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Coordinate transformation and conformal transformation?

  1. Mar 3, 2010 #1
    I'm confused by the relation of coordinate transformation and conformal transformation. I found a nice note about conformal field theory written by David Tong. It does contain the demonstration related to my question, but I still don't understand. Here it goes,

    The definition of conformal transformation is a coordinate transformation [tex]\sigma^a \rightarrow \sigma'^a(\sigma) [/tex], such that the metric changes by
    [tex] g_{ab}(\sigma) \rightarrow \Omega^2(\sigma)g_{ab}(\sigma) [/tex]

    What i'm confused is, Is the conformal transformation a subset of coordinate transformation? If this is true, then the conformal transformation is just part of the deffeomorphism(two-dimensional coordinate transformation is called deffeomorphism), but it seems that the answer is no.

    Tong talked about the dynamical background metric and fixed background metric, he said if the metric is dynamical, the weyl transformation is deffeomorphism; however, if the metric is fixed, the weyl transformation is not deffeomorphism, but a physical symmetry that takes [tex]\sigma[/tex] to [tex] \sigma'[/tex].

    And in a paragraph, he said, "any theory of 2d gravity which enjoys both deffeomorphism and Weyl invariance will reduce to a conformally invariant theory when the background metric is fixed." According to this, if the Weyl transformation is part of the deffeomorphism, then he could just say, ...enjoy deffeomorphism..., unnecessary to mention both deffeomorhpism and Weyl invariance. (I take Weyl transformation same as conformal transformation.)

    But, according to the definition, Weyl transformatio is evidently part of coordinate transformation. Anybody solves my confusion please?
    Thanks for any instructions!
  2. jcsd
  3. Mar 4, 2010 #2


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    This is a confusing topic because there are lots of definitions circulating about Weyl, conformal, and coordinate transformations. What you have written there is actually a Weyl scaling, not a conformal transformation (although the language is abused often, which adds to the confusion). In a Weyl scaling, you change the metric without changing the coordinates, and so it's distinct from a coordinate transformation. Now, I don't know the full definition of the conformal transformation off hand, but it incorporates both Weyl scalings and diffeomorphisms.
  4. Mar 4, 2010 #3
    I dont understand this stuff too well myself, so this may not be the appropriate level, but I do remember seeing a good explaination of conformal trans in one of the appendices of Sean Carrolls book Spacetime and Geometry which is available as a free PDF if you google it.
  5. Mar 4, 2010 #4


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    You can find what you need and more in this thread;


    In that thread, I neatly presented a self-containd introduction to this subject so that questions like the above do not arise on these forums. Obviously I have failed!


  6. Mar 4, 2010 #5
    Thank you so much, Sam! I will study it in detail, I'm eagerly to understand this stuff clearly.

    But allow me to ask a question in the beginning of your first post.:redface:
    According to your definition of conformal transformation, it is a nonlinear coordinate transformation,
    [tex] f:x\rightarrow \bar{x}=\bar{x}(x) [/tex]
    However, one of the definition of a tensor I learned is that, a tensor is a geometric object which transforms under coordinate transformation as follows, say, for example, the metric tensor should transform like this under coordinate transformation:
    \eta_{ab}(x) \rightarrow \bar{g}_{ab}(\bar{x}) = \frac{\partial x^c}{\partial\bar{x}^a}\frac{\partial x^d}{\partial\bar{x}^b}\eta_{cd}(x)
    So that, under general coordinate transformation, the line element is always invariant,
    ds^2 = \bar{g}_{ab}d\bar{x}^ad\bar{x}^b = \eta_{cd}dx^cdx^d

    Conversely, in the definition of a conformal transformation, we seem to keep the metric tensor unchanged while we change the coordinate, or seemingly, keeping the coordinate unchanged while changing the metric tensor.
    d\bar{s}^2 = \eta_{ab}d\bar{x}^ad\bar{x}^b = \bar{g}_{cd}(x)dx^cdx^d
    This seems to be in conflict with the definition of a metric tensor which I learned before. Could you explain this a little bit?

    Thank you so much!


    By the way, I just finished your first post, now I know why in 2D the conformal group is infinite dimensional! (if the above definition of conformal transformation accepted.)
    Last edited: Mar 4, 2010
  7. Mar 6, 2010 #6
    just to suggest some probably typos so as to make the posts perfect.

    (1) RHS of eq(3.2a) is probably [tex]\eta^{ac} [/tex]

    (2) Eq(5.8) is exact. RHS of eq(5.8) might be modified to be [tex] U(\alpha,b+ae^\beta-be^\alpha)[/tex]


    and, wow, you are very good at math, compact and rigorous. Hope I can digest more of latter part of math section tonight.

  8. Mar 14, 2010 #7


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    Last edited: Mar 14, 2010
  9. Mar 14, 2010 #8


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