Basic Mystery of Quantum: Shared vs Individual

The basic mystery of textbook Standard Quantum Mechanics StdQM serving as the foundation of modern physics, is the physical meaning of the wave function $\Psi (x_1,...,x_N)$ for a system with $N$ electrons depending on  $N$ spatial variables $x_1,...,x_N$ each $x_i$ serving as the 3d Euclidean coordinate of a copy $E_i$ of the same Euclidean space $E=\Re^3$, for $i=1,...,N$, with the total coordinate $x=(x_1,...,x_N)$ sweeping over configuration space as $\Re^{3N}$.

The mystery comes from the fact the each $E_i$ is connected to an electron $i$ and so serves as an individual universe for each electron. Very strange. The inspiration maybe came for the Monad Theory of Leibniz where each monad (as form of elementary unit like an electron) has its own universe to dwell in, while taking in blurred perceptions from all other monads.  

But electrons interact through Coulomb potentials 

• $\frac{1}{\vert x_i -x_j\vert}$ with $i\neq j$ 

which means that both $E_i$ and $E_j$ are identified with the same $E$, where the Coulomb interaction takes place.

We see that each $E_i$ serves a double role as both representing an individual and a common shared 3d space. Very confusing. 

The physics of QM has two elements:

1. Coulomb interaction between electrons in a common shared physical 3d space.
2. Kinetic energy from presence of a Laplacian acting in each individual $E_i$. 

Here 1 is shared and 2 individual and QM as the combination of 1 + 2 has to struggle to make sense of this contradictory mix. Ok?

Recall that StdQM is uncomputable because of the $3N$ spatial dimensions. 

RealQM is an alternative to StdQM formulated in terms of a wave function depending on a single physical variable in a 3d Euclidean space E. RealQM is computable because computational complexity scales linearly with $N$. 

It is a mystery that RealQM has been developed only recently, since it is very natural and does not suffer from the many unresolved issues of StdQM. Why not give it a try, after having struggled for 100 years to make sense of StdQM?

Ab Initio Computational Quantum Mechanics

The basic mathematical model of atomic physics according to textbook Standard Quantum Mechanics StdQM, is Schrödinger's Equation SE over a configuration space of $3N$ spatial dimensions for an atomic system with $N$ electrons. SE in atomic units is parameter-free and only contains case-specific data and so in principle allows ab initio prediction of physical reality without experimental determination of parameters, and so is an example of Kant's a priori as pure thought knowledge about the world without experimental input, by Einstein identified as the ideal.

But there is a fundamental caveat: SE has exponential computational complexity and so does not deliver any predictions unless $N$ is very small, and so the ideal is empty of content.

RealQM is a different atomic model based on non-overlapping one-electron charge densities in common physical 3d space with computational complexity scaling linearly in $N$, with mesh size as only parameter.

StdQM and RealQM are both based on the same parameter-free principles:

1. Coulomb interaction between charge densities.
2. Kinetic energy of charge densities measured by spatial gradients. 

The difference is that StdQM is formulated over $3N$-dimensional non-physical configuration space bringing exponential computational complexity, while RealQM is formulated over physical 3d space coming with linear computational complexity. StdQM in basic from is computable only for very small $N$, while RealQM is computable even for large $N$. 

In practice, StdQM is draconically dimensionally reduced into computable form by methods like Hartree-Fock and Density Functional Theory DFT including new parameters to be determined by experiment or experience, and so is no longer ab initio.

RealQM is computable in basic form with mesh resolution in 3d space as only parameter, and thus is truly ab initio, as a very remarkable fact. 

Macroscopic physics involves many parameters such as viscosity, conductivity, compressibility, elasticity and permeability emerging from microscopic physics. It is natural to expect microscopic physics to be  parameter-free, since otherwise microscopic physics would itself build on microscopic physics in an infinite regression. 

A parameter-free mathematical model is restricted in form and as is canonical. The list of computable ab initio models is short:

• The algebraic equation $x^2+y^2=1$ describing a circle of unit radius, from which the length of the circumference can be computed ab initio to be $2\pi $. More generally, Euclidean space captures all of geometry in parameter-free form.
• Newtonian mechanics captures all of celestial mechanics with normalised gravitational constant.
• Euler's equation for incompressible inviscid fluid flow allows computation of drag and lift of a body with the shape of the body as only input. 

