Let ψ(q) be a quaternionic probability amplitude distribution (QPAD). ψ(q) can be used as a quantum state function of a particle. It can affect its own parameter space. It does that when it couples to a second QPAD. That is why its parameter space is curved.
All quantum state functions share the affine versions of their parameter space. In the HBM this space is called "Palestra". The curvature of the Palestra is defined by a continuous quaternionic distribution ℘(x), which has a flat parameter space that is spanned by the quaternions. It connects a location in the Palestra to each location in its own (flat) parameter space.
Let Φ(x)= ψ(℘(x)) define the flat quaternionic probability amplitude distribution (FQPAD) that corresponds to ψ(q).
Φ(x) is also a quantum state function, but it has a flat parameter space. It is no longer a QPAD. It does not affect its parameter space like ψ(q) can do.
Φ(x)= ψ(℘(x)) unifies regular quantum physics with gravitation theory.
It converts "curved" quantum physics to "flat" quantum physics.
The distance function ℘(x) enables the specification of a "quaternionic GR". Its (full) derivative defines a quaternionic metric.
Note: the existence of black holes indicates that the distance function has no inverse. However, outside these singular locations it can have a local inverse.
It appears that quantum physics uses a flat kind of derivative in which partial derivatives that concern the functions values that belong to other quaternionic dimensions are ignored. This makes that in quantum mechanics the quaternionic nabla ∇ makes sense and the differential ∇f can be written as a product of two quaternionic objects ∇ and f of which the nabla ∇ is the operator.
In gravitation theory the full quaternionic differential df must be used. This differential concerns all sixteen partial derivatives of the quaternionic distribution f. In the HBM the full differential is used in the definition of the quaternionic metric d℘(x).
For more details see: