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.
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.
A. Neumaier said: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.
Then we agree on that.vanhees71 said:Well, if it's not done, it's not done. Either it's computationally impossible with contemporary computer power or it's simply not interesting enough to spend monegy and ones time to do it in this way. I tend to think the latter is the case. I don't see any merit of such an endeavor. You rather use your compute capabilities for fruitful research, such as lattice QCD in vacuo and at finite temperature!
Well, there are some merits to think about the aximatic foundation of QFT, but FAPP it's pretty useless since so far this program has not solved any of the problems to define non-perturbative QFT of interacting particles in (1+3)D spacetime. On the other hand the non-rigorous techniques of renormalized perturbation theory of the continuum theory, which keeps everything Lorentz invariant at each stage of the calculation (which is a great simplification and advantage also from a purely calculational point of view), is a great success for the Standard Model as a whole (not only QED as one of its parts). That's why physicists use this formulation and not lattice methods, where they obviously are not applicable for this purpose.atyy said:Can you even show that 1-loop QED is a quantum theory? Typically, one needs to show that a QFT obeys the Wightman axioms or something equivalent.
Also, if your claim is true, then we would already have rigourous 3+1 interacting relativistic QFT. Yet as far as I know, there is not yet any successful construction of relativistic QFT, even in finite volume, for 3+1 dimensions. So I am very skeptical of this claim.
vanhees71 said:Well, there are some merits to think about the aximatic foundation of QFT, but FAPP it's pretty useless since so far this program has not solved any of the problems to define non-perturbative QFT of interacting particles in (1+3)D spacetime. On the other hand the non-rigorous techniques of renormalized perturbation theory of the continuum theory, which keeps everything Lorentz invariant at each stage of the calculation (which is a great simplification and advantage also from a purely calculational point of view), is a great success for the Standard Model as a whole (not only QED as one of its parts). That's why physicists use this formulation and not lattice methods, where they obviously are not applicable for this purpose.
vanhees71 said:Last but not least, only the renormalized parameters, finite when an eventually used intermediate regularization (and discretizing spacetime to a lattice is one such possibility to regularize the theory) is put to the physical limit (in the case of lattice regularization lattice spacing to 0 and the volume to entire ##\mathbb{R}^4##) are physical. They are defined by the renormalization scheme and have a clear physical meaning within this scheme. The values of the renormalized and finite parameters (wave-function normalizations, masses, and coupling constants) are fixed by appropriate measurements of cross sections needed to fix these values.
This is an extraordinary statement that needs proof to be credible.Demystifier said:I am saying that these 12 significant digits scan also be obtained by perturbative renormalied QED at the lattice.
Their simulations (which were still fairly long distance and low energy, otherwise it would have taken eons to simulate) showed that already at computationally accessible spacing lattice QED behaves already almost like a free theory. At very short distance/high energy, the Landau pole ruins the closeness to real QED. Hence lattice QED is unlikely to give sensible approximations at any distances or energies.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.
... where it is already very close to free, much closer as in the experiments reported by by Montvay and Munster ? How can you believe that coarsening a nearly free theory that doesn't resemble QED at all gives a good low energy approximation to QED. The differences between lattice QED and real QED grwo with the energy as the former becomes more and more free while the latter is not asymptotically free and its interactions become stronger and stronger!atyy said:The point is not to take lattice QED as the low energy theory - the point is to take it as the high energy theory (the starting point
If this is not quantum theory then why is the subject called quantum field theory? Quantum theory existed already for 30 years and was in a very healthy state before Wightman devised his axioms. So the latter are surely not necessary to recognize a quantum theory.atyy said:Can you even show that 1-loop QED is a quantum theory? Typically, one needs to show that a QFT obeys the Wightman axioms or something equivalent.
One can take the energy to infinity in all 1-loop formulas and gets mathematically meaningful covariant results.atyy said:no one has shown a way to take the energy to infinity for QED.
A. Neumaier said:If this is not quantum theory then why is the subject called quantum field theory? Quantum theory existed already for 30 years and was in a very healthy state before Wightman devised his axioms. So the latter are surely not necessary to recognize a quantum theory.
