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.