Ah, I see that it is time to raise my objection again.
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.
Most people. Physics uses math to model the world, but is independent (math is not a science). At least we don't have any further evidence to say otherwise. There's a lot of interesting ideas around this in https://www.maths.ed.ac.uk/~v1ranick/papers/wigner.pdf and many of us believe that the universe itself is effectively an embedding of a large mathematical structure in a 3D-space 1D-time. But that would be effectively impossible to prove in any convincing way.
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)
> 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 today’s machines you can only get up to ~10 valence electrons
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.
I assume this is because most models are linear models and implicitely approximate a non-trivial amount which can become problematic from a stability perspective in many of the real-world settings?
No, that's not why. It's because models are not reality, they are a (mathematical) description of reality. And descriptions rarely perfectly match reality.
More like, quantum mechanics is the OS for 3 out of 4 fundamental forces in physics. After 100 years of trying, nobody has managed to express gravity in the language of quantum mechanics.
Because of emergence. Unintuitive properties that arise from the aggregate interactions of discrete entities. Temperature is an emergent property arising from the movement of all atoms within a space. There are no hot and cold atoms.
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).
Carlo Rovelli argues in "The Order of Time" that time itself emerges solely from thermodynamics since there is no arrow of time otherwise in physics. Given that, spacetime itself is not at all foundational but a result of limited human perception.
Gravity is a "consequence of the curvature of spacetime" in a particular theory that we know can't be the foundational description of reality, because all it has is mass and spacetime in it. Since it's a known-incorrect or known-incomplete theory (pick which one you like), we are not in a position to declare that gravity is absolutely, positively "curvature in spacetime" and nothing else.
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.
Recently in pbs spacetime I saw a video about entropic gravity, where some of the researchers in this field seems to explore wether gravity is just an entropic force resulting from thermodynamics. It was an entersting perspective I haven't heard of.
Who knows, maybe in a distant future we'll that that 4th fundamental force is not so fundamental after all.
A model that is able to perfectly describe and (more importantly) predict everything is effectively indistinguishable from “reality”. Yes, there might or might not be a completely different, and maybe even weird, process underneath that gives rise to that model, but assuming that nothing whatsoever of that underlying process “leaks” into the model, it does not matter and may as well not exist for us.
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.)
Science can never prove that it has reached the bottom, but it could hypothetically yield a model that makes predictions that always match the universe to within all relevant margins of error. It could be that that model is itself a simulation or that there are yet deeper layers we just can't see, whatever, you know, all that jazz. But if there was a complete model, with no internal contradictions, that matched everything we could see, then at least when someone said "in light of this theory, gravity is X", we would be making a coherent statement.
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.
I think that the same way that energy is shed when a particle steps from high energy to low energy is the way that "gravity aka possibility energy" is shed when quantum superposition collapses into position.
I have daydreamed of becoming a physicist just to try testing this.
To directly answer your question, by experimental results.
If you seek a foundational description of reality in physics, you are treating physics like religion.
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.
The curvature of spacetime is a fiction. A very useful fiction, as Einstein noted, but a fiction, nonetheless. It doesn't matter what the science popularizers say, it's a fiction. It's okay though, because we get very good results with Einstein's Field Equations.
The usual idea is that there’s something more fundamental. Threads to weave the fabric out of, to extend the analogy.
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.
Curved spacetime is a mathematical model Einstein used to model gravity. Einstein advised to be careful to not confuse the model with reality. To wit, GR doesn't explain how mass/energy curves spacetime, it merely(!) provides the tensor needed to explain gravity as the result of such a curvature.
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.
Taking a gravity and then performing an impossible physical modification so you can make conjectures on the model is nonsense.
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.
> 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.
The thing you are asking for, the "why does mass cause spacetime to bend" seems... perhaps impossible to answer? Or like, I don't see why there should necessarily be anything that satisfies what seems to be your reason for dissatisfaction.
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".)
