Identity Politics from Identity Physics?

Leibniz's famous Principle of the Identity of Indiscernibles PII states 

• No two things are exactly alike.
• Coexistence of two indiscernibles is metaphysically impossible.

A basic aspect of material physics is spatial extension as unique occupancy of some region of space over some period of time as unique identity. Material spatial overlap is impossible. Each cell of your body occupies its own region or volume in space, making it distinct or discernible from all other cells thus equipped with a unique identity. The same with the H2O molecules filling the Ocean even if they all have the same composition. Even point-like bodies of the same type carry unique identity by having unique positions in space-time.

Newton's Law of Gravitation and Coulombs Law state that force acting between two distinct point-like masses or charges, scales with $\frac{1}{r^2}$, where $r>0$ is their Euclidean distance. The Laws break down for $r=0$, which means that spatial overlap of masses and charges is forbidden. This is a fundamental principle of classical physics as an expression of PII. 

But in the modern atom physics of Standard Quantum Mechanics StdQM this fundamental principle is violated: Electrons are viewed to be indiscernible and to occupy space and satisfy Coulombs law only in a statistical meaning. 

Real Quantum Mechanics ReQM offers a new approach to atom physics where electrons appear as non-overlapping charge densities with unique spatial occupancy in accordance with PII which satisfy Coulomb's Law in a point-wise physical sense. 

The total Coulomb potential energy of two non-overlapping charge densities $\phi_1$ and $\phi_2$ takes the form of an integral: 

• $\int\int\frac{\phi_1(x)\phi_2(y)}{\vert x-y\vert}dxdy$ where $x\neq y$.       (E)

From strict mathematical point of view (E) can be viewed to be meaningful even if the charge densities overlap with formally $x-y=0$, since the volume where $\vert x-y\vert$ is small, is compensated by $dxdy$. In StdQM it is thus possible for two electrons to have the same charge density and thus overlap (if the electrons have different spin). 

Over-lapping point charges reflects self-interaction with infinite potential energy. This poses a serious problem to Quantum Field Theory for particles without identity requiring "renormalisation" to artificially remove infinities. 

Self-interaction is toxic and has to be prevented, but without identity how can you distinguish between interaction with yourself (toxic) and interaction with other people?

For a Hydrogen atom with the proton modeled as a positive point charge at $x=0$, Schrödinger's equation models the electron as a distributed density $\phi (x)$ of negative charge with finite (negative) potential energy balanced by the kinetic energy $\frac{1}{2}\int\vert\nabla\phi (x)\vert^2dx$ determining the size of the electron.    

Summary: StdQM violates PII. RealQM satisfies PII. Impossible to avoid self-interaction with unphysical  infinities if the identity of the self is not guaranteed.

Is this important? Is PII a basic principle of physical material existence, which cannot be violated? 

Is Identity Physics as violation of PII really physics? 

PS A central theme of this blog is that the roots of the present accelerating break-down of principles of civilisation can be traced back to the advent of modern physics in the beginning of the 20th century with quantum mechanics and relativity theory as a form of Identity Physics.