A Does QED Originate from Non-Relativistic Systems?

[atyy]

It is just a matter of belief, not of demonstration. Whereas it is a well-known fact that QED is Poincare invariant in each finite loop approximation. and is demonstated to have a predictive power far beyond QCD. What happens at energies above the Planck scale is completely irrelevant for QED; it is a matter for the future unified theory.

Perturbative QED does not suffer from the Landau pole; only a nonperturbative version possibly does, if one attempts to construct the theory using lattices (!) or using a cutoff. The Landau pole invalidates a construction only if the (lattice or energy) cutoff has to move through the pole in order to provide a covariant limit.

On the other hand, while the causal, covariant construction of QED also has a Landau pole, Landau's argument that a Landau pole invalidates perturbation theory no longer applies to this version. In the causal construction there are no cutoffs that must be sent to zero or infinity and move through the pole. The pole is only in the choice of the renormalization point (or mass $M$). But in the exact theory, the theory is completely independent of this renormalization point; so it can be chosen at low energy without invalidating the construction!

The associated Callan-Symanzik equation shows how the observables depend on the chosen renormalization mass $M$, giving an identical theory - apart from truncation errors, which are of course small only if $M$ is of the order of the energies at which predictions are made. Thus for any range of $M$ for which the Callan-Symanzik equation is solvable, one gets the same theory. The Landau pole found (nonrigorously) in causal QED in the Callan-Symanzik equation only means that one cannot connect the theory defined by a superhigh renormalization point irrelevant for the physics of the universe to the theory defined by a renormalization point below the Landau pole. This is a harmless situation.

It is quite different from Landau's argument that the cutoff cannot be removed because on the way from a small cutoff energy to an infinite cutoff energy perturbation theory becomes invalid since the terms become infinite when the cutoff passes the pole. This means that perturbation theory fails close to this cutoff. Thus any approximate construction with cutoff featuring a Landau pole cannot be made to approximate the covariant QED to arbitrary precision. In particular, the Landau pole believed to exist in lattice QED is of this fatal kind and proves (at this level of rigor) that lattice QED can never approach the covariant QED, hence is a fake theory. Your insistence on the Landau pole defeats even your basic goal!

So if QED does not even exist at energies above the Planck scale, how can it be Poincare invariant?

There is no need for lattice QED to reproduce QED to arbitrary precision, since QED does not have arbitrary precision as it is only a low energy effective theory.

There is of course a "Landau pole" in lattice QED, because the point is it acknowledges we only need a low energy effective theory. The point of the lattice is to provide a conceptually secure starting point for Wilsonian renormalization.

What you are assuming is that Wilsonian renormalization starts from a UV complete Poincare invariant QFT, whose low energy effective theory is covariant perturbative QED. That is fine. However, since we don't know of any such theory at the moment, it is just as good to start from lattice QED if one wants to start from a well-defined quantum theory.

 

[Demystifier]

It can be viewed as a rigorous definition of approximations only.

All theories in physics are approximations. If we are condemned to work only with approximations, it helps when, at least, the approximation can be defined rigorously.

 

[vanhees71]

Why be sloppy and leave room for confusing people when it is so easy being precise? In this context Poincare invariant refers to invariance under the proper orthochronous component of the Lorentz group and infinitesimal translations.

This is important since indeed the standard model is invariant only under the continuous part of the Poincare group, which is a semidirect product of the proper orthochronous Lorentz group with the group of translations. The weak interaction breaks P, T, and when extended by the charge-conjugation transformation also CP. As local relativistic QFT, of course CPT is a symmetry, and so far there's indeed no empirical evidence that CPT is broken anywhere.

Concerning the rest of the debate, from a strict mathematical view there's no conclusive answer to the question whether QED in (1+3) spacetime dimensions exists. It's quite likely that it doesn't exist because of a quite probable Landau pole. Arnold is right in saying that QED exists at any order of perturbation theory in the sense that you can define the renormalized effective action (and thus the proper vertex functions) truncated at any finite loop order. However, already the propagators, i.e., the resummation of the Dyson equation to any order is plagued by a Landau pole, but in the case of QED at very high energies. In this sense QED is indeed only an effective theory valid for low enough energies. However this energy limit is so high that it doesn't play a practical role at all for today's particle physics.

Of course, from a theoretical point of view, it's well justified to seek for a better theory than the Standard Model. Unfortunately so far, there's no success in this endeavor, neither from the axiomatic-QFT side nor from the phenomenological side since at the moment all the "signals indicating physics beyond the standard model" are disproven at the $5 \sigma$ or higher significance level.

