Exploring Bohmian Quantum Field Attempts

[atyy]

[Moderator's note: This thread was spun off from a previous thread since it was getting into technical issues beyond the scope of the previous thread. The quote at the top of this post is from the previous thread.]

I'm not familiar with Bohmian quantum field attempts, but I'll have a look.

But even before going into the technical details, here is a heuristic that may be helpful: relativistic QFT is not necessarily relativistic, and since Bohmian Mechanics can deal with non-relativistic QM, Bohmian Mechanics should be able to deal with some relativistic QFTs.

For non-relativistic attempts at relativistic QFT, we have the mainstream fields of lattice gauge theory (which is non-relativistic at any finite lattice spacing) and condensed matter physics (eg. relativistic massless Dirac particles in graphene). Also interesting are efforts like https://arxiv.org/abs/1503.02312. Lattice QED is generally thought to be ok, so probably the main problem for Bohmian Mechanics is that a lattice standard model is still lacking because of the chiral fermion problem (there are interesting proposals, but no consensus).

 

[A. Neumaier]

Lattice QED is generally thought to be ok

Maybe it is thought to be ok by you, but not by the community working on QED. In contrast to continuum QED, there are extremely few papers on lattice QED - primarily because it gives very poor numerical results, so that hardly anybody is motivated to use it.

 

[atyy]

Maybe it is thought to be ok by you, but not by the community working on QED. In contrast to continuum QED, there are extremely few papers on lattice QED - primarily because it gives very poor numerical results, so that hardly anybody is motivated to use it.

No, it is also accepted by the community working on QED. Calculational convenience is one thing, non-perturbative definition is another.

 

[A. Neumaier]

No, it is also accepted by the community working on QED. Calculational convenience is one thing, non-perturbative definition is another.

No, it is definitely not accepted. The continuum limit of lattice QED is (unlike the limit of a low fine structure constant defining perturbative continuum QED) most likely a noninteracting theory in which the electron has zero charge. This is called the triviality problem, and was demonstrated numerically by Kogut & Strouthos (Physical Review D, 2005), who also review earlier theoretical arguments. A paper published in Physical Review D, 2013 shows that nothing has changed in the mean time:

a result which suggests that the continuum limit is the one of noninteracting bosons, in agreement with earlier findings

Of course there are loopholes, but you are one of the very few who believe that lattice QED converges to true QED.

Please provide strong evidence justifying your opposite view!

 

[atyy]

No, it is definitely not accepted. The continuum limit of lattice QED is (unlike the limit of a low fine structure constant defining perturbative continuum QED) most likely a noninteracting theory in which the electron has zero charge. This is called the triviality problem, and was demonstrated numerically by Kogut & Strouthos (Physical Review D, 2005), who also review earlier theoretical arguments. A paper published in Physical Review D, 2013 shows that nothing has changed in the mean time:

That is not the relevant continuum limit.

 

[A. Neumaier]

That is not the relevant continuum limit.

There is no other continumm limit, or can you point to a source that gives one where the charge remains nonzero?

 

[atyy]

There is no other continumm limit, or can you point to a source that gives one where the charge remains nonzero?

The continuum limit you are talking about is a high energy limit. What we want is a low energy picture, in the same way that one can use classical statistical field theory for a lattice of atoms - the field is a continuum, but the lattice is discrete.

 

[A. Neumaier]

The continuum limit you are talking about is a high energy limit. What we want is a low energy picture, in the same way that one can use classical statistical field theory for a lattice of atoms - the field is a continuum, but the lattice is discrete.

If the lattice is discrete, there is no continuum field. How would it appear, and how would it show the continuum features we are accustomed?

Since you repeatedly stressed that your view is an accepted point of view in the QED community, it should be easy for you to justify this by pointing to a paper or book where, starting from lattice QED, this low energy picture is derived in sufficient detail to produce ((i) a reasonable value for the Lamb shift or the anomalous magnetic moment of the electron, and (ii) the macroscopic Maxwell equations. These are surely both low energy features.

