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Background info:

My understanding:

1.

  • Weyl transformation is a local rescaling of the metric tensor $$ g_{ab}\rightarrow e^{-2\omega(x)}g_{ab} $$
  • A theory invariant under this Weyl transformation is called conformally invariant, or Weyl invariance or with Weyl symmetry.
  • Diffeomorphism maps to a theory under arbitrary differentiable coordinate transformations (Diffeomorphism is an isomorphism of smooth manifolds. It is an invertible function that maps one differentiable manifold to another such that both the function and its inverse are smooth.)
  • A theory invariant under this Diffeomorphism transformation is called general covariance, also known as diffeomorphism covariance or general invariance.

Questions:

  1. What are the relations between: Weyl transformation vs diffeomorphism? conformally invariant vs general in/covariant?

Do the two concepts have some overlapped? (Say certain transformations contain the other transformations?) Or can we show they are totally independent from each other? Weyl transformation can never be diffeomorphism? Diffeomorphism can never be Weyl transformation?

  1. Do we have more general transformations beyond the Diffeomorphism and Weyl transformations for a physical theory? Or do we have a direct product Diff x Weyl, do they Diff x Weyl form a group? is this product x a direct product of groups?)

For example, it is commonly written that the transformation is Diff x Weyl, such as in Polchinski about gauging fixing, But do we have more general transformation than $$\text{Diff $\times$ Weyl}?$$ Are Diff x Wey really a direct product of "what groups (?)" without any dependence?

e.g. Polchinski book String Theory: enter image description here enter image description here

There are previous attempts to ask Conformal transformation vs diffeomorphisms but my concern is totally different.  

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  • $\begingroup$ Related: physics.stackexchange.com/q/38138/2451 $\endgroup$ – Qmechanic Oct 11 '20 at 18:33
  • $\begingroup$ They asked Weyl transformation vs Conformal, :) $\endgroup$ – annie marie heart Oct 11 '20 at 18:35
  • $\begingroup$ I ask Weyl transformation (~ Conformal) vs diffeomorphism $\endgroup$ – annie marie heart Oct 11 '20 at 18:35
  • $\begingroup$ Diffs change the coordinates, Weyl doesn’t; they are different. Whether there is a larger symmetry group surely depends on the theory. E.g. you can have internal symmetries. $\endgroup$ – Oбжорoв Oct 11 '20 at 20:41
  • $\begingroup$ thanks --- >> "E.g. you can have internal symmetries" -> I am asking only the spacetime or external symmetry. How does Poincare. symmetry relate to Diff × Weyl? $\endgroup$ – annie marie heart Oct 11 '20 at 22:01

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