If we consider the extra dimension(s) on which the amplitude of the wave function given to the Schrodinger Equation, the wave function instead defines a topology (or possibly another geometric object, depending on exactly what properties end up being invariant.)
If the topology can be evaluated over time by some alternative mathematical construct, that alternative mathematical construct may form the basis for a more powerful (in the sense of describing a wider range of potential phenomena) physics, because it should be constructable in such a way as to not possess the limitations of the Schrodinger Equation that the function returns a value for the entire dimensional space under consideration. (That is, observe that for y=sin(x), the waveform cannot be evaluated in terms of y, because it isn’t defined for all of y.)
Additionally, the amplitude of quantum waves are geometrically limited in a way that a geometric object possessing the dimension(s) of amplitude shouldn’t be; quantum waves have an extent from 0 to the amplitude, whereas a generalized geometric object should permit discontinuous extents, or extents which do not include the origin, or extents which cross the origin. If we treat the position but not the measure of the extent of the dimension(s) of amplitude as having topological properties, then with the exception of discontinuous extents / amplitudes, many of these geometries may be homotopic with the wave-function itself; however, there may be properties that can be described in terms of a geometric object / topology that cannot be described in terms of the homotopic wave function.
If we consider the extra dimension(s) on which the amplitude of the wave function given to the Schrodinger Equation, the wave function instead defines a topology (or possibly another geometric object, depending on exactly what properties end up being invariant.)
If the topology can be evaluated over time by some alternative mathematical construct, that alternative mathematical construct may form the basis for a more powerful (in the sense of describing a wider range of potential phenomena) physics, because it should be constructable in such a way as to not possess the limitations of the Schrodinger Equation that the function returns a value for the entire dimensional space under consideration. (That is, observe that for y=sin(x), the waveform cannot be evaluated in terms of y, because it isn’t defined for all of y.)
Additionally, the amplitude of quantum waves are geometrically limited in a way that a geometric object possessing the dimension(s) of amplitude shouldn’t be; quantum waves have an extent from 0 to the amplitude, whereas a generalized geometric object should permit discontinuous extents, or extents which do not include the origin, or extents which cross the origin. If we treat the position but not the measure of the extent of the dimension(s) of amplitude as having topological properties, then with the exception of discontinuous extents / amplitudes, many of these geometries may be homotopic with the wave-function itself; however, there may be properties that can be described in terms of a geometric object / topology that cannot be described in terms of the homotopic wave function.