A Is renormalization the ideal solution?

[Geonaut]

TL;DR Summary

Quantum gravity and GUTs are nonrenormalizable theories, but does this actually mean that these theories must be flawed, or does it mean that renormalization must be a flawed concept, or is this a not actually a problem? If it is impossible to produce a renormalizable quantum gravity theory then shouldn't we view renormalization as an effective, but flawed resolution? Moreover, is this the only problem that we have with renormalization?

Quantum gravity theories and GUTs are nonrenormalizable theories, but does this actually mean that these theories must be flawed, or does it mean that renormalization must be a flawed concept, or is this not actually a problem? If it is impossible to produce a renormalizable quantum gravity theory then shouldn't we view renormalization as an effective, but flawed resolution? Moreover, is this the only problem that we have with renormalization?

I'd like to get as many perspectives of this topic as possible since I am not entirely sure how I feel about it. At the moment I believe that the fact that quantum gravity theories are nonrenormalizable is definitive evidence that there "ought to be a better way of doing things" as Richard Feynman stated in his lectures, but I know that there are many experienced physicists out there that I can learn from and so I'm reaching out.

The fact that gravity is not considered to be part of the standard model leads me to believe that the consensus here is that nonrenormalizable theories can't possibly be anything more than flawed descriptions of nature. This seems to be a reasonable conclusion considering that these theories force us to employ an infinite number of parameters (which seems absurd), but is this conclusion nothing more than an opinion? Aren't we still able to produce accuracy within these theories despite all of the ugliness? Moreover, couldn't we just as easily argue that it is renormalization itself that is flawed rather than these theories?

It appears that we must have running coupling constants in a realistic theory (although, now that I think about it, I think that I may recall Peskin talking about an alternative concept), but that's not what I'm trying to argue. When I say "renormalization" I'm referring to our methods that we use to remove ultraviolet divergences from quantum field theories. What I'm imagining is that a better scheme must exist that produces finite loop diagrams while maintaining the idea of bare and physical quantities that works just as well for nonrenormalizable theories. I think that this method must exist if we consider nonrenormalizable theories to be flawed since gravity is a nonrenormalizable force.

Would you agree with this conclusion of mine? I realize that I've asked many questions here, and I am interested in opinions on all of them, but opinions on any of them would be very appreciated.

 

[atyy]

Both renormaizable and nonrenormalizable theories are nowadays considered ok. They are low energy effective theories. Although QED is renormalizable, it may not be valid at high energies due to the "Landau pole". So renormalizable theories like QED and nonrenormalizable theories like general relativity may both need the introduction of new degrees of freedom (like strings for quantum gravity) at high energies (unless the theories happen to be asymptotically safe).

https://arxiv.org/abs/hep-th/9210046https://arxiv.org/abs/1209.3511

 

[HomogenousCow]

Is renormalizability considered a physical quality of a theory or merely a mathematical property? If it's merely mathematical why was it such a crucial guideline in forming the SM?

 

[vanhees71]

The advantage of Dyson-renormalizable theories is that they have a finite number of parameters (masses of the involved particles and coupling constants) at any order in the perturbative expansion, while effective field theories need more and more "low-energy constants" in the expansion wrt. powers of energy and momentum relative to the cutoff.

 

[HomogenousCow]

Shouldn’t that be a predictive advantage then for non-renormalizable theories since there are more parameters to tune? What I don’t understand is how we can explain the success of the SM if we don’t view renormalizability as a physical property, that would be some coincidence wouldn’t it?

 

[atyy]

It is a coincidence, quite important historically. Another historically important coincidence is the "gauge principle", which is a principle of minimal coupling. One can think of these coincidences as reasonable and inspired constraints in the hunt for theories, although not as absolutely constraining if such theories don't exist. For example, there are non-renormalizable proposals for describing neutrino masses: https://webhome.weizmann.ac.il/home/yotams/notes/ep4.pdf

 

[Geonaut]

Both renormaizable and nonrenormalizable theories are nowadays considered ok. They are low energy effective theories.

When physicists call a theory "effective" they mean that the theory is accurate, but flawed, correct? In which case, you are saying that both renormalizable and nonrenormizable theories are accurate, but flawed, right? If that's what you are saying then I'd like to know what these flaws are.

Moreover, I figured that nonrenormalizable theories have slowly become more accepted in physics, but is this really do to the merit of these theories or is it due to physicists slowly giving up on the idea that the ugly nature of these theories can be circumvented?

Abdus Salam has stated "Field-theoretic infinities — first encountered in Lorentz's computation of electron self-mass — have persisted in classical electrodynamics for seventy and in quantum electrodynamics for some thirty-five years. These long years of frustration have left in the subject a curious affection for the infinities and a passionate belief that they are an inevitable part of nature; so much so that even the suggestion of a hope that they may, after all, be circumvented — and finite values for the renormalization constants computed — is considered irrational."

I find myself wondering if this is true, and if the fact that these infinities have been adopted by the community really means anything. More importantly, what did Salam think would be gained by constructing a standard model with finite values for the renormalization constants? If we created a new standard model that naturally produced finite loop diagrams then what would we gain from it? Would we gain a more predictive theory? Or would it just be creating a theory that's more aesthetically pleasing?

 

[meopemuk]

Hi Geonaut,

Renormalization allows us to calculate the S-matrix, i.e., the mapping from incoming to outgoing states in scattering events. The experimental high energy physics mainly deals with particle collisions, so it does not demand any deeper level of understanding from the theory. The renormalized QFT can calculate accurately scattering cross-sections and energies of bound states (also derivable from the S-matrix), so it is perfectly satisfactory from the modern viewpoint. Thanks to renormalization, there are no divergences in S-matrix elements in all perturbation orders. That's why most people are happy with the current state of affairs.

The trick of renormalization has shifted the divergences from the S-matrix to the Hamiltonian. So, strictly speaking, QFT does not have a well-defined (cutoff-independent) Hamiltonian. So, if you want to study the evolution of any interacting system with time resolution -- you are out of luck. The time evolution generator (aka Hamiltonian) in QFT is full of divergent counterterms, so it is useless for studying finite time intervals, but it is very accurate when used at infinite time intervals, because all infinities cancel out in the S-matrix.

