*Post by Kasper Jens Larsen*I was wondering, what is the reason for preferring the Polyakov over

the Nambu-Goto action in string theory? Of course, one can do the

quantization covariantly in the Polyakov action, but are there reasons

that make the Polyakov action unavoidable?

Dear Kasper, the main advantage of the Polyakov action is that the

corresponding action is bilinear in (the derivatives of) the physical

fields X(sigma). This allows one to quantize them using the standard rules

involving the Fock space etc. At the same moment, the auxiliary metric

tensor is introduced to the worldsheet together with Weyl symmetry (in

addition to diffeomorphisms) that makes the whole metric tensor locally

unphysical (3 parameters of symmetries vs. 3 numbers in the metric

tensor).

Clasically one can show that the Polyakov action is equivalent to the

Nambu-Goto action, and it is reasonable to define this statement to be

true in general at the quantum level, at least in the critical dimension.

Because it is hard to say what you exactly mean by a quantization of the

Nambu-Goto non-linear action (with a lot of square roots etc.), you may

consider the Polyakov quantization to be a refined and more accurate

answer to the question how to quantize the Nambu-Goto action. Once you

learn how conformal symmetry nicely works, you may very well consider the

Polyakov action to be the true answer and starting point while the

Nambu-Goto action is just a naive and heuristic motivation for the

Polyakov action.

Try to quantize the Nambu-Goto action without the auxiliary metric tensor

and without Weyl symmetry. Find its Hilbert space and justify it. You will

see that it is not easy. The Nambu-Goto action looks very nonlinear and

difficult to solve, but that's just an illusion. The Polyakov approach is

a way to show that the solutions of the Nambu-Goto action are actually

very simple and "linear" if you write them in the right coordinates.

What I say above may look like we're relying on some special choice of

variables, much like the guys in loop quantum gravity. The difference is

that once you show that the Nambu-Goto is classically equivalent to the

Polyakov action, you're done because the methods to solve the Polyakov

action are standard canonical methods from quantum field theory that no

reasonable physicist would ever question, and they lead to consistent

physical insights.

Moreover, the methods of conformal field theory agree with the well-tested

approaches in quantum field theory in general - and with the insights

about the Renormalization Group. The Nambu-Goto action may look "natural"

at the classical level but at the quantum level it is a very unnatural

beast. Once you write it in the Polyakov form, things become natural.

You know, the reason why we take the math that follows from the Polyakov

action seriously is not that it agrees with some fundamentalist doctrines

that the worldsheet area should be a natural thing to minimize. The true

reason is that it leads to robust and self-consistent mathematical

structure that agrees with the nature of physics, at least in the rough

picture. You are free to try to find a better theory or a better

quantization of the Nambu-Goto action but be sure that all non-Polyakov

approaches so far have turned out to be physically useless.

Best

Lubos