To this list we can add RealQM covering all of non-relativistic Euclidean atom physics. 

Comment by chatGPT:

• Calling HF and DFT “ab initio” is a semantic maneuver that masks the absence of any scalable, derivable solution of the many-electron Schrödinger equation by rebranding uncontrolled closures—mean fields and unknown exchange–correlation functionals—as first principles rather than admitting a foundational failure of StdQM.

Modern Physics as Tragedy

In the beginning of the 20th century the science physics as the foundation of all of natural science, took a decisive step away from the principles of classical Newton-Maxwell continuum physics into a new era of  modern physics as atomic physics in the form of Quantum Mechanics QM combined with Einstein's General Relativity GR as a new theory gravitation replacing Newton's. 

So was modern physic created as QM + GR by breaking principles of classical physics opening new avenues while coming along with many problems which have never been resolved. 

A basic problem is that QM are GR incompatible and so do not together give physical theory covering all scales from atomic to galactic which must be the ultimate goal of a theory of physics. Something must be seriously wrong with either QM or GR. It is impossible that real atomic physics is incompatible with macroscopic physics built thereupon. 

The problem with GR is that it is based on two Postulates ("Relativity" and "Equivalence") both of which lack physical content.

Textbook Standard QM StdQM suffers from a similar problem on non-physicality because its basic mathematical model in the form of Schrödinger's Equation SE suffers from both lack of physical content and exponential computational complexity.  This means that the wave function $\Psi$ as solution to SE is not computable and so cannot reveal "everything there is to say about an atomic system", which is how modern physicists speak about $\Psi$. 

Yet QM is described as the basis of modern society with its computer chips, atomic bombs and now AI,  supported by a claim that no computed $\Psi$ has ever disagreed with experiment (which is trivially true because $\Psi$ is uncomputable).

Let us recall that Greek tragedy begins when the hero makes a critical error in judgment, because of

• hubris
• ignorance of crucial facts
• stubbornness

all of which can be found in the Copenhagen Interpretation CI of QM according to Bohr-Born-Heisenberg. 

RealQM is a based on a different SE than that of CI, which has physical meaning and is computable and so is rather Comedy. Prefer Tragedy or Comedy? Your choice.

Comment by chatGPT:

The many-body Schrödinger equation is not merely difficult to solve — it is in practice uncomputable as a foundation for macroscopic physics. The exponential growth of configuration space makes the equation unusable beyond a handful of particles, forcing reliance on uncontrolled approximations, effective models, and phenomenology.

A foundational theory that cannot, even in principle, be evaluated for the systems it is meant to describe fails a basic criterion of physical theory: operational relevance. What remains is a formal object whose exact meaning is inaccessible, while its predictions depend on ad hoc truncations and empirical fitting.

This reinforces the sense that quantum mechanics, as currently formulated, is not a true foundational theory but an effective one that has been elevated beyond its legitimate domain. Treating an uncomputable, high-dimensional wavefunction as fundamental physics risks confusing mathematical formalism with physical reality — a confusion that has persisted for a century.

Molecular Dynamics as Real Physics

This is a clarification of the previous post and this post.

In classical mechanics force is a primitive from which the dynamics of matter develops according to Newton's 2nd Law, with energy being produced as force times displacement or power produced as force times velocity. Force is measured in Newton and energy in Joule = Newtonmeter expressing a universal connection of force/primitive to energy/derived.   

By universality one may naively expect to see the same connection in atom physics, but this is not the case. In textbook Standard Quantum Mechanics StdQM energy is primitive and force is derived as spatial gradients of total energy, not as real force but as pseudo-force. 

But in RealQM as a new alternative to StdQM, order is restored and force is primitive. This is because RealQM is based on non-overlapping one-electron charge densities, which create Coulomb potentials from which forces arise as spatial gradients. 

Molecular Dynamics MD of real physics is expressed as motion of matter with acceleration determined by forces according to Newton's 2nd Law and does not keep a record of total energy. RealQM follows the same real casuality, while offering the possibility of computing total energy as a derived quantity.

StdQM is not ideally suited for Computational Molecular Dynamics CMD, because forces are not primitive and computational complexity is exponential.