 

[atyy]

This is important since indeed the standard model is invariant only under the continuous part of the Poincare group, which is a semidirect product of the proper orthochronous Lorentz group with the group of translations. The weak interaction breaks P, T, and when extended by the charge-conjugation transformation also CP. As local relativistic QFT, of course CPT is a symmetry, and so far there's indeed no empirical evidence that CPT is broken anywhere.

Concerning the rest of the debate, from a strict mathematical view there's no conclusive answer to the question whether QED in (1+3) spacetime dimensions exists. It's quite likely that it doesn't exist because of a quite probable Landau pole. Arnold is right in saying that QED exists at any order of perturbation theory in the sense that you can define the renormalized effective action (and thus the proper vertex functions) truncated at any finite loop order. However, already the propagators, i.e., the resummation of the Dyson equation to any order is plagued by a Landau pole, but in the case of QED at very high energies. In this sense QED is indeed only an effective theory valid for low enough energies. However this energy limit is so high that it doesn't play a practical role at all for today's particle physics.

Of course, from a theoretical point of view, it's well justified to seek for a better theory than the Standard Model. Unfortunately so far, there's no success in this endeavor, neither from the axiomatic-QFT side nor from the phenomenological side since at the moment all the "signals indicating physics beyond the standard model" are disproven at the $5 \sigma$ or higher significance level.

But that is not really what is being discussed. The question being discussed is:

In the Wilsonian viewpoint, is lattice QED at fine but finite lattice spacing a conceptually adequate starting point for obtaining perturbative Poincare invariant QED as a low energy effective QFT?

 

[Demystifier]

But that is not really what is being discussed. The question being discussed is:

In the Wilsonian viewpoint, is lattice QED at fine but finite lattice spacing a conceptually adequate starting point for obtaining perturbative Poincare invariant QED as a low energy effective QFT?

Exactly! Or to put it in different words, the question is not whether lattice QED is more fundamental than Poincare invariant QED. The question is whether Poincare invariant QED at low energies can be derived from lattice QED in the low energy limit. Just because theory A can be derived from theory B does not imply that theory B is more fundamental than theory A.

Let me use an analogy. Fluid mechanics can be derived from a naive theory of atoms, in which atoms are hard balls. But it does not mean that such a naive theory of atoms is more fundamental than fluid mechanics.

U funny thing about QFT is that lattice QFT can be obtained as an approximation of continuous QFT, but also continuous QFT can be obtained as an approximation of lattice QFT. A string theorist would conjecture that this implies the existence of a mysterious M-theory, in which both QFT theories are special limits of the fundamental M-theory.

 

[A. Neumaier]

he is taking for granted that the million dollar question mentioned in the link has already been affirmatively answered.

No. The latter is about proving rigorously the existence of quantum Yang-Mills theory and is still an open problem. This problem has nothing to do with QED, which is the topic of the present discussion.

 

[A. Neumaier]

If full Poincare symmetry is required instead of the restricted one, then there would be no room for neutrinos and weak interactions in the Standard Model.

True but irrelevant for QED, which (in the version under discussion) by definition is only about electromagnetic fields, electrons, and positrons. We are not discussing the standard model.

 

[A. Neumaier]

The question is whether Poincare invariant QED at low energies can be derived from lattice QED in the low energy limit.

This was not the original question.

But even this question has a negative answer. Getting Poincare invariance requires the limit where the lattice spacing goes to zero, which is incurably problematic because of a Landau pole. The literature indicates that this limit gives a free theory only. Thus lattice QED is probably unrelated to true Poincare invariant QED as discussed in the textbooks. The whole question of triviality is discussed in detail on PhysicsOverflow.

 

[A. Neumaier]

if QED does not even exist at energies above the Planck scale, how can it be Poincare invariant?

Because the hypothesis of your question is invalid. Your argument from belief means nothing; it is a mere belief.

The question of the existence of QED is widely open, even the question of existence of $\Phi^4$ theory (see Section 8 of this paper, which discusses the state of the art from a rigorous point of view), for which the triviality arguments (i.e., the impossibility to construct it as a scaling limit from the corresponding lattice theory) are overwhelming.

The causal construction of QED avoids all these triviality arguments as it doesn't construct QED as a limit of theories with a cutoff but directly from a mathematical characterization by means of covariant axioms by Bogoliubov - not the Wightman axioms. Each loop order is valid at all energies. The only unsolved question is how to resum the series to make the construction nonperturbative. There is not the slightest argument indicating that this is impossible. And as mentioned, the constructive question is wide open.

