Even in isotropic models, difference equations of states can be difficult to analyze. And even if one finds solutions, analytically or numerically, one still has to face conceptual problems of interpreting the wave function correctly. In quantum physics there is a powerful tool, effective equations, which allows one to include quantum effects by correction terms in equations of the classical type. Thus, in quantum mechanics one would be dealing with effective equations, which are ordinary differential rather than partial differential equations. Moreover, the wave function would only appear indirectly and one solves equations for classical type variables such as expectation values with an immediate physical interpretation. Still, in some regimes quantum effects can be captured reliably by correction terms in the effective equations. Effective equations can thus be seen as a systematic approximation scheme to the partial differential equations of quantum mechanics.
Quantum cosmology is facing the problems of quantum mechanics, some of which in an even more severe form because one is by definition dealing with a closed system without outside observers. It also brings its own special difficulties, such as the problem of time. Effective equations can thus be even more valuable here than in other quantum systems. In fact, effective theories for quantum cosmology can be and have been derived and already provided insights, especially for loop quantum cosmology. These techniques and some of the results are described in this section. As we will see, some of the special problems of quantum gravity, such as the physical inner product and anomaly issues, are in fact much easier to deal with at an effective level compared to the underlying quantum theory, in terms of its operators and states. In addition to conceptual problems, phenomenological problems are addressed because effective equations provide the justification for the phenomenological correction terms discussed in Section 4. But they will also show that there are new corrections not discussed before, which have to be included for a complete analysis. Effective equations thus provide a means to test the self-consistency of approximations.
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