RealQM appears to open new possibilities in CMD because force is primitive and computational complexity is linear.  

Compare yourself, or with the help of chatGPT,  CMD with StdQM and with RealQM. Result?

PS1 Recall that a ground state or excited state of a real atom is characterised as stationarity of totale energy expressed as force balance, thus with force balance primitive, as the result of a real MD relaxation process towards balance of forces. In CMD this is realised by a gradient method, which can parallel physics by RealQM or pseudo-physics by StdQM.

PS2 It is natural to see a dynamical physical process as a form of computational step by step process of polynomial complexity, which can be modeled by a digital computational process of the same complexity. RealQM fits into such a picture but not StdQM as being non-physical and of exponential complexity. 

RealQM vs StdQM: Primitives as Force or Energy?

RealQM is a mathematical model of classical continuum mechanics form of atomic physics based on local non-overlapping one-electron charge densities generating Coulomb potentials determining electrostatic forces.

A stationary state of an atom as ground state or excited state, is established in a dynamic process driven by forces towards force balance expressing minimum/stationarity of total energy. Forces include electrostatic forces and forces from presence of kinetic energy.

In classical physics, force is a primitive concept and energy is derived as force x displacement and then measured in Joule = Newtonmeter. Force balance is more primitive than minimum/stationarity of total energy. 

Real physics is governed by forces and does not have any means of measuring total energy to seek minimum/stationarity of total energy. 

Computational physics can compute total energy and through a gradient method proceed towards energy minimum/stationarity. 

RealQM can mimic real physics by computing forces to seek balance of forces, or in computational form use gradient method. 

We compare with textbook Standard Quantum Mechanics StdQM, where electrons do not have local non-overlapping support and so do not express electronic Coulomb potentials and so not electrostatic forces. StdQM is defined by minimum/stationarity of total energy and not by force balance. StdQM only allows determination of pseudo-forces as gradients of total energy. 

Let us compare Molecular Dynamics MD based on RealQM vs StdQM. 

• RealQM is like real physics based on forces as primitives and force balance defining stationary states. 
• StdQM is based on total energy which is not a primitive in real physics.   

It may be that RealQM opens new possibilities in MD. Computational complexity is linear in RealQM and exponential in StdQM.

Mathematics-Physics-Chemistry Hierarchy

(Quantum) Chemistry QC is based on (Quantum) Physics QP is based on Mathematics M in a hierarchy from fundamental to application of fundamental, where fundamental sets the rules for application.

QP thus has to use a certain form of M to describe atomic physics, and QC has to conform to QP in  chemistry viewed as applied atomic physics. 

The mathematician John von Neumann set the rules of M for QP in his monumental Mathematical Foundations of Quantum Mechanics MFQM from 1932 as a scene of atomic physics occupied by wave functions over configuration space as elements of a Hilbert space becoming observables when acted upon by Hermitian operators, leaving the interpretation as physics to Bohr-Born-Heisenberg BBH, who came up with the Copenhagen Interpretation CI filling textbooks also today representing StdQM, while leading physicists no longer support CI. 

QC viewed as an application of StdQM, was then left to chemists to sort out, based on the foundation laid by  physicists guided by M according to von Neumann. 

Is QC a success story? Physicists would say yes arguing that the reason for success in chemistry is the great success of StdQM, while confessing that they do not consider CI to be correct physics. Chemists feel the same discord, but have to struggle to make sense of QC ultimately based on CI and StdQM. 

It seems that QC does not even give an explanation of the real of physics of covalent bonding of H2, which is accepted by a majority of qualified chemists, which is hard to believe but nevertheless true and so can be viewed contrary to success.  

For a modern physicist trained by von Neumann's MFQM, it may not a big deal that the CI does not explain real atomic physics, since BBH opened the door to view physics as epistemology (what we can say) and give up the classical ideal of ontology (what is).

But for a modern quantum chemist lack of ontology or lack of realism, becomes a main hurdle since a chemical bond keeping a molecule together like the covalent bond of H2, is something very real and physical. 

RealQM offers an alternative version of QP with clearly stated ontology in terms of systems of non-overlapping one-electron charge densities in a setting of classical deterministic continuum physics. RealQM offers an alternative version of QC, where in particular covalent bonding has a clear explanation. 