 

[Demystifier]

But even this question has a negative answer. Getting Poincare invariance requires the limit where the lattice spacing goes to zero ...

Note that my question contains "at low energies" caveat. For physics at low energies, it is not relevant to know what happens in the limit of zero lattice spacing.

Frankly, I don't give a damn how any QFT without quantum gravity behaves at very small distances (Planck distance and less) because I believe that, at such small distances, QFT without quantum gravity has nothing to do with reality.

 

[RockyMarciano]

No. The latter is about proving rigorously the existence of quantum Yang-Mills theory and is still an open problem. This problem has nothing to do with QED, which is the topic of the present discussion.

True but irrelevant for QED, which (in the version under discussion) by definition is only about electromagnetic fields, electrons, and positrons. We are not discussing the standard model.

You are permanently shifting your meaning of QED, but the posts are there to be read. We are discussing the theory that gives extremely accurate predictions, in your own words "in each finite loop approximation". This is not asymptotic and divergent QED, which you seem to refer to by QED at times to deliberately confuse the discussion, but the renormalized one that is indeed part of the predictions of the standard model (wich is indeed Yang-Mills), in other words, in the absence of a non-perturbative QED, perturbative QED rests on the renormalizable gauge quantum field theory $U(1)×SU(2)×SU(3)$, the mathematical base of the standard model . This is the topic of the present discussion since it is about accurate predictions in renormalized perturbative QED versus nonrelativistic lattice QED. The actual physics cannot be separated in independent compartments with QED separated from QCD or weak interaction theories, at enough precision different interactions concur in a given high energy particle physics process.

 

[vanhees71]

But that is not really what is being discussed. The question being discussed is:

In the Wilsonian viewpoint, is lattice QED at fine but finite lattice spacing a conceptually adequate starting point for obtaining perturbative Poincare invariant QED as a low energy effective QFT?

As I said in my previous posting in this thread, lQFT is a regularized version of continuum QFT, not more and not less.

 

[RockyMarciano]

This is important since indeed the standard model is invariant only under the continuous part of the Poincare group, which is a semidirect product of the proper orthochronous Lorentz group with the group of translations.

Exactly, mathematically is called the Poincare algebra, althought it is common among physicists to ignore the difference between Lie groups and Lie algebras.

 

[RockyMarciano]

What is a spacetime anomaly?

For instance the chiral anomaly leading to baryonic charge non-conservation and to violations of lepton number conservation.

 

[A. Neumaier]

For physics at low energies, it is not relevant to know what happens in the limit of zero lattice spacing.

But to get the predictions of the anomalous magnetic moment (a low energy quantity) agree to experimental accuracy one needs already the covariant version of QED. And to get the covariant version one needs a continuum limit. But the continuum limit of lattice QED is probably not covariant QED but as free theory.

Lattice QED as it exists is an extremely poor approximation to real QED. It has not given a single accurate contribution to the prediction of low energy physics.

 

[A. Neumaier]

You are permanently shifting your meaning of QED, but the posts are there to be read. We are discussing the theory that gives extremely accurate predictions, in your own words "in each finite loop approximation". This is not asymptotic and divergent QED, which you seem to refer to by QED at times to deliberately confuse the discussion, but the renormalized one that is indeed part of the predictions of the standard model (wich is indeed Yang-Mills).

This is the last time I discuss your confusion related to this thread.

The title and the OP define what is discussed in this thread. We mainly discuss textbook QED, which is a Poincare invariant theory at few loops, lattice QED, which is a nonrelativistic caricature of true QED (unlike lattice QCD which because of asymptotic freedom has a respectable status as an approximation to true QCD), and rigorous QED, of which it is unknown whether it exists. We discuss other theories such as the standard model or QCD not as topic in itself, but only only to obtain contrasting statements. (The standard model makes slightly different predictions than QED since it incorporates corrections from the interaction with the other fields.)

 

[vanhees71]

Exactly, mathematically is called the Poincare algebra, althought it is common among physicists to ignore the difference between Lie groups and Lie algebras.

Of course, usually physicists treat the Lie algebra, but the exponential map lifts it to unitary representations of the group. Since for QT it's sufficient to have unitary ray representations, instead of the classical propoer orthochronous Lorentz group you consider the covering group, which means to substitute $\mathrm{SL}(2,\mathbb{C})$ instead of the $\mathrm{SO}(1,3)^{\uparrow}$. There are no non-trivial central charges in the Poincare algebra. So at the end everything is deduced from unitary irreps. of the covering group.

