QM is Uncomputable + Unphysical but Still Fundamental?

Let me confront chatGPT with the following prompt:

• The basic model of Quantum Mechanics is Schrödinger's Equation SE.
• Solutions to SE as wave functions are uncomputable for multi-electron systems.
• SE is not derived from physical principles and so has unclear physical meaning.
• Wave functions have unclear physical meaning.
• Why is SE chosen as basic model when it is both uncomputable and unphysical?  

Below you get the response by chatGPT, which essentially says that SE was adopted as a basic model because this was the only alternative, not because it expressed any profound insight. I then inform about RealQM as an alternative and get the response that it sounds interesting, but will have to fight a tough match against QM as being the only alternative. I add that uncomputable and unphysical go together: A physical model is computable because physics is a form of computation.  

chatGPT:

You ask why physics chose a model that is “uncomputable” and “unphysical” as fundamental.The answer is simply this:

• Because nature forced it on us.

The Schrödinger equation was not adopted because it was philosophically clear or computationally convenient. It was adopted because no other model reproduced the observed facts: atomic spectra, electron diffraction, tunneling, chemical bonding, semiconductor behavior, and so on. Classical and semi-classical theories failed; the Schrödinger framework succeeded.

Regarding “uncomputable”:
Many-body equations in physics are frequently analytically intractable (Navier–Stokes, Einstein’s equations, turbulence). This does not disqualify a theory as fundamental. It only reflects the complexity of nature.

Regarding “unphysical”:
The wave function’s ontology is indeed unsettled, but conceptual ambiguity is historically common in fundamental physics. Maxwell’s fields and Newton’s gravity were both considered “unphysical” when introduced; their meaning became clearer only later. A theory can be empirically correct long before it is philosophically understood.

So the reason quantum mechanics is treated as fundamental is not because it is simple or fully interpreted, but because:

• It remains the only framework that correctly predicts the phenomena it describes, and no alternative “computable and physical” model has matched its empirical success.

Until such an alternative exists, Schrödinger’s equation stays fundamental by necessity, not by choice.

My prompt: 

• What about the alternative to QM in the from of RealQM by Claes Johnson?

ChatGPT:

RealQM is an interesting and ambitious alternative proposition. It addresses one of your original concerns: it offers a model that aims to be computable and physically interpretable, in contrast to how you characterised standard QM (uncomputable + unclear). If RealQM could be shown to reproduce all phenomena of standard QM (and beyond) while being simpler/clearer, it would warrant serious attention.

However, the burden of proof is high. Until RealQM demonstrates equivalence (or superiority) to standard QM across the full range of quantum phenomena, and is subjected to robust peer review and empirical tests (especially in areas where standard QM shines, like entanglement, quantum statistics, quantum field effects), we should treat it as a speculative alternative—not as yet a replacement of standard QM.

• RealQM is a model of atoms/molecules formulated as a deterministic continuum system in 3-dimensional physical space (plus time) with one-electron charge densities that occupy non-overlapping spatial regions, subject to Coulomb interactions, rather than a high-dimensional wave-function over configuration space. claesjohnson.blogspot.com+3Real Quantum Mechanics+3claesjohnson.blogspot.com+3

• In RealQM, the wave functions (or densities) live in ordinary 3-D space and satisfy boundary conditions (such as homogeneous Neumann conditions at boundaries between regions), thereby avoiding global support overlap and thereby avoiding the need for the Pauli Exclusion Principle (PEP) in the usual form. claesjohnson.blogspot.com+1

• Johnson argues that RealQM is “computable” in the same sense as classical continuum PDE models (like Maxwell, Navier–Stokes) and avoids what he calls the “curse of dimensionality” of the conventional many-body Schrödinger equation (which depends on 3N spatial coordinates for N electrons). claesjohnson.blogspot.com+1

• He claims RealQM has a clearer “physical meaning” (ontology) than standard quantum mechanics (StdQM) because the densities are spatially localised in 3D space rather than residing in abstract configuration space. claesjohnson.blogspot.com

• He extends the model even to nuclei (treating nuclei as analogous to atoms with a kernel and surrounding positive charge densities) under purely Coulomb interactions, in an attempt to bypass conventional nuclear/strong‐force models. claesjohnson.blogspot.com+1

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What RealQM tries to solve

Johnson presents RealQM as a response to several identified problems with conventional QM (StdQM):

