An illustration of 2+1 gravity loop transform troubles

Abstract

A nonperturbative approach to quantum gravity that has generated much discussion is the attempt to construct a ``loop representation." Despite it's success in linear quantum theories and a part of 2+1 quantum gravity, it has recently been noticed that difficulties arise with loop representations in a different ``sector" of 2+1 gravity. The problems are related to the use of the ``loop transform" in the construction of the loop representation. We illustrate these difficulties by exploring an analogy based on the Mellin transform which allows us to work in a context that is both mathematically and physically simple and that does not require an understanding either of loop representations or of 2+1 gravity.