 

[atyy]

As I said in my previous posting in this thread, lQFT is a regularized version of continuum QFT, not more and not less.

In other words, you agree that lQFT is a good starting point for continuum QFT (without taking the lattice spacing to zero)?

 

[atyy]

Because the hypothesis of your question is invalid. Your argument from belief means nothing; it is a mere belief.

The question of the existence of QED is widely open, even the question of existence of $\Phi^4$ theory (see Section 8 of this paper, which discusses the state of the art from a rigorous point of view), for which the triviality arguments (i.e., the impossibility to construct it as a scaling limit from the corresponding lattice theory) are overwhelming.

The causal construction of QED avoids all these triviality arguments as it doesn't construct QED as a limit of theories with a cutoff but dirsectly from a mathematical characterization by means of covariant axioms by Bogoliubov - not the Wightman axioms. Each loop order is valid at all energies. The only unsolved question is how to resum the series to make the construction nonperturbative. There is not the slightest argument indicating that this is impossible. And as mentioned, the constructive question is wide open.

Of course not - I agree with you that the construction problem is wide open - and that is the point! Since the construction problem is wide open, we cannot at the moment use a truly Poincare invariant QED as a starting point for Wilsonian renormalization.

Since the construction problem is open, the claim that Poincare invariant QED exists is not substantiated. Only when the construction problem is solved can one claim that Poincare invariant QED exists.

 

[vanhees71]

In other words, you agree that lQFT is a good starting point for continuum QFT (without taking the lattice spacing to zero)?

Hm, I don't know, whether it helps much in the case of QED. In QCD it's of course a great way to explore QCD (both in vacuo as well as finite temperature) from a first-principles numerical approach. For QED, there seems not to be gained much compared to the standard perturbative methods. Particularly when you use the modern techniques as dim. reg. to regularize the theory and then renormalize it. The Wilsonian point of view of the renormalization group is indeed equivalent to the older techniques developed by Stueckelberg&Petermann, Gell-Mann and Low, et al. and they are also better suited for practical calculations using the RG method as a way to resum leading log contributions.

The modern functional renormalization-group approaches (aka Wetterich equation) are closer in spirit to the Wilsonian and right now a hot topic in thermal-QFT applications in heavy-ion theory to explore the phase diagram of strongly interacting matter.

 

[atyy]

Hm, I don't know, whether it helps much in the case of QED. In QCD it's of course a great way to explore QCD (both in vacuo as well as finite temperature) from a first-principles numerical approach. For QED, there seems not to be gained much compared to the standard perturbative methods. Particularly when you use the modern techniques as dim. reg. to regularize the theory and then renormalize it. The Wilsonian point of view of the renormalization group is indeed equivalent to the older techniques developed by Stueckelberg&Petermann, Gell-Mann and Low, et al. and they are also better suited for practical calculations using the RG method as a way to resum leading log contributions.

The modern functional renormalization-group approaches (aka Wetterich equation) are closer in spirit to the Wilsonian and right now a hot topic in thermal-QFT applications in heavy-ion theory to explore the phase diagram of strongly interacting matter.

Yes, but the idea is not to use lattice for practical calculation. The idea is to be able to define a finite quantum theory that at least conceptually leads to the usual covariant perturbative QED as a low energy effective theory. Since at present we don't know how to make a Poincare invariant QEF that exists at all energies, if we want a well-defined quantum theory from which to start, we have to go with a quantum theory with a high energy cutoff such as lattce QED.

 

[Demystifier]

But to get the predictions of the anomalous magnetic moment (a low energy quantity) agree to experimental accuracy one needs already the covariant version of QED.

I have no idea why do you think that non-covariant (lattice) version cannot give the anomalous magnetic moment which also agrees to experimental accuracy. Reference?

 

[A. Neumaier]

I don't know, whether it helps much in the case of QED. In QCD it's of course a great way to explore QCD (both in vacuo as well as finite temperature) from a first-principles numerical approach.

It does not, because unlike QCD which is asymptotically free and hence presumably has a good continuum limit, the continuum limit of lattice QED is unlikely to be nontrivial because of the Landau pole.

For QED, there seems not to be gained much compared to the standard perturbative methods.

There is nothing to even seemingly gain but a lot is actually lost. There a no good predictions at all of lattice QED.

