accumulator in the image plane instead of a 3D simulation in object space. This allows a vastly more complex and detailed simulation to be calculated, which is important for the tiny wavelength of physical light instead of some non-diffracting idealisation of it.
For a physics problem the memory requirements of any non-trivial system would be gigantic, that's why everyone just ignores this approach even though it makes a lot more intuitive sense.
> that's why everyone just ignores this approach even though it makes a lot more intuitive sense.
- No one ignores the correspondence between QFTs and stochastic processes (which is what I assume you're talking about, since the claim that QED is just Monte Carlo is extremely false if taken literally), any halfway decent intro QFT class will cover it.
- It only "makes more intuitive sense" because you've papered over all the gory details. There's very little intuitive or natural about analytic continuation in C^{\infty}, for instance.
Stochastic processes is one thing. But most calculations in QFT is done using various perturbation theories (kind of like Taylor series expansion), instead of by using exact calculation.
That could be interpreted kind of like ray tracing with finite resolution and a finite number of reflections.
EDIT: Feynman diagrams serve as an attractive way to rationalize the various perturbations, but it may be that a "true" theory needs to be able to solve the (for us) unsolvable integrals directly.
Especially since the perturbation theory integrals (that we can solve) diverge, and require renormalization to avoid infinities.
While in principle you can evaluate certain path integrals by using MCMC, in practice you run into all kinds of issues like the sign problem, which even with things like complex Langevin dynamics can't be resolved to this day completely.
Even if the insult is unintended, or is plausibly deniable, if the person ends up being insulted anyway, then it is unlikely to go well.
In totally unrelated news, it might be worth noting that "being right" is not a relevant defence against the charge of "being an asshole". It's entirely possible to be one without being the other.
> I understand that you are fine being called an idiot and that the proper response is a smile and nod of understanding. Some do not. I do not.
Because the only possible responses are "smile and nod of understanding" and "call the professor an idiot in return", and no middle ground exists where you can draw attention to someone's mistaken beliefs without escalating the situation, or even without putting them on the defensive?
> you can draw attention to someone's mistaken beliefs without escalating the situation, or even without putting them on the defensive?
This requires a will to discuss (both sides). I have had plenty of such discussions where the other side arguments sinked in or not, but where the was will to actually listen and think.
I don't believe you only met such people, and if so congratulations (I am not being cynical - it would really be great).
Unfortunately this was not the case, not even close. He was an asshole using his power to dominate others, something which works in some places where there is a medieval structure.
I did not friends on him and he said that we would meet at my defence where he will remember that. To what I said that I don't do, but wish him all the best.
He was childishly ridicule at the defense where he absolutely wanted to be part of the jury.
The world is sadly what it is. My life motto is "always be nice twice" (stolen from Irving) and I like to be the one extending the hand after a failed first meeting. Some take this as weakness or submission, and well, b it does not go well indeed.
The path integral formulation is possibly the most local theory you could come up with - it's incredibly simple and does not require any nonlocal effects! The relativistic lagrangian is even more elegant than the classical one - it's just mass! What kind of theory would be more intuitive and straightforward?
When you look at quantum mechanics from Schroedinger's equation point of view, it is very simple. The implications of it are not, which is true of many simple systems. What is difficult in quantum mechanics is trying to understand how those equations have anything to do with what we see.
The path integral interpretation actually does give us an intuitive picture of why we see classical physics. If you zoom out, the stationary point of the Lagrangian is all that survives, with other things mostly canceling out as you mentioned. This explains why in lagrangian mechanics of clasical physics we are told, without reason, that the equations of motion are the stationary point of the Lagragian.
You don't say this, but I think people think quantum mechanics is weird just because it is different from what they see. People also had trouble with the idea of us moving around the sun, one reason being because it looked like everything was going around us. Of course, if we did go around the sun, it would still look like things went around us. In the end though neither is really true. The sun in the center is just a better model. Likewise, quantum mechanics is probably just a better model to "reality" than classical physics.
I'm tempted to go check, I read that too around the end of HS or very start of college, but isn't QED the one that starts off with a story about counting physical beans in jars to do familiar arithmetic operations like summing and multiplying? Then the rest of QED is presented as describing how QED works using a similar counting-beans approach that everyone can understand vs. the more sophisticated methods that working physicists use that require a few years of undergrad training at least. That both end up having infinities at the core is an important part of being able to accurately (mathematically) describe nature, like counting beans vs. just using a calculator both involve numbers at the core to accurately describe to other humans what's going on -- and numbers can seemingly be "canceled out" when removing beans from a jar! (Or perhaps adding a different color bean does the canceling.) Without thinking about it further I suppose this falls under the umbrella question of whether reality is fundamentally continuous (as we assume when we model things with calculus) or discrete.
Why wouldn't summing all possibilities be the simpler solution?
Thing about Monte Carlo simulations - it's simpler to just try all possibilities than to figure out what is the formula of the correct result.
Neural networks also work like that - you randomly do things and propagate errors with simple rules instead of figuring out the complex formula that would give you the answer.
There seems to be a lot of power in summing up all the things.
If our universe is a computation, our practical experience seems to suggest that a Monte Carlo like simulation of it would be more efficient than a closed-form formula.
