## String Theory, Part IV: Relativistic Point Particles and Strings

The Relativistic Point Particle

We want to calculate the action of a point particle under a boost. It is natural to begin with

$\displaystyle S = -mc\int_{\mathcal{P}} ds = -mc^2\int_{t_i}^{t_f}\sqrt{1-\frac{v^2}{c^2}} dt$

where $\mathcal{P}$ is the path of the worldline, and the Lagrangian,

$\displaystyle\mathit{L} =-mc^2\sqrt{1-\frac{v^2}{c^2}} \approx -mc^2\left(1-\frac{v^2}{2c^2}\right) = -mc^2 + \frac{1}{2}mv^2,$

which retrieves our classical interpretation at low order approximation. Using the Euler-Lagrange equation, we then find that

$\displaystyle\vec{p} = \frac{\partial\mathit{L}}{\partial\vec{v}}=-mc^2\left(-\frac{\vec{v}^2}{c^2}\right)\frac{1}{\sqrt{1-\frac{v^2}{c^2}}}=\gamma m\vec{v}$

and thus

$\displaystyle\mathit{H} =\vec{p}\cdot\vec{v}-\mathit{L}=\gamma mv^2 + \frac{mc^2}{\gamma}=\gamma mc^2$

which is precisely what we expect as a cursory approach.

The relativistic point particle requires more than simple boosts, but we will still remain in flat spacetime. As such, we have the metric $ds = \sqrt{-\eta_{\mu\nu}X^\mu X^\nu}$, however, for the action we want this in terms of some timelike value, $d\tau$, so we can instead say that $ds = \sqrt{-\eta_{\mu\nu}\dot{X}^\mu\dot{ X}^\nu} d\tau$, where $\dot{X}^\mu = \frac{\partial X^\mu}{\partial\tau}.$ Our action will then be

$\displaystyle S= -mc\int_{s_i}^{s_f} ds = -m\int_{\tau_0}^{\tau_1} \sqrt{-\eta_{\mu\nu}\dot{X}^\mu\dot{ X}^\nu}\; d\tau$

If we add in the electromagnetic field, it can be shown that the relativistic action of a charged particle is

$\displaystyle S= -mc\int_{\mathcal{P}} \left(\sqrt{-\eta_{\mu\nu}\dot{X}^{\mu}\dot{ X}^{\nu}}+\frac{q}{c}A_\mu\dot{X}^\mu\right)d\tau$

Next, it must be noted that

$d\tau =\frac{d\tau}{d\tau^\prime}\;d\tau\$ and so $\; \dot{X}^\mu = \frac{d\tau^\prime}{d\tau}\frac{dX^\mu}{d\tau^\prime}$

and thus

$\displaystyle S = -m\int \sqrt{-\eta_{\mu\nu}\dot{X}^\mu\dot{ X}^\nu}\; d\tau\\ \\ = -m\int \sqrt{-\eta_{\mu\nu}\frac{d X^\mu}{d\tau}\frac{d X^\nu}{d\tau}}\; d\tau\\ \\ = -m\int \sqrt{-\eta_{\mu\nu}\frac{dX^\mu}{d\tau^\prime}\frac{dX^\nu}{d\tau^\prime}\left(\frac{d\tau^\prime}{d\tau}\right)^2}\left(\frac{d\tau}{d\tau^\prime}\right)d\tau^\prime\\ \\ = -m\int \sqrt{-\eta_{\mu\nu}\frac{d X^\mu}{d\tau^\prime}\frac{d X^\nu}{d\tau^\prime}}\; d\tau^\prime$

which gives us reparameterization invariance. A good choice for $\tau$ that allows us to solve several equations is the equality $X^0 = c\tau = ct$. This is known as static gauge because, physically, we are examining slices of space at static moments.

The Relativistic String

If we construct the matrix

$\displaystyle g_{ij} = \frac{\partial\vec{x}}{\partial\xi^i}\cdot\frac{\partial\vec{x}}{\partial\xi^i},$

for arbitrary parameters $\xi^i$, we find that $g_{ij}$ is the induced metric on $S$, where

$ds^2 = \vec{x}\cdot\vec{x} = \frac{\partial\vec{x}}{\partial\xi^i}\cdot\frac{\partial\vec{x}}{\partial\xi^i}\;\xi^i\xi^j = g_{ij}\xi^i\xi^j.$

Furthermore, we find that for $g= \det(g_{ij})$, the area

$A = \int\sqrt{g}\;d\xi^{1}d\xi^{2}$

is also invariant under reparameterization.

We therefore define our two-dimensional worldsheet for a string by the parameters $\tau$ and $\sigma$, where $\tau$ is timelike, and $\sigma$ is spacelike.

Furthermore, we can define proper area on the worldsheet to be

$\displaystyle\mathit{A}=\int\sqrt{\left(\frac{\partial X^\mu}{\partial\tau}\frac{\partial X_\mu}{\partial\sigma}\right)^2-\left(\frac{\partial X^\mu}{\partial\tau}\frac{\partial X_\mu}{\partial\tau}\right)\left(\frac{\partial X^\nu}{\partial\sigma}\frac{\partial X_\nu}{\partial\sigma}\right)}\; d\tau d\sigma \\ \\ \\ = \int\sqrt{\left(\frac{\partial X}{\partial\tau}\cdot\frac{\partial X}{\partial\sigma}\right)^2-\left(\frac{\partial X}{\partial\tau}\frac{\partial X}{\partial\tau}\right)^2\left(\frac{\partial X}{\partial\sigma}\frac{\partial X}{\partial\sigma}\right)^2}\; d\tau d\sigma \\ \\ \\ =\int_{\tau_i}^{\tau_f}\int_0^{\sigma_1}\sqrt{(\dot{X}\cdot X^\prime)^2 - (\dot{X})^2(X^\prime)^2}\; d\sigma d\tau$

Here, the dot is the relativistic dot product. Now the action has units of  $\frac{[mass][length]^2}{[time]}$  (the same as $\hbar$, coincidentally), so by multiplying the proper area by $\frac{T_0}{c}$ we obtain the correct units and thus our action becomes

$\displaystyle S_{NG}= -\frac{T_0}{c}\int_{\tau_i}^{\tau_f}\int_0^{\sigma_1}\sqrt{(\dot{X}\cdot X^\prime)^2 - (\dot{X})^2(X^\prime)^2}\; d\sigma d\tau.$

This is called the Nambu-Goto action of a relativistic string.

Our reparameterization invariance allows us to construct the matrix

$\displaystyle\gamma_{\alpha\beta} = \eta_{\mu\nu}\frac{\partial X^\mu}{\partial\xi^\alpha}\frac{\partial X^\nu}{\partial\xi^\beta} = \frac{\partial X}{\partial\xi^\alpha}\cdot\frac{\partial X}{\partial\xi^\beta},$

such that the Nambu-Goto action becomes

$\displaystyle\mathit{S}_{NG}=-\frac{T_0}{c}\int\sqrt{-\gamma}\; d\sigma d\tau$

where $\gamma = \det(\gamma_{\alpha\beta}).$