I strongly doubt it too. I'm not aware of a single paper about lattice QED. Only lattice QCD is of course a standard method for several decades by now, and there are still serious problems to be solved (e.g., getting S-matrix elements, solving the finite-temperature case at finite baryo-chemical potential satisfactorily etc. etc., but indeed it's a very active field of research, while about lattice QED, I've not even seen a single paper).Demystifier said:So from the fact that nobody did it so far, you conclude that it cannot be done?
Well, as long as there don't exist some lattice perturbative results, we can't check, right?Demystifier said: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.
Only the construction problem for a QED satisfying the Wightman axioms or equivalents is wide open. The construction problem for a Poincare invariant QED at 1-loop that agrees with experiment was fully solved in 1948 and this solution was honored by a Nobel price for Feynman, Tomonaga and Schwinger. In fact, few physicists care about constructive QFT at all; it is a problem only for the small minority of mathematical physicists.atyy said:Of course not - I agree with you that the construction problem is wide open - and that is the point! [...] Since the construction problem is open, the claim that Poincare invariant QED exists is not substantiated.
This is the view of mathematical physicists. But for them, lattice QED is definitely not QED, and a mere effective theory is no theory at all. Thus to claim their view as the authoritative view completely defeats your goal to promote lattice QED.atyy said:Only when the construction problem is solved can one claim that Poincare invariant QED exists.
Wilsonian renormalization is virtually useless in QED. It is not responsible for any its successes. So not having this starting point is irrelevant in practice.atyy said: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.
Doubt what? It's technically more difficult, but I see no reason to expect that, if someone could solve the technical problems, one would not give results that agree with observations.vanhees71 said:I strongly doubt it too.
Don't take me for granted, but I think there are 1-loop lattice results which, for low energy phenomena, agree with standard 1-loop results. In any case, the book by Montvay and Munster is a very good reference. If you take some time to read it, you will see that the difference between perturbative lattice QFT and ordinary perturbative QFT can, to a large extent, reduce to the difference between Fourier sum and Fourier integral.vanhees71 said:Well, as long as there don't exist some lattice perturbative results, we can't check, right?
No, but for two other reasons:Demystifier said:So from the fact that nobody did it so far, you conclude that it cannot be done?
When I see a big picture that this calculation entails, I do.vanhees71 said:Well, I don't believe a calculation that doesn't exist.
There is a little trickle of them. Nothing at all in their results is inviting to work on the topic.vanhees71 said:I'm not aware of a single paper about lattice QED.
Of the 442 pages of main text excluding references, only 12 pages are on lattice QED (Section 4.5), and much of it summarizes results from continuum QED (e.g., Pauli-Villars regularization on p.227). The last 2 1/2 pages (pp.228-230) contain information on nonperturbative studies of lattice QED in very simplified scenarios (e.g., the quenched approximation). They conclude by stating thatDemystifier said:the book by Montvay and Munster is a very good reference.
This conclusion by experts on the subject is diametrically opposite to atyy's claims that lattice QED is a good approximation of QED, or a good starting point for low energy QED.Montvay and Munster said:the fermion always decouples in the continuum limit, leading to a trivial bosonic theory. These non-perturbative studies of renormalization suggest that the continuum limit of lattice QED is trivial.
Please share your thinking in enough detail that others can check its cogency.Demystifier said:I think there are 1-loop lattice results which, for low energy phenomena, agree with standard 1-loop results.
A. Neumaier said:This has nowhere be done; it is wishful thinking.
vanhees71 said: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.
vanhees71 said:Nobody claims that QED is valid at all energy scales (nor any other QFT for that matter). It is very likely, although not strictly proven, that QED has a Landau pole, where it breaks down, but it's at very high energies irrelevant for all practical purposes. Whether there exists some more comprehensive theory than relativistic QFT that is valid at all energy scales, is unknown today.
For the so modified interpretation of QFTs as low-energy effective theories, it is totally irrelevant which regularization you use. The regularized theory is just a step in the calculation for the S-matrix elements in terms of well-defined finite renormalized quantities.
It's not even necessary to use regularization at all. You can as well directly renormalize using the BPHZ formalism, it's only technical sometimes a bit less convenient than modern regularization techniques like dim reg or the heat-kernel methods or even Pauli-Villars (in some cases where dim. reg is not easily applicable). Finally, all that counts and all that's comparable to experiment are the results of renormalized perturbative QED.
