Controversial discussions about renormalisation, EFT, etc ...

Question

This thread is meant to contain general discussions and points of view about renormalisation and EFT which are not shared by the current mainstream physics community.

Answer 1

I was wondering if your complaint is simply about the form of the interaction Hamiltonian. In three dimensions, renormalization of superrenormalizable field theories is mathematically completely understood.

If you look at 3d field theory, the interaction Hamiltonian for scalar $\phi^4$ can be written with the counterterm absorbed in the interaction, so that the physical mass is the parameter in the equation. The equation for stochastic time evolution is explicitly written down by Hairer with the linear term equal to the mass of the physical state, and the interaction term includes the counterterm precisely.

The limiting procedure to define the theory then moves the counterterm linearly along to reproduce the correct results in the limit. The procedure is entirely rigorous, and you can't complain.

The same thing would be possible in 3d QED, which also has the same radiation reaction issues as in 4d QED, except that here the coupling is superrenormalizable, so one can define the theory nonperturbatively in a similar way. This explains what people would like to do to define renormalized theories, and that it is completely consistent. It also does what you want, in that the interaction term includes the counterterms inside naturally, and is understood to all orders in a sense, because you know how to take the limit physically.

This procedure is not applicable in 4d, because the interaction is not superrenormalizable. But your complaints don't depend on thedimension of space.

Comments

Yes, to a great extent it is a form of interaction Hamiltonian. To a great extent, but not only. The free Hamiltonian is essential too. In my "Toy Model" paper it is explained in details - what may be wrong, what one obtains after renormalization and what one obtains after the soft modes summation. The correct Hamiltonian has a physically reasonable free Hamiltonian, a physically reasonable interaction Hamiltonian, and the corresponding physics of permanently interacting constituents.

---

It is possible to do the perturbation theory using the "physically correct Hamiltonian" as a starting point, meaning a Hamiltonian where the physical mass and zero-wavelength charge are the terms in the bare Lagrangian. That's what everyone did in the 1950s and 1960s, before they introduced some convenient terms to allow them to do it without a fixed energy, to deal with the special cases of massless or confined particles, complications which don't appear in pure QED, at least not if you are concerned with low-energy physics only.

This changes nothing at all regarding the non-perturbative behavior, because the series doesn't approximate anything well-defined in the continuum limit as far as anyone can see, and you haven't said anything which can change this conclusion (you don't even know this conclusion).

The sum of the infinite series (or any truncation) is not sensible at large alpha. It is also not sensible at any alpha, and at a value of log(p/m_e) which is larger than 1/alpha. This means that it makes no difference how you write it down, you can't make the theory well defined, it will never be well defined, because it is fundamentally incomplete.

If you wish to complete QED, you should know that it is not clear you will succeed, because then you would need to understand QED with a large coupling. The only way we know to make sense of QED with large coupling is to add more stuff. The first thing is monopoles, the next thing is supersymmetry, and in certain cases, like Argyres-Douglas points, you can get interacting monopoles and charges which make a consistent theory because they are embedded as low-energy limits in a larger theory which makes sense. The extent to which this is possible and defines QED is still debated, because we can't make sense of things without monopoles and supersymmetry, and we certainly can't make sense of physical QED with just an electron and photon and any given nonzero fine structure constant. All we can do is write a perturbation series of dubious validity which agrees with experiment very well, but surely breaks down.

---

OK, Ron, I am not ready to answer, I am exhausted.

Answer 2

@RonMaimon: "Can you tell me what sense QED can fail to make sense ...?"

Yes, I can. For that, let me invoke a correcr CED as an example. As a good approximation to the correct CED, we can take a CED with $\dot{\mathbf{F}}_{ext}$ as a radiation reaction force. What does such a CED predict? The energy-momentum of the whole system is conserved (approximately with $\dot{\mathbf{F}}_{ext}$, but it is not essential at this moment). However, when we consider a couple of oppositely charged charges in a bound state, the radiated energy will be infinite in the end ( a collapse). To improve the agreement with experiment we are obliged to use QM which is change of equations rather than fitting constants within CED. Similarly in QED, it does not describe all particles and in this respect it fails. But as a model, it can be formulated correctly, without bare stuff/counter-terms.

Comments

@RonMaimon: The more I think of a strong coupling regime, the more I get confused. We know that an external field can create real pairs, for example, in a condenser. These pairs separate from each other and neutralize the opposite charges on the condenser plates, so the final system configuration is more stable (with a smaller electric field). One electron state in a QED with high $e$ cannot decay into pairs because of conservation laws, seems to me, but maybe I am missing something. For example, there is a formula for the ground state energy that contains $Z$ and $\alpha$ and which becomes imaginary at hight $Z$. Normally they say that one particle approximation in an external field is not valid anymore and such a state creates pairs. I do not know what we may expect as an exact solution for single charge in QED in the regime of strong coupling. I admit that such an imaginary energy may well appear somewhere and the theory will break. I have no clear idea.