In simple terms: QFT has an excellent S-matrix but a lousy Hamiltonian. If you care about time-resolved evolution and theoretical consistency you may want to find a different (finite) Hamiltonian for QFT without changing the S-matrix. This is possible, because in quantum mechanics the same S-matrix can be calculated from very different Hamiltonians. Among this variety of scattering-equivalent Hamiltonians one can actually find physically acceptable operators without divergences. This approach goes by the name of "dressed particle" theory. It is both aesthetically pleasing and more predictive than usual QFT, because in addition to scattering events and bound state energies, it also describes the detailed time evolution.

Eugene.

 

[atyy]

When physicists call a theory "effective" they mean that the theory is accurate, but flawed, correct? In which case, you are saying that both renormalizable and nonrenormizable theories are accurate, but flawed, right? If that's what you are saying then I'd like to know what these flaws are.

Effective field theories may be incomplete in the sense that they may not hold up to infinitely high energies. At such energies, a more accurate theory may require different degrees of freedom. For example, at low energies, gravity may be described by a theory of gravitons, while at high energies it may be that strings are more fundamental than gravitons.

Moreover, I figured that nonrenormalizable theories have slowly become more accepted in physics, but is this really do to the merit of these theories or is it due to physicists slowly giving up on the idea that the ugly nature of these theories can be circumvented?

Nowadays, after the ground-breaking work of Wilson, renormalization can be understood in a beautiful way, not as senseless subtraction of infinities, but as a way of getting predictions at low energies. I recommend the references in post #2 for expositions of the Wilsonian point of view. You could also try Srednicki's text (chapters 28 & 29), where he explains "The final results, at an energy scale E well below the initial cutoff Λ0, are the same as we would predict via renormalized perturbation theory, up to small corrections by powers of E/Λ0. The advantage of the Wilson scheme is that it gives a nonperturbative definition of the theory which is applicable even if the theory is not weakly coupled. With a spacetime lattice providing the cutoff, other techniques (typically requiring large-scale computer calculations) can be brought to bear on strongly-coupled theories. The Wilson scheme also allows us to give physical meaning to nonrenormalizable theories."

 

[A. Neumaier]

does it mean that renormalization must be a flawed concept, or is this not actually a problem?

No. it just means that when one renormalizes it one has much more freedom in choosing the details in the theory. it is somewhat analogous to the freedom in choosing functions analytic at zero. The renormalizable case corresponds to functions that are representable by quartics, while the nonrenormalizable case corresponds to functions that are representable by an arbitrary power series. In the latter case, many more parameters are to be chosen. For an effective theory, only the first few matter. The high order terms only affect the theory at energies too high to be deemed relevant.

Thus the standard model (being renormalizable) is completely fixed by fixing slightly over 30 constants, all of which are known to some meaningful accuracy. Whereas canonical gravity needs infinitely many constants to single out the unique true theory, and only the lowest order constant (the gravitational constant) is known. To determine the next constant we already need to observe quantum gravity effects, which is still beyond the capability of experimenters.

When physicists call a theory "effective" they mean that the theory is accurate, but flawed

No, they mean that for the purposes at hand, knowledge of the low order constants is enough to get the required accuracy.

strictly speaking, QFT does not have a well-defined (cutoff-independent) Hamiltonian

This is not true. Relativistic QFTs (at least those satisfying the Wightman axioms) have perfectly well-defined Hamiltonians, though these cannot be expressed in terms of linear operators on Fock space. You severely
limit your understanding by insisting on Fock spaces as the basis of QFTs.

after the ground-breaking work of Wilson, renormalization can be understood in a beautiful way, not as senseless subtraction of infinities, but as a way of getting predictions at low energies.

After the ground-breaking work of Epstein and Glaser and the popularization of their work by Scharf and others, renormalization can be understood in an even more beautiful way, neither as senseless subtraction of infinities nor as effective theory with a cutoff, but as a way of getting predictions at any energy, in terms of [URL='https://www.physicsforums.com/insights/causal-perturbation-theory/']causal perturbation theory[/URL]. Of course the theories obtained are not necessarily physically correct at all these energies, but in principle they give results that can be compared with experiments at these energies.

 

[meopemuk]

This is not true. Relativistic QFTs (at least those satisfying the Wightman axioms) have perfectly well-defined Hamiltonians, though these cannot be expressed in terms of linear operators on Fock space. You severely
limit your understanding by insisting on Fock spaces as the basis of QFTs.

Abdus Salam and his co-workers disagree with you:

"As is well known, the infinities result from a lack of proper definition of singular distributions which occur in field theory. One of the major obstacles to progress in the subject has been the uncertainty of whether these singularities have their origin in the circumstance that a perturbation expansion is being made or whether it is the form of the Lagrangian -assumed to be polynomial in field variables -which is at fault. An important suggestive advance in resolving this uncertainty has been the work of Glimm and Jaffe who, working with exact and mathematically well-defined solutions of polynomial Lagrangian field theories (in two and three space-time dimensions) have shown that infinities persist even in exact solutions. If their conclusions may be extrapolated to physical four-dimensional space-time, it would seem that the origin of the infinities is not so much in the inadequate mathematics of the perturbation solution. Rather, the fault lies with the inadequate physics of the assumed polynomial character of the electromagnetic interaction."

The dressed particle theory realizes the above idea of Salam:
1. The Lagrangian (in fact, Hamiltonian) of QFT is unitarily transformed and expressed not as a polynomial in fields, but as a polynomial in creation/annihilation operators. The coefficients of this polynomial are finite and cutoff-independent.
2. In doing so, the S-matrix of the theory is not affected, so all the achievements of QFT (Lamb shift, anomalous magnetic moment, etc.) are preserved.
3. Yes, this construction is performed in the Fock space, which means that particles keep their perfect meaning not only as asymptotic entities, but also within the region of interaction.

Eugene.

 

[DarMM]

Abdus Salam was writing before the more rigorous understanding of QFTs we possesses today. Renormalization does result in well-defined Hamiltonians that is a mathematically established fact.

In lower dimensions see any of the constructive papers by Glimm, Jaffe, Osterwalder, Schrader, Magnen, Sénéor and Rivasseau. In 4D see the papers of Balaban. Better yet is the monograph of Glimm and Jaffe which contain a very short proof in Chapter 8.

 

[A. Neumaier]

Abdus Salam and his co-workers disagree with you:

This means nothing as the quote you give completely misrepresents the achievements of Glimm and Jaffe:

"An important suggestive advance in resolving this uncertainty has been the work of Glimm and Jaffe who, working with exact and mathematically well-defined solutions of polynomial Lagrangian field theories (in two and three space-time dimensions) have shown that infinities persist even in exact solutions."