• The interpretational problem: what is the wave function? What is the physical meaning of Ψ in 3N-dimensional space? (He argues that StdQM lacks a clear ontology) claesjohnson.blogspot.com

• The computational problem: the standard many-body Schrödinger equation has dimensionality scaling as 3N for N electrons (leading to “exponential complexity”). Johnson views this as effectively uncomputable in practice. claesjohnson.blogspot.com+1

• The foundational “mystery” of quantum mechanics: measurement, superposition, indistinguishability, and the need for postulates like PEP are seen as symptoms of the model’s conceptual opacity. RealQM claims to provide a more “physical” classical-type description. claesjohnson.blogspot.com

• The incompatibility between quantum mechanics and relativity/gravity – Johnson argues RealQM as a 3D continuum system may align more easily with deterministic classical physics and therefore may provide a path toward unification. claesjohnson.blogspot.com

Strengths

• It is commendable that Johnson tries to bring clarity to the ontology: locating electronic densities in real 3D space is intuitively more “visualisable” than wave functions in 3N-dimensional space. That addresses a genuine foundational discomfort many have with standard QM.

• The motivation of reducing computational complexity and seeking a 3D spatial model rather than a configuration‐space model is interesting and resonates with efforts in quantum chemistry to develop tractable approximations (though RealQM claims much more than just an approximation).

• The attempt to build a unified view of atoms and nuclei within a continuum Coulomb‐only framework is bold and invites further exploration.

Challenges / Issues

• Empirical validation: Does RealQM reproduce all known empirical results of quantum mechanics (atomic spectra, scattering, entanglement, quantisation, exchange symmetry, spin, Bell inequalities, quantum nonlocality, quantum field effects)? Johnson makes claims for atoms/shell structure, but I haven’t found peer‐reviewed comprehensive studies showing full equivalence to the standard model of atoms, molecules, and quantum phenomena.

• Exchange symmetry, spin, entanglement: In standard QM, many phenomena (e.g., Pauli exclusion, spin‐statistics, entanglement, Bell inequality violations) rely on the full formalism of indistinguishable particles and multi‐particle Hilbert spaces. RealQM claims to bypass PEP by spatial non‐overlap of densities—but does it fully reproduce fermionic statistics/spinor structure? That is unclear.

• Quantum field effects / relativistic effects / many‐body quantum phenomena: Conventional QM (and quantum field theory) handles things like particle creation/annihilation, relativistic corrections, virtual particles, scattering amplitudes, quantum electrodynamics etc. RealQM appears focused on Coulomb‐bound atoms rather than full relativistic field theory – so how it handles high‐energy phenomena, quantum fields, and relativity is not clear.

• Justification of new boundary conditions and structure: RealQM introduces perhaps ad‐hoc conditions (non-overlapping densities, free boundaries, kernel radius, Neumann/Robin conditions) that need independent justification, and need to show why nature uses those particular structures rather than the standard formalism.

• Community acceptance / peer review: New fundamental models require extensive scrutiny, competing predictions (or at least matched predictions) and peer‐review. The mainstream physics community does not appear (at least so far) to have broadly adopted RealQM; that doesn’t invalidate it per se but means it has yet to demonstrate convincing superiority or broad explanatory range.

• Computability claims: While RealQM proposes “computable in principle” within 3D space, actual many‐electron quantum systems remain extremely challenging even for continuum models; the claim of linear or manageable scaling may be optimistic unless substantial computational evidence is provided for large N.

• Compatibility with known quantum phenomena: Many experiments verify non-classical features (superposition, interference, nonlocality, Bell violations). It is not clear how RealQM accounts for those in full generality. For example: how do non-local correlations or EPR‐type phenomena appear in a local continuum‐density model

Your conclusion?

The Dead-Lock of Modern Physics

The crisis of modern physics is the result of the following dead-lock of the theories of General Relativity GR of gravitation, Quantum Mechanics QM (including Quantum Field Theory QFT) and String Theory ST:

1. GR and QM/QFT are incompatible/contradictory and so do no serve as foundation of modern physics.
2. ST is an extension of QM/QFT to include gravitation.
3. ST is a failed project as extension of QM/QFT. 
4. Does failure of ST as extension of QM/QFT indicate failure of QM/QFT?
5. Incompatibility remains. Modern physics lacks foundation. Crisis.

As long as the dream of ST as extension of QM/QFT to gravitation could be upheld, critique of QM/QFT for lack of gravitation could be managed. But now?  

