Re: What exactly is wrong with Huygens' principle in two dimensions?

carlip-nospam@xxxxxxxxxxxxxxxxxxx wrote:

But why can't I go down more than one dimension? For example, a
point source in 1 dimension should look like a plane source in 3
dimensions -- why doesn't the strong version of Huyghens' principle
therefore fail in 1d?

(My first guess is that you must have destructive interference from
various points on the plane, but it's not obvious...)

Steve Carlip



The photon propagator in 1d isn't "on the light cone" only.
It is constant everywhere within the lightcone and zero outside
the lightcone. If you take a look at a paper I wrote recently:

http://chip-architect.com/physics/Higher_dimensional_EM_radiation.pdf

Then, you'll see in fig.1 the propagators for the first 5 dimensions.
Only the odd dimensions from 3d and higher are on the lightcone
only. It's only in 3d that the propagator is a nice Dirac pulse.
Higher odd dimensions have (higher) derivatives of the Dirac
pulse as propagator.


The paper contains a complete derivation of the propagators in
any dimension. The 1d propagator is particular simple: see
page 3, Sec. IV. at the top of the left column.

On page 6, halfway the left column, it talks about the propagator
from a plane in 3d which represents exactly the 1d situation you
mentioned.


Regards, Hans

.

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