Exploring Huygens' Principle: Odd # Spatial Dimensions & Beyond

[pervect]

A spin-off from another thread. I consulted a couple of my GR textbooks on Huygens' principle, and found little. Wiki had a little information, which said that it could be regarded as a consequence of the homogeneity of space-time, and "In 1900, Jacques Hadamard observed that Huygens' principle was broken when the number of spatial dimensions is even.".

So some of my questions are:

1) Does Huygens principle work in general space-times? (Wiki states that "Huygens' principle can be seen as a consequence of the homogeneity of space", but I wouldn't think a general space time would necessarily be homogenous).

2) How would one write the principle in an explicitly covariant manner?

3) In the limit of geometric optics, can Huygen's principle be logically be derived from Fermat's principle of "stationary optical path length" or Hamilton's principle principle of "stationary action"? And what about the reverse? (I ask about the geometric limit because I don't see how one would handle diffraction with Hamilton's principle).

On a related note, I am reminded of Feynman's "sum of all possible paths" approach to QED, I suspect there might be a relation. But I don't want to drag the thread too far afield.

4) Why does Huygen's principle only work for an odd number of spatial dimensions?

 

[fresh_42]

Does this help: http://www.mathpages.com/home/kmath242/kmath242.htm ?

 

[vanhees71]

Before going into curved spacetime look at usual electrodynamics. There you see that it works for massless fields in even-dimensional Minkowski spacetimes. For a derivation, see

S. Hassani, Mathematical Physics, Springer

 

[robphy]

Possibly useful (I haven't read any of it):

http://physics.stackexchange.com/qu...-valid-in-an-odd-number-of-spatial-dimensions
http://www.math.uAlberta.ca/~xinweiyu/527.1.08f/lec17.pdf
http://www.math.utk.edu/~freire/teaching/m435s14/WaveEquationExamples.pdf

http://www.lboro.ac.uk/microsites/maths/research/preprints/papers02/02-49.pdf (brief review)

http://physics.stackexchange.com/questions/129215/huygens-principle-in-curved-spacetimes
http://www.math.utah.edu/~chu/Talks/HuygensPrinciple/GravityGroup.pdf

https://books.google.com/books?id=R...pg=PA222&dq=huygens+friedlander+wave+equation
http://projecteuclid.org/euclid.bams/1183555741
https://books.google.com/books?id=X6viBQAAQBAJ Huygens' Principle and Hyperbolic Equations - Paul Günther

http://scitation.aip.org/content/aip/journal/jmp/18/11/10.1063/1.523192
JMP 18, 2125 (1977)
A necessary condition for the validity of Huygens’ principle on a curved space–time - Riccardo Goldoni

http://link.springer.com/article/10.1007/BF00760104
GRG 17, 15 (1985)
Cauchy's problem and Huygens' principle for relativistic higher spin wave equations in an arbitrary curved space-time - Volkmar Wünsch
(I had an interest in such a question a while back... but haven't really followed up on it.
https://groups.google.com/forum/#!s...ce/sci.math.research/zYMRH6ADAxc/Ba95dnyHy-IJ )