The Green's function of a continuum in isotropic conditions has a very simple shape. It declines as 1/r with distance r from the center. The spherical shock fronts that are solutions of a homogeneous second order partial differential equation integrate with these circumstances into this Green's function. The Green's function represents the reaction of a field on the interaction with a single shot isotropic artifact. The shock front quickly fades away and with it vanishes the deformation of the field that the volume of the Green's function represents. I have given the spherical shock front a name. I call them clamps.
Due to the fact that they temporarily deform their carrier field, clamps carry a standard bit of mass. Only a recurrently regenerated coherent and dense swarm of clamps can cause a significant and persistent deformation of the carrier field. Elementary particles hop around in a stochastic hopping path that causes such a coherent swarm.
This suggests that the volume of the Green's function represents an important physical constant.
https://en.wikiversity.org/wiki/Hilbert_Book_Model_Project/Quaternionic_Field_Equations/Solutions#Waves.2C_warps.2C_and_clamps
∇2 G(r-r’) = -4π δ(r-r’) is the definition of Green's function. It is potential at r due to unit charge placed at r'. In general for the meaning of volume of some function f(x); I guess the answer is ∫ |f(x)|2 dx (then for quantum mechanics volume of wave function is total probability); Volume of Green's function is then related with it's normalization and condition for it to be square integrable. Square integrable functions can form inner product spaces.
For a Gaussian location density distribution, the gravitation potential has the shape ERF(r)/r. Already at a small distance from the center it behaves as 1/r. The gravitation potential is a perfectly smooth function. The location density distribution equals the squared modulus of the wavefunction.
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