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Detecting and analyzing higher dimensions via the EM radiation field.

  1. Dec 1, 2006 #1

    Hans de Vries

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    This should be possible with table top experiments rather than LHC scale experiments:


    Electromagnetic radiation decays with [itex]1/r[/itex] in three dimensional
    space, while the non radiating Coulomb field decays faster with [itex]1/r^2[/itex].
    The general expressions for any dimension are [itex]1/r^{(d-1)/2}[/itex] for the
    Radiation and [itex]1/r^{(d-1)}[/itex] for the Coulomb field respectively, where
    d is the number of spatial dimensions.

    This means that there is a dimensional dependent ratio
    between the two, and one should expect, due to the [itex]1/r^{n}[/itex] nature,
    to be able to measure imprints of any propagation through higher
    dimensional structures at arbitrary scale down to Planck's scale.

    We present the rules for radiation resulting from the motion of
    charged objects at any dimension, checked by extensive numerical
    simulations. These rules are quite different from the 3d case and
    provide a toolset to analyze higher dimensional structures.

    We further present a very useful operator to transform any arbitrary
    propagator in an x-dimensional space into the corresponding
    propagator in any y-dimensional space.


    Regards, Hans
    Last edited: Dec 1, 2006
  2. jcsd
  3. Dec 1, 2006 #2
    I guess that you know that in an stringy secenary only gravitons (closed strings9 could see that extra dimentiosn while al vector bosons (open strings) would be tied up to the four dimensional (D-)brane.

    Of course it is still conceviable that culd be extra dimensions even withouth string theory. But in that escenary they look somewhat unnecesary.

    Anyway, as an excersice in field theory in extra diemnsions is ok.

    If i am misisng something tell me.
  4. Dec 1, 2006 #3

    Hans de Vries

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    Isn't what you are are describing a Brane world scenario.


    While in general there is no such limitation. After all Zwiebach discusses
    the higher dimensional EM fields in his introduction. This paper concentrates
    on the photon propagators but it's applicable as well on the massless
    graviton propagator, so it should be usefull even in the case of open string
    brane world scenarios with closed string gravitons.

    Regards, Hans
    Last edited: Dec 1, 2006
  5. Dec 2, 2006 #4

    Hans de Vries

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    Yes but, this paper should have been written 30 years ago. Not in 2006 by me.

    How can people seriously claim to study higher dimensional theories without
    first establishing the most basic physics involved? What can there be more
    basic as the EM propagation in higher dimensions?

    Regards, Hans
  6. Dec 4, 2006 #5

    Hans de Vries

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    I now see that my results are the same as Hassani [1] 1998 who has a
    complete derivation the Green functions in higher dimensional spaces.

    An interesting result as the magnetic vector potential in 1+9 dimensions (eq.31):

    {\vec A}_9\ =\ \rho\mbox{\Large
    P}_9 * {\vec v}_t \ =\ \frac{\rho}{2
    \pi^4} \left(\ \ \frac{3!}{0!3!}\ \frac{1}{2^4}\
    \frac{\partial^4 {\vec x}}{\partial t^4} \right. \ \ [/tex]

    - \qquad \qquad \qquad \qquad \ \ +\ \quad \frac{4!}{1!2!}\ \frac{1}{2^5}\ \frac{1}{r^5}\
    \frac{\partial^3 {\vec x}}{\partial t^3} \ \ \ [/tex]

    - \qquad \qquad \qquad \qquad \ \ +\ \quad
    \frac{5!}{2!1!}\ \frac{1}{2^6}\ \frac{1}{r^6}\
    {\vec x}}{\partial t^2} \ \ \ [/tex]

    - \qquad \qquad \qquad \qquad \ \ +\ \quad \left. \frac{6!}{3!0!}\ \frac{1}{2^7}\ \frac{1}{r^7}\
    \frac{\partial {\vec x}}{\partial t}\ \ \right) \nonumber


    corresponds with the propagator given by Gal'tsov [2] 2001. (eq. 3.5)
    Note that the slowest decaying term depends on the 4th derivative d4x/dt4
    rather than on the velocity. further connections can be found in
    Cardoso et. al.[3] 2003 for gravitons rather than photons.

