Usage of Lagrangian not Lagrangian density for quantum string in Polchinski

Question

In section 1.3 p. 18 while working out the energy spectrum of quantum string Polchinski uses Lagrangian instead of Lagrangian density while in QFT the usual convention is to use Lagrangian density (to keep the covariance intact). With the convention used there, calculations are done using expressions that are integral of spatial coordinate $int d sigma$. Is there any reason to choose this odd way of solving the field-action-variation problem?
This same problem is done using the usual density route in Tong's note and Becker book.

Qmechanic · Accepted Answer

The Lagrangian density ${cal L}$,

the Lagrangian $L=int_0^{ell}!dsigma~ {cal L}$,

and the action $S=int!dtau~ L$

contain the same information, and one is in principle free to use any of them. The reason why Polchinski chooses the Lagrangian $L$ in eq. (1.3.11) (which is integrated over $sigma$) is because it is more convenient when he (in the next step) wants to separate the string $X^-(tau,sigma)$ into a mean value
$$x^-(tau)~:=~frac{1}{ell}int_{0}^{ell}! dsigma~ X^-(tau,sigma)tag{1.3.12a}$$
(which is integrated over $sigma$), and the rest
$$Y^-(tau,sigma)~:=~X^-(tau,sigma)-x^-(tau). tag{1.3.12b}$$
See also this and this related Phys.SE posts.

Usage of Lagrangian not Lagrangian density for quantum string in Polchinski

One Answer

Add your own answers!

Ask a Question