Timeline for Why is Huygens' principle only valid in an odd number of spatial dimensions?
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9 events
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| Jan 2, 2018 at 6:33 | comment | added | tparker | As improbable as it sounds, I suspect that when dimensionally reducing from $(4+1)d$ to $(3+1)d$ space, there's a magical conspiracy such that after the initial positive wavefront passes, its infinite-duration negative wake is continuously exactly canceled by successive positive wavefronts arriving from sources that are ever further-away in the fourth spatial dimension. | |
| Jan 2, 2018 at 6:32 | comment | added | tparker | I think I might have figured this out. The Green's function for the $(4+1)D$ wave equation is indeed sign-definite strictly inside the light cone, but it has a delta-function of the opposite sign exactly on the light cone. See eq. (36) of aapt.scitation.org/doi/abs/10.1119/1.17230. | |
| Oct 1, 2017 at 20:00 | comment | added | WillO | @tparker: ah. That's very enlightening. I will have to give this some thought. Thank you. | |
| Oct 1, 2017 at 18:32 | comment | added | tparker | No, the Green's function for the wave equation in even spatial dimensions is positive-definite inside the light cone, so you can't get destructive interference. | |
| Sep 29, 2017 at 21:33 | comment | added | WillO | @tparker: I think --- but am not certain --- that the intuition I gave is correct but incomplete. The missing part is, as you say, to explain why we can't extend the same intuition more than one dimension downward. I believe --- but again am not certain --- that you can explain this by arguing that interference somehow magically erases this effect when you go down from dimension $n$ to dimension $n-2$, though I have at the moment no good intuitive story for why you should expect this. | |
| Sep 29, 2017 at 19:40 | comment | added | tparker | I don't think your physical intuition reason is correct, for the reason given by @user157879. Your picture suggests that there is some dimension where the influence is only on the light cone, and that for all lower dimensions, you also get influence inside the light cone, which is not correct. In general, dimensionally reducing a theory in the way you describe does not always reproduce the same theory in a lower dimension - it often introduces new fields as well. | |
| May 7, 2017 at 2:58 | comment | added | KF Gauss | Regarding the last comment on a $z$ independent solution, that is a nice way of thinking, but I think it's not clear why the same logic wouldn't apply going from 4d down to 3d. | |
| May 6, 2017 at 6:59 | history | edited | WillO | CC BY-SA 3.0 |
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| May 6, 2017 at 4:39 | history | answered | WillO | CC BY-SA 3.0 |