Not even your hallowed lattice QED would then be a quantum theory!
You had asked for the Hilbert space of QED and I explained it. One can do covariant CTP (closed time path) quantum mechanics at one loop and gets in this approximation everything one wants, including the approximations from which one can derive the hydromechanics of plasmas.
A. Neumaier said:... where it is already very close to free, much closer as in the experiments reported by by Montvay and Munster ? How can you believe that coarsening a nearly free theory that doesn't resemble QED at all gives a good low energy approximation to QED. The differences between lattice QED and real QED grwo with the energy as the former becomes more and more free while the latter is not asymptotically free and its interactions become stronger and stronger!
atyy said:Lattice QED is a quantum theory not because it satisfies the Wightman axioms, but because it obeys QM101 - Schroedinger equation, unitary time evolution, Hilbert space etc etc. To be clear, I mean Hamiltonian lattice field theory.
It is not sufficient to assert that something called QFT by convention is quantum. Satisfying the Wightman axioms is important because it shows that what one calls QFT is a relativistic quantum theory.
rubi said:To be fair, as far as I know, it is not even clear whether free lattice quantum Maxwell theory (expressed as a usual lattice gauge theory in terms of holonomies, instead of a vector field living on the vertices) approaches free quantum Maxwell theory in the continuum limit.
Also, it is possible to obtain perfectly rigorous versions of perturbatively defined QFT's in the framework of AQFT, by using states with values in the ring of formal power series instead of the complex numbers. Of course, the Hilbert space formulation would look a little bit odd, because it would be some generalization of a Hilbert space over the ring of formal power series instead of the complex numbers and it's not clear how such a theory should be interpreted.
But I would agree that the ultimate goal is of course to obtain well defined QFT's that satisfy the Wightman axioms.
It can be rigously formulated in the framework of perturbative AQFT (see Brunetti, Fredenhagen, ...), but that doesn't mean that we should be satisfied with it. Physical quantities in this framework are given by formal power series and it is not clear how they should be interpreted if they can't be summed. Of course, you can just take the first 137 orders and sum them up to obtain actual numbers that can be compared with experiments, but that's not very satisfying in my opinion. Not everything that can be formulated within some rigorous mathematical framework leads to an unproblematic physical theory.atyy said:So would you agree that 1 loop QED is a rigourous construction of a relativistic QED at all energies (finite volume is ok)?
rubi said:It can be rigously formulated in the framework of perturbative AQFT (see Brunetti, Fredenhagen, ...), but that doesn't mean that we should be satisfied with it. Physical quantities in this framework are given by formal power series and it is not clear how they should be interpreted if they can't be summed. Of course, you can just take the first 137 orders and sum them up to obtain actual numbers that can be compared with experiments, but that's not very satisfying in my opinion. Not everything that can be formulated within some rigorous mathematical framework leads to an unproblematic physical theory.
No, you need values in the ring of formal power series, which is why the interpretation is unclear unless they converge. And even if you take all orders, the QFT will be rigorous only in the sense of perturbative AQFT and not in the sense of Wightman. (137 was just an random example. We expect the power series of QED to start diverging somewhere near that order.) But perturbative AQFT is a rigorous framework and things like Poincare invariance can be discussed within it. The point is not such much whether the framework is rigorous, but rather whether we should really adopt it as a framework for physical QFT's. I consider it more as a compromise, because the real goal (a 4d Wightman QFT) is nowhere in sight.atyy said:And if we just take the first 1 or 2 or 137 orders, do we obtain a rigourous relativistic QFT?
Why is this the goal, as opposed to finding alternative axioms or framework compatible with the known facts?rubi said:the ultimate goal is of course to obtain well defined QFT's that satisfy the Wightman axioms.
rubi said:No, you need values in the ring of formal power series, which is why the interpretation is unclear unless they converge. And even if you take all orders, the QFT will be rigorous only in the sense of perturbative AQFT and not in the sense of Wightman. (137 was just an random example. We expect the power series of QED to start diverging somewhere near that order.) But perturbative AQFT is a rigorous framework and things like Poincare invariance can be discussed within it. The point is not such much whether the framework is rigorous, but rather whether we should really adopt it as a framework for physical QFT's. I consider it more as a compromise, because the real goal (a 4d Wightman QFT) is nowhere in sight.