Nobel Laureates like Penrose and 't Hooft can say that "QM is wrong" but do not really question QM as untouchable foundation. 

Comment by chatGPT:

You pose a bold thesis: that modern theoretical physics is in a “dead-lock” because of the incompatibility between General Relativity (GR) and Quantum Mechanics/Quantum Field Theory (QM/QFT), and that the hope that String Theory (ST) would reconcile them has failed — ergo, “Modern physics lacks foundation”.  Here are a few reflections:

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✅ Strong points

• It is true that GR and QM/QFT are currently not unified in a complete accepted theory, and that raises deep conceptual and technical issues (space-time + quantum, gravitation + fields).

• It is legitimate to question whether the failure (so far) of ST means we should revisit the foundations of QM/QFT and GR rather than simply extend them.

• The call to reflect critically on the foundational assumptions is healthy — science often benefits from re-examining its premises.

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🎯 My take

I find the blog post thought-provoking. It forces us to confront the question: “Are our biggest theories simply incomplete and waiting for extension, or are they fundamentally misguided in some way?” I lean toward the view that GR and QM/QFT are deeply reliable in their domains, but yes — the unification is unsolved. I’m less convinced that the foundations are entirely bankrupt; rather, I see them as incomplete. The term “dead-lock” is strong, but perhaps apt if progress toward unification has been stalled for decades.

PS List of Nobel Prizes to QM/QFT

The importance of QM/QFT as foundation of moder physics is expressed by the Nobel Committee. Who could seriously question this foundation? It is ok to ask about ST but not really about QM?

I. Quantum Mechanics – Foundational Awards

Year	Laureate(s)	Contribution
1918	Max Planck	Discovery of energy quanta (birth of QM).
1921	Albert Einstein	Photoelectric effect (quantum nature of light).
1922	Niels Bohr	Structure of atoms; early quantum atomic model.
1932	Werner Heisenberg	Creation of matrix mechanics.
1933	Erwin Schrödinger & Paul Dirac	Schrödinger equation; relativistic quantum theory.
1954	Max Born	Statistical interpretation of the wavefunction.
1963	Wigner, Goeppert-Mayer, Jensen	Nuclear shell model (quantum many-body concept).

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II. Quantum Chemistry – Electronic Structure & Methods

Year	Laureate(s)	Contribution
1935	Debye	Dipole moments, molecular structure (pre-QC roots).
1954	Linus Pauling	Quantum nature of chemical bonds; hybridization.
1966	Robert S. Mulliken	Molecular orbital theory.
1981	Fukui & Hoffmann	Frontier orbital theory.
1998	Walter Kohn & John Pople	DFT (Kohn); ab initio QC methods (Pople).

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III. Quantum Field Theory (QFT)

(QED, QCD, electroweak theory, Standard Model, Higgs mechanism…)

Quantum Electrodynamics (QED)

Year	Laureate(s)	Contribution
1965	Richard Feynman, Julian Schwinger, Sin-Itiro Tomonaga	Renormalized QED – first modern QFT.

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Electroweak Unification & Gauge Theories

Year	Laureate(s)	Contribution
1979	Glashow, Salam, Weinberg	Electroweak theory, gauge fields.
1999	’t Hooft & Veltman	Renormalization of gauge field theories.

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Spontaneous Symmetry Breaking / Higgs Mechanism

Year	Laureate(s)	Contribution
2008	Nambu	Spontaneous symmetry breaking (QFT).
1979	(also relevant)	SSB embedded in gauge theories.
2013	Higgs & Englert	Higgs field and mass generation (QFT → particle physics).

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Quantum Chromodynamics (QCD)

Year	Laureate(s)	Contribution
2004	Gross, Politzer, Wilczek	Asymptotic freedom in QCD (non-Abelian gauge theory).
2001	Cornell, Ketterle, Wieman	BEC experiments (field-theoretic many-body aspects).

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Neutrino Oscillations (field mixing)

Year	Laureate(s)	Contribution
2015	Kajita & McDonald	Neutrino mass & mixing (QFT flavour oscillation).

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IV. Quantum Information Science (QIS) — Modern QM

Year	Laureate(s)	Contribution
2022	Aspect, Clauser, Zeilinger	Bell inequality tests; entanglement; quantum information.

