Where are Non-Conformal maps encountered in Physics?

Question

We always learn about Conformal maps in Physics as they are easier to analyze in mathematical way. I am sure that Non-Conformal maps are also encountered but I have never heard of its analysis.

Are there methods to do so ?

and where are they encountered ?

In which areas of Physics are they a big hurdle ?

Comments

Most maps are not conformal, and they are analyzed without taking into account the additional properties available in the conformal case; hence they are not conspicuous in the literature. On the other hand, conformal maps are very special since they form the largest group of symmetries of a massless free field. This specialness implies the availability of additional techniques that makes, .e.g., conformal field theory a special chapter of QFT.

Answer 1

Most theories having physical scales involved (namely, almost all the theories in physics) are actually not conformally invariant, because, to start with, they cannot be scale invariant.

and where are they encountered ?

Classical mechanics, fluid dynamics, quantum mechanics and quantum field theory are all not conformally invariant, at least in their full generalisation (low dimensional massless bosons quantum field theories are, instead).

In which areas of Physics are they a big hurdle ?

Not sure what you really mean by that. All the above are, to most extends, fully investigated. Every field theory is expected to have a symmetry group, namely the action is supposed to be invariant under some set of transformations (although it needs not necessarily be so). Once the symmetry group is specified, standard tools like, for instance, the Noether theorem, provide insights on conserved quantities that might help solving the equations of motion by simplifying some particular features. Large symmetry groups very often result in pretty rigid theories, where rigid means that there are almost no degrees of freedom for the dynamics (see also topological field theories).