In fact, Lagrangian relativistic quantum field theories in 2 dimensions with polynomial Lagrangians are perfectly well-defined (rigorously) and finite in the sense of satisfying the Wightman axioms. They have well-defined Hamiltonians.

The only infinities that persist are those obtained by trying to keep the Fock space structure.

 

[meopemuk]

Abdus Salam was writing before the more rigorous understanding of QFTs we possesses today. Renormalization does result in well-defined Hamiltonians that is a mathematically established fact.

Then how should I understand the following quote? It reads as an admission of the absence of a well-defined (i.e., cutoff-independent) Hamiltonian in QED:

The more one thinks about this situation, the more one is led to the conclusion that one should not insist on a detailed description of the system in time. From the physical point of view, this is not so surprising, because in contrast to non-relativistic quantum mechanics, the time behavior of a relativistic system with creation and annihilation of particles is unobservable. Essentially only scattering experiments are possible, therefore we retreat to scattering theory. One learns modesty in field theory.

G. Scharf, Finite quantum electrodynamics. The causal approach. (Springer, Berlin, 1995)

Eugene.

 

[DarMM]

Then how should I understand the following quote? It reads as an admission of the absence of a well-defined (i.e., cutoff-independent) Hamiltonian in QED

Scharf seems to be making some sort of empiricist point about only discussing what you observe, which in QFT is scattering experiments.

However the exact meaning of Scharf's quote is irrelevant to the proven fact that well-defined Hamiltonians are formed via renormalization.

 

[A. Neumaier]

Then how should I understand the following quote? It reads as an admission of the absence of a well-defined (i.e., cutoff-independent) Hamiltonian in QED:

The more one thinks about this situation, the more one is led to the conclusion that one should not insist on a detailed description of the system in time. From the physical point of view, this is not so surprising, because in contrast to non-relativistic quantum mechanics, the time behavior of a relativistic system with creation and annihilation of particles is unobservable. Essentially only scattering experiments are possible, therefore we retreat to scattering theory. One learns modesty in field theory.

G. Scharf, Finite quantum electrodynamics. The causal approach. (Springer, Berlin, 1995)

Scharf only constructs the S-matrix, using causal perturbation theory for QED in 4 dimensions. There one has no finite time dynamics, hence the caveat. To construct a valid finite-time dynamics, one does not only need the time-ordered N-point functions but also the unordered Wightman functions.

Glimm and Jaffe construct interacting Wightman field theories in 2 dimensions, which have a well-defined dynamics at any time.

 

[meopemuk]

This is not true. Relativistic QFTs (at least those satisfying the Wightman axioms) have perfectly well-defined Hamiltonians, though these cannot be expressed in terms of linear operators on Fock space. You severely
limit your understanding by insisting on Fock spaces as the basis of QFTs.

Why are you so dismissive w.r.t. the Fock space and particles in general? When I look around I see particles everywhere: molecules, atoms, electrons, protons. Feynman told us that even light is nothing but a flow of particles-photons. Particles (both interacting and noninteracting) can be measured and counted. And the Fock space is a perfect mathematical tool for describing multiparticle states.

If the field theory of Wightman-Glimm-Jaffe has a difficulty in finding a Fock space particle interpretation for itself, so bad for the field theory. Weinberg was successful in formulating a relativistic quantum theory based fully on particle states, where quantum fields play only a subservient technical role.

Eugene.

 

[Geonaut]

Wow, look at all of these familiar names. Thank you @DarMM, @A. Neumaier, @meopemuk, and @atyy for taking the time to comment on this post. I'm actually glad to see a little bit of debate here since it's leading to a more in depth discussion. I hope you gentlemen can continue with it in a friendly manner. Of course, I have some further questions about your statements that would be helpful to clarify both for me and other readers that come across this post in their own studies.

Glimm and Jaffe construct interacting Wightman field theories in 2 dimensions, which have a well-defined dynamics at any time.

In 4D see the papers of Balaban.

So both of you gentlemen are saying that we can produce well defined hamiltonians in both lower dimensional and 4D space iff we abandon the Fock space structure? Are you also saying that because this is true there is no reason to doubt renormalization as it no longer creates any problems at all in modern physics?

@meopemuk, do you believe that we shouldn't abandon the Fock space structure? Why?

@DarMM, @A. Neumaier, if we assume that it's possible to create a version of QFT where these infinities never appeared, and we did just that then what would we have gained? Would we gain anything at all? Would we gain a more predictive/powerful theory? Moreover, if there really is nothing wrong with renormalization then why hasn't the community come to a consensus on a quantum gravity theory?

 

[meopemuk]

@meopemuk, do you believe that we shouldn't abandon the Fock space structure? Why?

I don't want to abandon the Fock space structure. For example, if I want do describe the Hydrogen atom I can choose the "1 proton+1 electron" sector of the Fock space and find the projection of the "dressed particle" Hamiltonian onto this sector. It is not difficult to show that this projection has the form

$H = \sqrt{m_p^2c^4 + p_p^2c^2} + \sqrt{m_e^2c^4 + p_e^2c^2} - \frac{e^2}{4 \pi |\boldsymbol{r}_p-\boldsymbol{r}_e|} + V_{2 rel} + V_{3 rad} + V_{4 rad} + \ldots$

All these terms have very clear physical meanings: the first three terms are responsible for the usual textbook Coulomb field spectrum of the hydrogen; $V_{2 rel}$ describe relativistic corrections responsible for the fine and hyperfine structures in the spectrum; $V_{3 rad}$ is the 3rd order correction describing the effects of photon absorption and emission, i.e., the lifetimes of the excited hydrogen states; $V_{4 rad}$ is the 4th order radiative correction, which results from 1-loop diagrams and is responsible for the Lamb shift. Analytical expressions for the corrections $V_{2 rel}, V_{3 rad}, V_{4 rad}$ are also available.

This can be continued to higher orders, but the overall message should be clear:
1. This Hamiltonian is finite, cutoff-independent and it captures all aspect of the physics of the hydrogen atom.
2. This Hamiltonian provides the same level of accuracy for the electron-proton scattering as the renormalized QED.
3. If necessary, this Hamiltonian can be used to describe the time evolution of hydrogen states, though high accuracy time-resolved experiments are not available yet.
4. This Hamiltonian is relativistically covariant: there exists a corresponding interacting boost operator such that all commutators of the Poincare Lie algebra are satisfied.

If there exists a comparable field-theory non-Fock-space Hamiltonian for the hydrogen atom, I would love to see it and compare its predictions.

Eugene.

