katch wrecki am starting to understand how the geometry of the Schrodinger equation relates to the drift-diffusion equation. most people know that the form of Schrodinger is exactly that of drift-diffusion, when the time variable is considered to be an imaginary number (complex with no real part). also, the unitarity of the Schrodinger operator is commonly known. And, many know that the square modulus of the wavefunction is proportional to a probability distribution. also, distributions are on the simplex.
recently, i learned that a simple square-root transformation can map points on the simplex onto patches of the sphere. essentially, the reverse of the transformation from wavefunction to probability distribution. to me, this suggests that the effect of imaginary time in the Schrodinger equation is to map the dynamics from the simplex, where a real-valued drift-diffusion would live, onto the sphere. since wavefunctions are the "square roots" of distributions, they live on spheres, not simplices.
from this geometric standpoint, you could explain the unitarity of Schrodinger operators: if you want to be able to map wavefunctions to probabilities by squaring the modulus, you have to stay on the sphere. hence, you need a unitary operator to describe the system's dynamics, in terms of state functions having this property.
i suppose the next logical question would be, what is it about quantum mechanics that necessitates running the drift-diffusion on the sphere, rather than on the simplex?
side note: since these concepts involve PDEs and therefore the state functions are in general infinite dimensional, there are probably some limitations to the geometric analogies discussed, due to the technical challenges associated of extending our intuition from finite-dimensional Hilbert spaces (e.g. the familiar Euclidean 3-space) to infinite-dimensional ones
i suppose since the sphere is defined in terms of an L2 norm, and the simplex is defined in terms of an L1 norm, you could probably think of squaring the wavefunction's modulus as mapping a manifold from L2 to L1
maybe what we can perceive using measurements lies in L1, while the underlying physics lives in L2
in that case, "decoherence" or "collapse" of the wavefunction entails instantiation of this L2-to-L1 mapping