Arnold, I mean QED with counter-terms where the physical constants remain the same and do not get any perturbative corrections.

---

Because if you claim it is mathematically well defined at one alpha, it should be well defined at all alpha. So show me how it is well defined for large alpha, and if you can't, stop claiming you see it is well defined at small alpha. The only difference between the two is that at small alpha perturbation theory takes a long time to crap out and give nonsense.

The way to see that the small alpha perturbation theory craps out at either extremely high order at normal energies, or else at first order at super-high energies even when alpha takes the physical value is to resum the terms with the most logarithms in the subtracted perturbative calculation of scattering.

This resummation gives the same exact nonsense for scattering at high energy (with small alpha) that you see for scattering in large alpha limit. The shorthand way of saying this is that the running coupling is large at large center-of-mass scattering, but the result does not require you to say the coupling runs. You can say the coupling is fixed at the physical value, as you like to do, and then you are simply summing alpha log(p/m_e) + alpha^2 log(p/m_e)^2 + alpha^3 log(p/m_e)^3 etc to find the scattering at large p.

The appearance of a log(p/m_e) term at one-loop order is experimentally verified, it can't be gotten rid of. The log(p/m_e)^2 term at two loops appears also experimentally, and the all orders extrapolation is fixed from unitarity and relativity, it is the only perturbation theory possible (and your formulation would have to reproduce it)

From these logarithms, when log(p/m_e) is order 1/alpha for any nonzero value of alpha, you have a sum with an effective alpha order 1, and when log(p/m_e) is 1,000,000/alpha the sum has an effective alpha of order 1,000,000. But you can't make sense of this perturbation series.

I am asking you to stop being irresponsible and stop claiming you can fix QED! If you can fix QED you would be able to say what happens at strong coupling, and you can't tell me what happens at strong coupling by your own admission . What that means is that you are doing the same stupid perturbation theory that everyone else has been doing with at best a philosophical renaming of terms.

---

OK, thanks, Ron, for explaining the QED failure at high energies of the projectile. I am so tired with discussions with Arnold that I give up.

Just for your information (from my experience): if you only sum up the leading logarithms (rather than all terms), you can obtain a wrong result.

---

Don't get me wrong, I don't think that this is the last word regarding QED. But to get beyond this, you should know the leading logarithm sum behavior. It's not something magical or complicated, it just is what it is.

To get beyond this, you need a non-perturbative formulation, and you can do this with a lattice. Then you can simulate pure QED very easily (people have been doing this in recent years). The nice thing is that on the lattice, you can also study Argyres-Douglas points, supersymmetric theories, and also just phenomenonlogical actions with monopoles of variable mass and spin.

Schwinger likely believed that adding monopoles would fix QED nonperturbatively, he worked on this in the 1960s, likely so did Dirac. But the justification for this belief only is getting strong in the 1990s with Argyres-Douglas work on Seiberg-Witten solutions of supersymmetric theories. You don't need to follow all the work to simulate monopole theories, and it is possible that there is a simple QED modification which is clearly sensible, but it would at least require monopoles to deal with strong coupling configurations.

The reason is that strong coupling QED on a lattice is also understood since Wilson and it is qualitatively different from weak coupling QED, in that it confines! So this means that there is a qualitative transition in QED on a lattice, between weak and strong coupling regimes, and this transition has not been studied or understood, and it can be, because  you can calculate it on a computer.

---

Thanks a lot!

---

Ron, as I said, I can guarantee nothing for high alpha.

Concerning Landau, he was trying to figure out what the value of a bare charge should be to observe the real charge $e$. Whatever finite value $e_B$ of the bare charge is, it is completely screened. That is why the only way to "choose" $e_B$ is to make it "running". It proves nothing but impossibility of such a screening physics. Counter-terms subtract it.

---

That's what you read in books. What Landau saw is exactly what you are seeing in attempting to do calculations at large alpha, and further that the inconsistencies don't go away as alpha gets small, because as you make alpha smaller, the inconsistencies only move to smaller distances.

If you can guarantee nothing for high alpha, you haven't guaranteed anything at small alpha either. If you make the claim that you can define physical scattering at some finite alpha, you should also at the same time sort out what happens at large alpha. If you can't, you don't know what is going on at small alpha either.

Answer 3

Continued from a tangential discussion at http://www.physicsoverflow.org/31682

The sentence

How can you "generally believe in theory" if the high energy behavior is certainly handled in a wrong way and you know it?

also contains (again) the unscientific neither theoretically nor experimentally founded claim that there is something wrong with how the high-energy behavior is handled in current modern (quantum field) theories.

Comments

I found the OP's (of the other post) approach very poorly presented (in spite of his second attempt) - too poorly to even make a constructive comment. This is enough to close it.

But even a better presentation would seems to me to be far too simple-minded, level $<$ graduate+ and hence closeable. (All infinities cancel exactly when handled properly, but because of nontrivial relations. The latter are not present if one doesn't take a proper limit, i.e., carry through the renormalization; so the PO's simple-minded  idea is seriously flawed.)

