Quantum mechanics is the operating system other physical theories run on (2007)
scottaaronson.com
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As I wrote before [0], I think that what Aaronson's calls "the theory" is a somewhat bastardized version of quantum mechanics that might suffice for quantum computing but, in my opinion, not for physics.
I discussed the difference here: https://news.ycombinator.com/item?id=38255476
QM/QFT is a particular model that has wide explanatory power in a certain realm. However, once you start getting further and further away from this realm, the model becomes increasingly less helpful, even though it might be "true"/correct in a sense.
Example: QFT should explain all chemical behavior, yet the wave equation only has "nice" solutions for a simple case: hydrogen. After that, the PDE is non-linear and non-separable, so. you have to resort to numeric methods. But the computational demands are very heavy, and even with todays machines you can only get up to ~10 valence electrons or so last I checked.
So modelling the behavior of say, a complex organic compound, such as a drug, first-principle QFT is of little aid. Which is why chemists have their own models which are loosely based on QM/QFT, and are simplifications (e.g the idea of electron "jumping" from one molecule to the other). Yes you have Pauli exclusion principle, Hund's rule, based on QFT, but there are exceptions, and I never got a good explanation it doesn't apply to the entire compound (or the entire object, or the entire universe)
Well as you mention, many of the models can be simplified at levels. For example regular QM and solid state physics like semiconductor assume nuclei’s are essentially stable compared to electrons. Applying this and other simplifications allowed us to build amazing semiconductors tech for example. DFT lets us estimate far more chemical interactions than pure QM, much less full QFT.
However fundamentally knowing the core rules doesn’t help us predict complex scenarios. Personally I don’t see it as too different to the halting problem in CS. It doesn’t prevent us from creating and understanding amazing things.
Actually I’m quite excited the rise of AI in quantum chemistry. These AI models can learn complex rules to simplify calculations like physicists figured out by hand, but can scale it out absurdly.
> Yes you have Pauli exclusion principle, Hund's rule, based on QFT, but there are exceptions, and I never got a good explanation it doesn't apply to the entire compound (or the entire object, or the entire universe)
It’s not clear to me exactly what you meant, but usually quantum tunneling and other effects are limited by decoherence. Yes an electron in your body can tunnel to say the moon. It’s just absurdly unlikely as the number of other electrons it would possibly interact with first is staggeringly large.
Even in a single molecule evolved for tunneling like chlorophyll, the probabilities of tunneling outside a few key paths quickly become too small to represent with 64bit floating point numbers. I did the computations in my physics days, and it was challenging to compute.
PHYS771 Lecture 9: Quantum - https://news.ycombinator.com/item?id=37198673 - Aug 2023 (31 comments)
PHYS771 Lecture 9: Quantum (2007) - https://news.ycombinator.com/item?id=33188276 - Oct 2022 (52 comments)
Quantum mechanics as a generalization of probability (2007) - https://news.ycombinator.com/item?id=8377680 - Sept 2014 (79 comments)
New straighforward approach to teaching quantum mechanics - https://news.ycombinator.com/item?id=4319276 - July 2012 (55 comments - top one just added to https://news.ycombinator.com/highlights)
Quantum mechanics for mathematicians - https://news.ycombinator.com/item?id=83594 - Nov 2007 (12 comments)
Moreover, quantum mechanics comprehensively lacks gravity. It isn't just "oh, well, maybe this system is also accelerating due to gravity but we can ignore it". It's, if atoms are in a superposition of "over there" and "over here", what is the gravity of the resulting system? We know we can produce such superpositions in the real world, and when those systems exist, there must be some answer to that question, even though we don't know what it is. What impact does very, very high gravity have on the non-superposition-related aspects of the standard model? How do energies of all the various bits of the proton change in such a field? And so on.
It doesn't help at all to simply reiterate the theory of relativity. The conflict between relativity and QM is deep, real, and very very challenging.
How can we ever really know that? Isn't the best we can do is just models?
I have daydreamed of becoming a physicist just to try testing this.
To directly answer your question, by experimental results.
Luckily, we’re still pretty far from a model which fully describes reality, and it may be entirely unobtainable for us. (Also, we only assume that spatial and temporal locality isn’t a thing, i.e. that the laws of physics are the same everywhere and everytime, so there’s that as well.)
Right now we can't say what gravity is. We know we don't have a complete description of it, because the contradictions between QM and relativity are something explainable to a motivated high school student, as they are pretty fundamental. You don't need to deeply understand either theory to understand enough of the conflict to know we don't have an answer.
