Massive and massless modes of the anchored string.

[Spinnor]

An anchored string is a simple modification of a vibrating string. We imagine a sideways restoring force applied to a vibrating string. The sideways force per unit length is proportional to the displacement of the string. This additional force gives the string the the same dispersion curve as that of a massive quantum. (see Iain G. Main, Vibrations and Waves in Physics, 3rd ed., Cambridge University Press, 1993, page 229.) What I also find interesting is depending on which plane the string vibrates we can have either "massive" or "massless" modes. Also, I'm guessing there should be an interaction between the massive and massless modes because the curvature of waves on the string will change the tension which effects velocity of wave propagation for both modes.

Could anyone write down the Lagrangian for this system?

If we can ever come up with a model or picture of the Universe which implies known physics I think some form of the anchored string will be part of that picture?

I think we can come up with a 3 dimensional analog of the anchored string but we will need two extra dimensions, one dimension for displacement with an additional restoring force giving us "massive" modes and one extra dimension for displacement of the "massless" modes.

Thank you for any thoughts.

 

[Spinnor]

I wrote,

...I think we can come up with a 3 dimensional analog of the anchored string but we will need two extra dimensions, one dimension for displacement with an additional restoring force giving us "massive" modes and one extra dimension for displacement of the "massless" modes. ...

A possible interesting variation on the above. Maybe we can get by with a single extra dimension and still have massive and massless modes. Let the extra dimension be a circle. Let there be a potential that goes as sin^2(theta). We will now have massless modes for small theta and we will have massive modes for large theta.

Any thoughts appreciated!