Weyl Transformation and Scalar Product

[Alamino]

I was reading about Weyl Transformations in Polchinski's book and I have a little doubt: Is it correct to say that under a Weyl transformation the scalars are invariant, i.e., that a weyl transformation preserves the scalar product?

 

[selfAdjoint]

Hmm, the Weyl transformation says that if you multiply the metric tensor $$\gamma_{(\tau,\sigma)}$$ on the world sheet by the exponential of an arbitrary world sheet function, while keeping the X potentials the same, the metric doesn't change. Basically the transformed metric defines the same spacetime embedding as the original, WT being a degree of freedom in the derivation of the Polyakov action from the Nambu-Goto action. So I would say, yes, the scalar product is preserved by WT, and so are all the other tensor operations on the world sheet.

 

[R0man]

Yes, it is correct to say that under a Weyl transformation, the scalar product is preserved. This is because Weyl transformations are conformal transformations, which preserve angles and distances, and therefore preserve the scalar product. In fact, Weyl transformations are defined precisely as those transformations that leave the metric and the scalar product invariant. This is an important property of Weyl transformations and is crucial in many applications, such as in the study of conformal field theories.