Again, the claim is that it is a good approximation at large distances. As I already said, I don't give a damn what any theory without quantum gravity say in the continuum limit.A. Neumaier said:This conclusion by experts on the subject is diametrically opposite to atyy's claims that lattice QED is a good approximation of QED, or a good starting point for low energy QED.
Why bother? Even if I find the reference (which would take some time because I'm not an expert), this will not change your general opinion.A. Neumaier said:Please share your thinking in enough detail that others can check its cogency.
True, but there is also a general feeling in the community that this does not affect predictions for long distance phenomena. The point of effective theories (lattice QFT, continuum Standard Model without quantum gravity, ...) is to study long distance phenomena. Those who worry much about the continuum limit are missing the point.A. Neumaier said:1. There is the general feeling in the community that the continuum limit of lattice QED is likely to be trivial
You are talking about Monte-Carlo simulations (of lattice QFT) at a computer, but it should be distinguished from lattice QFT as such.A. Neumaier said:2. What has been done anywhere in lattice QFT has always lead to very low accuracy (a few percent at best). Accuracy increases like ##N^{-1/2}## with the number of lattice points per dimension, and work increases like ##N^7##. Extrapolating to what is needed to get 10 significant did accuracy would require of the order of ##10^{20}## lattice points per dimension. Thus of the order of ##10^{80}## lattice points and of the order of ##10^{140}## floating point operations would be needed. This doesn't prove that it cannot be done. But surely it cannot be done during the time any of those participating in this discussion will live.
All formulas of the renormalized, textbook QED, existing since 1948 as witnessed by a Nobel prize given for its discovery, are fully Poincare invariant at every loop order, and make excellent predictions. Thus your assertion is a purist's claim (who apparently thinks that only satisfaction of the Wightman axioms may define what existence means) , irrelevant to the success of QFT since 1948.atyy said:It is no more wishful that asserting that Poincare invariant QED exists - no one has shown that it exists at all energies, and if it does not exist at all energies, then it cannot be Poincare invariant.
But unlike the first feeling, which is corroborated by quite a number of studies of QED and related theories whose lattice version has a Landau pole (and I pointed to one), you cannot point to a single paper where the second feeling is corroborated by making highly accurate predictions form lattice QED.Demystifier said:there is also a general feeling in the community that this does not affect predictions for long distance phenomena.
You should bother for the sake of the readers of this thread, who are surely interested whether your claims are substantiated or just irrelevant private thoughts. I write for the readers, not for the participants in the discussion - the latter rarely change their point of view due to the arguments.Demystifier said:Why bother? Even if I find the reference (which would take some time because I'm not an expert), this will not change your general opinion.
The whole point of a physical theory is to make good and testable predictions. A theory that makes predictions the content of which one can never find out is untestable and not worth its salt. Lattice QFT was created to contribute constructively to QCD through lattice computations, not to make empty statements (where those who make it don not even bother to justify them by references) about the power of a lattice QED with which nobody will ever be able to make competitive predictions! The paucity of papers on lattice QED is ample evidence of this fact.Demystifier said:You are talking about Monte-Carlo simulations (of lattice QFT) at a computer, but it should be distinguished from lattice QFT as such.
Let them speak for themselves!A. Neumaier said:You should bother for the sake of the readers of this thread,
Sure, and you keep mixing these two, while not clarifying atyy's query about what you mean by Poincare invariant in this context. The Poincare invariance you are talking about seems to refer to the global(spacetime) symmetry in the rigorous QED that is unknown whether it exists but it is confusely mixed with the local(at every loop order, at each energy) Poincare invariance that make to 10 or more decimal precision predictions that you mention here:A. Neumaier said:We mainly discuss textbook QED, which is a Poincare invariant theory at few loops,[...] and rigorous QED, of which it is unknown whether it exists.
A. Neumaier said:All formulas of the renormalized, textbook QED, existing since 1948 as witnessed by a Nobel prize given for its discovery, are fully Poincare invariant at every loop order, and make excellent predictions.
He didn't query about this meaning.RockyMarciano said:atyy's query about what you mean by Poincare invariant
He actually assumes the same meaning for Poincare invariance and yet he asks:A. Neumaier said:He didn't query about this meaning.
