Is there a general relationship between the conformal weight of a field and its (classical) scaling dimension?

The relationship between the conformal weight of a field and its classical scaling dimension is indeed a fundamental concept in conformal field theory (CFT). To unpack this, let’s first define what we mean by these terms and then explore how they relate to one another.

Defining Key Concepts

In the context of conformal field theory, the **scaling dimension** of a field describes how that field transforms under scale transformations. Specifically, if you scale the coordinates of your space by a factor of \( \lambda \), the field \( \phi(x) \) transforms as:

• \( \phi(x) \rightarrow \lambda^{-\Delta} \phi(\lambda x) \)

Here, \( \Delta \) is the scaling dimension of the field. It indicates how the field behaves under dilations, which are a specific type of conformal transformation.

Understanding Conformal Weight

On the other hand, the **conformal weight** (or conformal dimension) is a broader concept that encompasses how fields transform under the entire conformal group, which includes not only dilations but also special conformal transformations, translations, and rotations. For a primary field \( \phi \), the conformal weight is typically denoted as \( h \) and is related to how the field transforms under these transformations.

Connecting the Dots

Now, the relationship between the scaling dimension \( \Delta \) and the conformal weight \( h \) can be summarized as follows:

• For primary fields, the scaling dimension \( \Delta \) is equal to the conformal weight \( h \). That is, \( h = \Delta \).
• For descendant fields, which are obtained by applying the conformal generators to primary fields, the relationship can be more complex, but the scaling dimension still plays a crucial role in determining the behavior of these fields.

Illustrative Example

Consider a scalar field \( \phi(x) \) in two-dimensional CFT. If this field is a primary field with scaling dimension \( \Delta = 1 \), it will also have a conformal weight \( h = 1 \). This means that under a scale transformation, the field behaves predictably, maintaining its form while scaling appropriately.

In contrast, if we look at a descendant field, such as \( \partial \phi \) (the derivative of the primary field), its scaling dimension will be different, typically \( \Delta = 2 \), reflecting the additional complexity introduced by differentiation. However, it still relates back to the original primary field's properties.

Conclusion

In summary, for primary fields in conformal field theory, the scaling dimension and conformal weight are indeed equivalent. This relationship is crucial for understanding how fields behave under conformal transformations and plays a significant role in the broader framework of quantum field theory. By grasping these concepts, you can better appreciate the elegant structure of CFT and its applications in theoretical physics.