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V. Macroscopic Quantum Systems / Condensed Matter QFT

Year	Laureate(s)	Contribution
1972	Bardeen, Cooper, Schrieffer	BCS theory (many-body QM).
1987	Bednorz & Müller	High-Tc superconductivity (quantum many-body effects).
2003	Abrikosov, Ginzburg, Leggett	Superconductivity & superfluidity (quantum fields in matter).
2016	Thouless, Haldane, Kosterlitz	Topological phases (QFT + topology).
2025	Clarke, Devoret, Martinis	Macroscopic quantum tunneling & quantized circuits.

Periodic Table vs QM vs Chemistry

Consider the following questions still open after 100 years of debate:

1. Does the Periodic Table PT explain Chemistry?
2. Does Quantum Mechanics QM explain PT?
3. Does QM explain Chemistry?

Eugen Schwarz and Eric Scerri are leading chemists who hesitate to answer the YES of consensus, but still confess to believe in QM as the foundation of Chemistry, at least in principle if not in practice.

But the unanimous consensus is that QM is the canonical mathematical model of atom physics and chemistry in the form of Schrödinger's Equation SE in its original formulation given 100 years ago. The consensus is that physics of atoms is correctly captured by wave functions as solutions to SE, then supported the fact there is no QM prediction in contradiction to observation. The existence of a single contradicting example would shake the consensus. But there is no such thing.

There is a weakness in this argument coming from the exponential computational complexity of wave functions as depending on $3N$ spatial dimensions for a system with $N$ electrons, which make wave functions uncomputable and so impossible to inspect and compare with observation. The lack of contradicting example thus comes from lack of example. 

To make use of QM some form of approximate wave functions as approximate solutions to SE, must be computed. Any contradiction with observation will then be blamed on the approximation and not the exact solution, and so a fiction of no contradiction whatsoever can be maintained: 

• There is no prediction by QM contradicted by observation. 

Critics like Schwarz and Scerri thus focus on the role of PT in chemistry, rather than that of QM, specifically the Aufbau of PT based on periods complemented by Madelung's rule and Hund's rule for filling of electron orbitals, which show many exceptions. 

Despite the criticism of Schwarz and Scerri, physicists can continue to claim that QM explains Chemistry, and chemists that PT/QM explains Chemistry and so keep a facade that everything is in order, despite the criticism.  

The only disturbance to this comfortable status quo is RealQM offering a new form of QM based on non-overlapping one-electron charge densities, which is readily computable, and where atomic/molecule electron configurations are determined as electron packing. 

There is so far limited interest in RealQM as an alternative to the consensus of QM, which is understandable by the monumental investment made to cope with uncomputable aspect of QM, which is unphysical since real physics computes its own evolution.  

Comment by chatGPT

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Thank you for an incisive and thought-provoking piece. You raise three fundamental questions:

1. Does the periodic table explain chemistry?

2. Does quantum mechanics explain the periodic table?

3. Does quantum mechanics explain chemistry?

Your argument that the consensus — that quantum mechanics (QM) via the Schrödinger equation is the foundation of atomic physics and chemistry — rests on the absence of a contradictory prediction is interesting. You point out that because for many-electron systems the wave-functions live in $3N$ dimensions, they are in principle uncomputable and hence untestable in some sense:

“The lack of contradicting example thus comes from lack of example.” claesjohnson.blogspot.com

This is a valuable critique: the fact that we rely on approximations means that when things don’t match experiment, the blame is placed on the approximation rather than on the underlying theory. You also highlight how chemists like Eugen Schwarz and Eric Scerri are uneasy with the notion that quantum mechanics fully explains the periodic table and hence chemistry. claesjohnson.blogspot.com

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What I find persuasive

• Your point about computational intractability is well-taken: in many-electron atomic and molecular systems the practicality of QM is via approximation, and that leaves room for philosophical and foundational questions.

• The distinction between explaining the periodic table structurally (i.e., why the table has its shape) versus explaining chemistry (why substances behave as they do) is well drawn. It’s good to ask whether emergence plays a role here — whether chemistry has “laws” or patterns that go beyond what individual quantum mechanical interactions yield in a straightforward way.

• The invitation to consider alternative or complementary frameworks (you mention “RealQM” and electron packing) is intellectually healthy: science advances by examining assumptions.

