I've been very impressed to learn about kaluza-klein theory and compactification strategies. I would like to read more about this but in the meantime i'm curious about 2 different points. I have the feeling that there are no precise answers to these questions at the current moment but nonetheless i feel obligued to ask:

  1. Does a compactification strategy has to impose a fixed thickness in the compactification or this arises as a dynamic consequence of the evolution equations? and if i start with a small thickness, does the dynamic in all cases keeps the thickness in a bounded value? are there results that addresses this?

EDIT answer appears to be No, it is not assumed to be fixed. There is an effective scalar field that precisely describes this.

  1. Given that the argument is that vibrations in the compactified dimensions determine the internal states, any significant gradient ratio in the compactification length would have to be probably measurable as either spontaneous decay or spontaneous stimulation events of particles. My question is, would experimental effects of any thickness gradients would be as difficult to measure as expected effects from a fixed but non-zero compactified dimension?

EDIT a better rephrase of this question is that; since energy of non-gravitational fields like electromagnetism, etc. is stored in vibrations along the compactified dimensions, it would seem that even a small gradient in the radion field (even 0.1%) should be measurable as a effective diffraction index (since a 0.1% variation in the length of the scale would affect each non-gravitational mode by that amount)

I guess even another way to frame the question is: is the low-energy phenomena we are able to see currently, invariant/non-dependent under small changes of scale (from point to point in space-time) of the compactification? isn't scale of compactification itself a vibrating degree of freedom?

So bottom line: since we haven't see any low-energy consequences from small gradients in the radion field, can't we infer that the radion field, is, by whatever reasons (dynamics, broken symmetry, etc.) effectively fixed? can't we currently estimate a bound in the radion gradient given the negative results mentioned above it would have?

I hope i've made my questions clear and interesting enough.

  • $\begingroup$ Thickness? {filler} $\endgroup$ – pho Jan 24 '11 at 20:22
  • $\begingroup$ I suppose I am a bit uncertain what is being asked. Are you askinng about the scale of Calabi-Yau compactification? $\endgroup$ – Lawrence B. Crowell Jan 25 '11 at 2:12
  • $\begingroup$ well, Calabi-Yau is definitely the best known compactification, so if you think there is no significant loss of generality by restricting to those, then it works for me However, i would hope that, even if the concrete topology might determine actual spectras, the predictions of local inhomogeneities of the scale of the compactification would be largely independent from it $\endgroup$ – lurscher Jan 25 '11 at 3:20
  • $\begingroup$ What do you mean with thickness? Moduli? $\endgroup$ – WIMP Jan 25 '11 at 7:56
  • $\begingroup$ with 'thickness' i mean the scale of the compactified dimensions $\endgroup$ – lurscher Jan 25 '11 at 16:30

This is the problem of radion stabilization for KK theories. The radion is the effective 4D field which measures the size of the compactified dimension.

For Calabi-Yau compactifications with unbroken SUSY, the radion turns out to be a moduli, which is experimentally unsatisfactory.

There are many different radion stabilization mechanisms out there, e.g. fluxes, Goldberger-Wise, etc. . In general relativistic theories, the mechanism has to be dynamical.

After quantization, the radion would show up as a massive particle. The radion also doubles as a dilaton in the effective 4D theory as $M_{4D}^2$ (4D Planck mass squared) is proportional to the compactification volume.

  • $\begingroup$ thanks for your answer. What i was getting at with 'local inhomogeneities being easier to detect or not' is that, even if a local inhomogeneity changes that scale from 5*10e-37 to 10e-36 (a factor of 2), wouldn't such inhomogeneity cause great havoc, rendering those easily observable at low energies? i guess a better question is: is the low-energy phenomena we are able to see currently, invariant under small changes of scale (from point to point in space-time) of the compactification? $\endgroup$ – lurscher Jan 25 '11 at 19:06
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    $\begingroup$ Nitpick: the overall volume is a modulus (a true flat direction) in Type II string theories with ${\cal N}=2$ SUSY, but can be stabilized while leaving ${\cal N}=1$ SUSY unbroken (as in KKLT, for instance). $\endgroup$ – Matt Reece Jan 26 '11 at 7:52

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