A Is renormalization the ideal solution?

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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

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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).

 
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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

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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.
 
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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

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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
 
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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?
 
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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

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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

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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 causal perturbation theory. 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.
 
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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

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Abdus Salam was writing before the more rigorous understanding of QFTs we possess 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

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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.
 
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Abdus Salam was writing before the more rigorous understanding of QFTs we possess 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

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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

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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.
 
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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.
 
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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?
 
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@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.
 
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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.
 
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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.
 
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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

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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

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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

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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.
 
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