Effective constraint algebras with structure functions

This article presents the result that fluctuations and higher moments of a state, by themselves, do not imply quantum corrections in structure functions of constrained systems. Moment corrections are isolated from other types of quantum effects, such as factor-ordering choices and regularization, by introducing a new condition with two parts: (i) having a direct (or faithful) quantization of the classical structure functions, (ii) free of factor-ordering ambiguities. In particular, it is assumed that the classical constraints can be quantized in an anomaly free way, so that properties of the resulting constraint algebras can be derived. If the two-part condition is not satisfied, effective constraints can still be evaluated, but quantum effects may be stronger. Consequences for canonical quantum gravity, whose structure functions encode space-time structure, are discussed. In particular, deformed algebras found in models of loop quantum gravity provide reliable information even in the Planck regime.