The Physicist’s “Superpower” Over the Mathematician in the battle for our understanding of Quantum Field Theories: Transcendental Phenomenology II

At each point of our existence, the philosophy of science would always be inseparable from the advancement of physics and our overall technological capabilities: and at the foundation of this advancement is its phenomenological conceptions and possibilities. The same could be said of the advancement of mathematics, but with a pinch of restraint; for the applied mathematician, if it cannot be subjected to mathematical rigor and scrutiny, then it is not dependable, and probably wrong. So, in a way, the mathematicians are the big brother of all of the sciences and its philosophies, checking in to make sure everything is done according to acceptable logical standard. And the physicist, the immediate younger brother who thinks he knows best and flies off the handle phenomenologically; not allowing rigor or pedantry limits his mathematical imaginations and the possibilities of his description of nature and its processes, while at the same time respectful and cautions of the painful lessons his “big brother” – mathematicians – had thought him about rigor in the past.

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The mathematician still believes that regardless of the theories and accolades Quantum Field Theories (QFT) seem to be amassing, it should be tied down to a stable and consistent rigorous mathematical foundation, and has begun to vet the mathematical structure of QFT thoroughly, to rid it of every form of qualia and mathematical intuition. At this point, it appears they are fighting a losing battle because they started the phenomenological war with the physicist in the first place, won the war, then created new weapons which they cannot wield rigorously, and these weapons has been adopted by the physicists as their newly found “superpower” – transcendental phenomenology.

There has been a trend where ideas which were initially thought to be logical games invented/discovered by pure mathematicians find application in the explanation of real-world situation; but the pure mathematician, at the time wasn’t concerned with the real-world application of the ideas, rather their passion was the intellectual challenge and the aesthetic beauty of working out the logical consequences of basic principles. Just after a set-theoretical foundation was established for infinitesimals used in Newton’s Calculus, the pure mathematicians followed the logical consequences into proposing theories with counter-intuitive properties and paradoxes, eg non-Euclidean geometries, Cantor’s theory of infinite sets, Russell’s paradox, etc. 

The pure mathematicians are at the forefront of the battle of our understanding of the operations and processes of the world around us; but the theoretical physicist has built his armor from scraps of the logical extensions and consequences of the mathematical objects and propositions the pure mathematicians has provided. Quaternions, 3D rotation group, special unity group, spinors and many more pure mathematical ideas now find applications in QFT such as quantum gravity, minimal length, non-commutative geometry, theories with generalized uncertainty principle, violations of Lorentz symmetries, loop quantum cosmology, M-theory, string theory and many more.

The theoretical physicist uses a phenomenological approach to make quantitative predictions based upon pure mathematical theories, and to describe anticipated behaviors for phenomena in reality. These predictions are so abstract that it is difficult to think of experiments to test them. Regardless, it has given birth to lots of proposed theories for quantum phenomena with diversified and most times shocking predictions and possibilities. And, although the physicist basks on the glory of these predictions, there are some fundamental questions and conceptions that needs to be answered and clarified, respectively, in order to completely bridge the gap between the pure mathematical theories and its experimentation and engineering applications.

What is Planck’s constant? For now, understand that Planck’s constant is a mathematical object which has become the magic wand of the physicist in his understanding the quantum world. What is the origin of the Planck’s constant h? Well, nobody knows. It was postulated by Max Planck as a desperate fix to the observed spectral distribution of thermal radiation of a black body. It did fix the problem, and ever since then it has formed an essential part of the foundations of all quantum field theories. Other unanswered questions include; If quantum theory is more fundamental, why can’t it be obtained directly from an operator theory? Where do the probabilities come from? Is there a single dynamical framework that incorporates both the linear Schrödinger evolution part of quantum theory, and the nonlinear, stochastic state vector reduction part?

There are more unanswered questions than answered question in QFT. And even the answered question has stirred up more complications of interpretations. At this point the pure mathematicians need to extend the rigid and logically constructed framework with which the physicist has constructed the bridge between the mathematical conceptions that can be traced with rigor down to its set-theoretical foundations, and the phenomenological quantum conception whose set theoretical analysis is rigged with logical contradictions, paradoxes and probabilistic outcomes. In other words, in order for the mathematicians to be able to establish a relatively rigorous and complete framework for the phenomenological aspects of QFT, they would need to extend the implications, logical structure and equivalent relations of what can be considered to be a mathematical object, review and renew the meaning and logical implications of mathematical rigor. It is with these extended conceptions can we begin to make quantum computational problems like Shor’s algorithm fall within conceivable and controllable contingencies.

It is imperative to note that behind every phenomenological advancement of science by the physicist, the accruing rigor and pedantry brought in by the mathematicians are accompanied by the birth and explosion of mathematical ideas, theories and formalism that instigate experimentation, and all applications and technological advancements of science into a new realm which was before its time regarded as science fiction. Therefore, regardless of his “superpower”, it is expedient and necessary for the physicist and mathematicians to collaborate in finding an acceptable framework from which all of quantum field theory could be subjected to standard mathematical rigor. It would be from this point onward, if successful, that the beauty and strength of all the possibilities of quantum supremacy be properly understood, appreciated and put to applicable and tamable use.

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Next week’s publication: CLASSICAL TURING MACHINE VS. QUANTUM TURING MACHINE: THE CLASH OF TWO TITANS.