Mathematical Idealism of Quantum Mechanics 1926-2025

Mysterium Cosmographicum by Kepler (1596) describing the Solar system as being based on five Platonic solids as perfect expression of mathematical idealism. 

This is a follow up of the post on The Tragedy of Schrödinger and His Equation seeking the origin of the present crisis of modern physics disputed by few, as the departure from classical physics taken in 1926 when generalising Schrödinger's equation for the Hydrogen atom with one electron  to atoms with more than one electron as the foundation of modern physics. 

While Schrödinger's equation for one electron had the form of well understood classical deterministic continuum mechanics in 3 space dimensions, the Schrödinger equation for $N>1$ electrons took a completely new form as a linear differential equation in $3N$ spatial variables never seen before, which represented a formal mathematical generalisation which was easy to write down, but did not have any physical meaning. 

Mathematics thus presented a formal generalisation of Schrödinger's equation from one to many electrons, which modern physicists felt obliged to accepts because it looked so neat. But the generalisation was purely formal mathematical and so the physics had to be put in afterwards. That showed to be very difficult and so developed into the basic trauma of modern physics. 

The leading physicist Max Born took on the challenge and came up with the idea of giving solutions to Schrödinger's equation a statistical meaning thus changing atom physics into a casino of electrons without physics.

But physicists unable to come up with something better, were stuck with the linear $3N$ dimensiosnal Schrödinger equation in the hands of mathematicians like von Neumann which has taken over the scene as Standard Quantum Mechanics StdQM. 

The trouble with mathematics is that without proper understanding trivialities can be mistaken to be deep truths. Physicists were thus blinded by the mathematical formalism and the lack of physical meaning was rationalised as being natural because an atom was not really of this world, just a possibility in a world of statistics. 

RealQM offers a different generalisation of Schrödinger's equation for the Hydrogen atom, into a non-linear system of one-electron equations in the form of classical deterministic continuum mechanics with direct physical meaning in terms of Coulombic interaction between non-overlapping one-electron charge densities and atomic kernels. 

Paul Dirac (Nobel Prize in Physics 1933) was one of the key leading physicist who was carried away by the beauty of the Schrödinger equation for any number of electrons:

• The general theory of quantum mechanics is now almost complete, the imperfections that still remain being in connection with the exact fitting in of the theory with relativity ideas. 
• The underlying physical laws necessary for the mathematical theory of a large part of physics and the whole of chemistry are thus completely known, and the difficulty is only that the exact application of these laws leads to equations much too complicated to be soluble.
•  It there­ fore becomes desirable that approximate practical methods of applying quantum mechanics should be developed, which can lead to an explanation of the main features of complex atomic systems without too much computation.

This was an extension into the microscopic world of atoms of Laplace's grand vision for a macroscopic world fully described by mathematics: 

• We may regard the present state of the universe as the effect of its past and the cause of its future.
•  If an intelligence were to know, at a given instant, all the forces that govern nature and the positions of all its components, and if it were vast enough to analyze this data, then it could comprehend the entire past and foresee the entire future. 
• For such an intelligence, nothing would be uncertain; the future, just like the past, would be fully determined.

The extension represented a heroic break from classical physics of macroscopics into the modern physics of microscopics, but it came with severe caveats: Solutions of Schrödinger's  equation 

• lack physical meaning in a classical sense
• are uncomputable for many electrons. 

Great effort has been allocated to come to grips with these aspect for now 100 years and will certainly continue since no real relief has been attained. It is here RealQM comes in as a fresh restart from classical continuum mechanics. RealQM is a computable and parameter free model of real physics in 3d space and time. 

We may compare StdQM with the Pythagorean view of a World built from relations between natural numbers like the harmonics of the major musical scale created from overtones of a vibrating string with frequencies relating like 3 to 2 for the fifth tone. This was a grand vision based on a formal mathematical idea of the world as simply relations between natural numbers. The physics had to be put in afterwords which worked well for the vibrating string but in general new physics had to be brought for each new case. And then came the shock that ended the Pythagorean Society: The length of the diagonal of a square with side 1 is $\sqrt 2$, which is not a quotient of two natural numbers. Pythagoras grand formal mathematical idealism without physics did not capture real physics. Nor did the Platonic solids.

The same scheme is repeated with StdQM: A mathematical model is created from grand formal mathematical idealism without physics and the physics has to be put in afterwards in the form of new ad hoc ingredients such as

• electron spin
• Pauli Exclusion Principle
• Aufbau Principle
• antisymmetric wave functions 
• Hund's Rule
• Madelung Rule
• Octet Rule
• Spin-Orbit Coupling
• Russel-Saunders Coupling
• jj Coupling
• Lande's Interval Rule
• Selection Rule for Electronic Transitions
• Zeeman Effect
• Stark Effect
• Exchange Interaction
• Hyperfine Structure

while the 1926 Schrödinger equation itself remains intact. The idealism is even more extreme in the Standard Model and of course String Theory. 

Another aspect of the idealism of StdQM comes to expression as the now accepted factum that building a quantum computer by StdQM promised to be possible, lacks realism.