This isn't a criticism of either theory per se or the scientists involved. They're the best we've got and they're each on their own pretty good. But we are as justified in saying "gravity is definitely a curvature of spacetime because Relativity says so" as we are justified in saying "gravity doesn't exist because it doesn't appear in the standard model at all". Both are basically the same statement relative to the theory they relate to, it is just more obvious the latter is wrong.
Who knows, maybe in a distant future we'll that that 4th fundamental force is not so fundamental after all.
Physics, as a natural science, deals in “if we do this then we observe that”. Note the “we”: it is inherently subjective. The models resulting from that are, similarly, a product of our minds seeking analogies and stories. Those models are useful for coming up with more experiments but are necessarily simplified maps, and if you have a map that exactly corresponds to the 100% of the territory then the map you have is A) useless, and B) impossible (so if you think you found one, you ought to think again).
At best you will never find such a foundational description of reality. At worst you will believe you’ve found one and be angered at anyone who chooses to use a different map and dares to not treat yours as gospel.
Just shut up and compute!
Nobody has yet created a fully convincing model, but it certainly would be elegant if spacetime itself arises from something that doesn’t need four-dimensional semi-hyperbolic space to be assumed.
Consider this: you have two massless particles that are motionless with respect to one another. Suddenly both particles acquire mass at the same time (let's set aside this is physically impossible - this is a thought experiment). How does curved spacetime explain the sudden gravitational attraction between those two particles and how they're accelerating toward one another? It doesn't.
These problems are how we know GR is wrong, the problem is it's not flat-out wrong! Einstein's field equations are better than what Newton had provided and explains more observed phenomena - they even predict phenomena we have since observed! GR is a very good theory (model) for gravity, but it's not the whole picture - and we know it. We simply don't know how to improve upon it and I personally think that believing curved spacetime is actual "reality" is a big part of why we haven't made much progress. Well, that and some other issues QFT has when attempting to model a graviton, but this comment is already long!
EDIT: merely(!) - this isn't to understate Einstein's contribution to the matter which was quite considerable! But when the old genius himself is warning you to not get too caught up in the model, I suggest we take heed.
EDIT 2: QFT has issues modeling a graviton, not a gluon.
You can't take an model, apply an impossible scenario and then claim the model is invalid because it can't account for this scenario that you admit is impossible.
That's not what's being said. The fact is GR cannot explain why these two particles will suddenly start moving toward one another. All I've done in this simple thought experiment is eliminate every other externality.
The key point here is GR doesn't explain why curved spacetime causes objects to move. It only says that the movement can be modeled by curved spacetime. This thought experiment was just a way of expressing that.
Which is all to say GR doesn't explain what gravity is, it provides us a (complicated!) set of equations for determining an object's motion in a gravitational field. In my mind that's a fundamental shortcoming of GR as it was a fundamental shortcoming of Newton. NEITHER of their theories even tried to explain what gravity was, they provided a mathematical model for describing the effects of gravity. Newton's was good, Einstein's is better - but neither tell us much about gravity itself.
OTOH, quantum gravity tries to tell us what gravity is, but that has run into issues. Not only is it going to take someone of the caliber of Newton and Einstein to figure this out and people like that only come around every few centuries, but we have to have the means to test the theory. We have several theories, but we don't have the ability to test which theory is correct. We simply don't know which way to go.
Anyway, all that is to say GR is incomplete - and we know it.
Imagine we explained it as "Mass causes X, and because X causes spacetime to be curved, we have that, mass (indirectly) causes spacetime to be curved." . But then, why would you be any more satisfied with this? Why would you not then ask, "Well, why does X cause spacetime to be curved?" ? (Or, "why does mass cause X?")
Now, perhaps there is such an X, and perhaps it will be found. But, I don't see why this would satisfy you any more than "Mass causes spacetime to curve" satisfies you.
If you keep asking "why", you either eventually end up in a loop, have an infinite regress, or stop getting an answer. A non-empty directed graph either has a cycle, an infinite outgoing path, or a vertex without an edge away from it. (And also, either a cycle, an infinite incoming path, or a vertex without an edge going to it.)
Many are content to allow the laws of physics to just be, without explanation, others may say that the laws of physics are explained by God, who is without explanation. Personally, I go with the latter, but, unless one wants to go with infinite regress or a cyclic explanation, one has to allow that something is without explanation.