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What I’d challenge or ask for further clarification

• While it’s true that exact wave-functions for large $N$ are unattainable, quantum mechanics has been extraordinarily successful in predicting small system behaviour and chemical phenomena (bond energies, spectra, reaction rates) via approximate methods (e.g., Hartree-Fock, DFT, coupled cluster, etc.). It might strengthen the argument to engage more specifically with how far those methods go, where they falter, and to what extent that matters philosophically.

• On the periodic table: the “explanation” of the table via electron shell filling (Aufbau / Madelung / Hund) does have known exceptions (which you note). But one could argue that the table is a compact summary of periodicity arising from quantum structure. The question is: Does summarizing mean “explaining”? Your point that this may leave a gap is compelling—but how might one characterise the gap more precisely? For instance: Are the exceptions merely “edge-cases”, or do they highlight a structural deficiency in the quantum-mechanical explanation of the periodic table itself?

• The idea that QM in practice cannot be directly “inspected and compared with observation” because of dimension explosion: here I’d ask: At what point does this become a genuine theoretical deficiency rather than a practical limitation? If approximate QM works extremely well for the phenomena we test and use, then pragmatically we treat QM as the basis. So is your critique essentially a philosophical one (about understanding) rather than a scientific one (about prediction)? Clarifying this would help.

• The proposed alternative (“RealQM” with non-overlapping one-electron densities and electron-packing) is intriguing. But what empirical or conceptual advantages does it offer over standard QM? What predictions does it make differently (or more simply) that standard QM struggles with? That would make a stronger case for considering it seriously.

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Overall impression

This post is a worthwhile invitation to pause and reflect on the foundations of chemical science. It asks us not to take for granted the hierarchy: QM → periodic table → chemistry. Instead, you suggest that each of those links might have unresolved aspects. That’s valuable.

For someone working in chemistry or physics, you are nudging them to think about explanation not just prediction: Does our theory make sense, does it explain why, not only give numbers that match. And that’s a level of meta-reflection that’s often missing in everyday training.

If I may make a suggestion: it would be helpful to include a few concrete case-studies where QM-based methods fail to explain a chemical phenomenon (or fail to explain it satisfactorily) and show how the periodic table hints at something “more” or “different” than pure QM would suggest. That kind of grounding might make the critique more tangible.

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Thank you again for sharing these thoughts — they contribute to the ongoing dialogue on how we understand the interplay of physics and chemistry.

Computational Emergence: New Paradigm

Computation has turned the philosophical idea of emergence into a virtual laboratory for exploration of large scale complex structures developing in systems formed by small scale simple components. The laboratory is realised in efficient form by the Finite Element Method FEM covering all areas of continuum physics including fluids, solids and electro-magnetics modeled by the classical partial differential equations of Euler, Navier and Maxwell.

Computation thus brings new life into the classical models of continuum mechanics describing small scale simple local physics in terms of differential equations, by exhibiting the large scale global result as emergence by solving the equations typically by time stepping. In particular, the turbulent flow of a fluid with small viscosity like air and water can be simulated by computational solution of the Euler equations (expressing Newton's 2nd law and incompressibility in local form).

In general, emergence can be explored if solutions can be computed. With increasing computational power more of continuum physics can be explored as emergence in a FEM laboratory. 

Quantum Mechanics QM as the physics of atoms and molecules appears to fall outside this paradigm, because the basic mathematical model in the form of  Schrödinger's equation is uncomputable by involving $3N$ spatial dimensions for a system with $N$ electrons bringing in exponential computational complexity. Exploration of emergence in systems of atoms and molecules is thus not possible by computation because of exponential complexity, which can only be a big disappointment for a physicist seeking to understand emergence in atomic systems. 

There is however a version of QM named RealQM which is a computable because it has the form of classical continuum physics in 3 space dimensions. RealQM opens the possibility of exploration of emergence in systems of atoms as forms of chemistry and protein folding. 

Emergence emerges as a central concept of physics, open to  exploration by computation. Effective large scale models may be formed once emergence is uncovered.

Protein folding is an example of emergence in all forms of life based on proteins formed (from chains of simple amino acids specified by the genetic code) in a folding process into 3d structures determining the function of the protein. Protein folding has exponential complexity with QM, but only polynomial with RealQM which opens new possibilities of computational simulation of the emergence of life. 

Reductionism + Emergence vs Quantum Mechanics

Reductionism and emergence are two basic principles of science:

• Decomposition of a complex system into simpler parts.

• Composition of simple parts into complex system.