---

@VladimirKalitvianski: Scharf _is_ mainstream; no one in mathematical physics regards a theory with cutoff to be a good relativistic QFT. Poincare invariance is a must, and it is valid only if there is no cutoff.

---

@VladimirKalitvianski: We have lots of them  constructed rigorously in 2 and 3 dimensions, independent of Scharf.

We also have lots of them in 4 dimensions, though only on the level of formal series - the first one being QED as constructed by Feynman, Tomonaga, and Schwinger with three different approaches, for which they received the Nobel prize. Any form of renormalized perturbation theory constructs these in a Poincare invariant form. Some of them use cutoffs as a temporary scaffolding for construction, causal perturbation theory doesn't. in any case, the final result is in all cases free of cutoffs and bare stuff. 

---

@VladimirKalitvianski: Then your previous comment is misleading.

---

@VladimirKalitvianski: Sarcasm has no place in scientific discussions. By being sarcastic you only poison the atmosphere.

---

I just underlined sarcastically the difference between your causal position and QFT-with-cutoff ideology and lattice practice.

---

By sarcastic remarks I make you write better-thought responses.

Answer 4

This comment discussion originated below an answer to a question about Stückelberg renormalization which has as such nothing to to with integrating out any degrees of freedom ...

Comments

And I wonder whether one has the right to simply "integrate out" the high-energy physics if the low-energy processes affect the high-energy physics adiabatically?

---

@VladimirKalitvianski: In the Stueckelberg RG approach nothing at all is integrated out! It is just a parameter transformation.

---

@ArnoldNeumaier: That is why I took the words "integrated out" in double quote marks. Cutting off is even worse than integrating out. Because in a correct theory the integrating out happens automatically and in the limit of large number of the perturbation orders at a given Λ it is done more and more precisely (hopefully).

---

@VladimirKalitvianski: There is also no cutoff. Everything about the Stueckelberg renormalization group happens at $\Lambda=\infty$.

The cutoff is only used to construct the model in the first place by a limiting procedure. Just like $e^x$ is constructed using a cutoff polynomial $\sum_{k\le n} x^k/k!$ by taking a limit. Except that the renormalizing limiting procedure is a bit more complicated.

---

If it were just about limiting process, it would be fine. But renormalization means redefinition of what is what. It is a more serious matter ;-)

---

Renormalization redefines the parameters, which is just a parameter transformation, in such a way that the limit can be taken.

The Stueckelberg renormalization group is what removes the freedom left in the prescription of the renormalization conditions. Because the physics must be unique, the theory cannot depend on this freedom, as for the gauge freedom in a gauge theory. The Stueckelberg renormalization group is the analogue of the gauge group. The fact that this renormalization group exists, and gives correct predictions, is proof that the renormalization technique is not something arbitrary but an intrinsic feature - otherwise there would be no reason to expect that two different renormalization prescriptions should produce identical physical results.

---

Renormalization indeed redefines the parameters, but it is not just an innocent "parameter transformation in such a way that the limit can be taken". There is no freedom in parameters. Parameters are (re)defined in such a way that gives the desirable answer. This is what P. Dirac called "doctoring numbers".  And I say it even simpler: in order to obtain the right result from a wrong one, one need to discard the wrong part and leave the right one.

Any other choice of parameters leads to wrong results, so no freedom exists if you care about the correctness of the final result.

---

@VladimirKalitvianski: Well, you are almost alone in the world with your views; Dirac is long dead. Everyone else interested in the field understands the meaning and finds it innocent. 

---

What does mean "reparametrization"? It is a change of the numerical value, say, of mass me. You must change it; otherwise no word "renormalization" would have appeared.

---

No. Mass is an observable, defines in the theory as the location of a pole in the S-matrix.

The parameters in the defining action are just coupling constants and do not have a physical meaning as they are unmeasurable. It were different if the interaction could be switched off but it cannot. 

---

What you are saying is that the bare particles are the true (but modest) heroes of microworld, because their interactions make real particles. The bare particles are non observable exclusively due to their modesty - they are shy to be observed. But we happen to be clever enough to write nevertheless  the correct equations for them.

---

No. The bare particles are nonexistent - they are renormalized away.

Already for classical anharmonic oscillators such as the one given by the van der Pol equation, whose renormalization group treatment is given in https://arxiv.org/abs/1305.2372, the coefficients of the differential equations (which contain a bare frequency) - and hence the fictitious bare oscillator used to compute frequencies in perturbation theory - are physically completely meaningless, and only the renormalized frequencies - the one with which the anharmonic oscillator actually oscillates have a physical interpretation. Bare time-dependent perturbation theory produces secular terms that don't reproduce the correct long term behavior.

In the relativistic case, the rest mass takes the place of the nonrelativistic frequency. Again, the parameters in the field equations has nothing to do with the observed rest masses.

---

@ArnoldNeumaier: You readily resort to the non relativistic anharmonic oscillator, but reject my oscillator toy model as "non relativistic". You are biased, Arnold.