I am not a physicist, my sense though as a layman is that GR explains the motions with which it is concerned more effectively than QM or QFT do. As in, with Schrodinger's cat, what causes the poison's release? And why does that happen? The "shut up and calculate" thing I'd assumed referred to quantum theory.
:) - yes, the "shut up and calculate" was a reference to quantum theory applied to GR, since people neglect to mention that GR doesn't explain what gravity actually is. And really, if you "shut up and calculate" using Einstein's Field Equations, you'll get results that largely align with observations, more so than Newton's equations, anyway. But there are observations that can't be explained by GR and there are simple thought experiments that can't be explained by GR. As I keep saying, we know GR is not the final word on gravity. That doesn't mean GR hasn't been incredibly useful - especially if you just "shut up and calculate!"
An idea I’ve been toying with as an amateur physicist is to take the equal signs to mean “is” instead of “in proportion to”.
For example, take the famous equation:
E = mc^2
The common interpretation is that mass can be created from energy with the proportionality constant of c^2. That constant can be set to “1” using natural units. This just leaves:
E = m
But GR also have a similar equation basically saying that curvature = mass.
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.
You can't just set c = 1 as a pure number and conclude that E = m, the units won't match.
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.
That's a valid criticism, but the point is that you can have two types of curvature that you assign different "units" to when measured, but they're still both curvature.
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.
This doesn't appear to fit with other quantities like say, electric charge, or lepton number, or whatever. Like, there's more to matter than "how much is there here".
On the contrary, I got the idea from Kaluza Klein theory, which unifies general relativity and electrodynamics by converting the electric charge into spacetime curvature.
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.
I would substitute "may be" for "is" in your first sentence, but I've had similar thoughts. Gravity may be a force that acts in a way that is mathematically equivalent to having curved space. I think that may even be most likely. But I'm not sure it really matters.
In this banal set of metaphors it would have to be something else. There may arguably be some mysteries with respect to how deep learning is effective, but its precise mechanistic explanation as a series of operations on a processor is not mysterious. There is not presently a quantum theory of gravity (in fact, I'd argue that despite the practical successes of QFT there isn't really a theory of quantum mechanics _either_ in the sense that algebraic quantum field theories of non-trivial character are so far elusive and no one knows what quantum field theory at large actually is or why it works, but I digress). In the current account of the world there is literally know way for us to understand how gravity arises from the rules we presently believe must be true based on our successes with the other theories.
Complex numbers! I didn't expect complex numbers to have a deeper relation to quantum mechanics. It feels like negative probabilities is just the "sqrt(-1) doesn't make sense" topic all over again. Yeah, why is it Quantum Mechanics taught on how it was historically sequentially discovered instead of reorganizing it in a clear manner for future students? This may be long overdue and might even yield new perspectives on quantum mechanics from future researchers.
I would argue complex numbers don't really have a deeper relation to quantum mechanics. Quantum Mechanics is all about squaring the fact that we must find a representation for a system which simultaneously supports an expression of a probability distribution over measurement outcomes while also providing sufficient representational complexity to support all the symmetries that are found in the kinematic space of the system. Complex numbers happen to serve well for a lot of systems but when you have spinors, for example, you can think of them as just another sort of number the use of which is suggested by the symmetries in (or redundancies in the description of) the underlying system. In other words, the appearance of complex numbers is no more or less mysterious than the allowed transformations of the system itself. At least that is how I've come to understand the question.
Every time you're tempted to freak out because complex numbers, reflect that you should also be freaking out about negative numbers. Those don't exist either.
What does it even mean for a number to exist? In math we are happy if the things we talk about are well defined.
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.
Besides, they’re really not that scary once you get to know them well. Take a deep dive into digital signal processing for example, and you become well comfortable with them, and understand how “the real world” fits into them quite neatly.
There are straightforward interpretations of negative numbers that map intuitively to regular everyday situations, such as a distance in an opposite direction, moving in an opposite direction, or an amount of money owed.
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.
> On the other hand, there is no straightforward intuitive mapping for complex numbers.
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.
I'm very familiar with Needham, thank you. But perhaps I wasn't clear enough.
