Anomaly problem in a simple model analogous to the lightcone-gauge two-dimensional quantum gravity

It has been found in two-dimensional quantum gravity both in the de Donder gauge and in the lightcone gauge that one of the field equations breaks down at the level of the representation, though the breakdown is very slight. It is shown that this anomalous behavior occurs also in a very simple model analogous to the lightcone-gauge two-dimensional quantum gravity. This model, however, can be transformed into a free field theory by a nonsingular transformation. Of course, the latter has no anomaly. The cause of the discrepancy is analyzed.