String Tension Alpha Prime

Use units $\mathrm{\hslash }=c=1$. Adopt a primitive scalar parameter ${\alpha }^{\mathrm{\prime }}>0$ of dimension length squared, and define the string length ${\mathrm{\ell }}_s=\sqrt{{\alpha }^{\mathrm{\prime }}}$ and the string tension $T=1\mathrm{/}(2\pi {\alpha }^{\mathrm{\prime }})$, of dimension energy per length, or $L^{-2}$ in these units. Operationally, $T$ is the positive coefficient of the induced worldsheet area in the Nambu–Goto action functional: for the induced metric ${\gamma }_{ab}={\eta }_{\mu \nu }{\mathrm{\partial }}_a{X}^{\mu }{\mathrm{\partial }}_b{X}^{\nu }$ of the nondegenerate Lorentzian configuration branch, set

$S_{\mathrm{N}\mathrm{G},K}[X]=-T{\int }_K\sqrt{-\det \gamma }{d}^2\sigma \mathrm{.}$

Here $K$ is any compact worldsheet region with smooth boundary and the square root is positive. Equivalently, the Polyakov normalization names $-T\mathrm{/}2$ as the coefficient of $\sqrt{-\det h}{h}^{ab}{\gamma }_{ab}$ in its action density. These formulas fix coefficient conventions; adopting stationarity as dynamics is a separate physical postulate. Worldsheet coordinates are dimensionless and $X^{\mu }$ has length dimension, so both action functionals are dimensionless. This convention is fixed for the flat tensile bosonic string; it does not define the string coupling $g_s$ or a spacetime Planck scale.

Flat Bosonic String Model