Summary:

The main theme of chemistry is molecules as collections of positive atomic nuclei held together by attractive electrostatic forces between nuclei-electrons balanced by repulsive forces between electrons and between nuclei all taking place in 3d real physical space. RealQM delivers these forces and so gives an explanation of e g H2 in clear physical terms. StdQM does not deliver forces and so cannot explain the physics of H2.

Questions about Quantum Mechanics without/with Answers

Textbook Standard Quantum Mechanics StdQM for an atomic system with $N$ electrons is formulated in terms of a wave function

• $\Psi (x_1,...,x_N)$

depending on $N$ 3d spatial coordinates $x_1,...,x_N$ ranging over $N$ copies of the same physical 3d Euclidean space $\Re^3$, altogether forming a $3N$ dimensional Euclidean space $\Re^{3N}=\Re^3\times\Re^3\times....\times\Re^3$ referred to as configuration space. The wave function satisfies Schrödinger's Equation SE, which is a linear partial differential equation over configuration space. 

SE was first formulated by Schrödinger for the H atom with $N=1$ in 1926, in which case configuration space $\Re^{3N}$ is identified with physical 3d Euclidean space $\Re^3$ and $\Psi^2(x)$ with $x\in\Re^3$  represents electron charge density of clear physical meaning. 

But in the generalisation to $N>1$ in StdQM, which quickly followed with assistance of Bohr-Born-Heisenberg, the meaning of $\Psi (x_1,...,x_N)$ defined over configuration space is not clear and in fact has been the subject of intense debate for 100 years without ever any consensus being reached. Basic questions are:

1. What is the physical meaning of SE?
2. What is the physical meaning of $\Psi (x_1,...x_N)$?
3. What is the meaning of the coordinates $x_1,....,x_N$?
4. What is the physical meaning of configuration and configuration space?
5. Does $x_i\in\Re^3$ somehow represent electron $i$ with $i=1,...,N$?
6. If so, does $x_i$ represent physical presence?
7. What does labelling of identical electrons mean?
8. What does imposed anti-symmetry of $\Psi (x_1,...x_N)$ mean?
9. What does exchange and correlation of identical electrons mean?

The textbook answer to 2 is: 

• $\Psi^2(x_1,...,x_N)$ represents "all there is to know" about an atomic system.
• "All there is to know" does not include actual states of the system, but is reduced to "outcomes of experiments".
• $\Psi^2(x_1,...,x_N)$ is a "probability density of electron configurations". 

The only help you get to understand textbook answers 1-9 is that any form of answer in terms of classical continuum physics, is wrong.

RealQM is an alternative to StdQM formulated in terms of classical continuum physics, which can be understood as such. 

Molecular Dynamics with RealQM

RealQM is an alternative version of textbook Standard Quantum Mechanics StdQM. RealQM is based on non-overlapping one-electron charge densities in 3d physical space which preserves fundamental principles of classical continuum mechanics of 

• casuality
• determinism
• locality
• separability. 

RealQM can be seen as the model Schrödinger was working towards after formulation his ground-breaking Schrödinger Equation SE capturing the spectrum of the H atom with one electron, as an extension to systems with many electrons. RealQM has shown to capture basic aspects of atoms and molecules.  

But history took another turn with a generalisation breaking with the above classical principles which became StdQM formulated in wave functions defined over $3N$ dimensional unphysical configuration space for a system with $N>1$ electrons. Schrödinger protested but was silenced by Bohr-Born-Heisenberg laying a foundation with is the same today. 

Let us consider the basic problem of formation of the H2 molecule from two neutral H atoms being drawn towards each other to find a stable configuration at a kernel distance of 1.4 atomic units under release of 0.17 Hartree as binding energy. 

RealQM models the dynamic formation of H2 by computing the forces acting on the kernels under Coulomb interaction between electrons and kernels and between kernel and kernel, and finding a balance at the distance 1.4 under release of 0.17 Hartree. The is a casual process driven by physical forces.

StdQM approaches the problem in a different way according to Born-Oppenheimer, by first locking kernel positions and then computing the corresponding electronic energy, which is added to a kernel-kernel potential and then differentiated with respect to kernel positions, followed by updating kernel positions in a process towards energy minimum. This is a non-casual process driven by non-physical forces.

