Action and Polyakov

From Points to Strings

In classical mechanics, the path of a point particle through spacetime is the one that minimizes the action, which is proportional to the length of its worldline. The action $S$ for a free relativistic particle is given by its mass $m$ times the proper time $\tau$ it experiences, integrated along its path.

S = -m \int d\tau = -m \int \sqrt{-\eta_{\mu\nu} \frac{dX^\mu}{d\lambda}\frac{dX^\nu}{d\lambda}} d\lambda

String theory extends this idea. Instead of a zero-dimensional point particle tracing a one-dimensional worldline, we have a one-dimensional string tracing a two-dimensional surface in spacetime called a worldsheet (a world-line for each point on the string). The natural generalization of minimizing length is to minimize area. The action for a string should be proportional to the total area of the worldsheet it sweeps out.

The Nambu-Goto Action

The first attempt to write down this area-based action is the Nambu-Goto action . It measures the area of the worldsheet embedded in D-dimensional spacetime. Let's define the worldsheet coordinates as $\sigma^\alpha = (\tau, \sigma)$ , where $\tau$ is a time-like coordinate and $\sigma$ parameterizes the length of the string.

The spacetime coordinates of the string are then functions of these two parameters, $X^\mu(\tau, \sigma)$ . The action is proportional to the integral of the area element over the worldsheet.

S_{NG} = -T_0 \int d^2\sigma \sqrt{- \det(h_{\alpha\beta})}

While elegant, the Nambu-Goto action has a major drawback: the square root. This makes it notoriously difficult to work with, especially when trying to quantize the theory using path integrals. The square root leads to a non-polynomial and highly non-linear theory, which complicates calculations immensely.

Polyakov's Improvement

To solve this problem, Alexander Polyakov introduced an equivalent action that avoids the square root. The Polyakov action introduces an independent, auxiliary metric on the worldsheet, $\gamma_{\alpha\beta}(\sigma, \tau)$ , which describes the intrinsic geometry of the worldsheet itself, separate from its embedding in spacetime.

S_P = -\frac{T_0}{2} \int d^2\sigma \sqrt{-\gamma} \gamma^{\alpha\beta} \partial_\alpha X^\mu \partial_\beta X_\mu

At first glance, this seems more complicated. We've added a new field, the metric $\gamma_{\alpha\beta}$ . However, this action is quadratic in the $X^\mu$ fields, making it much easier to quantize. Classically, you can solve the equations of motion for $\gamma_{\alpha\beta}$ and substitute the solution back into the Polyakov action. When you do, you recover the Nambu-Goto action. The two are classically equivalent, but the Polyakov formulation is far more powerful for quantum calculations.

The Polyakov action trades a difficult square root for an auxiliary field, simplifying the path to quantization.

Worldsheet Symmetries

The Polyakov action has two crucial symmetries that are key to understanding string theory. The first is reparameterization invariance. This means the physics doesn't change if we choose a different coordinate system $(\tau', \sigma')$ on the worldsheet. It's a form of general covariance, but restricted to the two-dimensional worldsheet. It ensures that our choice of coordinates is just a label, with no physical meaning.

The second, and more subtle, symmetry is Weyl symmetry . This symmetry allows us to rescale the worldsheet metric locally without changing the action.

\gamma_{\alpha\beta}(\sigma, \tau) \rightarrow e^{2\omega(\sigma, \tau)} \gamma_{\alpha\beta}(\sigma, \tau)

Together, these two symmetries are powerful. They allow us to choose a convenient gauge, or coordinate system. Specifically, we can always choose coordinates such that the worldsheet metric is locally flat, meaning $\gamma_{\alpha\beta} = \eta_{\alpha\beta}$ , where $\eta_{\alpha\beta}$ is the 2D Minkowski metric. This is called the conformal gauge.

Working in this gauge simplifies the equations of motion for the string coordinates $X^\mu$ to a simple wave equation: $(\partial_\tau^2 - \partial_\sigma^2)X^\mu=0$ . However, fixing the gauge doesn't completely eliminate the freedom. The symmetries impose constraints on the physical states of the string, which are encoded in the stress-energy tensor of the worldsheet theory. Demanding that these constraints hold in the quantum theory leads to a remarkable conclusion: bosonic string theory is only consistent in 26 spacetime dimensions.

Now, let's review these concepts before moving on.

Test your understanding of the Nambu-Goto and Polyakov actions.

Quiz Questions 1/6

In string theory, what is the two-dimensional surface traced by a one-dimensional string as it moves through spacetime called?

Quiz Questions 2/6

What is the primary motivation for introducing the Polyakov action as an alternative to the Nambu-Goto action?

Understanding these actions and their symmetries is the mathematical foundation upon which the rest of string theory is built.

Advancing Through String Theory