# String Theory: Nambu-Goto or Polyakov?

1. Jan 19, 2010

### wam_mi

Hi there,

I 've recently looked at how the Nambu-Goto action of an open string can be derived from the proper area of the parameterised world-sheet, in the form of either the derivatives of space-time coordinates or the determinant of the induced metric.

However, may I ask what are the main advantages and disadvantages of adopting the Polyakov approach instead? Which one of these have a better application on other aspects of string theory?

Cheers!

2. Jan 19, 2010

### xepma

The Nambo-Goto action is the action corresponding to a classical vibrating string. However, because of the square root appearing in this action, quantizing this action turns out to be difficult -- it's not really known how to quantize such an action.

For this action it's possible to fix a gauge, called the light cone gauge, and this way you can obtain a formulation of the corresponding quantum theory. But a gauge independent formulation does not exist.

The Polyakov action, on the other hand, contains an auxillary field called the worldsheet metric. You can show that classically the Polyakov action is equivalent to the Nambu-Goto action (this follows by inserting the equation of motions of the auxillary field into the action). The Polyakov action has the advantage that it can be quantized using the gauge-invariant methods (i.e. BRST quantization) or gauge-dependent methods (Faddeev-Popov).

It should be noted that because a gauge-independent formulation of the quantum theory of the Nambu-Goto action does not exist, it is not clear wether the quantum theory of the Nambu-Goto action and the Polyakov action are truly the same theory. Classically they are, but quantum mechanically it's not known...

3. Jan 19, 2010

### tom.stoer

In GR you can use tetrads and a first order formalism; in addition you can change to Ashtekar-Barbero variables (which are quantized in LQG using the temporal gauge). Is something similar known for string theory?