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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^

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. So I'd argue that QM is probability in L^2.

richardjordan 12y ago
This is what the whole article says - the title is merely more layman-friendly wording.
tarblog 12y ago
Thanks. That makes a bunch of sense. (no sarcasm)
tagrun 12y ago
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.
- yummyfajitas 12y ago
  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.
  - tagrun 12y ago
    > It's L^2 in finite dimensional spaces also 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.
    - namlede 12y ago
      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.
      - tjradcliffe 12y ago
gone35 12y ago
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".

richardjordan 12y ago

This is what the whole article says - the title is merely more layman-friendly wording.

tarblog 12y ago

Thanks. That makes a bunch of sense. (no sarcasm)

tagrun 12y ago

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.

yummyfajitas 12y ago
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.
- tagrun 12y ago
  > It's L^2 in finite dimensional spaces also 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.
  - namlede 12y ago
    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.
    - tjradcliffe 12y ago

gone35 12y ago

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".