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Huygens principle in odd/even dimensional flat space

  1. Jan 14, 2016 #1
    A well known math theorem says that - if the spatial dimension is odd - D'Alembert equation gives rise to a solution containing a term which is completely supported on the light cone.
    A mathematical wrap up could be the following:
    "in dimension 3 (and in fact, for all odd dimensions), the solution of the wave equation at point x0, at time t0, depends only on the initial data in an infinitesimal neighbourhood of the sphere |x − x0| = ct0. This is a statement of Huygens principle. In particular, it says that information from a point source travels in the form of a sphere. The wavefront is thus sharp, with a sudden onset at the start, and sudden cutoff at the end. In dimension 2 (and all even dimensions), the behaviour is different. Wavefronts do have a sharp onset, but they decay with a long tail. This we see because the solution at x0, t0 depends on the initial conditions on the entire disk |x−x0| ≤ ct0. This behaviour can be observed in the ripples of a pond that are formed around a pebble that falls into the pond." from these lecture-notes
    From another point of view, the main mathematical point is the interpretation of the inverse Radon, the lack of the Hilbert transform in the even dimensions having a fundamental impact.
    Two questions:
    1) does the failure of the Huygens Principle in even spatial dimension imply a breakdown of global Lorentz invariance? (I don't think so, but the issue would be with the wave equation for the scalar field in odd-dimensional Minkowski spacetime [=> even spatial dimensions], where one naively expects Lorentz invariance to hold, therefore Huygens' principle to hold?
    2) is it true that "the Feynman propagator, for a massless field, in flat spacetime, is supported on the light cone, thereby realizing Huygens' principle, for all dimensions"? In other word is it true that "the distinction between even and odd number of space-like dimensions for the Huygens Principle is an artifact(*) of having made the choice of the retarded Green function (with causality plus natural boundary conditions at infinity) instead of the Feynman propagator? (I don't think so because I would expect that analogous expressions hold for the advanced propagator)

    (*) one way to show this would be by Wick rotating... (but, once more, as far as I know, the poles of the meromorphic resolvents in odd dimensional Minkowski space are responsible for Huygens principle)
    ref. from G+

    Note: please, move the thread to Quantum Physics forum if you deem that the discussion is more appropriate there. I've asked this here because I found a previous post in the Classical forum, explaining the difference between 2D and 3D.
     
    Last edited: Jan 14, 2016
  2. jcsd
  3. Jan 14, 2016 #2
    I think that the answer to both questions is no. My reference for this is in Ivan Avramidi's Heat Kernel Approach.
    In particular: "Since we only need Feynman propagators we can do the Wick rotation and consider instead of hyperbolic operators the elliptic ones. The Green functions of elliptic operators and their functional determinants can be expressed in terms of the heat kernel.". In this way we have a calculational scheme that is manifestly covariant, just to close the first question. Then, the exact heat kernel for the Laplacian in flat Euclidean gives a good framework for the solution, look at page 14 for the transport equation with the initial conditions. Finally, by integrating by parts and analytical continuation, we can recover the Hadamard coefficients (pag. 15-16), thus proving the theorem and so closing also the second doubt.


    I've also found the following statements, related to the fact that the retarded part is what gives solutions to wave. "More generally we allow for multiple sources of waves at various locations, and all these fields can be superimposed without interfering with each other. So, in order to fully define the wave function at a given point, we need to consider the retarded waves emanating from every direction on its past light cone." and "As explained in the note on Huygens’ Principle, the propagation of waves in a space with an even number of dimensions differs in a profound way from wave propagation in space with an odd number of dimensions, because with an even number of space dimensions the solution cannot be expressed purely in terms of scaled functions of r – t and r + t." from mathpages (that includes also the original link about the principle)

    Edit
    However it should be noted that the solutions of D'Alembert equation emitted by y propagates along null geodesics to reach x from y. This basically is Huygens' principle. So in flat 2n +1 D where the principle does not hold, further terms appear added to the one localized on the light cone and it seems that a massless scalar, that propagates along a time-like curve doesn't satisfy this condition of transforming appropriately under Lorentz transformations. So the answer of the first question could be "partially" yes...
     
