I show where the error is made. Everyone can follow it.
Many think that renormalization belongs to relativistic quantum non linear field theories, and it is true, but it is not all the truth. The truth is that renormalization arises every time when we modify undesirably coefficients of our equations by introducing somewhat erroneous “interaction”, so we must return to the old (good) values and call it renormalization. Both modifications of coefficients show our shameful errors in modeling and this can be demonstrated quite easily with help of a simple and exactly soluble equation system resembling the radiation reaction problem in Classical and Quantum Electrodynamics.
Let us consider a couple of very familiar differential equations with phenomenological coefficients (two Newton equations):
One can see that the particle acceleration excites the oscillator, if the particle is in an external force. In this respect it is analogous to the electromagnetic wave radiation due to charge acceleration in Electrodynamics. When there is no external force, the “mechanical” and the “wave” equations become “decoupled”.
The oscillator equation system can be equivalently rewritten via the external force:
It shows that the external force application point, i.e., our particle, is a part of the oscillator, and this reveals how Nature works (remember P. Dirac’s: “One wants to understand how Nature works” in his talk “Does Renormalization Make Sense?” at a conference on perturbative QCD, AIP Conf. Proc. V. 74, pp. 129-130 (1981)).
Systems (1) and (2) look like they do not respect an “energy conservation law”: the oscillator energy can change, but the particle equation does not contain any “radiation reaction” term. Our task is to complete the mechanical equation with a small “radiation reaction” term, like in Classical Electrodynamics. It is namely here where we make an error. Indeed, let me tell you without delay that the right “radiation reaction” term for our particle is the following:
If we inject it in system (2), we will obtain a correct equation system:
Here we are, nothing else is needed for “reestablishing” the energy conservation law. System (4) can be derived from a physical Lagrangian in a regular way (see formula (22) here). We can safely give (4) to engineers and programmers to perform numerical calculations.
Period. But it is not what we actually do in theoretical physics.
Instead, we, roughly speaking, insert (3) in (1) with help of our wrong ansatz on how “interaction” should be written. Let us see what then happens:
Although it is not visible in (5) at first glance, the oscillator equation gets spoiled – even the free oscillator frequency changes. Consistency with experiment gets broken. Why? The explanation is simple: while developing the right equation system, we have to keep the right-hand side of oscillator equation a known function of time or, more precisely, an external force, like in (2), rather than keep its “form” (1) (I call it “preserving the physical mechanism, the spirit, not the form”). Otherwise it will be expressed via unknown variable , which is coupled now to , and this modifies the coefficient at the oscillator acceleration when in the oscillator equation is replaced with the right-hand side of the mechanical equation. In other words, if we proceed from (1), then we will make an elementary mathematical error because we not only add the right radiation reaction term, but also modify coefficients in the oscillator equation, contrary to our goal. As a result, both equations from (5) have wrong exact solutions. If we insist on this way, it is just our mistake (blindness, stubbornness) and no “bare” particles are responsible for undesirable modifications of equation coefficients.
However, in CED and QED they advance such an “interaction Lagrangian” (self-action) that spoils both the “mechanical” and the “wave” equations because it preserves the equation “form”, not the “spirit”. In our toy model we too can explicitly spoil both equations and obtain:
with advancing a similar “interaction Lagrangian” for “decoupled” equations from (1):
Here in (6) – masses with “self-energy corrections”. Thus, it is the “interaction Lagrangian” (7) who is bad, not the original constants in (1), whichever smart arguments are invoked for proposing (7).
Moreover, there is a physical Lagrangian for the correct equation system (4). Therefore, we simply have not found it yet, so we are the main responsible for modifying the equation coefficients in our passage from (1) to (6), not some “bare particle interactions”.
In CED and QFT they perform a second modification of coefficients, now in perturbative solutions of (6) to obtain perturbative solutions of (4), roughly speaking. Such a second modification is called “renormalization” and it boils down to deliberately discarding the wrong and unnecessary “corrections” to the original coefficients in (6):
In other words, renormalization is our brute-force “repair” of spoiled by us coefficients of the original physical equations, whatever these equations are – classical of quantum. Although it helps sometimes, it is not a calculation in the true sense, but a “working rule” at best. A computer cannot do numerically such solution (curve) modifications. The latter only can be done in analytical expressions by hand. Such a renormalization can be implemented as a subtraction of some terms from (7), namely, a subtraction of
(called counter-terms) and it underlines again the initial wrongness of (7). It only may work by chance – if the remainder (3) is guessed right in the end, as in our toy model.
P. Dirac, R. Feynman, W. Pauli, J. Schwinger, S. Tomonaga, and many others were against such a “zigzag” way of doing physics: introducing something wrong and then subtracting it (physically we add an electron self-induction force that prevents the electron form any change of its state and then we discard its contribution entirely). However nowadays this prescription is given a serious physical meaning, namely, they say that no discarding we do, but it is the original coefficients who “absorb” our wrong corrections because our original coefficients in (1) are “bare” and “running”! Of course, it is not true: nothing was bare/running in (1) and is such in (4), but this is how the blame is erroneously transfered from a bad interaction Lagrangian to good original equations and their constants. Both modifications of coefficients (self-action ansatz and renormalization) are presented as a great achievement today. It, however, does not reveal how Nature works, but how human Nature works. Briefly, this is nothing else but a self-fooling, let us recognize it. No grand unification is possible until we learn how to get to (4) directly from (1), without renormalization.
Most of our “theories” are non renormalizable just for this reason: stubbornly counting that renormalization will help us out, we, by analogy, propose wrong “interaction Lagrangians” that not only modify the original coefficients in equations, but also bring wrong “radiation reaction” terms. Remember the famous leading to runaway exact solutions in CED and needing a further “repair” like or so.
We must stop keeping to this wrong way of doing physics and pretending that everything is alright.
P.S. Wilsonian framework, as any other, proceeds from an implicit idea of uniqueness and correctness of the spoiled (i.e., wrong) equations, and cutoff and renormalizations are simply and “naturally” needed there because “we do not know something” or because “our theory lacks something”. Such a “calming” viewpoint prevents us from reformulating the equations from other physical principles and “freezes” the incorrect way of doing physics in QFT. Wilsonian interpretation, as any other, is in fact a covert recognition of incorrectness of the theory equations (equations (6) in our case), let us state it clearly. First, one cuts off a correction under some “clever pretext”, and next, one discards it entirely anyway because this correction is just entirely wrong whatever cut-off value is, so the “clever pretext” for cutting off is put to shame.
And those who still believe in bare particles and their interactions, “discovered” by clever and insightful theorists despite bare stuff being non observable, believe in miracles. One of the miracles is the famous “absorption” of wrong corrections by wrong constants in the right theory (i.e., the constants themselves absorb corrections, without human intervention).