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?[/color]

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 dimension (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)?[/color]

Origing of the coformal anomaly is dependence of path integral measure
on metric of worldsheet. We need to construct natural infinite dimensional
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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