    Well, at least my work is correct :blushing:

    Regards, Hans

    [1] S. Hassani Mathematical Physics (Springer-Verlag, New York 1998)
    [2] D. V. Gal’tsov, Radiation reaction in various dimensions,
    Phys. Rev. D66, 025016 (2002). hep-th/0112110
    [3] Cardoso et. al. Gravitational Radiation in D-dimensional Spacetimes
    Physical Review D 67 064026 (2003) hep-th/0212168
    Last edited: Dec 4, 2006
  7. Dec 4, 2006 #6


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    A fascinating paper. Re propagators leaking outside the light cone, I suspect that the reason you're not seeing them while the various revered quantum physics sources such as P&S do is because you are making a classical calculation while theirs are quantum.
  8. Dec 4, 2006 #7

    Hans de Vries

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    That shouldn't be so. The question is per definition if the Green's function of
    the Klein Gordon propagator is non-zero outside the light-cone.

    P&S are throwing in creation and annihilation operators but Feynman and Zee
    don't do this at all. P&S use an anti-particle argument to restore zero pro-
    pagation outside the light cone.... thus, P&S don't agree with Feynman and
    Zee at the end!

    What we can do is go through all three (in 3 post) say in the order of:

    [1] Feynman, The Theory of Fundamental processes, chapter 18.
    [2] Zee, chapter I.3 , "free propagator" eqn 23.
    [3] P&S, chapter 2.4 "Causality" eqn 2.51 and 2.52.

    All three are very different!

    Let me now finish this post by telling what the simulator does:

    It simulates on a lattice. At time t=0, put a Dirac pulse on r=0.
    Then simply start simulating in time using:

    [tex]\frac{\partial^2 \phi}{\partial t^2}\ \ =\ \ \frac{\partial^2 \phi}{\partial x^2} + \frac{\partial^2 \phi}{\partial y^2} + \frac{\partial^2 \phi}{\partial z^2} - m^2\phi [/tex]

    This will produce the Green function which will:

    1- Never give any propagation outside the lightcone, not for any dimensional space.
    2- A real Dirac pulse will always give a real propagator (Green's function).

    Regards, Hans

    P.S I actually use the radial version of the Klein Gordon equation.
    Last edited: Dec 4, 2006
  9. Dec 5, 2006 #8


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    My skepticism is based on gravitational effects. When extra dimensions are added, leaky gravity appears to result in orbital instabilities on relatively short time scales.
  10. Dec 7, 2006 #9

    Hans de Vries

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    No propagation outside the lightcone: Part 1 - Feynman.

    As promised in post 7.

    First, Feynman on propagation outside the lightcone:

    From his book "The Theory of Fundamental Processes" chapter 18:


    At least Feynman comes up with a Green's function (The others just use
    approximations) He gives the following formula also found in his 1949 paper
    "Theory of positrons":

    {\cal F}^{1+3}\left( \frac{-1}{q^2-m^2}\right)\ =\
    \frac{1}{4\pi}\delta(s^2)\ -\ \frac{1}{8\pi}\frac{m}{s}\ \left(\
    J_1(ms) - iY_1(ms)\ \right)

    With [itex]J_1-iY_1 = H^{(2)}_1[/itex] is the Hankel function of the second kind and J and Y
    are the Bessel functions. Unfortunately, this formula is not error free.
    Besides minor scaling errors it is also complex since J and Y are real. The
    Green's function must be real since the response from the KG operator on
    a real Dirac pulse must be also real. The Hankel function is complex inside
    the light cone but real outside the light cone. This is the decaying exponent
    drawn in fig.18-4. The Hankel function is the "Bessel equivalent" of the complex
    exponential function: exp(-ims) = cos(ms) - i sin(ms), which also becomes real
    if m or s becomes imaginary.

