Quantcast
  • Register
PhysicsOverflow is a next-generation academic platform for physicists and astronomers, including a community peer review system and a postgraduate-level discussion forum analogous to MathOverflow.

Welcome to PhysicsOverflow! PhysicsOverflow is an open platform for community peer review and graduate-level Physics discussion.

Please help promote PhysicsOverflow ads elsewhere if you like it.

News

PO is now at the Physics Department of Bielefeld University!

New printer friendly PO pages!

Migration to Bielefeld University was successful!

Please vote for this year's PhysicsOverflow ads!

Please do help out in categorising submissions. Submit a paper to PhysicsOverflow!

... see more

Tools for paper authors

Submit paper
Claim Paper Authorship

Tools for SE users

Search User
Reclaim SE Account
Request Account Merger
Nativise imported posts
Claim post (deleted users)
Import SE post

Users whose questions have been imported from Physics Stack Exchange, Theoretical Physics Stack Exchange, or any other Stack Exchange site are kindly requested to reclaim their account and not to register as a new user.

Public \(\beta\) tools

Report a bug with a feature
Request a new functionality
404 page design
Send feedback

Attributions

(propose a free ad)

Site Statistics

205 submissions , 163 unreviewed
5,047 questions , 2,200 unanswered
5,345 answers , 22,709 comments
1,470 users with positive rep
816 active unimported users
More ...

  Weyl transformation vs diffeomorphism; conformal invariant vs general in/covariant

+ 4 like - 0 dislike
1089 views

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.  

This post imported from StackExchange Physics at 2020-12-07 19:32 (UTC), posted by SE-user annie marie heart
asked Oct 11, 2020 in Theoretical Physics by annie marie heart (1,205 points) [ no revision ]
Related: physics.stackexchange.com/q/38138/2451

This post imported from StackExchange Physics at 2020-12-07 19:32 (UTC), posted by SE-user Qmechanic
They asked Weyl transformation vs Conformal, :)

This post imported from StackExchange Physics at 2020-12-07 19:32 (UTC), posted by SE-user annie marie heart
I ask Weyl transformation (~ Conformal) vs diffeomorphism

This post imported from StackExchange Physics at 2020-12-07 19:32 (UTC), posted by SE-user annie marie heart
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.

This post imported from StackExchange Physics at 2020-12-07 19:32 (UTC), posted by SE-user Oбжорoв
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?

This post imported from StackExchange Physics at 2020-12-07 19:32 (UTC), posted by SE-user annie marie heart

Your answer

Please use answers only to (at least partly) answer questions. To comment, discuss, or ask for clarification, leave a comment instead.
To mask links under text, please type your text, highlight it, and click the "link" button. You can then enter your link URL.
Please consult the FAQ for as to how to format your post.
This is the answer box; if you want to write a comment instead, please use the 'add comment' button.
Live preview (may slow down editor)   Preview
Your name to display (optional):
Privacy: Your email address will only be used for sending these notifications.
Anti-spam verification:
If you are a human please identify the position of the character covered by the symbol $\varnothing$ in the following word:
p$\hbar$ysicsOverf$\varnothing$ow
Then drag the red bullet below over the corresponding character of our banner. When you drop it there, the bullet changes to green (on slow internet connections after a few seconds).
Please complete the anti-spam verification




user contributions licensed under cc by-sa 3.0 with attribution required

Your rights
...