 

[vanhees71]

If the lattice is discrete, there is no continuum field. How would it appear, and how would it show the continuum features we are accustomed?

Since you repeatedly stressed that your view is an accepted point of view in the QED community, it should be easy for you to justify this by pointing to a paper or book where, starting from lattice QED, this low energy picture is derived in sufficient detail to produce ((i) a reasonable value for the Lamb shift or the anomalous magnetic moment of the electron, and (ii) the macroscopic Maxwell equations. These are surely both low energy features.

Indeed, and another question I have is, why is "lattice QED" not existent even in the "practical sense", i.e., as a numerical tool to non-perturbatively evaluate physical quantities? This is the case for lattice QCD which is heavily used in both "vacuum QCD" (e.g., to evaluate the mass spectrum of hadrons) and in "hot QCD" (e.g., to evaluate the equation of state of strongly interacting matter at vanishing baryo-chemical potential and nowadays also extrapolated to ##\mu_{\text{B}} \neq 0##). The answer is of course that lQED seems not to have a non-trivial continuum limit!

 

[atyy]

Indeed, and another question I have is, why is "lattice QED" not existent even in the "practical sense", i.e., as a numerical tool to non-perturbatively evaluate physical quantities? This is the case for lattice QCD which is heavily used in both "vacuum QCD" (e.g., to evaluate the mass spectrum of hadrons) and in "hot QCD" (e.g., to evaluate the equation of state of strongly interacting matter at vanishing baryo-chemical potential and nowadays also extrapolated to ##\mu_{\text{B}} \neq 0##). The answer is of course that lQED seems not to have a non-trivial continuum limit!

Lattice QED is not existent in the practical sense, because perturbative QED is already very good. Rather the point of lattice QED (at small but finite lattice spacing) is to provide a conceptually valid starting point for the Wilsonian view of renormalization, in which the absence of a UV complete theory is irrelevant, and the Wilsonian flow to low energies recovers perturbative QED.

 

[A. Neumaier]

the point of lattice QED (at small but finite lattice spacing) is to provide a conceptually valid starting point for the Wilsonian view of renormalization, in which the absence of a UV complete theory is irrelevant, and the Wilsonian flow to low energies recovers perturbative QED.

The last statement is wishful thinking without the least shred of evidence in the literature.

At sufficiently small (and as the numerical results cited earlier show, not even very small but computationally accessible) lattice spacing,, lattice QED is already in the domain where triviality sets in, and there is no evidence that the Wilsonian renormalization flow would correct for this. On the contrary, there are studies involving the exact renormalization group equations to demonstrate the opposite.

 

[stevendaryl]

Indeed, and another question I have is, why is "lattice QED" not existent even in the "practical sense", i.e., as a numerical tool to non-perturbatively evaluate physical quantities? This is the case for lattice QCD which is heavily used in both "vacuum QCD" (e.g., to evaluate the mass spectrum of hadrons) and in "hot QCD" (e.g., to evaluate the equation of state of strongly interacting matter at vanishing baryo-chemical potential and nowadays also extrapolated to ##\mu_{\text{B}} \neq 0##). The answer is of course that lQED seems not to have a non-trivial continuum limit!

This is very interesting. The obvious difference between the two (QCD and QED) is the asymptotic freedom.

I have heard some people speculate that QED is just not a consistent field theory (because of the Landau pole), so perhaps the failure of lattice QED is just a reflection of this?

 

[A. Neumaier]

This is very interesting. The obvious difference between the two (QCD and QED) is the asymptotic freedom.

I have heard some people speculate that QED is just not a consistent field theory (because of the Landau pole), so perhaps the failure of lattice QED is just a reflection of this?

The majority of physicists interested in this question thinks that this is indeed the case. But I don't share their view. I have good reasons to believe that QED is a consistent theory and that only the lattice approach leads to triviality.