 

[Geonaut]

Renormalization does result in well-defined Hamiltonians that is a mathematically established fact.

In lower dimensions see any of the constructive papers by Glimm, Jaffe, Osterwalder, Schrader, Magnen, Sénéor and Rivasseau. In 4D see the papers of Balaban.

I just started taking a look at Balaban's work via another author, and I immediately noticed that his method uses a spacetime lattice instead of a continuous spacetime. Does this mean that 4D QFT in a continuous spacetime doesn't produce well-defined Hamiltonians, and that we need to adopt the idea of a spacetime lattice in order to produce the well-defined Hamiltonians that you speak of? I find that question to be extremely interesting as it would answer the question asked in the title of this post. If renormalization is not the ideal solution to our infinity problem then I'm lead to believe that it may be a sign that some other unknown physics exists that resolves this problem. Perhaps that physics is actually the idea of a spacetime lattice, perhaps not, regardless, it's very interesting.

 

[Geonaut]

If there exists a comparable field-theory non-Fock-space Hamiltonian for the hydrogen atom, I would love to see it and compare its predictions.

Eugene.

I would love to see that too. Do you have any good references for dressed particle theory? I'd like to read more about it.

 

[meopemuk]

I would love to see that too. Do you have any good references for dressed particle theory? I'd like to read more about it.

There are many works in dressed/clothed particle theory. They usually pay tribute to the seminal paper

O. W. Greenberg, S. S. Schweber, "Clothed particle operators in simple models of quantum field theory", Nuovo Cim. 8 (1958), 378.

You can use Google Scholar to find all citations of this work. Currently this search shows 113 results.

Eugene.

 

[vanhees71]

The

Why are you so dismissive w.r.t. the Fock space and particles in general? When I look around I see particles everywhere: molecules, atoms, electrons, protons. Feynman told us that even light is nothing but a flow of particles-photons. Particles (both interacting and noninteracting) can be measured and counted. And the Fock space is a perfect mathematical tool for describing multiparticle states.

Eugene.

Physically speaking the reason is that a particle (or if massless fields are present as in QED, QCD, and thus the Standard Model as a whole, strictly speaking an "infraparticle") interpretation and a Fock-space description is only possible for the appropriate asymptotic free states, while this is impossible in the transient states where interactions take place.

 

[DarMM]

So both of you gentlemen are saying that we can produce well defined hamiltonians in both lower dimensional and 4D space iff we abandon the Fock space structure? Are you also saying that because this is true there is no reason to doubt renormalization as it no longer creates any problems at all in modern physics?

Renormalization in all cases studied so far in constructive detail seems to render Hamiltonians finite, so there is no reason to view it as problematic.

@DarMM, @A. Neumaier, if we assume that it's possible to create a version of QFT where these infinities never appeared, and we did just that then what would we have gained? Would we gain anything at all? Would we gain a more predictive/powerful theory? Moreover, if there really is nothing wrong with renormalization then why hasn't the community come to a consensus on a quantum gravity theory?

It's not more predictive and powerful, it gives the same answers as normal QFT because it's just a rigorous version of regular QFT, not an alternate theory.
There's no real connection as such between this and quantum gravity, aside from that naively quantized gravity is non-renormalizable.

I just started taking a look at Balaban's work via another author, and I immediately noticed that his method uses a spacetime lattice instead of a continuous spacetime. Does this mean that 4D QFT in a continuous spacetime doesn't produce well-defined Hamiltonians, and that we need to adopt the idea of a spacetime lattice in order to produce the well-defined Hamiltonians that you speak of?

No. Balaban just uses a lattice as a method of proving continuum renormalized Yang-Mills is well-defined. The whole point of these type of lattice construction and the functional analytic methods that go with them (e.g. the cluster/Mayer expansion) is to show QFT on continuous spacetime has a well defined Hamiltonian.

 

[A. Neumaier]

Why are you so dismissive w.r.t. the Fock space and particles in general?

Because mathematical analysis of the 2D case implies the lack of the Fock structure. If you'd do your dressed particle construction with dressed particle states in a 2D massive QFT without bound states you'd reproduce power series approximations to the Wightman functions constructed rigorously.

However, in 4D QED, the dressed particle approach misses completely the infrared structure of the electron. In the dressed QED exposition of your book, the electron is not, as standard QED predicts, an infraparticle.

When I look around I see particles everywhere: molecules, atoms, electrons, protons.

When I look around I never see particles but light, colors, shapes. Particles are abstractions introduced to interpret these, not fundamental things. In Wightman field theory, particles appear as asymptotic constructs in scattering events. This is indeed the only situation where we notice the particle aspect of subatomic matter. Every QFT has an asymptotic Fock space in which scattering can be studied, while the Hilbert space at finite times is non-Fock..

Feynman told us that even light is nothing but a flow of particles-photons.

I don't take Feynman as the ultimate authority. He also said that nobody (which includes himself) understands QM, so he himself considers his views as provisional only.

Weinberg was successful in formulating a relativistic quantum theory based fully on particle states, where quantum fields play only a subservient technical role.

No. His book is called quantum field theory, and particles don't play a big role. Moreover, he formulated only perturbative scattering theory, which doesn't get rid of all infinities since the perturbative series diverges.

 

[A. Neumaier]

There are many works in dressed/clothed particle theory. They usually pay tribute to the seminal paper

O. W. Greenberg, S. S. Schweber, "Clothed particle operators in simple models of quantum field theory", Nuovo Cim. 8 (1958), 378.

You can use Google Scholar to find all citations of this work. Currently this search shows 113 results.

Eugene.

Why don't you say that the best exposition is in your [meomepuk = Eugene Stefanovich] 3 volume treatise on quantum electrodynamics ? I think this is a valid statement. Your treatise shows both the potential and the limitations of the dressed particle approach.

It uses perturbatively constructible (but rigorously ill-defined) ''unitary'' transformations to renormalize standard perturbative QED. It misses, like any purely perturbative treatment of QFT, the infrared aspects of the theory. The faults show up in your version by predicting small superluminal effects.

4. This Hamiltonian is relativistically covariant: there exists a corresponding interacting boost operator such that all commutators of the Poincare Lie algebra are satisfied.

No. You didn't prove the existence of the boost operator in your version of QED; you only give a perturbative construction for it, without showing its convergence. The superluminal effects you inherit are proof of the lack of true Lorentz invariance.

The devil is in the details, and getting the details right (as Glimm and Jaffe did in 2D) requires abandoning the Fock structure.