---

@VladimirKalitvianski: Yes, I am biased towards the truth and against speculation.

I explained that renormalization is already used for classical nonrelativistic anharmonic oscillators, and that there the bare frequency is physically meaningless, too. This confirms the large existing literature on classical and quantum, nonrelativistic and relativistic renormalization, and shows that your interpretation of the renormalization process is faulty.

You, on the other hand, define a nonrelativistic oscillator toy model (which is fine) but then use it to speculate on relativisitic issues where neither literature exists nor your model gives any insight for how to proceed in a covariant way. It is only these speculations that I reject. 

---

@ArnoldNeumaier: Nothing shows that my interpretation of the renormalization process is faulty. Calling it a speculation is not a proof, Arnold.

---

@VladimirKalitvianski: If your critique of the renormalization process were not faulty it would also apply to the classical anharmonic oscillator, because renormalization is there needed, too, to get rid of the secular terms.

---

In QFT we remove unnecessary corrections, not the secular terms.

The term " renormalization" is applied in statistics too, after changing the number of events in a sample, roughly speaking. There are different fields where the term "renormalization" is used and the procedure means namely it. It has nothing in common with my "interpretation".

---

The secular terms in relativistic QFT are infinite, hence must be removed. Renormalization is therefore necessary in relativistic QFT. If you consider QFT in a 1-dimensional space-time where space consists of a single point only you get precisely the anharmonic oscillator.

Thus the relation is immediate, and not far-fetched as your example from statistics.

---

My oscillator toy model is one soft QED oscillator rather than "far fetched" thing. OK. I do not want to continue to argue with you.

---

@Dilaton: Some light on whether Stückelberg and Wilson renormalization approaches have something in common:

Listen what David Gross says at t=20:33 about Dyson's interpretation of RG.

Answer 5

Discussions about one of t'Hooft's papers:

There was a period when renormalization was considered as a temporary remedy, working luckily in a limited set of theories and supposed to disappear within a physically and mathematically better approach. P. Dirac called renormalization “doctoring numbers” and advised us to search for better Hamiltonians. J. Schwinger also was underlying the necessity to identify the implicit wrong hypothesis whose harm is removed with renormalization in order to formulate the theory in better terms from the very beginning. Alas, many tried, but none prevailed.

In his article G. ‘t Hooft mentions the skepticism with respect to renormalization, but he says that this skepticism is not justified.

I was reading this article to understand his way of thinking about renormalization. I thought it would contain something original, insightful, clarifying. After reading it, I understood that G. ‘t Hooft had nothing to say.

Indeed, what does he propose to convince me?

Let us consider his statement: “Renormalization is a natural feature, and the fact that renormalization counter terms diverge in the ultraviolet is unavoidable”. It is rather strong to be true. An exaggeration without any proof. But probably, G. ‘t Hooft had no other experience in his research career.

“A natural feature” of what or of whom? Let me precise then, it may be unavoidable in a stupid theory, but it is unnatural even there. In a clever theory everything is all right by definition. In other words, everything is model-dependent. However G. ‘t Hooft tries to make an impression that there may not be a clever theory, an impression that the present theory is good, ultimate and unique.

“The fact that mass terms in the Lagrangian of a quantized field theory do not exactly correspond to the real masses of the physical particles it describes, and that the coupling constants do not exactly correspond to the scattering amplitudes, should not be surprising.”

I personally, as an engineering physicist, am really surprised – I am used to equations with real, physical parameters. To what do those parameters correspond then?

“The interactions among particles have the effect of modifying masses and coupling strengths.” Here I am even more surprised! Who ordered this? I am used to independence of masses/charges from interactions. Even in relativistic case, the masses of constituents are unchanged and what depends on interactions is the total mass, which is calculable. Now his interaction is reportedly such that it changes masses and charges of constituents and this is OK. I am used to think that masses/charges were characteristics of interactions, and now I read that factually interactions modify interactions (or equations modify equations ;-)).

To convince me even more, G. ‘t Hooft says that this happens “when the dynamical laws of continuous systems, such as the equations for fields in a multi-dimensional world, are subject to the rules of Quantum Mechanics”, i.e., not in everyday situation. What is so special about continuous systems, etc.? I, on the contrary, think that this happens every time when a person is too self-confident and makes a stupidity, i.e., it may happen in every day situations. You have just to try it if you do not believe me. Thus, when G. ‘t Hooft talks me into accepting perturbative corrections to the fundamental constants, I wonder whether he’s checked his theory for stupidity (like the stupid self-induction effect) or not. I am afraid he hasn’t. Meanwhile the radiation reaction is different from the near-field reaction, so we make a mistake when take the latter into account. This is not a desirable effect , that is why it is removed by hand anyway.

But let us admit he managed to talk me into accepting the naturalness of perturbative corrections to the fundamental constants. Now I read: “that the infinite parts of these effects are somehow invisible”. Here I am so surprised that I am screaming. Even a quiet animal would scream after his words. Because if they are invisible, why was he talking me into accepting them?

