The fact that the measurement of vanishing distances is physically impossible, which preempts continuity, may lead us to not consider renormalization as a proper procedure in particle physics.
Also, it may lead us to disconsider it, as not needed to define derivatives using infinitesimals, and use Galois fields instead.
We may be pushing our equations too close, to limits where they probably do not apply, and renormalization just tries to solve the symptoms -- to avoid infinities. But the "problem" remains -- there are no infinitesimals in Nature, nor can be created.
Can we not use a concept that we cannot find nor encounter? Infinitesimals do not exist? Then, is renormalization necessary?
To clarify, in Physics or in Constructive Mathematics, there is no place for infinitesimals. They cannot be constructed. They never existed nor will be created. In pure Mathematics, they can be postulated.
Therefore, one should not use Newton's calculus in Physics -- it needs infinitesimals. An effort has been made to use another basis in two different directions. By changing the arithmetic basis to Galois fields, finite integers, where all 4 operations of arithmetic can be defined, or by changing diferencial equations to difference equations. The second one is documented at
https://www.researchgate.net/publication/236420748_Solution_of_the_Schrodinger_equation_for_bound_states_in_closed_form
The question continues and extends to renormalization, also called normalization, in Physics. Would it still be needed, if any of the two alternatives to approach Newton's calculus is used? In other words, as we no.longer shove atoms onto one another, without limit, we may not need to account for that mistake. The Newtonian theory of calculus is incorrect in Physics and may be creating a second issue -- one of scale.
See related papers and comments at
https://www.quantamagazine.org/how-renormalization-saved-particle-physics-20200917/
https://www.researchgate.net/post/Can_QFT_replace_the_Standard_Model_particle_physics_and_classical_physics
https://www.researchgate.net/post/Do_we_need_to_abandon_the_Standard_Model_in_physics
https://m.youtube.com/watch?v=ATcrrzJFtBY
https://m.youtube.com/watch?v=GzbKb59my3U
Well, your question is very interesting and also very philosophical. In fact, the use of infinitesimals is in the foundation of field theory. If we take a simpler example, we can see that this kind of problem appears even in classical theories.
Look at the classical electrical field of a point, it is not defined at the origin, it diverges. The same occurs for the infinity plane at infinity. Then, when we are working with infinities or points without dimension, we are doomed to face such problems.
To solve these there are lots of approaches. String theory, for example, considers that the fundamental structure of nature is a string, not a point. Another example is the use of a cut-off to compute the integrals that we encounter in quantum field theory. And, perhaps the most famous, the effective field theory treatment of many theories. In this treatment, we restrict ourselves to a certain energy regime.
I hope that my comments could somehow help you.
Best wishes,
Huan Souza.
@EdGerck, maybe I did not myself clear, but I agree with you, such problems are present in classical theories. As I said, the electrical field of a point diverges at the origin. This problems are inevitable when we are dealing with infinities or infinitesimals.
HS: thanks. But infinities and infinitesimals do not exist, so they are a pseudo problem. That’s what the question also says, that renormalization, in spite of its Nobel prize fame, is a pseudo problem. They are caused by assuming them to exist, and then "solving" them -- while they are already solved when using the correct formalism to start wit -- which is Galois fields. Nothing is continuous in Nature, not even numbers.
HS: We all know the 3rd law of Newton, action and reaction forces. Besides being "orphan" in the formulation, it is unfortunately based on forces, and demons, outdated concepts. How to improve it?
We could consider that if a system obeys the PLA, then the system could exhibit action-reaction, not otherwise. This would allow the Euler-Lagrange equation to be used to describe the dynamical evolutions of action and its reaction in terms of a suitable PLA, such as the class of solutions that take a stationary, maximum time. Entanglement can be included, allowing geometry to emerge from suitable energy choices. Then, action-reaction can be linked -- as it seems Newton did -- to some form of consciousness, a demon, and even forces, as tangible causes.
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