- #1

- 42

- 0

Im studying Zweibach's "A first Course in String Theory".

From the Nambu-Goto Action he derives the wave equation which consists of the term

[tex] \frac{\partial \mathcal{L}}{\partial \dot{X} ^{\mu}} &= -\frac{T_{0}}{c}\frac{\dot{X} \cdot X')X'_{\mu}-(x')^{2}\dot{X}_{\mu}} {\sqrt{(\dot{X} \cdot X')^{2} - (\dot{X})^{2}(X')^{2}}} \equiv \mathcal{P}^{\tau}_{\mu} [/tex]

Now this [tex] \mathcal{P}^{\tau}_{\mu} [/tex] is referred to as the momentum density and is used as such later on in the book:

[tex] p_\mu(\tau) = \int ^{\sigma _{1}}_{0} \mathcal{P}^{\tau}_{\mu}(\tau , \sigma) d\sigma [/tex]

Im confused because I would expect [tex] \mathcal{P}^{\tau}_{\mu} [/tex] to have units of Mass divided by time, if \sigma has units of length, or units of momentum, if \sigma is dimensionless, so that when you integrate over \sigma you get units of momentum.

But as far as I can see [tex] \mathcal{P}^{\tau}_{\mu} [/tex] has units of mass times length. I get would think this becuase the lagrangian density has units of ML^2/T and it is differentiated with respect to velocity, thus it takes units of ML.

Can someone help with this?

Thanks.

B