# Is the choice of the automorphism of complex numbers relevant for Quantum Computing?

In Yale's paper, automorphisms on complex numbers $$\mathcal{C}$$ are shown to be wild, allegedly with cardinality $$2^{\mathfrak{C}^{\mathfrak{C}}}$$. There are two well-known ("trivial") automorphisms: identity $$id$$ and complex conjugation $$\bar{ }$$.

If we think of complex vector spaces as modules with action coming from the automorphisms of $$\mathcal{C}$$, then given a module $$V$$ we can identify $$V$$ as the domain of $$id$$ and $$\bar{V}$$ as the domain of $$\bar{ }$$. If we extend this idea to Hilbert spaces, we can write $$H$$ and $$\bar{H}$$ for the corresponding Hilbert spaces of these two vector spaces.

In quantum computing one tacitly assumes the existence of these two Hilbert spaces and their duals, that is, four spaces in total: $$H, \bar{H}, H^*, \bar{H^*} = H^{\dagger}$$.

If we now consider a non-trivial automorphism of $$\mathcal{C}$$ instead (i.e. different from $$id$$ and $$\bar{ }$$), say $$a$$, we can identify space $$H^a$$. Suppose we use this $$H^a$$ instead of $$\bar{H}$$. I think (I can't think why not) all calculations of quantum computing should remain the same, e.g.$$\langle v | v \rangle = 1$$.

So, my question is :

Does the choice of the $$\mathcal{C}$$-automorphism have any significance for quantum computing, or quantum mechanics in general?

If all operations in QM remain the same with this change of automorphism, then I think it is fair to say the answer is no. But I wanted to post this question to hear of anyone else's thoughts on it.

Moreover, could there be a physical or even philosophical significance to the choice of the automorphism? Could it amount to a choice of the measuring apparatus? This of course sounds too far-fetched, but since I mentioned the word "philosophy" I'm going to leave it.

• I have no idea what "domain of action" means, but if your question comes down to whether twisting by an arbitrary automorphism of ${\mathbb C}$ preserves all the mathematical structure we care about in quantum mechanics, the answer is clearly no, because (among other things) a typical automorphism is not continuous and (among other other things) does not preserve the real numbers. – WillO Oct 19 '17 at 14:03
You cannot "choose" an automorphism other than the identity and complex conjugation and get a Hilbert space $H^a$ as you seem to imply.
The difference between $H$ and $\bar{H}$ is the choice of complex structure on the underlying real vector space, but no other automorphism of the complex numbers preserves the reals (this is theorem 3 in the paper by Yale you refer to), so no other automorphism induces such a choice of complex structure, and these automorphisms have no obvious relevance at all to the formalism of quantum mechanics on complex Hilbert spaces.