For your complaint to be compelling, I think you should give criteria for, what conditions would something have to satisfy, in order for it to satisfy you?
(Also, you seem to assume that if the two particles suddenly acquired mass, that they would immediately begin to feel a force between them. While the hypothetical is presumably impossible, still, what I imagine happening would instead by that there would be a light-speed delay between when they gained mass and when they began to accelerate towards one-another. Though, I'm not sure if there is a fact of the matter as to "what would happen in this impossible hypothetical".)
For example, take the famous equation:
E = mc^2
E = m
Soo… by my interpretation:
Mass, energy and spacetime curvature are the same thing! They’re not “proportional” to each other and one doesn’t “generate” the other like an electric field by an electron. Instead, everything is literally made up entirely of space time curvature. That’s what matter and energy are.
This is why all forms of matter have masses — the only other option is empty space with a flat curvature, but that’s just the absence of matter. If everything else is curvature, then they must cause long-range distortions — which we call gravity.
Roughly, my concept is conserved quantities are topological defects, which is why they seem to have a neat “algebra” and integral quantities.[1] Conversely, mass comes in fixed but non-integer quantities because the total curvature of some complicated knot doesn’t have a simple algebraic expression.
[1] Makimg the fractions disappear is easy. Just multiply by the denominator. E.g.: we say the electron has a charge of -1 because we discovered it before the quarks historically. If the quarks were discovered first, we would assign it a charge of +3.
Natural units are very useful when doing math, but they don't reduce constants down to plain numbers, they still retain their units. Once you factor this in the rest of your argument falls apart.
As a hand-wavey example, one could claim that energy in the form of bosons is just a "wiggle"[1] of the spacetime fabric, and that matter in the form of fermions are topological defects or knots.
That way they're different enough that you'd want to use different units to represent them, but at their core, they're both distortions in spacetime that must inherently cause a distortion in spacetime at a distance (GR).
It also explains how they're inter-convertible. E.g.: energetic gamma rays can be converted into electron-positron pairs. If they're both "made of distortions" then it's like a very strong wave creating a pair of vortices spinning in opposite directions. You can't count the waves (it's smooth and continuous), but you can count the vortices.
[1] For the wiggle, imagine taking a huge sheet of cloth laid out flat over a smooth surface. If you tried to put a wave into the middle, the edges of the cloth would be pulled in. Contrast this with the typical view of fields as "vectors on top of a base (flat) spacetime", much like a mathematical function graph.
quantum mechanics is the operating system that other physical theories run on [...] with the exception of general relativity [...]
The graviton doesn't exist. Gravity emerges from the interaction of matter (energy?) with spacetime. They are literally inseparable in our version of the universe.
For those still looking for the graviton, the reasoning has led to the realization that you would create a black hole if you made the device that could detect it (as I understand it).
In standard quantum mechanics, the state of a quantum system is represented by a complex-valued wave function or a vector in a complex Hilbert space. Observables are represented by self-adjoint operators acting on this complex Hilbert space. The complex nature of the wave function gives rise to the phenomenon of quantum interference and the probabilistic interpretation of quantum mechanics.
However, in this real quantum mechanics, the complex Hilbert space is now replaced by a real Hilbert space, and the complex wave function is replaced by a real-valued wave function or a vector in this real Hilbert space. The observables are represented by self-adjoint operators acting on the real Hilbert space.
One way to achieve this is by using real algebra or real matrix representations. Now we let the complex numbers represented by 2x2 real matrices of the form:
a + ib = [a -b]
[b a]
i = [0 -1]
[1 0]
More importantly (as is pointed out in the lecture), every time-evolution of a quantum state from A to B can be represented as a unitary operator, and if you want to take the square root any such operator, you must in general use complex numbers (or something equivalent). Taking the square root like that is a very simple operation that answers the question "what is the operator for the evolution to the point halfway between A and B?"
Numbers are just mathematical objects which satisfy a few interesting properties that lead to very rich structures we can study. Some of those structures map onto concepts in nature, which makes them actually useful.
Instead of real and imaginary or complex we could have chosen entitirely different adjectives. They're completely arbitrary. They don't matter.
On the other hand, there is no straightforward intuitive mapping for complex numbers. Geometrically, they arise out of the spiral motion needed to provide a continuous solution to an oscillating function such as y=(-1)^x, but there is no regular everyday situation that mirrors that. Algebraically, they arise in solving certain polynomials, but in practical situations they are generally only a required intermediate calculation, and in practical (engineering) usage it is usually only the real roots that have any meaning.
