Well, it depends somewhat on what you would accept as ‘physical reality’. Complex numbers are essential to the standard formulation of quantum theory, in that there exists no way to replace them and obtain agreement with all of its theoretical predictions. There are scenarios somewhat akin to Bell-tests (with three parties instead of two) where complex quantum mechanics predicts correlations that can’t be reproduced in the real-valued version. This is somewhat surprising: in all other physical theories, such as electromagnetism, while you can use complex numbers to neatly calculate some things, you can also always do away with them, and for a lot of scenarios (involving single systems or two-player ‘games’) that’s also true in quantum mechanics: you can just replace the state space of the system by one that’s twice as large, to accommodate the extra degree of freedom inherent in a complex number, and that’ll get you there. But for three parties, scenarios exist where that’s not true anymore.
So if you’re happy with standard quantum mechanics, there’s a good case you should accept complex numbers as ‘real’. But of course, there’s always a catch: there are ways to modify the quantum formalism so as to work only with real numbers, such as the one suggested by Ernst Stückelberg in the 1960s—which changes the way systems are combined into larger systems (basically, implementing structure ‘naturally’ present in complex tensor products with one over the real numbers). But this is somewhat artificial, to my mind.
I can nit that if we didn’t have complex numbers, the same things could all be modeled by 2x2 matricies of the form:
\left[\begin{array}{cc}a & b \\ -b & a \end{array}\right]
Where a and b are the real and imaginary parts of the equivalent complex number.
So complex numbers are not “essential” to the formulation (although they are certainly convenient).
Yeah, you can always come up with some new mathematical structure that has all of the same relevant properties, and just call it by a different name. It’d be a headache, and there’s no good reason to do so, but it can be done.
Is there any other field that has the relevant properties that isn’t isomorphic? I suppose since you only really care about the field properties, you can make the topology funky in some way, but as a field I would expect it to be isomorphic to the complex numbers, otherwise some of your calculations would be wrong.
I suppose isomorphisms are what we mean when we talk about simply renaming something, but I’m used them being a bit more meaty, like isomoprhisms that exist because of the First Isomorphism Theorem, or similar, where the construction of the two sets is very clearly different, but it can be shown that the structure is the same. I suppose in this instance you could construct R[x]/<x^2+1> as isomorphic to C, which is supposedly constructed in a different way than the complex numbers, but it’s not meaningfully different to me in the same way isomorphisms found from various homomorphism relationships can be.
Sure, and that’s how it works in the Stückelberg-formalism. But in order to get things to work out that way, you have to add something to the ‘vanilla’ real vector space of dimension 2n, namely, a linear transformation J such that J^2 = -1, in other words, a complex structure. Otherwise, the 2n-dim real vector space has too many parameters, as compared with an n-dim complex one. Think about a spin-\frac{1}{2} particle: it lives in a 2-dimensional complex vector space, and you need two real parameters to define an arbitrary pure state (one being removed due to the norm-constraint, and one due to the arbitrary overall complex phase); but in a real 4-dimensional vector space, you have three free real parameters for a pure state, since you only have the norm constraint.
To remove the additional freedom, you then add the above complex structure, in the form of an operator
J = \begin{pmatrix}
0 & -1_n \\ 1_n & 0
\end{pmatrix},
where 1_n is the n-dim identity matrix, and require that all operators commute with it. This constrains them automatically to be of the form
\begin{pmatrix}
A & -B \\ B & A
\end{pmatrix},
with A and B being n-dim matrices.
So there is more to do than just replace the complex amplitudes by real ones, and this constitutes a change to the usual formalism of quantum mechanics. You basically have three formulations:
- Standard, complex quantum mechanics in n (complex) dimensions
- Standard quantum mechanics with complex amplitudes replaces by real ones in 2n dimensions
- Quantum mechanics with real amplitudes in 2n dimensions with added-on complex structure
1 and 3 are equivalent by construction, but it’s difficult to motivate where the additional structure should come from in the case of 3. But it was an open question whether there are any predictions of 2 that differ from 1—this is what the above Bell experiment has demonstrated: there are correlations present in 1 that have no model in 2.
To get this back somewhat to the OP, suppose we’d started out with real quantum mechanics (2): then, just changing to complex numbers would yield differing predictions that could be experimentally tested.
There are plenty of algebraically closed fields of characteristic 0. However, if you start from the real numbers and take the algebraic closure you get the complex numbers.
Not that you cannot come up with weird fields (that will be isomorphic to the complex numbers as a field) but with a completely different absolute value: for example, start with the p-adic absolute value for a prime p instead of the usual absolute value; then you get p-adic numbers \mathbb{Q}_p. This has an algebraic closure \overline{\mathbb{Q}_p}, then you can pass to its completion. This may be thought of as some sort of analogue of the complex numbers, however we did not start with the real numbers which you get by completing the rational numbers with respect to the usual absolute value (measure of distance), so yes it will have a “funky” topology that may be a problem (is it?) for some physical reason, but not that there will be a problem abstractly enumerating the finite-dimensional representations of \mathrm{SU}(2) or \mathrm{SL}_2.
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