Does topology distinguish between real and imaginary dimensions?

[WWGD]

Maybe if the OP can give us a few examples of what s/he wants to do, we can figure out and suggest something.

 

[ohwilleke]

Heuristically, "real" and "imaginary" dimensions ought to be the same because "imaginary" dimensions are those involving a real number times the square root of negative one in each dimension. But the designation of something as positive or negative is a purely arbitrary choice of units.

For example, if you are dealing with electromagnetic charge, the fact that an electron has a negative one charge, and a proton has positive one charge is purely an arbitrary assignment of directions on the real number line and could have been different with a different arbitrary assignment of units. So, nothing "physical" should depend upon how positive v. negative units are assigned.

 

[Feynstein100]

Heuristically, "real" and "imaginary" dimensions ought to be the same because "imaginary" dimensions are those involving a real number times the square root of negative one in each dimension. But the designation of something as positive or negative is a purely arbitrary choice of units.

For example, if you are dealing with electromagnetic charge, the fact that an electron has a negative one charge, and a proton has positive one charge is purely an arbitrary assignment of directions on the real number line and could have been different with a different arbitrary assignment of units. So, nothing "physical" should depend upon how positive v. negative units are assigned.

Thank you for the analogy. I found it really helpful. However, I think you misunderstood me. In the proton vs electron example, I'm not arguing that one should be labeled positive and the other negative. That's completely arbitrary and simply a matter of choice. What I am saying, however, is that the proton and electron have different charges and thus should be treated as such. To me, saying that real and imaginary numbers are same is like saying the proton and the electron have the same charge. Or worse, that they're the same particle. Might as well take an apple and an orange and say they're the same fruit. I don't care what you call each fruit, as long as you acknowledge that they're different fruits and give them different names.
I'm sorry, I'm feeling quite frustrated. I recently learned that I have autism and one of the consequences of having autism, apparently, is that I can distinguish between things where non-autistic people cannot. I treat things individually instead of making generalizations. I thought that in a forum full of physicists, most of whom are likely autistic like me, I'd find sympathetic ears. And yet, I feel just as different and alien as always.
Not that I'm saying I'm better than everyone else. It's just that I feel like I'm trying to describe the difference between colors to someone who's colorblind. It makes me wish I was colorblind too.

 

[ohwilleke]

Thank you for the analogy. I found it really helpful. However, I think you misunderstood me. In the proton vs electron example, I'm not arguing that one should be labeled positive and the other negative. That's completely arbitrary and simply a matter of choice. What I am saying, however, is that the proton and electron have different charges and thus should be treated as such. To me, saying that real and imaginary numbers are same is like saying the proton and the electron have the same charge. Or worse, that they're the same particle.

The thing is that if you reverse the positive and negative charges of the proton and electron, then equations involving the square root of proton charges will now be in the imaginary plane, while equations involving the the square root of electron charges will now be in the real plane.

As you correctly posed the question in the first place, it isn't that real numbers and imaginary numbers are the same. But, they have the same topology.

You can, in general, map points in the real plane to the imaginary plane on a one to one basis. You can likewise map points in the imaginary plane to the real plane on a one to one basis.

 

[PeroK]

Not that I'm saying I'm better than everyone else. It's just that I feel like I'm trying to describe the difference between colors to someone who's colorblind. It makes me wish I was colorblind too.

I'm sorry to disabuse you of the notion that you are seeing deeply into mathematics. You are seeing far less than those of us with mathematical training. You lack a basic understanding of abstract mathematical concepts. Your question about "real" and "imaginary" dimensions misses the point about how real and complex numbers relate to each other mathematically.

The real plane $\mathbb R^2$ and the complex plane $\mathbb C$ are isomorphic when considered as real vector spaces. I.e. vector spaces over $\mathbb R$. In this case, we are simply considering a complex number (vector in the complex plane) as an ordered pair of real numbers with no further structure or properties.

The complex plane, however, is usually considered not as a real vector space, but an algebraic field. That is to say, we define multiplication of complex numbers, which (as @Pikkugnome has pointed out) distinguishes $\mathbb C$ from $\mathbb R^2$.

Note that a set in mathematics can be given algebraic and/or analytic structure. This means that the same underlying set can be the basis of different mathematical objects. Algebraic generally refers to addition and/or multiplication and analytic refers to properties of some fundamental notion of distance or length. Ultimately, leading to the notion of analytic topology underpinned by the notion of a collection of open sets.

It's perfectly acceptable, therefore, to see both $\mathbb R^2$ and $\mathbb C$ as built on the same underlying set of ordered pairs of real numbers. The difference is that $\mathbb C$ is endowed with a rule for multiplication that produces a closed field.

If you introduce the same multiplication rule for $\mathbb R^2$, then you do not create a new, distinct mathematical object, but simply $\mathbb C$ in a different notion. This is called an isomorphism. And, to claim that this somehow avoids introducing complex numbers misses the point. You have created an object with the same properties as the complex numbers and have the complex numbers in all but name.

In fact, an interesting and sometimes useful isomorphism of the complex numbers is to a subset of real $2 \times 2$ matrices:
$$a + ib \leftrightarrow
\begin{bmatrix}
a& b\\
-b&a
\end{bmatrix}$$Finally, if we define distance between two points in $\mathbb R^2$ by the usual Euclidean metric:$$|(x_1, y_1) - (x_2, y_2)| \equiv \sqrt{(x_1 - x_2)^2 + (y_1 - y_2)^2}$$And we define the distance between two compelx numbers as$$|z_1 - z_2| = \sqrt{(z_1 - z_2)(z_1 - z_2)^*}$$Then, purely as simple topological spaces, the two are isomorphic.

However, when we look at complex-valued functions of a complex variable, we move into complex analysis, which has no analogue in real analysis or multi-dimensional real vector spaces.

 

[martinbn]

Not that I'm saying I'm better than everyone else. It's just that I feel like I'm trying to describe the difference between colors to someone who's colorblind. It makes me wish I was colorblind too.

Your original question was about topology. I asked, and you didn't answer, if you know what topology was. Have you studied topology and do you know what it means for a set to be given a topology? If the answer is "no", then you are the one who has not seen some colors yet. The good news is that it is not like colorblindness and it can be overcome by studying. The bad news is that you have to put in the effort.