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> 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

by 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
  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.
  - tagrun 12y ago
    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.
  - yummyfajitas 12y ago
    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. I explored this idea in more detail here: http://www.chrisstucchio.com/blog/2014/why_to_reject_complex_theories.html http://www.chrisstucchio.com/blog/2014/why_to_reject_complex...
  - TeMPOraL 12y ago
    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).

by tagrun 12y ago

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
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.
- tagrun 12y ago
  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.
- yummyfajitas 12y ago
  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. I explored this idea in more detail here: http://www.chrisstucchio.com/blog/2014/why_to_reject_complex_theories.html http://www.chrisstucchio.com/blog/2014/why_to_reject_complex...
- TeMPOraL 12y ago
  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).