The meaning you refer to below is only compatible with the existence of the rigorous QED(i.e. at all energies).atyy said:If we don't know whether QED exists at all energies, then doesn't it mean we don't know whether it is truly Poincare invariant (since an energy cutoff spoils Poincare invariance)?
Perhaps is not so pointless because the local Poincare(algebra) invariance guaranteed loop by loop (therefore already at one loop) doesn't inmediately generalize to the full group global invariance, you need to assume the rigorous asymptotically valid QED at all energies that you admit might not exist. If it doesn't exist the exponential map from the algebra to the group is not guaranteed to be injective.The meaning of Poincare invariance is everywhere the same, with a choice of 4 different Poincare groups which have the same Lie algebra. In case of QED, the biggest (the universal cover) applies, as anyone can check who understands 1-loop QED. Nothing in my or atyy's arguments depends on whether one has a more restricted notion of Poincare group, considering only the connected component of unity. Thus is is pointless to dwell on these details.
1-loop QED exists rigorously at all energies. It is invariant under exactly the same Poincare group as the full theory. Only the accuracy is not the same at high energies as that of fully nonperturbative QED, whose existence is in doubt. But it is already much better than anything what lattice QED has to offer.RockyMarciano said:The meaning you refer to below is only compatible with the existence of the rigorous QED(i.e. at all energies).
You mean that it is compatible with approximations at other energies, 1-loop is a low energy approximation, but what I'm saying is that the fact of obtaining finite accurate predictions at different orders doesn't imply Poincare group invariance at all orders, because that precisely what you need a rigorous QED for. The Landau pole is just a possible obstruction to that, there are others.A. Neumaier said:1-loop QED exists rigorously at all energies.
No, only under the local identity component algebra, unless you assume a Minkowski spacetime from the beginning, but then why would you say the existence of a rigorous QED is in doubt. Only if there was a rigorous QED in the first place we could assume the full symmetry of Minkowski spacetime as the exact one for the physical theory and also derive it from the algebra using the exponential map.It is invariant under exactly the same Poincare group as the full theory.
Of course, I'm not discussing this.But it is already much better than anything what lattice QED has to offer.
Well, that's close to the question I've been trying to pose. Admitting the possibility that if the input is not rigorously proved, we must admit the possibility that the output is not globally Poincare invariant and belongs to a different input.dextercioby said:This debate might look interesting, but is in fact pointless. QED as a quantum field theory in 4D Minkowski spacetime has as input parameters the Lagrangian action of the coupled Dirac field (psi, psi-bar) and the electromagnetic field (A). This is obviously Poincare invariant. As in any other Quantum Theory, the question poses as: the input of this theory is the Lagrangian action, but what are the states and observables of this theory (output) ? Are they truly Poincare invariant ?
All of my arguments in this thread are good and don't need to be repeated.RockyMarciano said:I've got the feeling that you have run out of good arguments
Well, all I am saying (and I think also Arnold is saying) is that lattice QED is not the right formulation of the low-energy (long-distance) effective theory described by QED, leading to results that agree to experiment at 12 significant digits, but that's good old perturbative renormalized QED, formulated around 1948 and finally proven to exist at all orders as a Dyson renormalizable relativistic QFT in the mid 1960ies, when the problem of overlapping divergences was finally solved by the work of BPHZ.Demystifier said:True, but there is also a general feeling in the community that this does not affect predictions for long distance phenomena. The point of effective theories (lattice QFT, continuum Standard Model without quantum gravity, ...) is to study long distance phenomena. Those who worry much about the continuum limit are missing the point.
And I am saying that these 12 significant digits scan also be obtained by perturbative renormalied QED at the lattice. Nobody did it explicitly, but since Feynman rules for lattice QED are essentially the same (except that we have sum over momenta instead integral over momenta in loop diagrams), nobody expects significant deviations for g-2.vanhees71 said:Well, all I am saying (and I think also Arnold is saying) is that lattice QED is not the right formulation of the low-energy (long-distance) effective theory described by QED, leading to results that agree to experiment at 12 significant digits, but that's good old perturbative renormalized QED, formulated around 1948 and finally proven to exist at all orders as a Dyson renormalizable relativistic QFT in the mid 1960ies, when the problem of overlapping divergences was finally solved by the work of BPHZ.