Symmetry restriction and its application to gravity

Abstract

In the Hamiltonian formulation, it is not a priori clear whether a symmetric configuration will keep its symmetry during evolution. In this paper, we give precise requirements of when this is the case and propose a symmetry restriction to the phase space of the symmetric variables. This can often ease computation, especially when transcending from the infinite dimensional phase space of a field theory to a possibly finite dimensional subspace. We will demonstrate this in the case of gravity. A prominent example is the restriction of full Hamiltonian general relativity to the cosmological configurations of Robertson-Walker type. We will demonstrate our procedure in this setting and extend it to examples which appear useful in certain approaches to quantum gravity.

Algorithmic approach to Cosmological Coherent State Expectation Values in LQG

“…Note, that we have not peaked on p, but on p which includes the gauge-covariant corrections. This is congruent with general relativity on a graph restricted to cosmology, see[37] for further details.…”

“…As it was shown in [37] if not at least one of the arguments of the Poisson bracket is invariant under the symmetries of the lattice, the symmetry restriction to the reduced phase space (spanned by coordinates (p, c)) does not work.…”

“…Therefore, the third bullet point becomes paramount: in which situations is it possible to restrict the full dynamics of a Hamiltonian on the lattice phase space to some subspace, without loosing information? In [37], this procedure was proven on a classical level to work, given the reduced phase space consists of those points that are left invariant by the action of some group of symplectomorphims and if the Hamiltonian is invariant under action of the same group as well. In [37] it was presented how gravity on the graph can be symmetry restricted to the phase space of isotropic cosmology parametrised by p, c as outlined before.…”

“…In [37], this procedure was proven on a classical level to work, given the reduced phase space consists of those points that are left invariant by the action of some group of symplectomorphims and if the Hamiltonian is invariant under action of the same group as well. In [37] it was presented how gravity on the graph can be symmetry restricted to the phase space of isotropic cosmology parametrised by p, c as outlined before. However, this only proves the validity of (C) in the limit t → 0.…”

“…Notice, that this is different to the situation of the Gauss constraint: (A.2) and its flow on the phase space given by the vector field exp { G[ Λ], • } can be translated into a discrete version, given by G ǫ J (v) = e;e∩v =0 se v P J (ev), see[37] for further details.…”