Combination of these principles allows simulation and control of complex systems. The Finite Element Method FEM is a realisation of this combination covering the vast area of Continuum Mechanics CM. See also this recent post. The canonical example is the formation of a moving large scale coherent wave from small scale motion up and down of water particles. 

FEM decomposes a structure like a bridge into finite elements as beams, columns and cables with simple behaviour captured by analytical mathematics, which are then put together into the structure represented by a system of equations describing the coupling of the finite elements. The action of the structure under loads can then be simulated by computing solutions to the system of equations. 

The finite elements represent reductionism and emergence comes from assembly into structure. FEM is a powerful methodology covering all of CM by digital computing made into a very powerful tool for scientists and engineers. The key is that finite elements are described by simple analytical mathematics while the the structure is made to emerge by powerful computing, as a synthesis of analysis and computation. 

It is essential that the physics of the element is simpler to describe mathematically than that of the whole structure composed of elements. Elements more complicated than the whole structure destroys the whole idea of combined reduction and emergence. 

CM represents macroscopic physics while microscopic physics of atoms and molecules is described by Quantum Mechanics QM. Modern physics consists of CM + QM.

Does QM represent a reduction of CM into elements in the form of atoms and molecules of simpler mathematical form? No, it is the opposite: The QM mathematical model of atoms and molecules is  Schrödinger's equation in $3N$ spatial dimensions for a system with $N$ electrons, which contains immensely more of complexity than the 3 spatial dimensions of CM.  

This means that QM does no appear by reduction of CM, and CM does not emerge by assembly of QM. In other words, the grand scheme of reduction-emergence so successful in CM cannot be applied when including QM to the picture. 

Real Quantum Mechanics RealQM is a reduced form of QM with the same complexity as CM which opens to 

• reduction of molecules to atoms 
• emergence of molecules from atoms 
• reduction of CM to QM molecules 
• emergence of CM from QM molecules.  

RealQM thus (in principle) connects to CM into a synthesis covering (in principle) all scales from micro to macro following the scheme of reduction-emergence. 

QM does not combine with CM in the same constructive way since the models of QM are vastly more mathematically complex and computationally demanding than CM. 

RealQM thus offers an alternative to QM which combines with CM into a synthesis over all scales. 

In particular, RealQM represents Structural Mechanics of the Atom.

QM was loaded from start in 1926 with complications which have never been resolved including exponential complexity defying computation and wave-particle contradiction. To the already long list of complications, we can now add the gulf between CM and QM preventing the use of the principle of reductionism + emergence.

Here is a comment to the post by chatGPT:

Key Claims of the Post

From what I can tell (based on the blog post and previously quoted material) the major claims include:

1. Reductionism (breaking things down into simpler parts) has been undermined in modern physics because the microscopic theory (quantum mechanics) is more complex than the macroscopic (classical/continuum) theory.

2. The usual expectation of reductionism (“the part is simpler than the compound”) fails: in that sense, QM is not “simpler” than classical or continuum mechanics.

3. Therefore the standard reductionist/emergent framework—that macroscopic phenomena emerge from simpler microscopic laws—doesn’t hold in the straightforward way often assumed.

4. The author points toward an alternative theory (he calls it “RealQM”) that would restore a simpler microscopic basis to make the reduction → emergence ladder more conventional.

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✅ Where the Argument Has Strength

• It’s true that the microscopic quantum description (especially for many-body systems) is extremely complex and difficult to solve. The sheer mathematical/or computational complexity of going from many interacting quantum particles to a full continuum description is daunting. That fact backs up the observation that the “part” (quantum many-body) can be harder to handle than the “whole” (classical continuum) in practice.

• It’s legitimate to highlight the practical gap between microscopic laws and macroscopic descriptions — the “how do you get from A to B” question is real and non-trivial.

• The post raises a useful philosophical point: just because a theory is “fundamental” doesn’t guarantee we can easily derive all higher-level behaviour from it in practice. This aligns with mainstream philosophy of science (see e.g., discussions of emergence + reduction). arXiv+1

Towards Computational Protein Folding?

Modern physics has focus om micro-scales of atoms and molecules while classical physics concerns macro-scales of fluids, solids and electro-magnetics. 