 

[Vanadium 50]

I have heard some people speculate that QED is just not a consistent field theory (because of the Landau pole)

That is a problem that is not a problem. It occurs at crazy high energy scales (like mass of the universe crazy high) and long before that you have electromagnetism give way to electroweak.

 

[atyy]

I have heard some people speculate that QED is just not a consistent field theory (because of the Landau pole), so perhaps the failure of lattice QED is just a reflection of this?

This idea is discussed by https://www.uni-muenster.de/Physik.TP/~munsteg/buch.html (last part of Chapter 5).

 

[stevendaryl]

That is a problem that is not a problem. It occurs at crazy high energy scales (like mass of the universe crazy high) and long before that you have electromagnetism give way to electroweak.

But the discussion is about the failure of lattice QED (which I assume does not take into account electroweak).

 

[Vanadium 50]

which I assume does not take into account electroweak

But then it's a failure of a theory known to be a low energy approximation at very high energies. I would characterize this as "the theory gets the wrong answer outside the range where we know it is no good". Hard to get excited about that.

 

[stevendaryl]

But then it's a failure of a theory known to be a low energy approximation at very high energies. I would characterize this as "the theory gets the wrong answer outside the range where we know it is no good". Hard to get excited about that.

I wasn't claiming to be excited about it. I was just trying to understand whether the failure of lattice QED says something about lattice methods, or whether it just reflects the problems that QED has, no matter how you try to do it.

 

[king vitamin]

That is a problem that is not a problem. It occurs at crazy high energy scales (like mass of the universe crazy high) and long before that you have electromagnetism give way to electroweak.

Electroweak theory still has a U(1) gauge coupling (not to mention the Higgs quartic self-coupling which has the same UV behavior), so the problem doesn't actually go away.

 

[king vitamin]

The continuum limit you are talking about is a high energy limit. What we want is a low energy picture, in the same way that one can use classical statistical field theory for a lattice of atoms - the field is a continuum, but the lattice is discrete.

I am pretty sure triviality is a problem in that limit. You can either take the UV cutoff finite and flow to small energy scales, or you can take the renormalized energy scale constant (but always much smaller than your initial cutoff) and take the cutoff to infinity. These two limits are believed to commute* and leads to the same triviality problem: the QED coupling flows to zero in the IR.

This is not a problem in the effective field theory perspective because the cutoff and the bare couplings are kept finite (the cutoff is given by some relevant physical high energy scale), and the ratio between the renormalized energy scale and the cutoff has some physical meaning. In the case of QED being UV completed by electroweak theory, the low energy value of the electric charge has some relation to the scale of electroweak symmetry breaking. In this approach relativistic symmetry is just approximate (as it always is in condensed matter systems).

*At least it is usually assumed they do. If A Neumaier thinks lattice QED gives different results than some other definition of QED then he probably disagrees.

 

[atyy]

I am pretty sure triviality is a problem in that limit. You can either take the UV cutoff finite and flow to small energy scales, or you can take the renormalized energy scale constant (but always much smaller than your initial cutoff) and take the cutoff to infinity. These two limits are believed to commute* and leads to the same triviality problem: the QED coupling flows to zero in the IR.

This is not a problem in the effective field theory perspective because the cutoff and the bare couplings are kept finite (the cutoff is given by some relevant physical high energy scale), and the ratio between the renormalized energy scale and the cutoff has some physical meaning. In the case of QED being UV completed by electroweak theory, the low energy value of the electric charge has some relation to the scale of electroweak symmetry breaking. In this approach relativistic symmetry is just approximate (as it always is in condensed matter systems).

*At least it is usually assumed they do. If A Neumaier thinks lattice QED gives different results than some other definition of QED then he probably disagrees.

Yes, I am simply saying that lattice QED defines a quantum theory that is a good starting point for the effective field theory picture.

 

[A. Neumaier]

I was just trying to understand whether the failure of lattice QED says something about lattice methods, or whether it just reflects the problems that QED has, no matter how you try to do it.

