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Adjoint of functional derivative

  1. Feb 27, 2008 #1
    In the space of Riemannian metrics Riem(M), over a compact 3-manifold without boundary M, we have a pointwise (which means here "for each metric g") inner product, defined, for metric velocities [tex]k^1_{ab},k^2_{cd}[/tex] (which are just symmetric two-covariant tensors over M):
    is the Wheeler-Dewitt supermetric.
    Now, let [tex]f:Riem(M)\ra\R[/tex] be a functional of the metric which is of the form
    i.e. it is represented by a local function [tex]f_x:T_xM\otimes_ST_xM\ra\R[/tex]
    (where the subscript S indicates symmetrized). It is known that the functional derivative [tex]\delta{f}[/tex] can be seen as a differential form in Riem(M). It can be defined at [tex]g_{ab}[/tex] as:
    Let us suppose a one-form in Riem(M) can be defined pointwise in M by
    and that we have the inner product as above:
    Now, my question is, in this instance, where the functional derivative basically acts as a exterior derivative, can we define it's adjoint with respect to the above supermetric? I.e., can we find a functional differential operator [tex]\delta^*[/tex] such that
  2. jcsd
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