Scaling dimensions of operators in AdS/CFT

[jdstokes]

I have a question about Witten's original 1998 paper on AdS/CFT

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),

$ds^2 \to d\widetilde{s}^2 = f^2 ds^2$

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

$\varphi \sim f^{-\lambda}\varphi_0$

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

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

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

$\varphi \sim g^{-\lambda}\varphi_0$

where in general $g\neq f$.

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

 

[AndyW]

it is important to always question and critically analyze the work of others in order to gain a deeper understanding of the subject. In this case, your question about Witten's use of the same function f in both equations (2.2) and (2.36) is valid and worthy of further discussion.

Firstly, it is important to note that Witten's paper on AdS/CFT is a highly influential and groundbreaking work in the field of theoretical physics. It has been extensively studied and cited by many other researchers, and its predictions and implications have been tested and confirmed in various experiments.

Now, in regards to your question about the use of the same function f in both equations, it is important to understand the context in which Witten is using it. The AdS/CFT correspondence is a duality between a conformal field theory (CFT) living on the boundary of an Anti-de Sitter (AdS) space and a gravitational theory living in the bulk of the AdS space. The key aspect of this duality is the equivalence between the two theories, which means that any physical quantity in one theory can be mapped to a corresponding quantity in the other theory.

In this case, the function f is used to map the metric and scalar field from the AdS space to the boundary CFT. This means that the metric and scalar field in the CFT are not independent, but are related through this function f. So, in order to preserve the duality between the two theories, it is necessary to use the same function f in both equations.

Furthermore, the choice of f is not arbitrary, but is determined by the AdS geometry and the boundary conditions. In fact, the function f is uniquely determined by the AdS space and is not a free parameter. Therefore, it is not possible to choose a different function g in equation (2.36) and still maintain the duality between the two theories.

In summary, Witten's use of the same function f in equations (2.2) and (2.36) is justified by the duality between the AdS space and the boundary CFT. The choice of f is not arbitrary, but is determined by the AdS geometry and the boundary conditions. Therefore, the scaling dimension cannot depend on the choice of an arbitrary function, as it is uniquely determined by the AdS space. I hope this helps to clarify your concerns.

 

[Entropia]

Thank you for your question about Witten's original 1998 paper on AdS/CFT. This is a valid concern and I will try to address it in my response.

Firstly, it is important to understand that the AdS/CFT correspondence is a duality between two different theories, the Anti-de Sitter space (AdS) and a conformal field theory (CFT). In this duality, the AdS space is seen as the bulk theory and the CFT is seen as the boundary theory. The AdS space is described by the metric given in Eq. (2.1) of the paper, while the CFT is described by the metric given in Eq. (2.2). These two metrics are related by the conformal transformation given in Eq. (2.2), which introduces the function f.

Now, when we talk about operators in the AdS space, we are actually talking about fields in the CFT. This is because the AdS/CFT correspondence states that there is a one-to-one correspondence between operators in the AdS space and fields in the CFT. So, the operator \varphi in the AdS space corresponds to a field in the CFT, which we also denote by \varphi for simplicity.

In order to define the scaling dimension of the field \varphi in the CFT, we need to understand how it behaves under conformal transformations. This is where the function f comes into play. The function f is chosen such that the metric in the CFT (Eq. (2.2)) is finite at the boundary of the AdS space. This is important because it allows us to define a conformal transformation (Eq. (2.2)) that relates the two metrics.

Now, let's consider your suggestion of using a different function g in the definition of the scaling dimension. This would mean that we are considering a different conformal transformation, which would lead to a different metric in the CFT. This would also mean that the operator \varphi in the AdS space would correspond to a different field in the CFT, which we would denote by \psi for example. In this case, the scaling dimension of \psi would be different from that of \varphi, as it would depend on the choice of the function g.

In conclusion, the choice of the function f in both (2.2) and (2.36) is justified because it allows