Yes, they are very visible, and yes, it is we who should make them invisible and this is called renormalization. This is our feature. Thus, it is not “somehow”, but due to our active intervention in calculation results. And it works! To tell the truth, here I agree. If I take the liberty to modify something for my convenience, it will work without fail, believe me. But it would be better and more honest to call those corrections “unnecessary” if we subtract them.

How he justifies this our intervention in our theory results? He speaks of bare particles as if they existed. If the mass and charge terms do not correspond to physical particles, they correspond to bare particles and the whole Lagrangian is a Lagrangian of interacting bare particles. Congratulations, we have figured out bare particles from postulating their interactions! What an insight!

No, frankly, P. Dirac wrote his equations for physical particles and found that this interaction was wrong, that is why we have to remove the wrong part by the corresponding subtractions. No bare particles were in his theory project or in experiments. We cannot pretend to have guessed a correct interaction of the bare particles. If one is so insightful and super-powerful, then try to write a correct interaction of physical particles, – it is already about time.

“Confrontation with experimental results demonstrated without doubt that these calculations indeed reflect the real world. In spite of these successes, however, renormalization theory was greeted with considerable skepticism. Critics observed that ”the infinities are just being swept under the rug”. This obviously had to be wrong; all agreements with experimental observations, according to some, had to be accidental.”

That’s a proof from a Nobelist! It cannot be an accident! G. ‘t Hooft cannot provide a more serious argument than that. In other words, he insists that in a very limited set of renormalizable theories, our transformations of calculation results from the wrong to the right may be successful not by accident, but because these unavoidable-but-invisible stuff does exists in Nature. Then why not to go farther? With the same success we can advance such a weird interaction that the corresponding bare particles will have a dick on the forehead to cancel its weirdness and this shit will work, so what? Do they exist, those weird bare particles, in your opinion?

And he speaks of gauge invariance. Formerly it was a property of equations for physical particles and now it became a property of bare ones. Gauge invariance, relativistic invariance, locality, CPT, spin-statistics and all that are properties of bare particles, not of the real ones; let us face the truth if you take seriously our theory.

I like much better the interaction with counter-terms. First of all, it does not change the fundamental constants. Next, it shows imperfection of our “gauge” interaction – the counter-terms subtract the unnecessary contributions. Cutoff-dependence of counter-terms is much more natural and it shows that we are still unaware of a right interaction – we cannot write it down explicitly; at this stage of theory development we are still obliged to repair the calculation results perturbatively. In a clever theory, Lagrangian only contains unknown variables, not the solutions, but presently the counter-terms contain solution properties, in particular, the cutoff. The theory is still underdeveloped, it is clear.

No, this paper by G. ‘t Hooft is not original nor accurate, that’s my assessment.

Comments

 You are very surprised about effects visible already in ordinary quantum mechanics.

You quote ‘t Hooft saying

“The fact that mass terms in the Lagrangian of a quantized field theory do not exactly correspond to the real masses of the physical particles it describes [...].”

 It is indeed as little surprising as that an anharmonic oscillator has a frequency (ground state energy divided by Planck's constant) different from the harmonic oscillator obtained by dropping the anharmonic terms in the oscillator. 

This is the precise 1+0-dimensional quantum mechanical analogue of what happens in 1+3-dimensional quantum field theory, where $p^2=m^2$ reduces to $\hbar\omega=p_0=p=m$. the mass renormalization counterterm is just the frequency shift due to the interaction.

The harmonic oscillator is in many contexts (e.g., quantum chemistry) a very convenient but fictional bare oscillator, just used to be able to formulate physical anharmonic oscillators (bond stretchs and bends) and to be able to do perturbation theory.

The only dimensionally induced difference is that in 1+0D and 1+1D, the mass renormalization is finite, while in 1+2D and 1+3D one needs a limiting process that if presented sloppily (as in many textbooks) leads to infinite mass renormalization.

You also write:

Let us consider his statement: “Renormalization is a natural feature, and the fact that renormalization counter terms diverge in the ultraviolet is unavoidable”. It is rather strong to be true. An exaggeration without any proof.

But that renormalization is natural can be seen from the anharmonic oscillator, and a proof that renormalization counter terms diverge in 1+3D can be found in any QFT textbook, so that 't Hooft need not repeat it.

---

''What I am speaking of is, for example,'' - But this is not what is done in QFT renormalization. one just does the same things that are done for an anharmonic oscillators, but since there are now infinitely many of them one must take an appropriate limit of theories with finite running coupling. The running coupling is needed to make theories at slightly different cutoffs give only slightly different physical results. Any hokuspokus beyond that is just poor presentation, not a relevant feature.

And it is irrelevant talking (to Ron Maimon) about CED in a context ('t Hooft) where only QED matters.