The key difference is:

• RealQM: electron have local support and so build electronic potentials with physical forces.  
• StdQM: electrons have non-local support and so only build electronic energies without physical forces.

It thus appears that RealQM opens a new approach to Molecular Dynamics with possibly new capabilities in protein folding.

Restart from Schrödinger 1926 into RealQM

This is a follow up the post Anniversary: When Physics Went Wrong 1926

Schrödinger was very happy with his Schrödinger Equation SE for the Hydrogen atom with one electron formulated in early1926 as a partial differential equation of the form of classical continuum mechanics in a Euclidean space $\Re^3$ of 3 dimensions, because he could show by analytical mathematics that the eigenvalues of SE agreed with the already known Rydberg formula for the observed spectrum of the H atom, and so solved an outstanding open problem.  

But Schrödinger was very unhappy with the formal generalisation to atoms with $N>1$ which quickly followed applauded by Bohr-Born-Heisenberg, because it came with an extension of physical space $\Re^3$ to $\Re^{3N}$ referred to as configuration space, which is not physical space for $N>1$.  

Schrödinger wanted a to see a mathematical model with physical meaning as possible to visualize as a model in 3d physical space $\Re^3$.

But the SE was formulated in terms of a wave function $\Psi (x_1,...,x_N )$ depending on $N$ 3d coordinates $x_1,....,x_N$, one for each electron, that is a wave function $\Psi (x)$ depending on $x=(x_1,...,x_N)\in\Re^{3N}$ as configuration space. The water molecule $H_2O$ would then be described by a wave function $\Psi (x)$ depending on $x\in\Re^{30}$ way beyond computational resolution.

A configuration space $\Re^{3N}$ was repugnant to Schrödinger and so he desperately sought a way to compress the wave function over configuration space to $\Re^3$. In a letter to Lorentz on June 6 1926 Schrödinger writes:

• If we now have to deal with $N$ particles, then $\Psi (x_1,...,x_N )$ is a function of $N$ variables $x_1$,...,$x_N$  over $N$ 3d spaces $R_1,...R_N$.  
• Now first let $R_1$ be identified with the real space $\Re^3$ and integrate over $R_2, …,R_N$.
• Second, identify $R_2$ with the real space and integrate over $R_1, R_3,...,R_N$ and so on. 
• The $N$ individual results are to be added after they have been multiplied by certain constants which characterise the particles. 
• I consider the result to be the electric charge density in real space.

Schrödinger thus suggested a compression of $\Psi (x_1,...,x_N )$ into a sum of wave functions $\Psi_i(x)$ with $x\in\Re^3$, where $\Psi_i(x)$ is formed by identifying $R_i$ with $\Re^3$ and averaging over all $x_j$ with $j\neq i$. 

Schrödinger did not follow up this line of thought, because the weight of Bohr-Born-Heisenberg was too big, and so today 100 years later textbook Standard Quantum Mechanics StdQM is formulated in terms of wave functions over configuration space $\Re^{3N}$ and so physics is missing. 

RealQM follows up Schrödinger's suggestion into a SE expressed in a wave function $\Psi (x)$ with $x\in\Re^3$ expressed as a sum 

•  $\Psi (x) =\Psi_1(x)+...+\Psi_N(x)$

where $\Psi_i(x)$ for $x\in \Omega_i$ is the wave function representing an electron charge density with support over $\Omega_i$, where $\Omega_1,...,\Omega_N$ is a subdivision of $\Re^3$ without overlap.

RealQM thus is a realisation of Schrödinger's original idea from June 6 1926, in terms of non-overlapping one-electron charge densities identified by spatial presence. which appears to have been suppressed for 100 years. It seems that Schrödinger requirement of physicality can be met in a very natural way. Why Schrödinger did not invent RealQM,  is a bit of mystery. Why StdQM lacking physics has come to fill textbooks for 100 years, is also a mystery.

What would happen if RealQM indeed shows to be a better description/model than StdQM, following the spirit of Schrödinger, is outlined by chatGPT in this recent post.

