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The Lebesgue integral can be explained as intuitively as the Riemann integral (https://en.wikipedia.org/wiki/Lebesgue_integration#mediaviewer/File:RandLintegral

by ssivark 12y ago

The Lebesgue integral can be explained as intuitively as the Riemann integral (https://en.wikipedia.org/wiki/Lebesgue_integration#mediaviewer/File:RandLintegrals.png https://en.wikipedia.org/wiki/Lebesgue_integration#mediaview...). One can certainly have the intuition for both without too much knowledge of real analysis. And in general, I believe that analogies are very useful. The author explains in the article how to understand the behaviour of light. It turns out quantum particles behave in exactly the same way. Feynman (building on Dirac's observations) built up the theory for quantum particles. With regards to light, this understanding comes from a couple of centuries ago (Ref: [Huygens principle](https://en.wikipedia.org/wiki/Huygens%E2%80%93Fresnel_principle https://en.wikipedia.org/wiki/Huygens%E2%80%93Fresnel_princi...) and [relations to quantum mechanics](https://en.wikipedia.org/wiki/Huygens%E2%80%93Fresnel_principle#Huygens.27_principle_and_quantum_electrodynamics) https://en.wikipedia.org/wiki/Huygens%E2%80%93Fresnel_princi...)

abstrakraft 12y ago
"The author explains in the article how to understand the behaviour of light." That isn't the argument. What I'm arguing is the (lack of) utility of the explanation. In what way does it enhance, support, or in any way contribute to the article? His argument is essentially: A. We think of light as traveling in straight lines. B. But because quantum physics, they don't! They travel along infinitely many paths. C. But all of these paths cancel out except the straight line path. D. And that's how beamforming works. Except it's beam-un-forming. So he brings up a quantum phenomenon to complicate the scenario, then immediately reduces it back to the original scenario, without ever explaining how that original scenario works (i.e. how waves add together to create constructive and destructive interference: note that the words "wave" and "phase" never occur in same paragraph), then claims that this is somehow elucidating. So what's the benefit?
- judk 12y ago
  As stated in the article, the point is to give an intuition for how energy appears to "move" from troughs to peaks: by going everywhere. This coincides with the Pilot Wave model, where particles travel along paths modeled by the peaks of mathematical waves.

by ssivark 12y ago

abstrakraft 12y ago

"The author explains in the article how to understand the behaviour of light." That isn't the argument. What I'm arguing is the (lack of) utility of the explanation. In what way does it enhance, support, or in any way contribute to the article? His argument is essentially: A. We think of light as traveling in straight lines. B. But because quantum physics, they don't! They travel along infinitely many paths. C. But all of these paths cancel out except the straight line path. D. And that's how beamforming works. Except it's beam-un-forming. So he brings up a quantum phenomenon to complicate the scenario, then immediately reduces it back to the original scenario, without ever explaining how that original scenario works (i.e. how waves add together to create constructive and destructive interference: note that the words "wave" and "phase" never occur in same paragraph), then claims that this is somehow elucidating. So what's the benefit?

judk 12y ago
As stated in the article, the point is to give an intuition for how energy appears to "move" from troughs to peaks: by going everywhere. This coincides with the Pilot Wave model, where particles travel along paths modeled by the peaks of mathematical waves.