Physicists sell the idea that modern physics is fundamentally different from classical physics in the sense that: 

• Micro-scales are described mathematically by Quantum Mechanics QM and macro-scales by Classical Mechanics CM.
• QM is fundamentally different from CM which makes modern physics fundamentally different from classical physics.  
• CM is computable as model in 3 space dimensions.
• QM is uncomputable as model in $3N$ spatial dimensions for atom/molecule with $N$ electrons.
• The connection from micro-scale QM to macro-scale CM is unclear. 

CM is based on an idea of reductionism in the sense that the parts are simpler than the whole, so that the whole can be modeled as a system of parts with simple behaviour. This is the idea of the Finite Element Method FEM as the computational form of CM allowing simulation of very complex macro-scopic systems.

CM contains turbulence as a special macroscopic phenomenon, which can be captured in mean-value sense by computing solutions to the Navier-Stokes Equations NSE on macro-scales. This makes CM computable on macro-scales for both non-turbulent and turbulent physical systems and as such a formidable success story of computational mathematical modeling.

QM is fundamentally different from CM as a mathematical model in $3N$ spatial dimensions in the form of Schrödinger's Equation SE, which is uncomputable even for a small number of electrons such as $N=4$. There is no clear connection between QM and CM because reduction from $3N$ to 3 space dimensions is unclear.  

Modern physics is in a state of crisis witnessed by many prominent scientists, which can be viewed to be rooted in the fundamental difference between uncomputable QM as SE and computable CM as NSE with no bridge between QM ans CM. This is a failure of reductionism. 

RealQM is based on a different mathematical model than QM which has the form of CM in 3 space dimensions on atomic scales. RealQM thus offers a computable model of atom physics of the same form as CM effectively making QM into CM on micro-scales. 

RealQM thus restores the lost feature of reductionism in QM and thus opens to a synthesis of physics on all scales in the same form of computable conservation laws in 3 space dimensions. 

Can RealQM in 3 dimensions replace QM in $3N$ dimensions? If so that would open to computational simulation of microscopics of atoms of the same complexity as macroscopic turbulence, thus opening to in particular ab initio computational protein folding today impossible by QM.  

Comment by chatGPT:

Really interesting post. Your critique of how modern physics handles micro vs. macro systems is thought-provoking, especially the point that classical mechanics is computationally workable in 3D while quantum mechanics becomes impractical in $3N$ dimensions. The idea that this signals a deeper conceptual “crisis” is bold, and your proposal of a RealQM formulated directly in 3-space is intriguing.

It would be fascinating to see how such a model could reproduce key quantum phenomena like entanglement and tunneling while remaining computable. If RealQM can match standard QM’s empirical success with better computational tractability, that would be a major development. Thanks for raising these important questions about the foundations of physics.

Reductionism vs StdQM and RealQM

Reductionism is a basic classical scientific principle of breaking down phenomena of complex geometry into constituent parts of simpler geometry. But in modern physics this principle has been turned around into its opposite.

Macroscopic objects of physics described by Classical Mechanics CM are formed by parts in the form of atoms and molecules which are described by Quantum Mechanics. 

Reductionism expects microscopic parts to be simpler than macroscopic compounds, and so QM to be simpler than CM, but textbook StandardQM is immensely more complicated than CM by having a new multi-dimensional probabilistic form way more complex than the continuum mechanical form of CM.  Reductionism thus has failed in modern physics: The part is much more complex than the compound. 

RealQM is a more recent alternative to StdQM, which has the same form of CM and thus models atoms  in terms of continuum mechanics of simple geometry. 

The failed reductionism of StdQM has remained an unresolved issue of modern physics since the formation of StdQM 100 years ago, and can be viewed to be the basic reason for the present crisis of modern physics. 

The reaction by physicists to the reductionism crisis has since the 1960s been to double down seeking a reduced simpler theory on yet smaller scales than atoms in the form of QED, Standard Model and String Theory in a grandios effort to find a simple Theory of Everything ToE on subatomic scale. Nobody claims that anything like a simple ToE has been found. The complexity appears to increases on smaller scales into an impossible situation for the science of physics. 

Here is an illuminating discussion with chatGPT on reductionism vs StdQM and RealQM.

The Curse of Dimensions in Schrödinger's Equation

The basis of modern physics is viewed to be Schrödinger's Equation SE as a linear time-evolution equation in $3N$ spatial dimensions for a system with $N$ electrons. Numerical digital solution with a resolution of 100 in each spatial variable involves $100^{3N}=10^{6N}$ mesh points in space, already with $N=4$ beyond thinkable computational power. 

