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!

 

[Entropia]

I find this concept of anchored strings to be intriguing and potentially useful in understanding vibrations and waves. The idea of a sideways restoring force applied to a vibrating string is a simple but effective modification that can produce a similar dispersion curve as that of a massive quantum. It is interesting to note that depending on the plane of vibration, the string can exhibit either "massive" or "massless" modes, which suggests a potential for different behaviors and interactions between these modes.

In terms of further understanding this system, it would be helpful to have a mathematical representation such as a Lagrangian, which would describe the dynamics of the anchored string system. This would provide a more quantitative understanding of the interactions between the massive and massless modes and could potentially lead to new insights and applications.

I agree that the anchored string concept could potentially play a role in our understanding of the Universe. With further research and development, it is possible that this model could be extended to a 3-dimensional analog, incorporating extra dimensions to account for the different modes of displacement. This could provide a new perspective on the behavior of vibrations and waves in our world.

Thank you for bringing up this interesting concept and I look forward to seeing how it evolves in the future.