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

Abstract

For interacting classical field theories such as general relativity exact solutions typically can only be found by imposing physically motivated (Killing) {\it symmetry} assumptions. Such highly symmetric solutions are then often used as {\it backgrounds} in a {\it perturbative} approach to more general non-symmetric solutions. If the theory is in addition a {\it 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 non-symmetric degrees of freedom up to higher order. If one is interested in the problem of {\it backreaction} between symmetric and non-symmetric dgrees 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 {\it first} perform the {\it exact} gauge reduction on the {\it full} phase space which results in the reduced phase space of observables and physical Hamiltonian respectively and {\it secondly} expand that physical Hamiltonian perturbatively. Surprisingly, this strategy is not only practically feasible but also avoids the above mentioned tensions. We also show how to perform the partial reduction with respect to only the asymmetric constraints but that theory is not quantisable at finite orders unless there is only one symmetric constraint.