    Last edited: Jan 14, 2016
  4. Jan 14, 2016 #3
    In odd dimensional Minkowski space (even spatial + time) the retarded propagator fills the interior of the forward light cone even when the field is massless. That seems to shift also the second answer to a partial yes, since the theorem looks based on the retarded propagator while Huygens-like solutions could be found using Feynman propagator?
     
    Last edited: Jan 14, 2016
  5. Jan 14, 2016 #4
    In conclusion the answer could be a double yes. I have to concede that Stam Nicolis is convincing in saying that the Feynman propagator eludes Huygens principle dimensional constraints.
    1) Huygens principle failure implies Lorentz violation: support in the light cone would allow a massless field to propagate along a time-like curve. Therefore it doesn't realize a representation of the Lorentz group, since a representation of the Lorentz group, that's massless, can't do that. It's a representation of other groups, relevant for other applications (the water in the pond's example)
    2) On the contrary, as written above, Feynman propagator determines Lorentz invariant solution that are not subject to Huygens principle dimensional contraints (validity only in odd spatial dimensions): " In flat spacetime. It's unfortunate that the Feynman propagator is associated with quantum fields, when it's about the properties of the solutions of the classical equations of motion..."

    But the final math check that the Feynman propagator vanishes in the lightcone interior also in even spatial D is still missing, so the question remains open till the exact calculation is done or found or referenced for sure.
     
    Last edited: Jan 14, 2016
  6. Jan 15, 2016 #5
    My final remarks:
    1. The propagators of massless Bosons are only zero inside the lightcone if the number of spatial dimensions is odd and larger than 1 as you can prove as exercise
    2. From the Late-time tails of a self-gravitating massless scalar field paper: "This illustrates how viewing the dimension of a spacetime as a parameter may help understand which features of general relativity depend crucially on our world being four dimensional [3 space-like + 1 time-like] and which ones are general."

    Another paper on the subject is "Self-force and synchrotron radiation in odd space-time dimensions"
    Classical electrodynamics in flat 3+1 space-time has a very special retarded propagator ∼ δ(x2) localized on the light cone, so that a particle does not interact with its past field. However, this is an exception, and in flat odd-dimensional space-times as well as generic curved spaces this is not so... As already mentioned in the introduction, the main new feature in odd spacetimes compared to the conventional flat 3+1 dimensional space-time is that the massless retarded propagator from a given source has support inside the light-cone. This implies that at a given moment/position of the charge, the electromagnetic field acting on it obtain some contributions from the past trajectory of the charge.
     
    Last edited: Jan 15, 2016
  7. Jan 15, 2016 #6

    mfb

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    That is interesting. So "massless particles move at the speed of light" is only true in odd (>1) dimensions? I wonder how that would affect gravity and other forces mediated by massless bosons.
    There is certainly some frame where the field coming from a point-like emission is symmetric - but how can it be symmetric in all other frames? If it is not, it would mean that you cannot write down quantum field theory (or even relativistic quantum mechanics) in those dimensions.
     
  8. Jan 15, 2016 #7
    Yes, it is interesting.

    In fact I need help in understanding the relativistic implications. For example "... this divergence is proportional to the covariant acceleration µ and is readily absorbed by renormalizing the mass".

    I will try to formulate some questions, if I'm able to make a mathematically/physically sound statement. Indeed, I'm wondering if those dimensions are not permitted by QFT"
     
    Last edited: Jan 15, 2016
  9. Jan 15, 2016 #8
    A possible physical explanation is described here:

    http://arxiv.org/abs/1309.2996

    It would be a charge, moving along the light cone and hidden by Lorentz contraction, the origin of the tail...
     
  10. Jan 17, 2016 #9
    I've posted further comments here about
    1. implications of even dimensional space-time => chiral observable algebras => Huygens principle
    2. the symmetry property of global conformal invariance that implies Huygens principle
    3. Lorentz invariance and Maxwell equations
    4. waves (d'Alembert equations => speed of light) versus signals (field generated by charge, velocity<=c)
    5. etc..
     
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