How to visualise complex vector spaces of dimension 2 and above

[pellis]

TL;DR Summary

Although the algebra of complex vector spaces (Dimension >= 2) makes sense to me, I’m not sure about how to imagine the geometry.

According to e.g. Keith Conrad (https://kconrad.math.uconn.edu/blurbs/ choose Complexification) If W is a vector in the vector space R2, then the complexification of R2, labelled R2(c), is a vector space W⊕W, elements of which are pairs (W,W) that satisfy the multiplication rule for complex vectors. Therefore R2(c) is isomorphic to the 2D complex space C2.

The elements of R2(c), i.e. of C2, can be thought of, by what appears to be a slight abuse of notation, as (W+iW).

And the elementary algebra of complex vector spaces treats elements as linear combinations of basis vectors having complex coefficients (i.e. the basis vectors need not themselves be complex).

Does this mean that:

- Every vector in C2 has its own complex plane?

OR

- Only the basis vectors in C2 are complex planes?

OR

- Something different from either of the above.

Advice appreciated.

Thank you in advance

 

[fresh_42]

It all depends on what you want to do, i.e. in which branch of mathematics you are.

A finite-dimensional vector space over any field is always the same. You have basis vectors, then its multiples of any scalar, real, finite, or complex, and all you can reach from there with vector additions.

Only the scalar multiples might be a bit unusual in the finite or complex case. It all starts with what you want to do. A vector space for me is $\mathbb{F}v_1+\ldots+\mathbb{F}v_n$ regardless of what $\mathbb{F}$ is.

The complexification suggests that you want to consider real and imaginary parts separately. Why?

If $W$ is a real vector space, then $W\otimes_\mathbb{R}\mathbb{C}$ is its complexification.

 

[pellis]

Thank you for your reply.

I'm a chemist trying to make more sense of spinors and especially C2 where 2-component spinors such as Pauli and Weyl spinors "live". I've long been aware of this, but not how to think of C2 geometrically.

You wrote "A vector space for me is Fv1+…+Fvn regardless of what F is." And I'm fine with that, but what does it look like for n = 2 and F = C?

Trivially, one visualises C1 as the standard Argand diagram. But if you extend C1 to C2, can the result be thought of as two real axes and two imaginary axes? Or two intersecting complex planes? Or two real axes sharing one imaginary axis, or a space in which every vector has "its own" complex plane as the (W+iW) in my question above, seems to suggest?

As a chemist I'm used to building a visual sense of something, even for abstractions. But I'm also well aware of the quote about the difficulty of understanding spinors, from the recently deceased Sir Michael Atiyah.

Hopefully, this makes my query clearer.

 

[fresh_42]

If we draw a line, then it is a one-dimensional, real vector space.

But $\mathbb{C}^2$ has a four-dimensional realization, something we cannot imagine. At least most of us.

So all we can do is visualize $\mathbb{C}^2$ as a plane, with the strange property that $\vec{v}$ and $i\cdot \vec{v}$ point in the same direction. However, we cannot draw $i\cdot\vec{v}.$ As soon as we try we have real numbers.

The imagination of the complex numbers as a real plane is a crutch. One that fails as soon as we make analysis with it. That is why I like the abstract algebraic visualization: All $\mathbb{C}\cdot \vec{v}$ point into the same direction. We can do algebra and analysis with it, but we have to make painful compromises if we try to draw something.

And as a chemist, you will know that you cannot draw an electron. A ball is a crutch, too. Nevertheless, we can work well with electrons. Even if visualizations are always a compromise.

 

[pellis]

Thank you for that.

I had wondered whether one could consider slices or projections of C2 to gain more geometric insight.

But for now I'll try to think about "C2 as a plane, with the strange property that v→ and i⋅v→ point in the same direction."