Detecting and analyzing higher dimensions via the EM radiation field.

1. Dec 1, 2006

Hans de Vries

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

abstract:

Electromagnetic radiation decays with $1/r$ in three dimensional
space, while the non radiating Coulomb field decays faster with $1/r^2$.
The general expressions for any dimension are $1/r^{(d-1)/2}$ for the
Radiation and $1/r^{(d-1)}$ 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 $1/r^{n}$ 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. Dec 1, 2006

Sauron

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.

3. Dec 1, 2006

Hans de Vries

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

http://superstringtheory.com/experm/exper51.html

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
4. Dec 2, 2006

Hans de Vries

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

5. Dec 4, 2006

Hans de Vries

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{1}{r^4}\ \frac{\partial^4 {\vec x}}{\partial t^4} \right. \ \$$

$$- \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} \ \ \$$

$$- \qquad \qquad \qquad \qquad \ \ +\ \quad \frac{5!}{2!1!}\ \frac{1}{2^6}\ \frac{1}{r^6}\ \frac{\partial^2 {\vec x}}{\partial t^2} \ \ \$$

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

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

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
6. Dec 4, 2006

CarlB

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.

7. Dec 4, 2006

Hans de Vries

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:

$$\frac{\partial^2 \phi}{\partial t^2}\ \ =\ \ \frac{\partial^2 \phi}{\partial x^2} + \frac{\partial^2 \phi}{\partial y^2} + \frac{\partial^2 \phi}{\partial z^2} - m^2\phi$$

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
8. Dec 5, 2006

Chronos

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.

9. Dec 7, 2006

Hans de Vries

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:

http://www.chip-architect.com/physics/KG_propagator_Feynman.jpg

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 $J_1-iY_1 = H^{(2)}_1$ 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 $\Theta$. The formula can be derived as follows.

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

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

This translates to the following in configuration space for the Fourier
transform:

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.

et-cetera.

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:

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

Which then gives gives:

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

Which has explicitly zero propagation outside the light cone.

Regards, Hans

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

http://functions.wolfram.com/BesselAiryStruveFunctions/
http://en.wikipedia.org/wiki/Bessel_function

Last edited: Dec 7, 2006
10. Dec 8, 2006

arivero

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.

11. Dec 8, 2006

Hans de Vries

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.