Understanding Nambu-Goto String Action

[wam_mi]

Hi there,

I recently read about the construction of the Nambu-Goto action by looking at the proper area of the world-sheet. However, when one varies the action with an arbitrary space-time coordinate (here I treat it in Minkowski space-time X^(\mu)), there appears three terms and some of them would go to zero.

What I don't understand is why is that

(i) The momentum density running along the string P^(\mu)_(\sigma) at the open string endpoints \sigma (0, \pi) is zero?

(ii) Why is it that the momentum density transverse to the string, i.e. P^(\mu)_(\tau) \neq 0?

Has this got something to do with the conservation law, that energy is conserved?


Thanks

 

[AndyW]

for your post and for sharing your thoughts on the Nambu-Goto action and its implications. I can provide some insight into the questions you have raised.

Firstly, when we vary the Nambu-Goto action with an arbitrary space-time coordinate, we end up with three terms because the action is a functional of the world-sheet coordinates X^(\mu). These terms correspond to the variation of the world-sheet metric, the variation of the world-sheet coordinates, and the variation of the Lagrange multiplier. The Lagrange multiplier term ensures that the world-sheet is parametrized by a proper time, which is an essential requirement for the action to be invariant under reparametrization.

Now, to address your questions about the momentum density along and transverse to the string. The momentum density running along the string, P^(\mu)_(\sigma), is zero at the open string endpoints because the endpoints have fixed positions in space-time. This means that the variation of the action with respect to X^(\mu) at the endpoints is zero, resulting in a zero momentum density.

On the other hand, the momentum density transverse to the string, P^(\mu)_(\tau), is non-zero because the string is free to move in this direction. This momentum density is related to the transverse vibrations of the string, and it is not conserved. It can change as the string moves and interacts with its surroundings.

Lastly, your intuition about energy conservation is correct. The conservation of energy is related to the invariance of the action under time translation. This means that the action remains unchanged when we shift the world-sheet coordinate corresponding to time. This is a fundamental principle in physics, and it is also reflected in the Nambu-Goto action.

I hope this helps clarify your questions about the Nambu-Goto action and its implications. Keep exploring and asking questions – that's what science is all about!