Hilbert space can’t really be represented visually, but it is defined mathematically in terms of the vectors and inner products that can be integrated across all of the dimensions. But look at the conformal fields and the orthogonal two-dimensional information spaces of electromagnetism and you can see how you need an infinitely dimensional Hilbert space (THE Hilbert space) to conform the quantum information spaces of different particles, each with its own orthogonal electromagnetic fields, into a mutually intelligible Minkowski space, let alone a generally intelligible three-dimensional space.
This means that each photon, which has only two degrees of freedom, exists in two orthogonal planes, or 2D spaces, but that there are an infinite number of these spaces, so that there is no limit on the actual polarisation or orientation of the photon’s vectors in real space. At any given moment, an individual photon exists in a plane of electrical charge and a perpendicular plane of magnetic moment, but this 2D topology is not privileged, so it is related to all of the possible 2D topologies through an infinite number of conformal adjustments in direction and orientation. The conformal fields require an infinitely dimensional space so that all 2D topologies are valid. 3D space sort of approximates the possibilities, but it fails to account for the actual constraints on the degrees of freedom of each photon. An actual photon can’t exist in 3D space because it doesn’t have 3 degrees of freedom. Rather, it exists in an infinitely dimensional conformal field of orthogonal 2D spaces.
To put it another way, we don’t live in 3 dimensional space, we live in an infinite dimensional space that can be integrated over a minimum of three dimensions. 3d is the lowest common denominator of infinite dimensional 2d topologies, and 4d is the lowest common denominator that includes conformal transformations over time.