Problems of Self-Interaction in Standard Quantum Mechanics

Self-interaction in Density Functional Theory

In Newton's theory of gravity gravitational potential $\phi (x)$ and mass distribution $\rho (x)$ with $x$ a spatial coordinate are coupled through the differential equation: 

• $-\Delta\phi (x) = 4\pi\rho (x),$                                                       (1)

or equivalently through the integral equation: 

• $\phi (x)=\int\frac{\rho (y)}{\vert x-y\vert }dy.$                         (2) 

We here assume a setting of a continuum with no smallest scale in space, in which the integration variable $y$ in (2) is never equal to $x$, and we can argue that a mass at an isolated point $x$ does not contribute to the gravitational potential $\phi (x)$ present at $x$ and so that there is no self-interaction on a microscopic scale. 

In the setting of $N$ distinct mass points with location $x_i$ and mass $m_i$ for $i=1,...,N$, (2) takes the form 

• $\phi (x_i) =\sum_{j\neq i}\frac{m_j}{\vert x_i-x_j\vert}$,         (3)

which does not involve self-interaction because $j\neq i$. Choosing $j=i$ would cause break down of (3) into infinity by dividing by 0.  

But on a macroscopic scale there is self-interaction in the sense that Earth's gravitational potential is present everywhere on and inside the Earth. Similarly the Universe interacts with itself by gravitational forces. But that is because the Earth and the Universe are made up of elementary particles (protons and electrons) which must be free of self-interaction as in (3) to avoid break-down into infinity.  

Standard quantum mechanics stdQ for an atom/molecule with $N$ electrons is governed by an electronic wave function $\Psi (x_1,...,x_N)$ depending on $N$ 3d spatial variables $x_1, x_2,...,x_N$ satisfying a Schrödinger equation with an electron-electron repulsion term of the following form connecting to  (2) and (3):

•  $\sum_i\sum_{j<i}\frac{1}{\vert x_i-x_j\vert}\Psi (x_1,...,x_N)$.       (4)

This expression does not carry any infinity since $j\neq i$, but is it free of self-interaction? This is not clear since the coordinates $x_i$ do not represent actual presence of individual electrons which be tracked by index, but only possible presence in a $3N$ dimensional configurations space without electron individuality where a specific coordinate is not tied to a specific electron, thus without Exclusive Occupancy.  The wave function $\Psi (x_1,...,x_N)$ of stdQM is anti-symmetric (assuming for simplicity that all electrons have the same spin) thus asking $\Psi (x_1,...,x_N)$ to vanish if $x_j=x_i$ for some $i$ and $j$, but that is not enough to prevent self-interaction.  

In fact, in density functional theory derived from the standard Schrödinger equation self-interaction is present and has to be handled by some trick. More precisely, the wave function is built from products of overlapping electron orbitals generating a contribution to Coulomb repulsion energy from exchange correlation expressing self-interaction.   

RealQM offers a different version of quantum mechanics keeping electron individuality as non-overlapping charge clouds with Exclusive Occupancy, which does not suffer from self-interaction. 

Reality of Space: Exclusive Occupancy

Exclusive Occupancy in Europe 2021

The era of modern physics was initiated at the turn to the 20th century as a break away from the basic principles of classical Newtonian macroscopic continuum physics covering all of mechanics and electromagnetics, viewed to be useless to describe the microscopic physics of the world of atoms built from protons and electrons, and so a new form of physics based on new principles named quantum mechanics QM was developed by Bohr, Born, Heisenberg and Schrödinger.  

The classical principles of realism with a direct connection between model and reality and determinism were given up and replaced by mathematical formalism in multi-dimensional configuration space combined with indeterminism/statistics into a new quantum world beyond realistic interpretation. This was a step away from rationality with far-reaching consequences for modern society/politics following the same tragical path away from rationality. Compare with the sequence of posts on Corruption of Modern Physics.

But is it really true that the microscopic world of atoms is conceptually so different from the macroscopic world we can model and understand, like a planetary system governed by Newtonian mechanics?. In RealQM I describe such a scenario based on principles of classical continuum mechanics. In RealQM an atom is described as a kernel surrounded by non-overlapping shells of electrons, each electron like unit charge density cloud occupying a certain region of its shell.   

A basic principle of continuum mechanics is the concept of density like mass density as amount of matter per volume, like number of atoms or particles per $m^3$. It is then assumed that each particle occupies a certain region/volume in space altogether filling space and that particles do not overlap. Allowing particles to overlap would ruin the idea of density. We can refer to this principle as Exclusive Occupancy, which we meet in so many ways in ordinary life and Family Law. Each passenger has Exclusive Occupancy of a seat in an airplane. Each country occupies its own territory, unless it is occupied by someone else. Each atom or particle is the sole occupant of its region in space. But a passenger cannot occupy two different seats at the same time, nor a particle.  

Non-overlap in space is thus a basic principle of real physics as physics with direct connection to reality. It is a principle of RealQM, but not of the standard form of QM, where electrons are not assigned different regions of real space, only in configuration space. 

We can view Exclusive Occupancy as a basic feature of space. We know that a 2d chip has room for only a finite number of circuits each occupying a certain area, and that stacking on top into 3d is the only possibility to give room for more, but we also know that stacking into 4d is impossible. 

But of course in the quantum world of configuration space of any dimension there are no such limitations which is now funnelling venture capital into quantum computing suggesting endless fiction, but probably nothing real, like solving the housing problem for home-less by opening one single apartment for occupancy without exclusivity.  

Main question to answer: Is Exclusive Occupancy a basic principle of physics?   

In RealQM electrons meet with continuity of charge density combined with zero normal derivative (homogeneous Neumann condition), while a proton meets an electron with zero charge density from both sides. Electrons meet shoulder by shoulder like friends, while a proton and electron meet as antagonists divided by a neutral (zero) boundary zone.