An Illustration of 2+1 Gravity Loop Transform Troubles

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.