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.
> 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.
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]]
These 2x2 matrices operate over 2D vectors, obviously. Furthermore, notice that
<e,i> = [[0 0] the inner product of e and i
[0 0]]
in other words e and i are orthogonal, meaning that their span is a 2D subspace of the underlying matrix space. 2D... Meaning every matrix like this can be written as v = a×e + b×i. Also, um, notice that i^2 = -1×e, or more on the nose, i = sqrt(-1×e). We've just rediscovered complex numbers!
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]]
where the repetition is obvious. Why not throw away the slop and just directly use e and i? Also, notice that e acts just like the identity 1, meaning that a+bi as a notation makes sense. Then we end up writing that i^2 = -1 and unfortunately invite all sorts of confusion about the meaning of "imaginary" numbers.
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.
Complex numbers are simply a mundane promise to multiply out your i at some future date, arguably. Much in the same way that a negative number is a promise to square your books eventually at the end of the day. If you have a balance of -10 dollars the implication is that you'll give someone 10 dollars some day before they try to spend that money (for example).
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.
Negative numbers are very natural to me. Math isnt just about counting real objects, but also transformations through space/time. The notion of duality or things going in reverse is extremely natural. -5 is just the inverse of 5, such that if you go 5 and go back 5, you end up at the same place.
Complex numbers you just go sideways as well, and if you multiply you not only stretch or squish your path to the origin, you also change your angle to it by adding both angles together. Exponentiation is then doing this continuously, so you end up walking in a circle. That’s it.
> Complex numbers! I didn't expect complex numbers to have a deeper relation to quantum mechanics.
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.
Quantum states are represented as vectors of length 1 (unit vectors), or corresponding complex numbers, so really just positions on a circle or n-dimensional sphere. To me that makes more sense than talking about roots of negative numbers or negative probabilities, and complex numbers are just a mathematically convenient representation.
For that to make sense, the n-dimensional sphere must not live in an Euclidean n-dimensional space, but a space over the complex numbers (a Hilbert space) -- that is, coordinates for the space are complex numbers. Also note that (this is a little beside the point) the n-dimensional sphere picture only works for quantum states that can be described with a finite number of dimensions (like in quantum computing): to talk about even simple things like position and momentum of particles, states must live in a infinite-dimensional space.
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?"
Yes, it is a very convenient representation, and it is possible to formulate quantum mechanics without using complex numbers at the fundamental level of the mathematical representation.
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]
now a and b are real and the imaginary unit i is represented by the matrix:
i = [0 -1]
[1 0]
One also can use quaternions, but now it will be more complicated without much to gain.
Complex numbers are just one example of more general Clifford Algebras. Matrices, quaternions, and bivectors can all be used to represent the same stuff.
QM and GR look like two equally fundamental views on reality: GR is a spatial representation of energy distribution, while QM is a sort of frequency representation of the same energy. QM feels like the Fourier transform of GR.
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.
For a more lighthearted approach to the topics in this lecture you might want to check out the comic Aaranson created with Zach Weinersmith: http://www.smbc-comics.com/comic/the-talk-3.
I couldn't really follow all the math in the article, but "the mathematics of quantum mechanics is based on probabilities but with negative values" just blew my mind. Never heard that before.
My mistake. His specialty, as a computer scientist, is quantum computing. I maintain that this is a baseline skill, to be expected of anybody claiming his credentials.
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.
If I didn't know this was from 2007 I'd have said it shows that he's been hanging out with OpenAI kids. (He worked there working on watermarking and stuff.)
(Altman having said they're working on an AI OS or something.)
The relevancy is that at its heart, quantum mechanics really only has a few key different phenomena from classical, and you can understand those phenomena more easily using qubits than esoteric particle canon experiments.
You know how students raise their hands and ask when they will ever use some math? This is essentially suggesting you don't tell them, and entirely skip the history of why it's relevant.
I'd personally just walk out of class and cancel the course.
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
[0] https://news.ycombinator.com/item?id=39627618