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
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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