When SE was formulated in 1926 when digital computation was not an issue, and so the fact that SE effectively is uncomputable did not enter the minds of its creators Born-Heisenberg-Schrödinger, although Schrödinger was not happy with the many dimensions lacking physicality. It was sufficient that an analytical solution was found for $N=1$ leaving $N>1$ into terra incognita waiting to be explored until digital computation became available, but then was found hit the wall from the curse of dimensions.

This is where we stand today: SE is the basic mathematical model of atom physics but SE is not a computable model. It is thus impossible to make a prediction of the evolution in time of an atomic system with more than 3 electrons by computational solution of SE. It is thus impossible to check if SE correctly models physics by comparing SE predictions with observations of real physics.

Yet SE serves as the canonical model of atom physics in its original formulation, as uncomputable today  as 100 years ago, because of its many dimensions also without physical meaning. 

What can be the value of an uncomputable mathematical model of some physics?  A physicist will tell that it still has a value because SE can be (drastically) dimensionally reduced to computable form and so allow computation of (drastically) simplified approximate solutions. SE would then serve as a suitable starting point for dimensional reduction into a computable model with physical meaning. But it would be the dimensional reduction which would carry the physics.

The alternative would be to start instead directly with a dimensionally reduced model with physical meaning, and thus leave SE to history as no longer useful. This possibility is explored as RealQM. 

Physicists speak with large ease about multi-dimensional wave functions $\Psi$ as solutions to SE, as if they are computable and have physical meaning. The consensus is the "SE works but nobody understands why". Philosophers of physics study the (lack of) meaning of SE, theoretical physicists have turned to more fundamental models such as QED and String Theory, chemists seek to understand what SE offers for molecules, while computational physicists solve other equations, and there is no synthesis in sight.   

Standard QM vs Real QM: Physics and Computability

Let us compare the textbook Standard Quantum Mechanics StdQM from 1926 with the recent alternative Real Quantum Mechanics RealQM as concerns the two basic aspects of physical meaning and computability. 

Both seek to model a collection of $N$ negatively charged electrons subject to Coulomb interaction with a collection of positively charged nuclei together forming atoms and molecules. 

The RealQM model is expressed in terms of a wave function $\psi (x,t)$ depending on a 3d spatial coordinate $x$ and a time coordinate $t$ defined over a subdivision of 3d space into non-overlapping regions acting as supports of one-electron charge densities $\vert\psi (x,t)\vert^2$ meeting with continuity. The corresponding Schrödinger equation has the form of a classical non-linear continuum model for a collection of non-overlapping electron charge densities and thus describes precisely specified real physics. The model is computable in the same sense as classical continuum mechanics with equations such as Maxwell's and Navier's. Computational complexity is (in principle) linear in $N$.

The StdQM model is expressed in terms of a wave function $\Psi (X,t)$ where $X$ is a $3N$ dimensional spatial variable with 3 independent dimensions for each electron. The corresponding Schrödinger equation has non-classical form as a linear equation in multi-dimensions with exponential computational complexity. The physical meaning of $\Psi (X,t)$ has been debated for 100 years without any agreement. 

We now compare:

• RealQM: Computable with precise physical meaning.
• StdQM: Uncomputable with unknown physical meaning.

Does this invite to a further study of what RealQM can deliver to chemistry as Coulomb interaction between electrons and nuclei? 

Quantum Computer as Test of Standard Quantum Mechanics

Quantum computing is suddenly booming with many start-ups after 50 years of brooding. The main objective of quantum computing is to solve problems of quantum mechanics which are not tractable by digital computing because of exponential computational complexity. The prospect is that a quantum computer will deliver exponential computational capacity meeting exponential complexity. 

Quantum computing can also be seen as test of the physicality of Standard Quantum Mechanics StdQM based on a multi-dimensional Schrödinger Equation of exponential complexity allowing superposition of states with a potential of exponential capacity in the form of analog quantum computing. 

If a quantum computer based on StdQM can be constructed capable of computing/simulating real physical systems described by StdQM as the expectation of investors, this will give support to the validity of StdQM as a functional model of real physics. 

But there is no quantum computer yet and skeptics believe that controled superposition as key feature of StdQM will be impossible to realise because the physics is missing. 

So the quest for a quantum computer can be seen as the ultimate test of physicality of StdQM. 

What are the odds today? Will there by a quantum computer in 10 years, in 50 years or ever?