[eigenspace]

I used to think of it like another basis too, but nowadays I think this basis analogy is a bit fraught, or at least not the whole story.

In particular, for multidimensional spaces, the usual multidimensional Fourier transform only really works if you have a flat metric on that space (I.e. no curvature). That’s a bit of a warning signal given that our universe itself is curved.

There was some very interesting work recently where it was shown how to generalize Fourier series to certain hyperbolic lattices [1], and one important outcome of that work is that the analog of the Fourier space is actually higher dimensional than the position space.

Furthermore, the dimensionality of the ‘Fourier space’ in this case depends on the lattice discretization. One 2D lattice discretization may have a 4D frequency-like domain, and another 2D lattice might have a 8D frequency-like domain.

[1] https://arxiv.org/abs/2108.09314 or https://www.pnas.org/doi/full/10.1073/pnas.2116869119

[mananaysiempre]

Not the whole story indeed, but you have to dive into representation theory somewhat to get more: the Fourier transform is more or less the representation theory of the (abelian) group of the translations of your space, thus the homogeneity requirement. The finite-lattice version[1] (a discretized torus, basically) may serve to hint what’s in stock here.

[1] https://www-users.cse.umn.edu/~garrett/m/repns/notes_2014-15... (linear algebra required at least to the degree that one is comfortable with the difference between a matrix and an operator and knows what a direct sum is)

[eigenspace]

If you like this topic, I strongly recommend you read the references I attached to my comment.

In uniformly curved 2D hyperbolic spaces, it turns out that there is a higher dimensional non-Abelian Fuchsian translation group defined on a higher genus torus.

[Jensson]

> That’s a bit of a warning signal given that our universe itself is curved.

What does this has to do with whether they are a different basis for cases where we don't account for curvature? This seems completely irrelevant, sure the tool can't be used in some cases but it can be used as a basis change in other cases.

[messe]

It's not an analogy. It's literally just another basis.

> In particular, for multidimensional spaces, the usual multidimensional Fourier transform only really works if you have a flat metric on that space

What the hell does the metric of space-time have to do with this? When computing a fourier transform, we're not working in 3+1 dimensional space-time, we're working in either an N-dimensional (in the discrete case) or \infty-dimensional (in the continuous case) vector space; while that term contains the word "space" they DO NOT, in this context, have anything to do with Euclidean space or the Pseudo-Riemannian manifold that GR treats space-time as.

[nextaccountic]

I wanted to know more about this too, and I hate to make meta comments, but I'm afraid your confrontational approach may make the other person think this conversation isn't worth the hassle

Which would be a bad thing, reading this kind of conversation is what makes this site worthwhile

[eigenspace]

For what it’s worth, I did reply, but I probably wouldn’t have if you hadn't expressed frustration at the prospect of not reading further.

[eigenspace]

> What the hell does the metric of space-time have to do with this?

Maybe calm down for a moment and try not being such a hot-headed ass. You seem to have missed the point entirely.

I’m well aware that these functions can be described as vectors in an infinite dimensional Hilbert space.

The problem I’m bringing up is that the domains of these functions (i.e. not the vector itself) typically have geometric properties we care about.

The problem is that if one has a manifold with a non-trivial intrinsic geometry, then functions defined on that manifold cannot be faithfully Fourier transformed without losing pretty much all geometrically relevant information.

It turns out that in some cases, there are generalizations of the Fourier transform of a function on a curved manifold, but in those cases, the domain of the transformed function is very different, typically having a higher dimensionality.

This is particularly relevant and problematic in physics, where the Fourier transforms of functions on spacetime are really important and useful, but dont work in curved spacetimes.

E.g. it’s a big problem when doing QFT on a curved spacetime that one cannot separate positive frequencies of a field from negative frequencies.