Symmetry reduction, gauge reduction, backreaction, and consistent higher order perturbation theory

For interacting classical field theories, such as general relativity, exact solutions typically can only be found by imposing physically motivated (Killing) symmetry assumptions. Such highly symmetric solutions are then often used as backgrounds in a perturbative approach to more general nonsymmetric solutions. If the theory is in addition a gauge theory such as general relativity, the issue arises how to consistently combine the perturbative expansion with the gauge reduction. For instance, it is not granted that the corresponding constraints expanded to a given order still close under Poisson brackets with respect to the nonsymmetric degrees of freedom up to higher order. If one is interested in the problem of backreaction between symmetric and nonsymmetric degrees of freedom, then one also must consider the symmetric degrees of freedom as dynamical variables which supply additional terms in Poisson brackets with respect to the symmetric degrees of freedom and the just mentioned consistency problem becomes even more complicated. In this paper, we show for a general theory how to consistently combine all of these notions. The idea is to first perform the exact gauge reduction on the full phase space, which results in the reduced phase space of observables and physical Hamiltonian, respectively, and secondly, expand that physical Hamiltonian perturbatively. Surprisingly, this strategy is not only practically feasible but also avoids the above mentioned tensions. There is also a variant of this strategy that employs only a partial gauge reduction with respect to some of the nonsymmetric degrees of freedom on the full phase space. We show that in perturbation theory,z the left over constraints close up to higher orders but not exactly, unless there is only one of them such as in cosmology. Since such classically anomalous constraints are problematic to quantize, the full gauge reduction for which these issues are absent is preferred in this case.