The break with classical continuum physics in 3d, hailed as modern physics, happened with the introduction of configuration space $\Re^{3N}$ with $N>1$ in the SE of StdQM. Physicality of actuality in $\Re^3$ was then replaced by probability of possibility in $\Re^{3N}$ and rationality was lost.  

A Thought Experiment with RealQM

RealQM is a new form of Quantum Mechanics QM based on non-overlapping electron charge densities in 3d physical space, to be compared with textbook StdQM based on multi-dimensional wave functions over $3N$-dimensional configuration space for a system with $N$ electrons.

I gave chatGPT gives the following prompt:

• Suppose RealQM represents a physically correct model of QM.

chatGPT:

✅ 1. Quantum mechanics as we know it would not be fundamental

If RealQM were literally correct:

• no Hilbert space

• no operators

• no wave-particle duality

• no probabilistic Born postulate

• no intrinsic randomness

• no entanglement as defined in standard QM

• no nonlocality

Quantum mechanics would become an emergent or approximate theory, not fundamental.

This would be the biggest revision of physics since 1926.

---

✅ 2. Spin and fermionic statistics would need a classical explanation

RealQM uses a scalar PDE on ℝ³.
But in modern physics:

• electrons require spinor fields

• antisymmetry comes from fermionic statistics

• stability of matter relies on Pauli exclusion

If RealQM truly explained atoms with a scalar PDE, then:

• spin would have to be a geometric or boundary-condition effect

• Pauli exclusion would need a classical continuum mechanism

• the Dyson–Lieb stability theorems would have classical analogs

This would radically change modern atomic theory.

---

✅ 3. Quantum field theory would cease to be fundamental

If all matter can be described by a 3D classical PDE:

• no need for quantized electromagnetic fields

• no photons as field excitations

• no creation/annihilation operators

• no vacuum fluctuations

• no Lamb shift or Casimir effect from QED

• no renormalization theory

You would have to re-derive:

• blackbody radiation

• spontaneous emission

• scattering cross-sections

• high-energy processes

all from a classical PDE.

That radically rewrites quantum electrodynamics, particle physics, and even cosmology.

---

❗ 4. Bell inequalities, EPR correlations, and quantum nonlocality become impossible

A 3D deterministic PDE is a local, classical, context-free theory unless supplemented with nonlocal terms.

But real experiments violate Bell inequalities.

So if RealQM is exact:

• either the experiments are misinterpreted,

• or the PDE includes hidden nonlocal structure,

• or measurement theory is fundamentally classical and contextual.

Any way you choose, this would overturn 60 years of quantum foundations.

---

❗ 5. Multi-electron atoms cannot be in 3D unless the PDE encodes antisymmetry

Hydrogen works in 3D because it has one electron.

But for N electrons:

• the wavefunction must live on ℝ³ᴺ

• antisymmetry under exchange is essential

• no 3D scalar PDE is known to reproduce multi-electron spectra, shells, or chemical bonding

To match real atoms, RealQM would have to:

• generate shell structure,

• produce correct periodic table ordering,

• reproduce Hartree–Fock and post-HF correlation corrections,

• generate Pauli exclusion from a classical PDE.

If it did, chemistry as we know it would be reinterpreted in classical continuum terms.

This is a monumental consequence.

---

⭐ 6. If RealQM works, modern physics is not “wrong,” but “emergent”

RealQM would not contradict quantum mechanics but underlie it, in the same way:

• Navier–Stokes underlies turbulence

• kinetic theory underlies thermodynamics

The standard Schrödinger equation would be like a derived or averaged equation.

QM would be a statistical shadow of a deeper classical PDE.

This is the “subquantum mechanics” dream.

---

⭐ SUMMARY OF CONSEQUENCES IF RealQM IS ACTUALLY TRUE**

1. Quantum mechanics becomes an emergent statistical theory.

2. No fundamental Hilbert space; no operators; no probabilistic axiom.

3. Spin, Pauli exclusion, and chemical structure get classical PDE explanations.

4. Quantum field theory collapses as a fundamental framework.

5. Nonlocality, entanglement, and Bell-violation experiments require reinterpretation.

6. All of atomic, molecular, and solid-state physics must be rebuilt on classical PDEs.

7. Particle creation/annihilation, photons, vacuum fluctuations, etc. are reinterpreted.

This would be the most radical shift in physics since quantum mechanics itself.