- #1

jdstokes

- 523

- 1

http://arxiv.org/abs/hep-th/9802150

Since the AdS metric diverges at the boundary, the boundary metric is only defined up to a conformal class Eq. (2.2),

[itex]ds^2 \to d\widetilde{s}^2 = f^2 ds^2[/itex]

Similarly, the solution for a massive scalar [itex]\varphi[/itex] is divergent. In order to define the scaling dimension of [itex]\varphi[/itex], Witten writes it as Eq. (2.36)

[itex]\varphi \sim f^{-\lambda}\varphi_0[/itex]

where [itex]f[/itex] is the same function used to give a finite metric in (2.2) and [itex]\varphi_0[/itex] depends only on the boundary.

Thus if we fix some appropriate [itex]f[/itex] in (2.2) then we can easily determine the scaling dimension [itex]\lambda[/itex] from (2.36).

What concerns me is why Witten is justified in using the same function [itex]f[/itex] in both (2.2) and (2.36). One would have thought that it is possible to be more general by defining say

[itex]\varphi \sim g^{-\lambda}\varphi_0[/itex]

where in general [itex]g\neq f[/itex].

Suppose, for example, that [itex]f = e^{-y}[/itex] and [itex]g = e^{-2y}[/itex], then all of the scaling dimensions will be shifted from their values had we chosen [itex]g = f[/itex]. How can the scaling dimension depend on the choice of an arbitrary function?