 

[atyy]

https://www.physicsforums.com/threads/collection-of-science-jokes-p2.847743/post-6237164

 

[vanhees71]

No. His book is called quantum field theory, and particles don't play a big role. Moreover, he formulated only perturbative scattering theory, which doesn't get rid of all infinities since the perturbative series diverges.

Well Weinberg in his books takes explicitly the particle picture as fundamental. It's not so simple though. He does not discuss the IR problem from the "infraparticle picture" either but uses the conventional "ladder summation" approach to deal with the "soft-photon problem". It's also not a book about axiomatic QFT but a physics book with all the sloppiness used by physicists to use perturbative QFT to describe particles, but indeed he presents the "particle point of view" right from the very beginning.

 

[A. Neumaier]

Well Weinberg in his books takes explicitly the particle picture as fundamental. It's not so simple though. He does not discuss the IR problem from the "infraparticle picture" either but uses the conventional "ladder summation" approach to deal with the "soft-photon problem". It's also not a book about axiomatic QFT but a physics book with all the sloppiness used by physicists to use perturbative QFT to describe particles, but indeed he presents the "particle point of view" right from the very beginning.

He presents this view, but as an asymptotic one, not as a fundamental one. This is obvious from his preface, where particles appear only in two sentences - as ingredients for the unitary representations of the Poincare group, and as a collection of free particles in the far past.

Fundamental are the quantum fields, but they need a motivation, which is given through the picture of free particles. See also his paper ''What is quantum field theory, and what did we think it is?'', where he says on p.2,

In its mature form, the idea of quantum field theory is that quantum fields are the basic ingredients of the universe, and particles are just bundles of energy and momentum of the fields.

 

[Geonaut]

I don't take Feynman as the ultimate authority.

I love Feynman, but the man didn't believe in brushing his teeth or even washing his hands after using the bathroom... so I'm with you there, otherwise, I wouldn't be very hygienic. Actually, I don't think even the physics community should be seen as the ultimate authority. Humans are smart, but we're not all knowing. Which is why I'm still not 100% sure about renormalization, although it seems like I'll come to accept it.

It's a shame that these papers that you gentlemen have been referring to cost money to access as I am very poor, I'll have to read them once I become less poor lol. For the time being I'll have to just go by what you are saying to me which has made a lot of sense. When renormalization was first introduced to me it was done so using the cut-off procedure, and I immediately thought it was a bunch of nonsense. Then I was introduced to dimensional regularization and the pauli-villars procedure. The pauli-villars procedure also seemed like unnatural nonsense to me, but dimensional regularization made more sense. However, the problem I had then was that it seemed very bizarre to just introduce counter terms and arbitrarily claim that they are divergent in exactly the right way so as to make certain things finite. After that I read about Wilson's approach, but I still wasn't completely convinced... in retrospect, I need to go back and read more about it. I decided to make this post to get more perspectives, and look at that, there's apparently a better approach out there than Wilson's. I still have doubts that I'll believe in it, but the fact that you gentlemen are so certain is reassuring that it's not something that I need to worry about. Although I'm sure I'll worry about it anyway. I need to go investigate it more on my own, but your statements have been very persuasive.

@A. Neumaier so to be clear, it sounds like you're saying that we should abandon the idea of a Fock space, and that this means that particles don't exist, right? If particles are "just bundles of energy and momentum of the fields" then are you claiming that bundles of energy and momentum don't exist? Perhaps the better question is: How does constructing a QFT in a non-Fock space change our interpretation of things?

naively quantized gravity is non-renormalizable.

Wow, are you saying that there is a way to quantize gravity that gives us a renormalizable theory? How have I never heard of that before. What's this theory called, and does it have any problems within it?

 

[meopemuk]

Why don't you say that the best exposition is in your [meomepuk = Eugene Stefanovich] 3 volume treatise on quantum electrodynamics ? I think this is a valid statement. Your treatise shows both the potential and the limitations of the dressed particle approach.

Thank you for the generous introduction of my humble book.

It uses perturbatively constructible (but rigorously ill-defined) ''unitary'' transformations to renormalize standard perturbative QED. It misses, like any purely perturbative treatment of QFT, the infrared aspects of the theory. The faults show up in your version by predicting small superluminal effects.

I deliberately avoided infrared aspects. Otherwise I would have to write a 4th volume of the book. The handling of IR divergences in QED is well understood, so I do not see any problem with applying the same ideas within the dressed particle approach. For experimentally measured things like the anomalous magnetic moment or the Lamb shift, IR divergences do not show up or cancel out. I've explicitly demonstrated this cancellation in the book.

The superluminal effects are not bugs, but unavoidable features of the dressed particles approach. In this approach we eliminated fields as carriers of interactions. Then the energy-momentum conservation implies that the energy-momentum exchange between particles (=interaction) occurs without retardation. This is not an approximation, but a solid prediction of the theory. It appears that this prediction was confirmed in the experiment of G. Pizzella et al. If you think that instantaneous Coulomb potentials violate relativity or causality, then please read this paper, which proves otherwise:

E. V. Stefanovich, "Does Pizzella's experiment violate causality?", J. Phys. Conf. Series, 845 (2017), 012016.
PDF file

No. You didn't prove the existence of the boost operator in your version of QED; you only give a perturbative construction for it, without showing its convergence. The superluminal effects you inherit are proof of the lack of true Lorentz invariance.

In Appendix E.2 of volume 2, I copied Weinberg's proof that 10 Poincare generators of QED $(\boldsymbol{P}_0, \boldsymbol{J}_0, \boldsymbol{K}_0 + \boldsymbol{Z}, H_0+V)$ satisfy the required commutation relations. This proof is non-perturbative. Ten generators of the dressed theory are obtained from QED generators by means of a unitary "dressing" transformation. This means that all commutators are preserved and the dressed theory is rigorously relativistic.

In practice, this unitary transformation can be performed only at low perturbation orders, so -- you are right -- some violations of the Poincare invariance are to be expected. This is not different from the Poincare-non-invariance of QED in finite orders. However, as I explained above, predictions of superluminal effects would remain in the dressed theory even non-perturbatively.

Eugene.

 

[meopemuk]

Renormalization in all cases studied so far in constructive detail seems to render Hamiltonians finite, so there is no reason to view it as problematic.

Do I understand it correctly that this field-based theory has not advanced enough to handle the Hamiltonian of the hydrogen atom?

Eugene.

 

[haushofer]

I feel another Insight being born here :P

 

[DarMM]

Do I understand it correctly that this field-based theory has not advanced enough to handle the Hamiltonian of the hydrogen atom?

