Defining the derivative of a Hermitian inner product symbolically [duplicate]

Question

I have various $\mathbb{C}^{n}$ valued function $f[z,\overline{z}],g[z,\overline{z}]$ with $z \in \mathbb{C}$ and I wish to represent their Hermitian inner product and furthermore the derivative of this inner product symbolically.

The rules I wish to include would be the complex product rule; $\frac{\partial}{\partial z} \langle f,g \rangle = \langle \frac{\partial f}{\partial z} , g \rangle + \langle f, \frac{\partial g}{\partial \overline{z}} \rangle$ as well as the conjugate linear property of the bracket $\langle f , \alpha g \rangle = \overline{\alpha} \langle f, g\rangle$

However, I've been unsuccessful when trying to define such a product as the expression;

H[a_,b_] = a.Conjugate[b]

when differentiating, will result in Conjugate'[f[z,Conjugate[z]]] and Conjugate'[z] terms.

What would be the best way to represent these symbols, I've tried defining the differential directly

H[a_,b_]:=a.Conjugate[b]; Dz[A_,z_]:=H[D[A[[1]],z],A[[2]]]+H[A[[1]],D[A[[2]],Conjugate[z]]];

But this will result in the same problem as above, as breaks down once you apply it twice.

Answer 1

I'm not sure this will solve your problem entirely, but my suggestion would be to leave the Hermitian operator unevaluated usually and then define manipulation rules on it specifically.

For your cases we can try something like:

Clear@H;
H /: HoldPattern[D[H[a_, b_], o___]] := 
  H[D[a, o], b] + H[a, D[b, o]];
H[a_, c_*b_] := Conjugate@c*H[a, b];
HEvaluate[expr_] := (expr /. (H[a_, b_] :> a.Conjugate[b]));

Where H /: HoldPattern[D[H[a_, b_], o___]] means set an UpValue on H such that when it is being differentiated, apply the Hermitian differentiation definition. And we also define an evaluation wrapper (HEvaluate) so that we can do something like:

In[1213]:= 
D[H[y^2, x^2], {x, 2}]
HEvaluate@D[H[y^2, x^2], {x, 2}]

Out[1213]= H[0, x^2] + H[y^2, 2]

Out[1214]= 0.Conjugate[x]^2 + y^2.2

Hopefully this is at least somewhat helpful.