---

We can subtract the counterterms exactly in 3d. While we can't write down the closed form expression for the "correct interaction" (physically renormalized interaction) in renormalized QED, I can tell you what it is:

$j\cdot A - A \bar\psi\psi - B F^2 - C \bar\psi \gamma\cdot{\partial}\psi$

This is your "sensible interaction", with appropriate constants A, B, C determined numerically on a lattice as a function of the lattice size a, or order by order in perturbations as a function of the cutoff energy $\Lambda$, or more or less exactly (asymptotically exactly) in superrenormalizable theories in 3d. The condition that the interaction "does as little as possible to the free theory" is simply the physical renormalization condition, and it fixes A,B,C uniquely at any a or any $\Lambda$.

Your quest to find the correct interaction is just to find A,B,C precisely in the continuum limit, where there is no a or $\Lambda$. This quest is totally quixotic because as far as we can see there is no continuum limit. In those cases where there is a continuum limit, superrenormalizable 3d theories or asymptotically gauge theory, we can write down good enough formulas for the values of A,B,C at short cutoffs to do the calculations completely, so we know a good enough approximation to the correct perturbation.

But you don't have to be so careful. The problems in renormalization don't change if you take such care to minimize the impact of the perturbation, it doesn't help to do this, except for some infrared questions.

---

I am not demoralized! I am very moralized. There is a cutoff dependent coefficient in front of j\cdot A too, I just forgot to write it down.

There is nothing you can do about this interaction, this is the correct interaction on the lattice, and it has all the properties you demand--- it fixes the physical mass and charge, it does nothing to single particle states (photons/electrons), and there is nothing to change about it, because it is a perfectly reasonable interaction term on the lattice or in a Pauli-Villars regulator, or wherever.

The question of taking a continuum limit, getting rid of the cutoff, is entirely separate from formulating the interaction, because the continuum limit only exists if there is a second order phase transition with appropriate properties on the lattice. This transition is nearly certainly not there for the theory of pure QED, from simulations, thinking, and perturbative calculations within the standard theory.

But there is no demonstration that there is no second order transition in the same theory with monopoles, and there are strong indications from Argyres Douglas points that it is possible to make an nontrivial continuum interacting theory with both monopoles and charges, but these examples are only for cases where we have some analytic understanding. You shouldn't need any analytic understanding, we can simulate all these things on a computer, and check to see if the appropriate transition is there.

---

The particle momentum is not gauge invariant, because of the transformation of the wavefunction.

$$ \psi(x)\rightarrow e^{ie\phi(x)} \psi(x) $$

$$ A \rightarrow A + \nabla \phi $$

Only $(p-eA)^2$ interaction is gauge invariant modification of the nonrelativistic Hamiltonian. If you introduce interactions using only E, you don't get the proper gauge invariant Hamiltonian. You have disguised this in your formalism somewhat by calling the E the position and the A the momentum, so that now the field "positions" are gauge invariant, but whatever, you still have to check that the Hamiltonian is gauge invariant (it isn't unless you use the normal interaction, that's the principle that fixes $p-eA$).

The gauge invariance condition is stringent, because it modifies the wavefunction for the electron. It's not by purely coupling to E, because the electron wavefunction is altered under gauge transformations. This is why your formalism is well suited for condensed matter applications, phonons, where this issue is absent.

---

Ron, take your time, I do not diagonalize something. I advance a diagonalized thing as an ansatz in a complicated case. It was just an idea and my development stopped at this point. I just wanted to show that one can construct something different and more reasonable. I never had time to finish my study.

By the way, reviews are not about this particular article. And if you want to discuss my papers, let us open another thread for it. This one is already too long.

---

Ok, I don't need to take my time anymore, I didn't understand anything you were doing before outside the simple models, and you wrote a lot about the simple models, so I got stuck there, but I get it completely now. It's an interesting idea, but it's not so new, people often do things like this in Lagrangian language, the proper analog is choosing the quadratic piece in a Lagrangian to make a best fit to the interacting problem. For your case of fixed electron number QED, this can help with soft-radiation.

The articles you gave for review with their models are (probably, I didn't check every line) totally fine, the models you give are correct, they get an improvement from your procedure, the improvement you see is real, it is from rotating kinetic terms, it happens.

The analog for nonrelativistic (or relativistic) QED doesn't exist because the interaction is not approximated well as any kind of kinetic term mixing except for extremely long wavelength photon emission, where you can ignore the size of the things that make the photons, in short, infrared physics. If you look at the short wavelength electron-photon interaction, a mixing of modes of your sort cannot and will not ever simplify the zeroeth order. I understand everything now.

I should point out that the complete lack of effect is made obvious in physical renormalization, where if you use physical counterterms, the interaction does nothing to the physical mass and charge, there is no change at all to the quadratic part of the "true free Lagrangian". Did you really think that people spent three decades studying something that can be fixed by a mode rotation?