Eugene.

The non-relativistic Hamiltonian is well defined and self-adjoint. In QED there isn't really a Hamiltonian for hydrogen, just the QED Hamiltonian.

 

[DarMM]

Wow, are you saying that there is a way to quantize gravity that gives us a renormalizable theory? How have I never heard of that before. What's this theory called, and does it have any problems within it?

It's possible that gravity is asymptotically safe, so that gravity quantized with extra terms may be well defined. However it is currently unknown if that is true.

 

[meopemuk]

The non-relativistic Hamiltonian is well defined and self-adjoint. In QED there isn't really a Hamiltonian for hydrogen, just the QED Hamiltonian.

I am OK with that. In the dressed theory there is also a full Hamiltonian $H= H_0+V$ in the entire Fock space. This Hamiltonian is obtained by a unitary transformation from the "QED Hamiltonian". $H$ is a normally-ordered polynomial in particle creation and annihilation operators, like in eq. (4.4.1) of Weinberg's vol. 1. This polynomial is Poincare invariant, cluster separable and has huge (infinite) number of different terms. However, if we are interested only in the hydrogen atom (which I define as a bound state of two particles -- an electron and a proton), we don't need the full interaction $V$, we can choose only few terms that can be relevant to this physical system.

Let me introduce the following creation/annihilation operators: $a^{\dagger}, a$ for electrons, $d^{\dagger}, d$ for protons, $c^{\dagger}, c$ for photons. Then in a pretty decent approximation, the part of the polynomial $V$ that is relevant to the description of the hydrogen atom is

$V = d^{\dagger}a^{\dagger}da + d^{\dagger}a^{\dagger}dac + d^{\dagger}a^{\dagger}c^{\dagger}da + \ldots$

For brevity, I left only operator symbols and omitted coefficient functions, integral signs and other paraphernalia. If necessary, I can supply the missing info. These three terms have clear physical meanings:

$d^{\dagger}a^{\dagger}da$ is the direct electron-proton interaction potential. Its leading part is the usual Coulomb potential, plus there are relativistic (contact, spin-orbit, spin-spin) corrections, plus -- starting from the 4th perturbation order -- there are corrections responsible for the Lamb shift. If we diagonalize $H_0 + d^{\dagger}a^{\dagger}da$ in the electron+proton sector of the Fock space, we will get a decent energy spectrum of hydrogen.

$d^{\dagger}a^{\dagger}dac$ is interaction by which the atom can absorb a free photon and jump to a higher energy level

$d^{\dagger}a^{\dagger}c^{\dagger}da$ is interaction by which photons are emitted from excited atomic levels. So, the energy widths of the excited levels can be properly described.

In brief, the dressed approach provides a clear physical picture of the hydrogen atom with all important interactions. The relevant part of the Hamiltonian is resembling the familiar non-relativistic hydrogen Hamiltonian. However, there are also relativistic and radiative corrections, which ensure that the whole theory is Poincare invariant and cluster separable.

Now, you are saying that the "proper" field-based QED does not use the ideas of particles, their creation/annihilation operators and their Fock space, when interacting systems (such as the hydrogen atom) are involved. On the other hand, you claim that this theory possesses a finite well-defined cutoff-independent Hamiltonian. I am really interested in learning how this Hamiltonian looks like and how one can extract from it useful info, like the energy spectrum, lifetimes, wave functions, etc?

Eugene.

 

[DarMM]

Ten years ago I went through all of this with you, giving you the relevant papers, links etc that mathematically prove these facts and how Fock space and creation and annihilation operators will not work and explaining every point in detail. I'm not repeating it.

 

[A. Neumaier]

@A. Neumaier so to be clear, it sounds like you're saying that we should abandon the idea of a Fock space

No. A Fock space is alright for approximations, but not for the limit of these approximations.

 

[A. Neumaier]

I deliberately avoided infrared aspects.

You had to, because you only work perturbatively. As a result, your 2-point function for the electron has a pole at the electron mass while the correct QED 2-point function has there a branch point (as found by a renormalization group analysis). This conclusively proves that your dressed particle QED is not equivalent to QED proper.

In Appendix E.2 of volume 2, I copied Weinberg's proof that 10 Poincare generators of QED $(\boldsymbol{P}_0, \boldsymbol{J}_0, \boldsymbol{K}_0 + \boldsymbol{Z}, H_0+V)$ satisfy the required commutation relations. This proof is non-perturbative. Ten generators of the dressed theory are obtained from QED generators by means of a unitary "dressing" transformation. This means that all commutators are preserved and the dressed theory is rigorously relativistic.

Weinberg has no dressing transform, hence his derivation is alright. But your dressing transforms are only perturbatively defined, and do not have a limit when the order goes to infinity. Thus they are no true unitary transformations in the limit relevant for true QED. This is indeed impossible by Haag's theorem.

In practice, this unitary transformation can be performed only at low perturbation orders, so -- you are right -- some violations of the Poincare invariance are to be expected.
This is not different from the Poincare-non-invariance of QED in finite orders.

It is very different from standard renormalized QED, which is Poincare invariant at every loop order.

 

[meopemuk]

...you only work perturbatively. As a result, your 2-point function for the electron has a pole at the electron mass while the correct QED 2-point function has there a branch point (as found by a renormalization group analysis). This conclusively proves that your dressed particle QED is not equivalent to QED proper.

The 2-point function is a field concept, but the dressed particle theory is about particles, not fields. It deals with particle positions, momenta, spins, wave functions, etc. But n-point functions is a foreign concept for the dressed particle theory.

As for the non-equivalence to QED, I can argue that one can make the dressed particle theory as close to QED as one wishes. For example one quick and dirty way to get the dressed Hamiltonian is simply to fit its interactions to the S-matrix calculated at the specific loop order in QED.

But your dressing transforms are only perturbatively defined, and do not have a limit when the order goes to infinity. Thus they are no true unitary transformations in the limit relevant for true QED.

You claimed that the renormalized QED has a finite cutoff-independent Hamiltonian. However this is not what textbooks say. You can find the Lagrangian of QED in eqs. (11.1.6) - (11.1.9) of Weinberg's vol. 1 and in eq. (10.38) of Peskin & Schroeder. (Obviously, the QED Hamiltonian $H_{QED}$ has a similar structure.) These expressions have counterterms that depend on divergent (=cutoff-dependent) constants $Z_1, Z_2, Z_3, \delta m$. The counterterms are divergent, but (in Weinberg's words) "these terms just suffice to cancel the ultraviolet divergences that arise from loop integrals." That's the whole idea of renormalization, as far as I understand it.

