I'd disagree with the title of this a bit. Rather than describing it as "probability to allow minus signs", I'd describe it as "probability in L^2 instead of L^1". The author actually discusses this a bit towards the end.
It makes sense to ask what is the distance between two continuous probability distributions. It's given by:
\int | p1(x) - p2(x) | dx
L^1 (the space of all functions for which \int |f(x)| dx < \infty) is a weird space, and does not admit concepts like "what is the angle between two vectors".
Quantum mechanics changes this to:
\int |p1(x) - p2(x)|^2 dx
Functions like this are called L^2. Once you put the square in, you can immediately derive a lot of geometry, inner products, angles between vectors, etc.
The author actually discusses this a bit towards the end.
Far from it. As early as the first subsection, "A Less Than 0% Chance", Scott writes:
"Now, what happens if you try to come up with a theory that's like probability theory, but based on the 2-norm instead of the 1-norm? I'm going to try to convince you that quantum mechanics is what inevitably results."
So discussing that distinction is the entire point of the lecture, and not some minor point "discussed a bit towards the end".
It is L^2 for infinite dimensional Hilbert spaces like position or momentum, where quantum states become functions.
For finite dimensional Hilbert spaces such as spins, quantum state is a complex vector rather than a function, and the "braket" is not an integral but vector product.
But I don't think that's even the point. The whole article is a rationalization of basic rules of quantum mechanics; when I look back at the history of physics, rationalization of known --at the time-- physical laws (which are often replaced by "better laws" after a while when get a better we understanding of nature) is most often counter-productive.
It's L^2 in finite dimensional spaces also - you can just as well define || v || = \int |v_i| dC(i) (where C is the counting measure). Finite dimensional spaces can also be interpreted as functions, i.e. C^3 is the same as {1,2,3} => C.
This is a rationalization but it's useful. We have a physical theory in L^2, and we know that you can't have such a theory in L^2.1 or L^7. Our theories might be totally wrong. But we are confident they can't be only a little wrong.
The concern of "is this thing square-integrable" arises only in infinite dimensional spaces. Inner product of all $\mathbb C^N$ where N in finite is finite. There is no point in saying "L^2" when talking about spins etc. (forcefully casting a finite sum of finite terms to integration and then introducing the concept of "square integrable" isn't helpful, if not trivial).
> This is a rationalization but it's useful.
You can do that kind of rationalization when you "know" the answer ahead. It has zero prediction power.
Newton had a rationalization about the independent nature of time. Descartes also had a rationalization about how action-at-a-distance works for gravity. Aether was also a popular rationalization once.
None of those rationalizations held any value at the end.
Such "rationalizations" are better cut by Occam's razor and left for philosophers.
At least, I prefer not waste my time with such stuff as a physicist.
I also disagree; the article's rationalization is productive.
Your mention of Occam's razor is key. The point of this rationalization is that it shows quantum mechanics (or at least a subset of the theory) is mathematically simple, thus serving a threefold purpose:
1) Due to Occam's razor, a simpler theory is more likely to be correct than a more complicated theory with the same explanatory power. If quantum mechanics can be expressed more simply, then the estimated probability of its correctness should be increased (by only a slight amount).
2) Quantum mechanics, explained this way, is interesting from a purely mathematical standpoint. Even if we knew quantum mechanics doesn't describe reality, mathematicians (and theoretical computer scientists like Aaronson) would perhaps still investigate it. Of course, this may not interest physicists.
3) The article presents a novel way to teach quantum mechanics. Mathematical simplicity can (to some extent) replace intuition as a way for learners to grasp the theory. As Aaronson remarks, quantum mechanics is often taught by following the historical order in which the ideas discovered. Starting from the "conceptual core" (if Aaronson is correct about the conceptual core) is arguably a superior pedagogical technique.
The problem with Occam's razor is that there is no generally agreed-upon notion of simplicity. Occam himself believed it proved the existence of God, and only God. After all, what did you need material reality for when it could be "explained" as God's dreams and imaginings? To Occam that was far "simpler" than the messy reality of matter, and the messy reality we know of today would have been anathema to him.
"Simpler" theories are not ab initio more likely correct. The world is full of "simpler" theories that are wrong: the four elements, the caloric theory of heat, Newtonian dynamics and gravity, and so on.
"Simpler" is a purely human notion, and a heavily culturally laden one at that. For this reason, Occam's razor is best left in the dust-heap of philosophy. It is never useful in doing actual science, except now and then by accident.
Occam's razor doesn't say "pick the simplest theory", it says "pick the simplest theory out of all that agree with all discussed experimental evidence". We didn't throw away four elements because they weren't simple enough; we did it because that theory didn't explain the things we saw.
As for the notion of simplicity itself, we have Kolmogorov complexity and Solomonoff induction, they capture the essence pretty well (though I heard that with some caveats).
That's actually not correct. From what I can tell, most theories with god in them that also predict reality can all be converted to a simpler theory without god.
The problem with all your examples is that Occams razor says "choose the simplest theory that works", not the simplest of all possible theories.
If you going to pick nits like amateur philosophers and don't like the name Occam's razor, let me put it in more direct terms less open to "weird" interpretations so that hopefully you'll see what I'm saying: "just because I can" is not a good justification to put unnecessary concepts in a theory. We already know about probability amplititudes and we do understand them quite well (path integral formulation). There is no need to introduce crazy things such as "negative probabilities" into the theory, because 1) it doesn't make sense as a fundamental concept (and it just isn't) 2) more importantly, it is not needed 3) and it doesn't add anything new to our understanding of nature or predicts anything at all.
There is nothing simpler or more elementary than the quantum mechanics we already know on that page.
Don't take his word on how quantum mechanics is thought. While we don't teach undergrads it, there is a conceptually simpler approach to quantum mechanics: Feynman's space-time approach. Not only it is conceptually simple and intuitive, it most importantly gives an elementary understanding to the principle of minimal action (sum over all possible paths in space-time) but it also offers a way of calculation that is much better suited to certain class of problems. Now, that is useful.
That is how quantum mechanics is made simple and intuitive (and that is superior pedagogically).
Not by introducing new and strange additional concepts such as negative probabilities as basic things just because you can.
The reason we don't teach path integrals in the undergrad is, we expect students to actually use it for calculations, and unfortunately the mathematics is much more involved in comparison to matrix mechanics or wave equations where you can get away with "basic" maths.
If you have a very good intuition and understanding of quantum mechanics, it is doable though, see Feynman Lectures on Physics Vol. III (not sure how many percentage of the students will actually be able to absorb the intuition along with the new information though).
It makes sense to ask what is the distance between two continuous probability distributions. It's given by:
\int | p1(x) - p2(x) | dx
L^1 (the space of all functions for which \int |f(x)| dx < \infty) is a weird space, and does not admit concepts like "what is the angle between two vectors".
Quantum mechanics changes this to:
\int |p1(x) - p2(x)|^2 dx
Functions like this are called L^2. Once you put the square in, you can immediately derive a lot of geometry, inner products, angles between vectors, etc.
So I'd argue that QM is probability in L^2.