The improvements you do are also understood and incorporated into modern perturbation theory. They are all geometric series resummation. This is something you do all the time in perturbation series work, and it is understood as modifying the quadratic initial approximation. But it never gets rid of the interaction, because the interaction is not quadratic. It also never makes the interactions less divergent, nor can it ever give a better starting approximation to electron structure at short distances, because the quadratic approximation at short distances is already best-possible. Such a rejiggered initial approximation can be used to include extremely soft-radiation compared to any scale in the problem, because there the modes of the EM field are coupled to the electrons in a dipole approximation, i.e. momentum minus momentum, and that's amenable to mode-mode rotation.

The interaction is not anywhere approximated better by a different quadratic piece, except to the small extent that the coupling runs, which can be thought of as the correct version of your procedure applied by resumming geometrically the leading logs into corrections of alpha. This changes the quadratic piece slowly to make a best-fit to scattering at energy p.

You need to understand that your methods are understood, and incorporated into modern perturbation theory, they are not changing the conclusions, and they can't change the conclusions.

Answer 6

"Introducing an interaction into the interpretation would presuppose introducing an unphysical notion of something noninteracting to which interactions are added. This something does not exist. Thus the interaction you introduce into the interpretation is erroneous. Do not fool yourself - it is your error, not an error in QED."

(taken from here)

This "something" physically exists - always together with the right interaction. It can be approximately described by an equation for the center of mass or for an average position due to averaging. The latter equation is an equation where you do not see this interaction. Such an equation is physical, but approximate. That is why all equations in external fields work well. As simple as that.

Comments

It doesn't exist in QED, and it is only QED that I am discussing.

It might exist in your attempted approximate formulations but these are too far from being realistic to be taken seriously. They cannot compete with QED,  the best physical theory we have.

---

I had read all your papers. It was enough to lose interest, as you consider ad hoc interactions, prove something about these, and use your results as a justification for denouncing QED, which is (and was already in 1949) far stronger than all you did. 

Your approach is doomed because you are not following up - the slightest advance is already too difficult -, and nobody will do it in your place. 

At another place you excused yourself by complaining that you are alone whereas the physics community worked on QED for many years. But you are now (though having access to all the insight of the physics community) still far less advanced than the physics community was already in 1949.

In 1949, the progress was achieved independently by three different people (Tomonaga, Feynman, Schwinger), which shows that it could be achieved by single people with a moderate amount of work, if they only asked the right questions and tried to answer them! The fact that three very different independently found approaches were soon found to be equivalent (by Dyson) proved that QED is the correct way to go. This wouldn't have happened if any of the approaches had been a crutch.

You seem to have endless time to defend your criticism and your ideas but no time to advance your ideas into something that would answer even the questions answered in 1949. Because it is a dead end, not because one would need more time - no amount of time would make your method fruitful for QED. It may be fruitful for toy problems, but who cares? 

Rather than guess new equations that lack all of the basic principles proved to be correct through QED, you should - if you want to contribute to a new QED - take the regularized equations of QED, solve these by perturbation around one of your conjectured improved starting points (if you can find one), and show that one can get answers with less work than before. 

---

It's about time to read carefully what I propose.

The most careful reading doesn't help. In  http://arxiv.org/pdf/0811.4416.pdf you propose an equation (60) as the '' trial relativistic Hamiltonian of the Novel QED'' and praise it with the attributes ''It replaces the wrong “minimal coupling” (self-action) ansatz ...  the problems of IR and UV divergences do not exist in the Novel QED thanks to using the notion of electronium. No bare constants are introduced, no renormalization is necessary, and there is no such a feature as the Landau pole''. This is all true (and indeed you didn't introduce new parameters) but your marvellous new QED doesn't make any correct predictions! At least not any you advertise.

You reformulate a very successful theory (Old Honorable QED) into one (Novel QED) that is completely useless, at least as far as your powers to exploit your marvels are concerned. But you feel encouraged by your failure to reproduce known results to suggest 

that the other “gauge” field theories should be reformulated in the same way

If your scientific ethos displayed in that paper (throw away old theories just because someone has ideas that he was not able to develop into a calculus where one can make some testable prediction)  were followed, theoretical particle physics would be dead soon.

Perturbation theory is very well developed and poses not a single challenge. If nevertheless the necessary perturbative calculations for establishing predictions from your Novel QED are so horrendous that you cannot perform them in your spare time within a period (2008-2014) far longer than the year (1947-48) the fathers of QED needed for theirs (filling only a few pages in a journal or textbook), your Novel QED is worthless since too difficult to use.

Novelty alone is not a quality sign. You need to convince others that what you do is better in a serious respect - nobody will do it for you; if you don't, your ideas are doomed. 

---

I was saying that Vladimir denounces QED (e.g., by calling the very coupling wrong that guarantees gauge invariance). The newest version of  his reformulation paper ([v13] from Tue, 20 May 2014) still says exactly the same statements I just quoted from the original 2008 version, including the ''wrong coupling'' of QED. Compared to 2008, there is some additional bla bla in the appendix but not the slightest supporting perturbation calculation. But the abstract promises 

For example,  ... in QED ... it means obtaining the energy corrections (the Lamb shift, the anomalous magnetic moment) quite straightforwardly and without renormalizations.