Now, the dressing approach says: I would like to make a unitary transformation of the QED Hamiltonian

$H_d = e^{i\Phi} H_{QED} e^{-i \Phi}$

where the operator $\Phi$ is chosen to satisfy 3 conditions:

1. It is Hermitian in each perturbation order. This means that the dressing transformation $e^{i\Phi}$ is unitary in each perturbation order, thus your objection about the non-unitarity is not valid.
2. It is sufficiently smooth. Then one can prove that the S-matrix computed with $H_d$ is exactly the same as the S-matrix computed with $H_{QED}$
3. It cancels order-by-order certain "bad" terms in $H_{QED}$. The "bad" terms are those which act non-trivially on the vacuum and 1-particle states. They are the ones responsible for self-interactions and ultimately for the divergences in the traditional QED.

At the end of this procedure one obtains the dressed Hamiltonian $H_d$ without self-interactions. This Hamiltonian makes exactly the same predictions as $H_{QED}$ about scattering and bound state energies. Not surprisingly, $H_d$ resembles the familiar Hamiltonian of the non-relativistic theory. For example, the leading interaction term is the Coulomb potential between charged particles. As a little bonus, $H_d$ is finite, i.e., all the divergent renormalization constants cancel out in all orders. More precisely, they get absorbed into the transformation operator $\Phi$, which becomes divergent now, but we shouldn't be bothered, because this operator has no physical meaning.

Now we can use $H_d$ to make all kinds of electrodynamics calculations without ever thinking about cutoffs, counterterms and renormalization. All integrals are ultraviolet-convergent.

Eugene.

 

[A. Neumaier]

The 2-point function is a field concept, but the dressed particle theory is about particles, not fields. It deals with particle positions, momenta, spins, wave functions, etc. But n-point functions is a foreign concept for the dressed particle theory.

The 2-point function is the integral kernel of the inner product in the renormalized 1-particle Hilbert space (which exists since a single electron does not scatter), hence can be compared with the inner product in the dressed particle theory. They do not agree.

As for the non-equivalence to QED, I can argue that one can make the dressed particle theory as close to QED as one wishes. For example one quick and dirty way to get the dressed Hamiltonian is simply to fit its interactions to the S-matrix calculated at the specific loop order in QED.

You claimed that the renormalized QED has a finite cutoff-independent Hamiltonian.

This is assuming that QED exists as a Wightman field theory. (You need to assume as well that QED exists since your symmetry arguments depend on it...)

However this is not what textbooks say. You can find the Lagrangian of QED in eqs. (11.1.6) - (11.1.9) of Weinberg's vol. 1 and in eq. (10.38) of Peskin & Schroeder. (Obviously, the QED Hamiltonian $H_{QED}$ has a similar structure.)

This is only the approximate, cutoff-dependent unrenormalized Hamiltonian in Fock space describing a noncovariant approximation of QED; without cutoff the Hamiltonian and the other generators satisfy formally the Poincare commutation relations, but mathematically these expressions are meaningless. The limiting Hamiltonian of true QED is an operator on the renormalized Hilbert space, and cannot given in terms of Fock space but in terms of the Wightman reconstruction theorem, as the infinitesimal generator of the translations on the constructed states.

Now, the dressing approach says: I would like to make a unitary transformation of the QED Hamiltonian
$H_d = e^{i\Phi} H_{QED} e^{-i \Phi}$

which is a cutoff-dependent approximation to QED only. No Poincare invariance there; the generators don't satisfy exact Poincare relations.

where the operator $\Phi$ is chosen to satisfy 3 conditions:

1. It is Hermitian in each perturbation order. This means that the dressing transformation $e^{i\Phi}$ is unitary in each perturbation order, thus your objection about the non-unitarity is not valid.
2. It is sufficiently smooth. Then one can prove that the S-matrix computed with $H_d$ is exactly the same as the S-matrix computed with $H_{QED}$
3. It cancels order-by-order certain "bad" terms in $H_{QED}$. The "bad" terms are those which act non-trivially on the vacuum and 1-particle states. They are the ones responsible for self-interactions and ultimately for the divergences in the traditional QED.

At the end of this procedure one obtains the dressed Hamiltonian $H_d$ without self-interactions.

Well, you proceed formally without any consideration of limits, thus you only prove formal relations, not true ones. Note that in general, limits are not interchangeable, which is one of the reasons why subtle issues like the existence of QED are technically so difficult.

This Hamiltonian makes exactly the same predictions as $H_{QED}$ about scattering and bound state energies.

No. It makes exactly the same noncovariant predictions as the noncovariant approximations used at the start of the procedure. You don't even try to analyze the limit. Indeed, the limit does not exist, by Haag's theorem.

 

[bhobba]

Abdus Salam was writing before the more rigorous understanding of QFTs we possesses today. Renormalization does result in well-defined Hamiltonians that is a mathematically established fact.

Indeed. Interestingly as was shown by Hawking Zeta Function renormalization is equivalent, and in some cases maybe better than dimensional regularization. What is more, no counter-terms are required and it's basis is a mathematically rigorous and a very well understood area of math called analytic continuation:
http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.570.4579&rep=rep1&type=pdf
Evidently something similar was tried by Schwinger but he couldn't make it work.

Thanks
Bill

 

[A. Neumaier]

There are many works in dressed/clothed particle theory. They usually pay tribute to the seminal paper

O. W. Greenberg, S. S. Schweber, "Clothed particle operators in simple models of quantum field theory", Nuovo Cim. 8 (1958), 378.

You can use Google Scholar to find all citations of this work. Currently this search shows 113 results.

Why don't you say that the best exposition is in your [meomepuk = Eugene Stefanovich] 3 volume treatise on quantum electrodynamics ? I think this is a valid statement. Your treatise shows both the potential and the limitations of the dressed particle approach.

Thank you for the generous introduction of my humble book.

The interesting part about your books is the extent to which the dressed particle approach can be made to work, as a (though somewhat pedestrian) application of what nowadays is usually called similarity renormalization. (The linked Google scholar search for this turns up over 1400 results.)

The problematic part of your books is that they lack a self-critical attitude and do not discuss the (to experts obvious) limitations of the approach, but sell them as nonexistent or even progress:

The superluminal effects are not bugs, but unavoidable features of the dressed particles approach.

4. This Hamiltonian is relativistically covariant: there exists a corresponding interacting boost operator such that all commutators of the Poincare Lie algebra are satisfied.

