Quantum Mathematical Objects: Are the mathematical objects used in describing Quantum Mechanics invented or discovered? Part 2.

The wave function is synonymous with the classical description of a particle, only that this particle is described in a complex space (a Hilbert space for example). Newton’s equations usually describe particles in our usual space-time environment whose location in space and time can be assigned any value from our real number system; conversely, the description of a wave particle assigns a complex number to each point x at time t. Remember that the initial discovery/invention of the complex number was shrouded in mystery and fear of it being a portal/access to the description of a mystical environment. With the invention/discovery of the wave function, it seems like we can finally describe what a particle is, and how it would behave, in this new dimension. The initial question at this point is, are these wave particles our inventions, or did we discover them? The next question is, how do we play with these particles? And most importantly, how do we watch/observe them play without our interference?

If you missed the first part of this newsletter, please read for continuity: https://www.dhirubhai.net/pulse/quantum-mathematical-objects-used-describing-invented-felix-wejeyan-uxkqc

We have learned to “play” mathematically with particles in Newtonian space by measuring some physical observable about these particles. These observables include position, momentum and energy amongst others. To play mathematically with a wave particle described in complex space we would need a quantum corollary for physically observables such as position, momentum, energy, spin etc. These corollaries are called eigenstates, and each possesses its own eigenvalue.

Note at this point that the momentum and position eigenstates are not square-integrable, and are technically not considered as functions at all. What does this mean? It means that although the Hibert Space is an abstract space of complex numbers, these eigenstates (position and momentum) are not and cannot be particles in it; they are rather convenient mathematical inventions/abstractions we use for calculational convenience or purposes, and they do not represent physical states.

Unencumbered by these abstraction of abstractions, the Schrodinger’s equation describes the evolution of a wave function over time in an isolated physical system. ?It is imperative to note that Schrodinger’s equation describes a space and time that are not on equal footing: it contains a first derivative in time and a second derivative in space, and therefore it is considered to be a none relativistic equation. ?And there goes another story of how Paul Dirac incorporated special relativity into this quantum mechanical description, but not for today.

The already ... mentioned psi-function (wave function) .... is now the means for predicting probability of measurement results. In it is embodied the momentarily attained sum of theoretically based future expectation, somewhat as laid down in a catalog. —?Erwin Schr?dinger

While the Schrodinger’s equation was successful in helping us understand the time evolution of these particles and play with them, it doesn’t tell us anything about the nature of the wave particles it describes. The physical meaning and reality of a wave particle is thus open to different interpretations of quantum mechanics; and oh boy, there are many and diverse interpretation. These interpretations range from the Copenhagen interpretation, through relational quantum mechanics, QBism, multi-worlds interpretation (Multiverse), down to the deterministic formulation of Bohmian mechanics.

It reminds me of an old note that mathematics is a supermarket where physicists just choose tools which suit them the best…new products are more and more sophisticated than previous ones and require you to read a voluminous manual to understand them – Michal Grochol.

Just as Max Planck shopped from the Statistical Mechanics section of the mathematics supermarket to create a foundational backing for his invention/discovery the Planck’s constant, other physicist has visited other sections of the supermarket to pick out tools for their diverse interpretations of quantum mechanics; Weiner Heisenberg et al derived the Matrix Mechanics from the Matrices section, Richard Feynman derived the path integral formulation from Functional integral Calculus section, and others even had to create their own section such as the Dirac delta function, Dirac equations and notations.

After the foundational quantum mathematical objects had been invented/discovered, there had been many more interpretations and descriptions of quantum mechanics requiring new, very complicated, and abstract quantum mathematical objects whose list and description one would not be able to exhaust in a simple newsletter; but at this point the bottom line is, are these quantum mathematical objects invented or discovered? And of what use is it to us whether they are invented or discovered?

The question is whether a sufficiently complex logical path of which the outcome is not "trivial" is invented by the person laying down the starting axioms, or discovered by the person following the reasoning – Patrick Van Esch

The foundations of quantum mechanics were initially purely phenomenological: as although there was no rigorous justification for quantization, following the complex logical path of solving the ultraviolet catastrophe with a heuristic method, Max Planck’s invention of the Planck’s constant, which he claimed to be a purely heuristic fix, opened doors to more complex quantum phenomena and subsequent interpretations, abstractions and complexities.

Although, the constant pi can be regarded as non-trivial and a direct consequence of Euclidean axioms, can Planck’s constant be considered likewise? In other words, if there were another civilization somewhere with the same initial conditions given to us humans on earth given to them as well, would their logical path of reasoning inevitably lead them to heuristic conclusions like Planck’s constant as well? Or would any slight twist of interpretation, description or representation of axiom lead their scientific advancement away from what we already know as humans? What then would be fundamental and symmetrical between our scientific inventions/discoveries and that of the said civilization?

But even the "invention" of fundamental axioms is not without "discovery" because you might discover that your invention is inconsistent. So, if you "invent" even a set of axioms, you still "discover" that they are consistent or not. There's finally a lot of discovery, and the amount of free invention is rather limited. The freedom resides more in what part of that Landscape we chose to explore, rather than inventing arbitrary stuff I'd say – Patrick Van Esch

The implications of the word “consistency” have been many times misrepresented by the word “determinacy”: this misrepresentation is at the heart of the confusion most people have about quantum mechanics and its results. While we have lost determinacy of quantum mechanical event, which we once had with Newtonian mechanical description, the theories used to describe quantum mechanics has proven to be consistent.

While we can claim that the consistency in result and approach that we have been able to attain with our quantum mechanical descriptions and interpretations might stem from some foundational axioms, with a complete logical mathematical structure, we haven’t been able to prove that yet. This is why most people believe that we have lost determinacy of quantum events to probability; But it is imperative to note that there hasn’t been any scientific endeavor or pursuit that has required our apparent “invention” of many mathematically consistent non-physical formulations (momentum and position eigenstates for example) in other to describe a physical phenomenon as we have encountered with quantum mechanics. So, of a truth, it is a frontier of its own.

In search for a consistent structure for quantum mechanics, we have extended our description into quantum field theories and beyond (string theory for instance), and discovered lots of mathematical objects which can be considered to be non-trivial, but whose axiomatic foundations still eludes us; along the way, depending on the landscape we chose to explore, we have as well invented lots of mathematical objects founded on prior axiomatic structures, and our test of the triviality of these objects is whether they are consistent with the current framework or not. ?

I would be presumptuous to believe that I have exhausted in this simple newsletter all the possible and important quantum mathematical objects that were invented/discovered till date. So please do feel free to discuss on the comment any that comes to mind, and their possible implications.

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Next week’s publication: A quantum computational perspective of consciousness: How cool would it be to store your consciousness in the Cloud and live forever?