Using left-invariant vector fields obeying
and with Lie brackets
the momenta canonically conjugate to
can be written as
with
being
canonically conjugate to
. Here,
is the determinant of the left-invariant metric
on
, which is used to provide the density weight of
. The symplectic structure
can be derived from
to obtain
with the volume It is convenient to absorb the coordinate volume into the fields by redefining
and
. This makes the symplectic structure independent of
in accordance with background
independence. These redefined variables automatically appear in holonomies and fluxes through coordinate
integrations.
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