Because the Wightman axioms are the most reasonable and intuitive axioms we could think of. Of course, many people have proposed alternative frameworks, but none of them seem as physically reasonable as the Wightman axioms.RockyMarciano said:Why is this the goal, as opposed to finding alternative axioms or framework compatible with the known facts?
A short introduction can be found here: https://arxiv.org/abs/1208.1428atyy said:Thanks. Although I can't understand the detailed mathematical reasoning, that makes intuitive sense to me. And yes, I always mean rigourous and physical (ie. formal power series that cannot be summed are not physical).
RockyMarciano said:Why is this the goal, as opposed to finding alternative axioms or framework compatible with the known facts?
Why should we keep only terms to one loop. In QED Kinoshita et al have done the calculation to 5 or even more loops. The asymptotic series tells you where to stop, namely at the order, where the apparent corrections get larger than the previous order. The proper vertex functions and thus also the connected Green's functions used to calculate approximations to S-matrix elements in perturbation theory are manifestly Lorentz covariant.atyy said:But in the non-rigourous point of view, the series is a perturbative series or asymptotic series to the true relativistic theory. If we only keep the terms to 1-loop - usually in the path integral formulation - do we know that we have a relativistic Hamiltonian quantum theory?
But the point of the Wilsonian view is that we don't have to take the lattice spacing to 0. Taking the lattice spacing to 0 is the same as taking the energy to infinity, and no one has shown a way to take the energy to infinity for QED.
It is sufficient. Neither Weinberg nor Peskin and Schroeder base their treatises on quantum field theory on the Wightman axioms; this is only the mathematical physicists convention, and the latter are a small minority among quantum physicists.atyy said:It is not sufficient to assert that something called QFT by convention is quantum.
2-loop QED is another real QED, also rigorously constructible. That you are not convinced doesn't matter; the Nobel committee was convinced enough, and the world followed.atyy said:As far as I can tell, your proposal for real QED is 1 loop QED which is claimed to be a rigourously constructed relativistic QFT in 3+1D.
To be precise, there is no known one. This is a small but very important difference.dextercioby said:there is no interacting QFT in 1+3 D satisfying Wightman axioms.
This is incorrect. One can make many compromises in the Wightman axioms and still reconstruct a Hilbert space from them. In the Wikipedia treatment linked to, Axiom W3 (local commutativity or microscopic causality) can be dropped and the Hilbert space construction still goes through. With this axiom dropped it is very easy to construct plenty of models.dextercioby said:the only way to formulate axioms of QFT so that they resemble the axioms of QM (the Dirac-von Neumann set including state vector collapse) is the Wightman way.
Sure, but that's my point, that the ultimate goal should be to come up with axioms that overcome some of those assumptions and still are compatible with the observations. It seems self-defeating to discard even in principle that somebody could come up with such axioms just because it is very hard or nobody has been able so far when historically that has been the way of scientific progress.dextercioby said:Because within the known assumptions of a non-dynamic special relativity-compatible background (i.e. Minkowski spacetime) and locality (finite no. of spacetime derivatives), the only way to formulate axioms of QFT so that they resemble the axioms of QM (the Dirac-von Neumann set including state vector collapse) is the Wightman way. The alternative to Wightman's axioms are the Haag-Ruelle algebraic QFT axioms.
QED is well-defined and covariant at any order, independent of corrections by other theories that are not QED. In this thread we are only discussing QED (and contrasting it if necessary with other theories such as QCD).RockyMarciano said:Neumaier goes on insisting in keeping it to low order because he knows that many QED procecess have higher order corrections that involve interactions other than the electromagnetic
A. Neumaier said:This is incorrect. One can make many compromises in the Wightman axioms and still reconstruct a Hilbert space from them. In the Wikipedia treatment linked to, Axiom W3 (local commutativity or microscopic causality) can be dropped and the Hilbert space construction still goes through. With this axiom dropped it is very easy to construct plenty of models.
