Quantum Mathematical Objects: Are the mathematical objects used in describing Quantum Mechanics invented or discovered?

The fussy distinction between whether quantum mathematical objects were discovered or invented lays at the foundation of the non-intuitionistic, confusing, complicated and unorthodox nature of our description of quantum mechanics. A mathematical object has been seen by some philosophers and mathematicians as abstractions of physical objects; others see them as realities existing independent of physical objects. Some think that they are related to the transcendental realm of Forms, while others think of them as intermediaries between physical objects and souls. In simple terms, whatever mathematical conception a mathematician thinks about and is able to describe and write out as a physical form of logical expression can be considered as a mathematical object.

Mathematical objects form the building blocks and structure of all mathematical theories, from number systems and arithmetic, through geometry and abstract algebras, to category and model theories: We have been able to build an entire universe made of abstract mathematical conceptions that has formed many beautiful structures, designs and models, for which many has found correspondence and correlations with nature’s design and processes. Although, most of these structures, still beyond the comprehensive capabilities of many, and regardless of their untenable mathematical foundations, hasn’t yet found any physical correspondence with nature; However, over time, some new physical phenomena find mathematical correspondence with nature, and a new field of physical enquiry is born.

"A new scientific truth does not triumph by convincing its opponents and making them see the light, but rather because its opponents eventually die, and a new generation grows up that is familiar with it" – Max Planck

When the mathematical object called Planck’s constant was discovered/invented by Max Planck in 1900, it opened our eyes to see a new description of nature and natural processes: the Quantum world. Ever since then, many more quantum mathematical objects have been discovered/invented, and with the aid of many already established mathematical theories, structures and models, we have been trying to explore quantum interactions and hopefully subject these natural processes to our industrial/computational control. So, what then are these quantum mathematical objects? Have they always been there, and we happen to discover them? Or do we invent them as a tool/weapon with which to explore deeper into our understanding of the processes of nature?

The debate as to whether mathematical conceptions are discoveries or inventions has reached a stale mate where both sides has come to agree to disagree; others take a stand in the middle with the fact that 1 + 1 = 2 is a discovery, and the square-root of minus 1 (complex numbers) is an abstract mathematical invention, implying that mathematical concepts are immutable universal facts/laws, but we invent the mathematical objects we use to explore these immutable universal facts. During our mathematical exploration, one of the most important mathematical objects that makes physics so unique in describing the universe around us are physical constants. These physical constants form immutable number points where our mathematical manipulations of other mathematical objects find correspondence with natural processes.

“The quantum hypothesis will eventually find its exact expression in certain equations which will be a more exact formulation of the law of causality” - Max Planck.

The shore from which the bridge between classical mechanics and quantum mechanics was built was first proposed by Ludwig Boltzmann in 1877, when he extended the works of James Clark Maxwell in developing models of probability distributions of states/systems, that the energy levels of a physical system could be discrete rather than continuous as it was then believed. With such thinking, the foundations of statistical thermodynamics were established, further giving birth to statistical mechanics theories.

In order to solve the problem of blackbody radiation (the ultraviolet catastrophe) which arose in the late 19th century, when Max Planck fixed Wigner’s equation to account for both short and long wavelengths of light of blackbody spectrum by proposing the Planck’s constant h, he called it “an act of desperation”. A heuristic fix. Although the new equation worked well, it became obvious soon enough that its resulting outcomes isn’t unique: different solutions gave different values of entropy. To solve this problem, he had to turn to Statistical Mechanics for support; and yes, it worked, but at what cost? This was the foundations of the probabilistic structure and design of all of quantum mechanics. Initially, Planck considered his approach as heuristic, and was hoping there would be a more fundamental process of arriving at similar conclusion soon enough; sadly, his heuristic and desperate approach is all we’ve got thus far.

Whether this actually occurs in nature one can, in the last analysis, prove only by experience. Max Planck.

With the invention/discovery of Planck’s constant, many more classical phenomena which defied consistent mathematical description began to fall into a grand and new picture; the bridge between classical physics, and quantum physics began construction. First was Albert Einstein’s publication on the photoelectric effect in 1905, which used Planck’s constant to calculate the quantum energy of light photon. And, this was closely followed by Neil Bohr’s atomic model which used multiples of reduced Planck’s constant to calculate the angular momentum and magnetic momentum of the electron, then later the electron spin was proposed. Then Louis deBroglie proposed the notion of matter waves whose wavelength is proportional to Planck’s constant and the inverse of matter momentum, and the conception of wave/particle duality was established.

With an improved quantum atomic model, the next quantum mathematical object that was discovered/invented was the Schrodinger’s wave equation, which Erwin Schrodinger himself considers to be an intellectual approach to divinity. Until now, all formal proofs of Schrodinger’s equation are heuristic; it seems to be divine equation which for reasons beyond our current comprehension, works well in describing the wave mechanics of a quantum particle. One would ask oneself, was this equation invented by Schrodinger? Or has it always existed, and was only discovered by him?

Before we go further into more quantum mathematical objects that were discovered/invented, it is imperative to note that Schrodinger’s equation …

To be continued…

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