At the end of this procedure one obtains the dressed Hamiltonian $H_d$ without self-interactions. This Hamiltonian makes exactly the same predictions as $H_{QED}$ about scattering and bound state energies.

No. It makes exactly the same noncovariant predictions as the noncovariant approximations used at the start of the procedure. You don't even try to analyze the limit. Indeed, the limit does not exist, by Haag's theorem.

And, most sadly, you seem to be de facto uninterested to learn what is outside your horizon:

Now, you are saying that the "proper" field-based QED does not use the ideas of particles, their creation/annihilation operators and their Fock space, when interacting systems (such as the hydrogen atom) are involved. On the other hand, you claim that this theory possesses a finite well-defined cutoff-independent Hamiltonian. I am really interested in learning how this Hamiltonian looks like and how one can extract from it useful info, like the energy spectrum, lifetimes, wave functions, etc?

Ten years ago I went through all of this with you, giving you the relevant papers, links etc that mathematically prove these facts and how Fock space and creation and annihilation operators will not work and explaining every point in detail. I'm not repeating it.

 

[Geonaut]

@bhobba that's interesting, but I lost hope once I read the passage "This form of re-calculate a series has been widely used in Physics, one of the most famous examples being the calculation of the force due to “Casimir Effect” using the Zeta regularized value zeta (-3)=-1/120, or as another example, the value zeta(-1)=-1/12 for the divergent sum 1+2+3+4+5+6+7+…. That appears in theoretical physics."

I've seen people use the zeta function to claim that the sum of all positive integers is equal to -1/12, but it's my understanding that this is actually incorrect. It seems this paper does something similar at first glance, and if this paper uses that same "form of re-calculate a series" then it should contain the same error. With that in mind I stopped reading it, but if this isn't the case then please let me know.

 

[A. Neumaier]

I've seen people use the zeta function to claim that the sum of all positive integers is equal to -1/12, but it's my understanding that this is actually incorrect.

This caricature is of course incorrect, but the sum of $kx^k$ for all positive integers defines an analytic function with the value of -1/12 at x=1 after removal of the singularity, and its Taylor expansion around zero at x=1 agrees with the series you mention. Something similar happens in all cases where the series of interest is a special case of a power series expansion of an analytic function, and then one can use the method to find the intended sum, though the literal sum diverges. (Note that the singular terms in individual infinite contributions to Feynman diagrams cancel by design of any renormalization procedure, so dropping them in each term of the sum is justified!)

 

[Geonaut]

@A. Neumaier I wish I was as educated as you, I'm not sure what you mean by "intended sum" and "literal sum" (Edit: Oh, OK). More importantly, are you saying that you think that paper provides a legitimate technique?

 

[Geonaut]

@A. Neumaier regarding everything else, I only have a Bachelor's in physics at the moment, but I understood enough to find your arguments to be very persuasive. I admire your willingness to go to the lengths that you have here to contribute to sites like this one for the sake of spreading the truth.

The fact that I didn't understand everything makes me want to learn more. You have me asking myself the question: What are all of the ways that we can test the limitations of a quantum theory? I know of many, but clearly, I'm not aware of all of them. Which is very worrisome, not knowing all of these methods can determine whether or not someone spends years confused about an important topic. Unfortunately, it might be impossible to adequately discuss such a broad question on Physics Forums, but maybe it would be appropriate to make a new post regarding that question to avoid going too far off topic. Do you think this question is too broad for Physics Forums?

 

[Geonaut]

Fyi, if you don't ignore me at some point then I will never shut up. I have a divergent number of questions about physics (I also have subtle jokes), and a desire to avoid making myself seem uneducated in public in order to find the answers to them is nothing in comparison to my desire to find those answers.

 

[bhobba]

@bhobba I've seen people use the zeta function to claim that the sum of all positive integers is equal to -1/12, but it's my understanding that this is actually incorrect. It seems this paper does something similar at first glance, and if this paper uses that same "form of re-calculate a series" then it should contain the same error. With that in mind I stopped reading it, but if this isn't the case then please let me know./

Dr Neuaimer explained it correctly. If its true or not is often argued about. Its simply how you define infinite sums. If you replace the terms as the special case of a series where the terms involve a variable, and interpret the variable as a complex number, you can often sum it for some values of that variable - let's call it x. Then you can use analytic continuation to define the sum for a much larger range of values of x. Then you substitute the values you started with and viola - you get an answer. There are a number of ways of doing it - my favorite is Borel Summation because explaining it is so simple (it will not work on the Zeta Function though). There other ways to sum it - the general one I like best is Ramanujan Summation - despite Hardy waning against it - he wrote a textbook on summing divergent series.

As an example let's look at Borel Summation. ∑an = ∑an*n!/n! = ∑an*∫(t^n)*e^-t/n!
= ∑∫(an*t^n)*e^-t/n! where we have used n! = ∫(t^n)*e^-t.

The above is rigorously correct but sometimes in physics and applied math we get sloppy and reverse integrals and sums without showing you can do it. This gives ∫∑(an*t^n)*e^-t/n! and is called the Borel Sum. It turns out for series that converge normally you can reverse the sum and integral. So we have a more general summation method allowing the summation of things like 1-1+1-1... A deeper analysis than I will give shows its really a form of analytic continuation. The great mathematician, Borel, when an unknown young man, discovered that his summation method gave the 'right' answer for many classical divergent series. He decided to make a pilgrimage to Stockholm to see Mittag-Leffler, who was the recognized lord of complex analysis. Mittag-Leffler listened politely to what Borel had to say and then, placing his hand upon the complete works by Weierstrass, his teacher, he said in Latin, 'The Master forbids it'. So even experts get fooled by this. Mittag-Leffler eventually came around and cam up with his own method, Mittag-Leffler summation.

We are very lucky having Dr Neumaier and others like him post here - you will find you will look back after a while and be surprised at what you have learned.

Thanks
Bill

 

[Geonaut]

The fact that we're talking about debatable math is crazy to me. Math is supposed to be the one thing that is black and white, but I guess we've reached the realm of theoretical math where that's no longer the case.

What is more, no counter-terms are required

This is very interesting, honestly, I expected someone to shoot this down. Now I'm extremely curious about this, and I'm reading the paper very closely. If this is legit then what happens to renormalization conditions? I've never been a fan of their presence in QFT, and if this means that we no longer need them then that would be interesting. That's currently what's going through my mind... I'm going to have to think about this some more. Thank you for your input Bill.