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
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
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
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.