Graduate The Polyakov Action & Weyl Transformations

[Higgsono]

The Polyakov action is invariant under Weyl transformations, that is local rescaling of the metric tensor on the world sheet. However, I don't really understand the physical meaning of this. What would it mean for the action to not have this symmetry?

I also have another question concerning reparametrization invariance. Because of this gauge invariance, it means that we have a redundancy in the system and we actually have fewer degrees of freedom than what it appears to be in the action. But why does the presence of gauge invariance reduce the number of degrees of freedom?

 

[haushofer]

The Polyakov action is invariant under Weyl transformations, that is local rescaling of the metric tensor on the world sheet. However, I don't really understand the physical meaning of this. What would it mean for the action to not have this symmetry?

I also have another question concerning reparametrization invariance. Because of this gauge invariance, it means that we have a redundancy in the system and we actually have fewer degrees of freedom than what it appears to be in the action. But why does the presence of gauge invariance reduce the number of degrees of freedom?

1) It would mean that the worldsheet-metric can not be gauge-fixed entirely, and hence contains its own dynamics. This in contrary to the Nambu-Goto action, where the worldsheet metric is induced by the target spacetime. So it would mean that the Nambu-Goto formulation and Polyakov formulation aren't equivalent already on the classical level.

2) Because you can use a gauge transformation to put a field component to zero.

A 2-dim. metric has three components, because it can be written down as a symmetric 2X2 matrix. But gct's allow you to gauge-fix two of those three components. If you don't believe me: try it by transforming the metric and choosing the vector field as such to eliminate two of the three components! With this you can gauge fix the metric to the 2-dim Minkowski metric diag(-1,+1). So that would be it normally; another gct would take you out of this gauge choice again. But then you remember you also have Weyl-rescalings. So if you first perform a gct on your Minkowski-metric, taking you out of the gauge diag(-1,+1), and then perform a Weyl rescaling to get back to the diag(-1,+1) gauge, then you're still fine! The restriction this condition gives you on the Weyl-parameter gives you the conformal group in 2D and is the very basic reason why String theory is a 2D conformal field theory.

Hope this helps.

 

[tom.stoer]

But why does the presence of gauge invariance reduce the number of degrees of freedom?

Look at QED as the simplest example.

You start with the four components of A_μ. A₀ has no associated canonical momentum, that means it acts as a Lagrange multiplier; its e.o.m. is just the Gauß law constraint. You can set A₀ = 0 using gauge invariance, and you can use the Gauß law to eliminate the longitudinal polarization of the vector potential A. That means you reduce the four d.o.f. of A_μ subtracting two unphysical ones related to local gauge symmetry, resulting in two physical d.o.f. = the two transversal photon polarizations A_⊥.

Of course the details are different in string theory, but the basic idea is the same.