Conformal dimension of a scalar function?

[ismaili]

Actually, I'm not fully understand what the meaning of conformal dimension is. But I know how to read off the conformal dimension of a tensor, say, [tex]t^{++}{}_+[/tex], then the conformal dimension is -2 + 1= -1, where the lower index carries conformal dimension 1 and upper index carries conformal dimension -1. The + index denotes the index for the light-cone coordinate [tex] z = \sigma^+ = \tau + i\sigma[/tex].
In other words, the conformal dimension is defined by the power of the transformation factor, for example,
[tex]t_+ \rightarrow \left(\frac{\partial z}{\partial\tilde{z}}\right)^1t_+[/tex]
hence, [tex]t_+[/tex] has conformal dimension 1.
However, I read from a text that the conformal dimension of [tex]\frac{1}{\sigma^+}[/tex] is 1.
But I only know the definition of conformal dimension for tensors, how can I extend the definition of conformal dimension to the scalar function like [tex](\sigma^+ - \sigma'^+)^{-n}[/tex]??
Thanks in advance.

 

[eljose79]

thank you for your question. Conformal dimension is a concept used in conformal field theory, which is a branch of theoretical physics that studies the symmetries and properties of physical systems that are invariant under conformal transformations. In simple terms, conformal transformations are transformations that preserve the angles between points, but can change the distances between points.

The conformal dimension of a tensor or scalar function is a measure of how it transforms under a conformal transformation. In your example, t^{++}{}_+ has a conformal dimension of -1, which means that it transforms with a factor of (\frac{\partial z}{\partial\tilde{z}})^{-1}. This can be seen from the fact that the upper index carries a conformal dimension of -1, and the lower index carries a conformal dimension of 1.

For a scalar function like \frac{1}{\sigma^+}, the conformal dimension is 1. This is because it transforms with a factor of (\frac{\partial z}{\partial\tilde{z}})^1, as you mentioned. This can also be seen from the fact that the conformal dimension of a scalar function is equal to the number of light-cone coordinates it depends on. In this case, \frac{1}{\sigma^+} depends on one light-cone coordinate, hence its conformal dimension is 1.

To extend this definition to a function like (\sigma^+ - \sigma'^+)^{-n}, we can use the same logic. This function depends on two light-cone coordinates, \sigma^+ and \sigma'^+, and the conformal dimension is equal to the number of light-cone coordinates it depends on. Therefore, its conformal dimension is n.

I hope this helps clarify the concept of conformal dimension for a scalar function. Feel free to ask any further questions or for clarification. Best of luck with your studies.

 

[denian]

The conformal dimension of a scalar function is a measure of its scaling behavior under conformal transformations. In general, the conformal dimension of a scalar function is given by the power of the transformation factor that it carries. In the case of a tensor, this is simply the sum of the conformal dimensions of its indices. However, for a scalar function, the conformal dimension is defined by the power of the transformation factor itself.

For example, in the case of \frac{1}{\sigma^+}, the conformal dimension is 1 because it transforms as \frac{1}{\tilde{\sigma}^+}, which has a transformation factor of 1. Similarly, for (\sigma^+ - \sigma'^+)^{-n}, the conformal dimension is -n because it transforms as (\tilde{\sigma}^+ - \tilde{\sigma}'^+)^{-n}, which has a transformation factor of -n.

In general, the conformal dimension of a scalar function can be extended by considering the transformation behavior of the function under conformal transformations. This can be done by looking at the transformation factor and determining its power. However, it is important to note that the conformal dimension of a scalar function is not the same as the conformal dimension of a tensor, as the transformation behavior is different.