> 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.
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.