I have recently tried to make sense of the concept on an antiunitary passive transformation on a complex Hilbert space $H = \mathbb{C}^N$. I still do not know whether the concept even makes sense since I haven't been able to find anything about it online.

As is understand it, given a vector $\phi \in H$ an active transformation is just a change in the vector obtained by applying an operator:

$$\phi \rightarrow A \phi$$

Meanwhile a passive transformation does not change the vector $\phi$ at all. Instead a passive transformation refers to a change of basis in the Hilbert space. It changes the components of the vector $\phi$ by

$$[\phi]_a \rightarrow [A]_{ab}[\phi]_b$$

Where I have explicitly written the component forms of $\phi$ and $A$ w.r.t to a given basis as $[\phi], [A]$. Additionally components of operators $M$ transform as

$$[M]_{ad} \rightarrow [A]_{ab}[M]_{bc}[A^{-1}]_{cd} $$

I also understand that in the case of a passive transformation the matrix [A] can not be identified with an operator on the Hilbert-space since, after all, the objects do not truly change, only their components do.

Now on to the case of an antiunitary passive transformation. Consider the complex conjugation operator $K$ w.r.t to a basis $\{e_i\}$. I.e. K acts as $$K ( \alpha e_i + \beta e_j) =\alpha^* e_i + \beta^* e_j$$

I have seen it stated that a "transformation" given by such an operator acts like $O\rightarrow K O K^{-1}$ on operators and as above on vectors. This looks to me very much like it should be interpreted as a passive transformation. However, there is no corresponding matrix $[K]$ that would yield the sought-after complex-conjugation behavior of the components upon a passive transformation.

I have thought about two ways of making such a thing possible but am unsure about both of them. The possibilities are

1) The antiunitary passive transformation given by $K$ changes the orientation of the underlying complex planes used to parametrize the Hilbert space $H$. I.e. on a complex plane $i$ is now relabeled as "-i".

2) We could map the complex Hilbert space $\mathbb{C}^N$ to a real space $\mathbb{R}^{2N}$ of twice the original dimension. There the sought after basis transformation would exists, which would just send every second basis vector to minus itself (the basis vectors corresponding to the purely imaginary parts in the complex space). We then map the real space back onto the complex space.

In a sense 2) is just a more detailed explanation of how to go about 1).

After all this my Questions are:

I: Is my understanding of the concepts of active and passive transformations correct?

II: Does it make sense at all to think about the concept of a passive transformation corresponding to an antilinear map?

III: Does the complex plane carry an orientation of some sort and does it makes sense to think about an antiunitary passive transformation as a "switching" of this orientation. I.e. a reparametrization of the Hilbert space using "flipped" complex planes?


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