Summing all the things is similar to just a big "OR" right? It lets the important signal through (you don't know which it is) and the other ones just have to be suppressed or suppressed enough to not interfere. For example all the other ones might as well be as if stochastic, so on average you get it right.
>is there a conceptual reason I'm missing why this fundamentally isn't possible, why the sum of all possible paths is the only way
It's not so much a conceptual reason it isn't possible so much as that's just the way the universe is. If you have your particle going from A to B, changing things at some possible other path C simply does change the way it arrives at B, like it or not.
Feynman's take was:
>If you will simply admit that maybe [nature] does behave like this, you will find her a delightful, entrancing thing. Do not keep saying to yourself, if you can possible avoid it, "But how can it be like that?" because you will get 'down the drain', into a blind alley from which nobody has escaped. Nobody knows how it can be like that.
Personally though I actually like pondering how it can be like that as an interesting puzzle. But it's the way the universe is, not a conceptual thing. My guess as to why is that reality at the most basic level is more like a path integral than like objects whizzing around.
Many-Worlds probably fits what you're looking for, that conforms with orthodox quantum mechanics. How to bootstrap probability seems like the only open question there.
If you want something less orthodox, then superdeterminism is seeing a bit of a revival. For instance, Tim Palmer's Invariant Set Theory (Invariant Set postulate on Wikipedia).
Born rule postulates that probabilities exist at sample size 1, which is not supported by reality: in reality probabilities are unobservable at sample size 1, which better corresponds to MWI than copenhagen. MWI shoves it in your face why probabilities are unobservable at sample size 1.
> So now that I'm on a forum with smart people, the kind I didn't have access to back in high school... has there been any progress towards something like this at all? Are there any directions for this? Or is there a conceptual reason I'm missing why this fundamentally isn't possible, why the sum of all possible paths is the only way to account for certain behaviors?
It certainly not the only way. One easy trick: you put it under multiple layers of definition, and at the lowest layer you just do the integral without calling anything "paths" or so. (Equivalent theories can be seen as bastardization of each other, that's why none of them says anything meaningful about ontology.)
It kind of feels like the epicycles that were used to make geocentrism work. Sure, they make the math work out correctly, but the equations for the motion of the planets become needlessly more complex than if you just take the sun to be the center.
The canceling out part is the reason quantum computing works. We basically just avoid computing a bunch of bad answers making the actual problem faster to solve. So it has some observable examples.
The summing over all paths is basically already there when you do the two slit experiment. In that case you sum over two possible paths. So it is already right there in the basis of quantum mechanics.
I think the fact that the oddest approaches work aren't exactly a measure of how the particles actually work in physics. I think it proves that math isn't the language of the universe no matter how much mathematicians and physicists want to say it is. It just proves we're good at modeling but not good enough to actually know what we're seeing/measuring (a natural limit to our knowledge). I don't know why this position is considered controversial or out of the mainstream when it seems to be the logical answer.
I don't see why it's logical to conclude that difficulties modeling hard problems means we can't model them using math. These difficulties are not uncommon in the history of science, and we've eventually solved them all before with math, and we should expect such problems to become more and more difficult as the low hanging fruit has been plucked. I see literally no reason to jump to the conclusion that the core problem is trying to use math at all.
>I don't see why it's logical to conclude that difficulties modeling hard problems means we can't model them using math.
That's not what I'm stating though. I'm stating that the math involved is a model but it isn't ever going to be identical to the thing being modeled. Meaning that math isn't the "language of nature" as we don't have a direct means to truly comprehend it (direct realism has a whole has been discarded by philosophers for a long time now).
>I see literally no reason to jump to the conclusion that the core problem is trying to use math at all.
Again, that's not what I said, please read my post again.
> I'm stating that the math involved is a model but it isn't ever going to be identical to the thing being modeled.
I don't know what "identical" means in this context. Either a mathematical model can capture all of the information in the system, or it can't. We know that we can reproduce a function on a long enough timeline just by observing its outputs via Solomonoff Induction.
The only escape hatch here is if reality has incomputable features. There's no evidence of this at this time. That's why it's confusing that you would go from "we have persistent hard problems" to "mathematical models can't exactly correspond to reality".
The only progress I would say is more and more people realising it's a waste of time. Why should we expect reality on a scale very different to everyday experience to behave like everyday experience?
(And frankly I've never found it particularly counterintuitive. It's just a wave bro. It behaves like any other wave)
Flash storage uses QM [1]. Who knows what other advancements could be made with a better grasp of reality, at these scales. It doesn’t seem like a waste of time.
Grasping reality is better done by grappling with it directly than by trying to force it into your everyday intuitions. I'd bet the people who made flash work were many-worldsers.
Sure the math works out by summing all possible paths, and then most of them usually cancel each other out, and so you're left with the ones that mostly contribute to the actual probabilities, as complicated as those can still be.
But I always suspected... surely that just has to be one mathematical way of looking at it. That at some point we'd discover that turns out to be mathematically identical to something else that would be a bit more local, a bit more intuitive, a bit more straightforward. That never even had to deal in the first place with the infinity of paths that all wind up cancelling each other out anyways.
So now that I'm on a forum with smart people, the kind I didn't have access to back in high school... has there been any progress towards something like this at all? Are there any directions for this? Or is there a conceptual reason I'm missing why this fundamentally isn't possible, why the sum of all possible paths is the only way to account for certain behaviors?