Can somebody explain the conformal anomaly?

[heinrich_neumaier@yahoo.com]

String theory takes a lot of its motivation fron the conformal anomaly,
and the way to get rid of it. Can somebody explain a non-string physicist
what the conformal anomaly is? (Wikipedia has an article about this, but
is has almost no content. Google search is not very helpful on the topic.)

Since an anomaly is a quantum effect that breaks a classical symmetry, is
there a way to make this effect tangible in the case of conformal
symmetry?

Is there a simple toy model, or a way to specify a simple conformal
transformation that is broken? The conformal group is quite large after
all; what is the simplest transformation that shows the anomaly (or do all
of them)?

The background of the question are a few comments from Lubos' blog, which
are intriguing for every interested physicist. I'm just thirsty to learn
more about the issue.

Heinz

 

[Alexey Popov]

heinrich_neumaier@yahoo.com wrote:

> String theory takes a lot of its motivation fron the conformal anomaly,
> and the way to get rid of it. Can somebody explain a non-string physicist
> what the conformal anomaly is?

I am not sring physicist but will try to explain. Situation in same sense issimilar to QED. When you
change scale (it equivalent to comformal
transformation) the action must be renormalized. QED - renormalizable
theory without any additional conditions (action changes can be absorbted).
Is the string theory we have two way: 1) critical dimention (zero conformal
anomaly). 2) to introduce new terms in action - Liouville field.

> Is there a simple toy model, or a way to specify a simple conformal
> transformation that is broken? The conformal group is quite large after
> all; what is the simplest transformation that shows the anomaly (or do all
> of them)?

Origing of the coformal anomaly is dependence of path integral measure
on metric of worldsheet. We need to construct natural infinite dimentional
form in functions space. To construct this form we can take scalar product
defined as int u(x) v(x) sqrt(g) dx. So path integral measure naturally
depend on metric.

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[Entropia]

The conformal anomaly is a phenomenon in theoretical physics that arises when attempting to quantize a theory with conformal symmetry. Conformal symmetry is a symmetry that preserves angles and ratios of distances, and it is often found in theories of gravity, such as string theory.

In string theory, the conformal anomaly is a crucial aspect that motivates the theory and also poses a major challenge. It arises due to the quantization of the theory, and it breaks the classical conformal symmetry at the quantum level. This means that the theory is no longer invariant under conformal transformations, and this can lead to various issues such as non-renormalizability and inconsistencies in the theory.

To explain the conformal anomaly to a non-string physicist, it can be thought of as a quantum effect that breaks the classical symmetry of conformal transformations. It can be visualized as a distortion or deformation of the original symmetry, similar to how a perfect circle can become an ellipse when viewed from a different angle.

There are various ways to make the conformal anomaly tangible, such as through calculations and simulations in theoretical models. However, there is no simple toy model that can fully capture the complexity of the conformal anomaly. It is a highly technical and mathematical concept that requires a deep understanding of quantum field theory and string theory.

In summary, the conformal anomaly is a crucial aspect of string theory that poses challenges and motivates further research in the field. It is a complex and technical concept that is still being studied and understood by physicists.