The modern functional renormalization-group approaches (aka Wetterich equation) are closer in spirit to the Wilsonian

But the Wetterich equation and other exact RG equations are based on the continuum version and not the lattice.

 

[A. Neumaier]

a finite quantum theory that at least conceptually leads to the usual covariant perturbative QED as a low energy effective theory.

This has nowhere be done; it is wishful thinking.

 

[vanhees71]

Yes, but the idea is not to use lattice for practical calculation. The idea is to be able to define a finite quantum theory that at least conceptually leads to the usual covariant perturbative QED as a low energy effective theory. Since at present we don't know how to make a Poincare invariant QEF that exists at all energies, if we want a well-defined quantum theory from which to start, we have to go with a quantum theory with a high energy cutoff such as lattce QED.

As I said, I consider it also not very advantageous to use the lattice regularization to define QFT. There are better ways to regularize perturbative QFT like Pauli Villars or dim. reg. or a stupid old cutoff (in Euclidean QFT maybe the most simple idea). Conceptually regularization is even a bit unintuitive for my taste, and my great "aha feeling" came when I learned the BPHZ formalism, which uses always physical masses, couplings and fields order by order in PT (either loop-wise, i.e., in powers of $\hbar$ or in some coupling constant(s) or any other counting scheme appropriate to the definite model under investigation) and the counter-term approach. Then you simply subtract the divergences, introducing the renormalization scale etc. Then you get the RG equation from the independence of S-matrix elements on the choice of the renormalization scale (and even the renormalization scheme). This more conventional approaches are all well defined ways to work in renormalized perturbation theory with only finite quantities at any step of the calculation. For me it's way more intuitive than an artificial space-time lattice with quite nasty properties (fermion doublers and other kinds of artifacts, the lost Poincare, Lorentz, rotation invariance etc. ect.) and very delicate convergence issues if you want to get to the continuum limit. I'm also not familiar with the calculational techniques to evaluate even a simple one-loop diagram as vacuum polarization and electron self-energy on the lattice. With the other techniques it's not such a big deal. For a nice introduction also to calculational techniques see P. Ramon, QFT - A Modern Primer.

In my opinion, the best way to understand the quite involved ideas behind renormalized perturbation theory is to sit down and do calculations like the one-loop structure of QED to the end. For a beginner, I'd recommend to use the modern approach, using dim. reg. as regulator and then discussing various renormalization schemes like minimal subtraction and/or modified minimal subtraction, the usual on-shell scheme.

 

[A. Neumaier]

I have no idea why do you think that non-covariant (lattice) version cannot give the anomalous magnetic moment which also agrees to experimental accuracy. Reference?

A reference is needed for the claim that it can!

For all alternatives to established successful theories, the alternatives must be proved to be at least as good in order to be taken seriously. This is not the task of the mainstream physicist but that of the promotors of the alternative as a viable way to go! As long as no such proof is given, the alternative is left to rest in peace and ignored.

 

[vanhees71]

I have no idea why do you think that non-covariant (lattice) version cannot give the anomalous magnetic moment which also agrees to experimental accuracy. Reference?

To the contrary, I'd like to ask you for a reference, where even only the famous one-loop result by Schwinger has been achieved using the lattice regularization. I've no clue, how one would do such a calculation, let alone to get analytical results as with the standard continuum approaches like Pauli-Villars or (my favorite) dim. reg. as an intermediate step leading finally to a way to renormalize and get the physical result first derived by Schwinger (I think using a kind of Pauli-Villars reg., but I'd have to look that up, and Schwinger's papers are not easy to read ;-)).

 

[Demystifier]

A reference is needed for the claim that it can!

So from the fact that nobody did it so far, you conclude that it cannot be done?

 

[vanhees71]

But the Wetterich equation and other exact RG equations are based on the continuum version and not the lattice.

Of course. I'd never recommend to use the lattice approach except for where it is used extensively, namely in QCD as a way to numerically address the non-perturbative theory, where of course also the physics only comes out in the continuum limit, which to get is an art of its own. I think, in QED nobody ever has gotten anything via the lattice approach, not even the one-loop corrections, which are available even analytically with continuum-PT methods.

 

[Demystifier]

I'd like to ask you for a reference,

I'm not an expert in lattice QFT, so I cannot give a reference. I know that perturbative analytic calculations at lattice are more difficult than those in the continuum, but this in no way indicates that lattice perturbative results don't agree with observations.