    The Fourier transform can be done analytically and the correct result is:

    {\cal F}^{1+3}\left( \frac{-1}{q^2-m^2}\right)\ =\
    \frac{1}{2\pi}\delta(s^2)\ +\ \frac{1}{2\pi}\frac{m}{s}\Theta(s^2)\
    J_1(ms)\ \qquad \qquad \mbox{for}\ t \ge 0}

    The analytic result is explicitly zero outside the light cone because of
    the Heaviside stepfunction [itex]\Theta[/itex]. The formula can be derived as follows.
    First we start with the 1+1d case and write:

    \frac{1}{q^2-m^2}\ =\ \frac{1}{q^2} + \frac{m^2}{q^4} +
    \frac{m^4}{q^6} + \frac{m^6}{q^8} + ..... \nonumber

    =\ \frac{1}{(E-p)(E+p)}\ +\ \frac{m^2}{(E-p)^2(E+p)^2}\ +\
    \frac{m^4}{(E-p)^3(E+p)^3}\ +

    This translates to the following in configuration space for the Fourier

    first term: start with a Dirac pulse at t=r=0 and integrate first over the
    (t-r) direction and then integrate over the (t+r) direction.

    second term: integrate twice over the (t-r) direction and twice over the
    (t+r) direction.

    See also http://chip-architect.com/physics/Higher_dimensional_EM_radiation.pdf section IV

    The Fourier transform thus becomes:

    {\cal F}^{1+1}\left( \frac{-1}{q^2-m^2}\right)\ =\
    \frac{1}{2}\Theta(s^2)\left(1 - \frac{m^2s^2}{(1!)^2} +
    \frac{m^4s^4}{(2!)^2} - \frac{m^6s^6}{(3!)^2} + ..... \right)

    The series represents the Bessel function J of zero-th order:

    {\cal F}^{1+1}\left( \frac{-1}{q^2-m^2}\right)\ =\
    \frac{1}{2}\Theta(s^2)\ J_0(ms)

    We can get the 1+3d Green's function from the 1+1d version via the
    Inter-dimensional operator:

    {\cal F}^{1+3}\left( \frac{-1}{q^2-m^2}\right)\ =\
    \left\{\frac{1}{2}\Theta(s^2)\ J_0(ms)\right\}

    Which then gives gives:

    {\cal F}^{1+3}\left( \frac{-1}{q^2-m^2}\right)\ =\
    \frac{1}{2\pi}\delta(s^2)\ +\ \frac{1}{2\pi}\frac{m}{s}\Theta(s^2)\

    Which has explicitly zero propagation outside the light cone.

    Regards, Hans

    P.S. The epsilon prescription can be included with [itex]m \rightarrow \sqrt{m^2-i\epsilon}[/itex]
    but seems rather irrelevant here.

    useful links:
    Last edited: Dec 7, 2006
  11. Dec 8, 2006 #10


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    There is a typo in the exponent of r in formula (3). It can even be internally checked, as it does not coincide with formula (18).

    I think that the key point, the expansion for formulae 16, 17 (typo d <--> a in 16, btw) was discused time ago here in this forum, related to a question of Bee about extra dimensions. Perhaps you can remember the exact point and provide a link!

    The nomenclature "non radiating Coulomb field" and "Electromagnetic radiation" sounds elementary, but it could be worthwhile to define it in the introduction. I am still not sure what do each term refers to.
  12. Dec 8, 2006 #11

    Hans de Vries

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    The exponent d of r in the denominator is 1 higher as you might expect
    because of the x, y, z's in the numerators... :^)

    The result here is indeed a follow up on this earlier thread:
    I checked 16, It's OK although I see that I still have to specify parameter b.
    Well, one can see what b is in 17.

    The keyword is energy here. Radiating fields radiate energy away. This
    means that there must be a radiation reaction which results in a force
    opposing the motions of the charge which cause radiation. In higher
    dimensions this force isn't simply proportional to the acceleration
    anymore but also dependent on various higher derivatives (in time) of
    the acceleration.

    Regards, Hans
    Last edited: Dec 8, 2006
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