Empty talk, pure wishful thinking. If it were straightforward, he would long have done it as an exercise.

The 13th revision of the paper then wouldn't contain the classical prelude but would start by presenting the novel Hamiltonian with a 1-page motivation, followed by 24 pages of details of the perturbative calculation (more than enough to give complete detail), and a conclusion that every reader can check by comparing the predictions with the textbooks. 

For comparison, the original 1949 paper (11 pages only) computes the Lamb shift quite straightforwardly and with renormalization, ready to check for anyone with access to a library:

N.M. Kroll and W.E. Lamb, Jr., On the Self-Energy of a Bound Electron, Phys. Rev. 75 (1949), 388

---

The feedback you get here consistently is that you take your mouth far too full concerning QED and renormalization in QFT, while your ideas might have some validity for simpler problems and in the classical domain (where you actually provide substantial support through your derivations). But in the realm of QED you demonstrated nothing at all but still make big claims.

Except in your personal views that convinced no one, there is nothing at all wrong with modern perturbative QED if presented correctly (e.g., within causal perturbation theory or with a fix huge but finite cutoff) - it is fully consistent without any infinities, and fully predictive. (The doubts about existence at very large energies or coupling are of a completely different nature and don't affect any of the testable predictions.) 

Your grounds for claiming the opposite are based on nothing but wishful thinking, nourished by grand extrapolation from simple toy problems. Toy problems are useful to illustrate more complex existing theories, but they are not a scientifically acceptable way of making claims about nonexisting or not yet existing theories. Extrapolation from ideas successful in simple cases may be a sensible guide for choosing a research topic, but it ruins your reputation if you use it to make claims that everyone except you finds unsupported by the demonstrated evidence. The evidence must be on the real thing, not on the toy example, before you can begin to modestly make claims.

---

@ArnoldNeumaier, what do you mean by denouncing QED, I thought it works rather well when applied inside its domain of validity?

---

To be fair, people were breaking their head on the problem of QED from 1930 to 1949, before it was solved, and Feynman and Schwinger both spent a lot longer than a year on this. Stueckelberg proposed the renormalization program in 1941, Bethe started thinking about it in 1947, so the comparison is not fair. When it was one person working alone (Stueckelberg from 1934 to 1947), nobody got anywhere. So blaming his rate of progress is not particularly apropos.

Grand extrapolation is natural when you think you have a new idea that nobody else knows. In this case, it is best to understand exactly what VK has done. Reading the paper you linked, this time I finally understood exactly what calculations he is doing, and it is really weird and not totally off base for case of particle coupled to continuum oscillators (for example, a polaron, electron coupled to phonons). But his "new interaction" has an arbitrary choice in it $\alpha(k)$, and this choice reflects the scale of the vector potential entering the kinetic term.

There are two possibilities, either his choice is equivalent to adding p-eA, or else his choice is not gauge invariant. I can determine which it is and write a review for this paper.

The intuitions are fine for polarons, but really irrelevant to QED (really, VK's condition that the interaction is not shifting the one-particle properties, is, as far as the ultraviolet renormalization is concerned, nothing more than a physical renormalization condition--- you can implement this perturbatively easily. He isn't implementing this as a physical renormalization condition, he isn't doing ultraviolet renormalization at all. Rather he is figuring out the long-wavelength nonrelativistic radiation production from a slowly shaken nonrelativistic electron, in the case where the field and electron degrees of freedom are mixed properly from the start).

He is using a Hamiltonian formalism (no covariant perturbation theory, so forget about pair-creation or reproducing modern perturbation theory, it would be a nightmare) just to rewrite the nonrelativistic one-quantum-particle system coupled to QED (cut off at the mass of the electron) in different variables, rediagonalizing the kinetic terms. I didn't know you could do this, and perhaps you can't--- the transformation is strange in the infrared modes, and perhaps doesn't reproduce QED coupling. But this transformation moves the coupling to the electromagnetic modes from the momentum of the particle to the position of the particle, and then the produced radiation is from the shaking of the particle from the action of the external potential directly, instead of from an interaction where the particle shakes the radiative field. It is really the exact quantum analog of the rewrite of the radiation reaction as the derivative of the force, and it is perhaps useful for the infrared issues in QED, or for polaron descriptions. It's irrelevant for renormalization (which is still just fine as always).

I don't understand how the heck he has a free choice in his rewrite--- the $\alpha_k$ are (he claims) a free choice, but they aren't. These are the scale of the vector potential, but the vector potential scale in the p term is determined from minimal coupling.

To VK: you need to check gauge invariance.

$$\psi(x) \rightarrow e^{i\phi(x)} \psi(x)$$

$$A(x) \rightarrow A(x) + \nabla \phi  $$

Needs to leave the equation invariant. This determines your $\alpha(k)$, and probably not at the value you chose for them. I can't vouch for the accuracy of your method at all, I just finally got exactly what you are doing. You wrote it pretty unclearly, using all sorts of annoying conventions.