Complex equations are used greatly for modeling waveforms, but that's mainly just because it's more mathematically convenient than dealing with a bunch of sin() functions. Not because waveforms are inherently complex/imaginary -- they're not.
So the idea that negative numbers are just as unintuitive as complex numbers, is an idea I think should be firmly rejected. Negative numbers make easy, intuitive, real-world sense in a way that complex numbers simply don't.
There's still an ongoing philosophical debate as to the "reality" of complex numbers, and the formulation of QM plays a part in that debate. But it doesn't answer it -- similar to waveforms, we can argue whether the math behind QM is "essentially" complex, or if we use complex representations merely for convenience. It's entirely possible to express QM without complex numbers at all, obviously.
Au contraire! Complex numbers are just rotations, translations, and scalings in the plane. In the same sense that real numbers capture 1D rigid transformations and scaling, complex numbers capture 2D rigid transformations and scaling.
When you treat negative numbers as "distance in an opposite direction", you're more generally treating real numbers as a transformation of the 1D line, i.e. translation (addition) or scaling (multiplication) by some amount. Try thinking about the same but for the 2D plane and complex numbers. Addition is 2D translation and you'll see that multiplication corresponds to a rotation and a scaling.
The first section of Needham's "Visual Complex Analysis" walks you through this perspective in detail, if you're interested and also want lots of good exercises.
When engineers need to handle rotations, translations, and scaling, they don't use complex numbers. They use vectors, usually.
There's no intuitive mapping for the concept of the square root of negative one in real life. Not in the way there is for negative numbers.
And complex numbers are not about 2D geometric representation in general, the way vectors are. They are much more specifically about rotation or spiral motion.
I really think you might be a victim of bad pedagogy. Admittedly, the notation confuses complex numbers as points vs. complex numbers as operations. Let me drive home the point. You can represent complex numbers as matrices:
e := [[1 0] our identity element, i.e. what you typically write as 1.
[0 1]]
i := [[0 -1] our imaginary unit
[1 0]]
<e,i> = [[0 0] the inner product of e and i
[0 0]]
Now, using the above matrix representation uses 4 parameters, but you really only need 2. Indeed, any general matrix in our 2D algebra looks like this
a×e + b×i = [[a -b]
[b a]]
However! Even though these are 2D things, it's very important to not confuse them with the 2D vectors they manipulate. The matrix representation above makes that more manifest. Said another way, it doesn't make sense to compose points in space, but it does make sense to compose operations which operate on that space.
It's an unfortunate and confusing quirk that we often write a+bi to mean a 2D point in space, a 2D translation operation, or a 2D rotation and scaling, despite all these really being completely different things.
If you are trying to factor a polynomial with complex roots, is not a coincidence they appear in pairs which cancel out eventually, if you want to get back to a "real" value implied by the polynomial.
1. Do numbers exist? Have you seen a number in your life?
2. What does exist, can you name one thing? Because everything isn't what it seems.
Can you elaborate more on what you mean?
Why are complex numbers “unexpected”?
So much of contemporary physics is about distillation. You care more about things like consistency and simplicity over readability.
Our current methods of solving these equations, vector calculus, requires certain operations, specifically, square roots and derivatives.
To be able to utilize the well studied methods of linear algebra we need to make ample use of the Pythagorean theorem that ultimately uses a square root. To ensure that our model works in all domains we have to use a form of the square root that allows negative numbers. Complex numbers.
It’s why we use e^(iHt/h_bar) in the Schrödinger equation. Rather than see it as “nature works by using exponentiation!” I see it as “exponentiation is ‘stable’ in the face of complex derivatives”, ie the derivative of the exponential function is the exponential function f(x)=e^x == f’(x)=e^x. So this formulation simplifies our calculations.
These things can be calculated with any coordinate and numbering system, the ones we choose just help make the problems more tractable for us.
I suspect that another mental roadblock that prevents further progress is how we see time: a straight arrow slightly bended by local gravity. It's quite possible that a proton naturally exists in 3-dimensional time that on average appears 1-dimensional to us.
It's not the OS, but spells out the rules of the universe.
I get that he's got a lay-fanbase, but it's a little cringe when folks are impressed by the mundane. It's honestly a little insulting.
(Altman having said they're working on an AI OS or something.)
I'd personally just walk out of class and cancel the course.