Adjoint of functional derivative

[HenryGomes]

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 $$k^1_{ab},k^2_{cd}$$ (which are just symmetric two-covariant tensors over M):
$$\int_MG^{abcd}k^1_{ab}k^2_{cd}d^3x$$
where
$$G^{abcd}=\frac{1}{2}\sqrt{g}(g^{ac}g{bd}-g^{ab}g^{cd})$$
is the Wheeler-Dewitt supermetric.
Now, let $$f:Riem(M)\ra\R$$ be a functional of the metric which is of the form
$$f[g_{ab}]=\int_Mf(g_{ab}(x))d^3x$$
i.e. it is represented by a local function $$f_x:T_xM\otimes_ST_xM\ra\R$$
(where the subscript S indicates symmetrized). It is known that the functional derivative $$\delta{f}$$ can be seen as a differential form in Riem(M). It can be defined at $$g_{ab}$$ as:
$$\delta{f}_{g_{ab}}[k_{ab}]=\int_M\frac{d}{dt}_{t=0}f(g_{ab}(x)+tk_{ab}(x))d^3x$$
Let us suppose a one-form in Riem(M) can be defined pointwise in M by
$$\lambda^{ab}_x:T_xM^*\otimes_ST_xM^*\ra\R$$
and that we have the inner product as above:
$$\int_MG_{abcd}\lambda_1^{ab}\lambda_2^{cd}d^3x$$
where
$$G_{abcd}=\frac{1}{det{g}}(g_{ac}g_{bd}-\frac{1}{2}g_{ab}g_{cd})$$
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 $$\delta^*$$ such that
$$\int_MG_{abcd}\lambda^{ab}(x)\delta{f}^{cd}(x)d^3x=\int_M\delta^*\lambda_g^{ab}(g(x))f(g(x))d^3x$$
Thanks,

 

[jvicens]

Thank you for your question. I am always happy to discuss mathematical concepts and their applications. In this case, you are asking about the possibility of defining an adjoint for the functional derivative in the space of Riemannian metrics over a compact 3-manifold.

To answer your question, we first need to understand the concept of an adjoint operator in functional analysis. In general, an adjoint operator is defined as the operator that satisfies certain properties with respect to a given inner product. In this case, the inner product you have defined is the Wheeler-Dewitt supermetric, which is a pointwise inner product over the space of Riemannian metrics.

Now, for an adjoint operator to exist, two conditions need to be satisfied: the operator must be linear and the inner product must be Hermitian. In your case, the operator \delta is linear, but the inner product is not Hermitian. This is because the supermetric G^{abcd} is only symmetric, not Hermitian. Therefore, we cannot define an adjoint operator with respect to this inner product.

However, it is possible to define an adjoint operator with respect to a different inner product. For example, if we consider the inner product defined by the inverse of the Wheeler-Dewitt supermetric, i.e. G_{abcd}, then we can define an adjoint operator for the functional derivative. This is because G_{abcd} is both symmetric and Hermitian.

In summary, the existence of an adjoint operator for the functional derivative depends on the choice of inner product. In the case of the Wheeler-Dewitt supermetric, we cannot define an adjoint operator, but it is possible with respect to other inner products. I hope this answers your question.