Does QED Originate from Non-Relativistic Systems?

[Demystifier]

Please share your thinking in enough detail that others can check its cogency.

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.

 

[Demystifier]

1. There is the general feeling in the community that the continuum limit of lattice QED is likely to be trivial

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.

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.

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]

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.

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.

there is also a general feeling in the community that this does not affect predictions for long distance phenomena.

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.

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.

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.

You are talking about Monte-Carlo simulations (of lattice QFT) at a computer, but it should be distinguished from lattice QFT as such.

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]

You should bother for the sake of the readers of this thread,

Let them speak for themselves!

 

[RockyMarciano]

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.

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:

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.

Instead of using a patronizing tone towards everyone you could address this.

 

[A. Neumaier]

atyy's query about what you mean by Poincare invariant

He didn't query about this meaning.

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.

 

[RockyMarciano]

He didn't query about this meaning.

He actually assumes the same meaning for Poincare invariance and yet he asks:

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)?

The meaning you refer to below is only compatible with the existence of the rigorous QED(i.e. at all energies).

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.

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.

 

[dextercioby]

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 ?

 

[A. Neumaier]

The meaning you refer to below is only compatible with the existence of the rigorous QED(i.e. at all energies).

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]

1-loop QED exists rigorously 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.

It is invariant under exactly the same Poincare group as the full theory.

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.

But it is already much better than anything what lattice QED has to offer.

Of course, I'm not discussing this.

 

[RockyMarciano]

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 ?

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]

Rocky, the input is clear, we're speaking about a specially-relativistic field theory (hence it admits a Lagrangian density in flat Minkowski spacetime) with known assumptions: locality (the no. of space-time derivatives is finite), global Poincare invariance (the action in terms of integrated fields and their derivatives stays the same under a global Poincare transformation of the fields up to a hypersurface term). These two assumptions uniquely define the (gauge-fixed) Lagrangian density. Now what do we do with it? Define the quantum observables: energy spectra of bound states could be an example. Can this be calculated exactly (within formal mathematics) ? If so, it must be Poincare invariant.

 

[A. Neumaier]

I've got the feeling that you have run out of good arguments

All of my arguments in this thread are good and don't need to be repeated.

 

[vanhees71]

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.

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]

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.

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]

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!

 

[atyy]

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.

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 for arguing that textbook perturbative covariant QED is a low energy effective theory) - ie. the same as perturbative QED with a cutoff. And once you have a cutoff, you don't have Poincare invariance.

 

[atyy]

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.

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.

 

[Demystifier]

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!

Then we agree on that.

 

[vanhees71]

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.

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.

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.

 

[atyy]

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.

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?

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.

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.

 

[A. Neumaier]

I am saying that these 12 significant digits scan also be obtained by perturbative renormalied QED at the lattice.

This is an extraordinary statement that needs proof to be credible.

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.

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]

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

... 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!

 

[A. Neumaier]

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.

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]

no one has shown a way to take the energy to infinity for QED.

One can take the energy to infinity in all 1-loop formulas and gets mathematically meaningful covariant results.

That they differ from potential theoretical physics at extremely high energies that can never be realized experimentally is completely irrelevant. It neither affects the Poincare invariance of the formulas nor the fact that 1-loop QED is quantum field theory, even one of its highlights, discussed in every book on quantum field theory.

 

[atyy]

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.

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.

 

[atyy]

... 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!

The problem with your statements is you seem to believe there is something called real QED. 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. I'm open to being convinced, but as far as I know, there is no rigorously constructed QFT in 3+1D at the moment, so I have to be skeptical.

My position is that without either lattice or a theory satisfying the Wightman axioms (or equivalent), QED cannot be said to be a quantum theory. It is only a collection of calculation fragments. Of course, they are very important fragments, and the Nobel to Schwinger, Tomonaga and Feynman was deserved, but it doesn't change the fact that it remained to be shown by Wilson, Wightman, Nelson, Osterwalder and Schrader why the standard perturbative renormalization deserves to be understood as a quantum theory.

 

[dextercioby]

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.

Just get over it, there is no interacting QFT in 1+3 D satisfying Wightman axioms.

 

[rubi]

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.

 

[atyy]

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.

So would you agree that 1 loop QED is a rigourous construction of a relativistic QFT in 3+1D at all energies (finite volume is ok)?

 

[rubi]

So would you agree that 1 loop QED is a rigourous construction of a relativistic QED at all energies (finite volume is ok)?

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]

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.

And if we just take the first 1 or 2 or 137 orders, do we obtain a rigourous relativistic QFT?

 

[rubi]

And if we just take the first 1 or 2 or 137 orders, do we obtain a rigourous relativistic QFT?

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.

 

[RockyMarciano]

the ultimate goal is of course to obtain well defined QFT's that satisfy the Wightman axioms.

Why is this the goal, as opposed to finding alternative axioms or framework compatible with the known facts?

 

[atyy]

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.

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).