There are no logical relations between the failure of a particular construction technique and the existence problem in itself.

##\phi_4^4##-theory shares with QED the features of being not asymptotically free and having a Landau pole at 1- or 2-loop perturbation theory.

For this theory, the status of the existence problem (and possible disproofs of existence) is discussed in detail in Section 8, ''IS DESTRUCTIVE FIELD THEORY POSSIBLE?'' of a paper by Gallavotti and Rivasseau from 1984. They call your suspected equivalence the ''Super-strong Triviality Conjecture'', and state:

we do not see at the moment any compelling reason to believe it at all

That this is still the state of the art is confirmed by a remark of Gallavoti at the end of p.14 in a 2014 paper:

the conjecture that it is impossible to obtain nontrivial Schwinger functions in a scalar quantum field theory in dimension 4 is still (wide) open

For further information see the discussion and references in my posting at PhysicsOverflow.

 

[A. Neumaier]

I am simply saying that lattice QED defines a quantum theory that is a good starting point for the effective field theory picture.

You are repeating that without giving a proof that this starting point leads to an effective QED. The information I gave in this thread points to the very opposite. Already at computationally accessible lattice spacings (let alone at very small spacing), lattice QED shows signs of triviality, and no high energy averaging (as done in Wilson renormalization group flows) will undo that.

 

[Vanadium 50]

Electroweak theory still has a U(1) gauge couplin, so the problem doesn't actually go away.

A. The fact that some other theory has that problem is not relevant.
B. Hypercharge is not conserved. What is the hypercharge of an electron?

 

[A. Neumaier]

This idea is discussed by https://www.uni-muenster.de/Physik.TP/~munsteg/buch.html (last part of Chapter 5).

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 that

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.

Thus this conclusion by experts on the subject is diametrically opposite to your 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 almost like a free theory. The closeness to triviality becomes even stronger at smaller lattice spacing, since the zero spacing limit is trivial.

This has been discussed extensively in another PF thread last year, see especially #108 and #174 there.

 

[king vitamin]

A. The fact that some other theory has that problem is not relevant.

Then why did you bring up electroweak theory? If we are interested in pure QED as a mathematical theory, let's just discuss QED. But you suggested that the UV completion to electroweak theory made that question moot, so the fact that similar mathematical issues arise there is certainly relevant.

B. Hypercharge is not conserved. What is the hypercharge of an electron?

Sure, hypercharge isn't conserved below the electroweak scale where electrons exist. How does this fix the UV behavior of the U(1) gauge coupling or the quartic Higgs coupling?

 

[jordi]

This thread is old. But I am surprised to read that lattice QED does not need to be the "correct" non-perturbative quantum version of QED (apparently, lattice QED is trivial, but perturbative QED gives correct results, and Neumaier argues that it could be QED is a perfectly fine quantum theory, irrespective of Landau poles present at perturbation theory).

I always thought that lattice would be always the "right quantum theory", but very hard to compute. But it seems this does not need to be the case.

How can one then distinguish (apart from checking experimental data, of course) which is the "right" quantum, non-perturbative theory, among all the possible quantization routes?

(I always implicitly thought that all "reasonable" quantizations would give the same results, but it seems this does not need to be the case).

 

[A. Neumaier]

(I always implicitly thought that all "reasonable" quantizations would give the same results, but it seems this does not need to be the case).

This is true, but in the absence of asymptotic completeness, lattice QFT is most likely not reasonable, since triviality of the limit seems to follow.

How can one then distinguish (apart from checking experimental data, of course) which is the "right" quantum, non-perturbative theory, among all the possible quantization routes?

By comparison with experiment, which is the final arbiter. Properly extrapolated lattice QCD produces results in agreement with experiment, while lattice QED results suggest triviality, i.e., the wrong limit.

 

[jordi]

Thank you for the explanation, Neumaier.

Do you have any way to reconcile the idea that lattice QCD seems to be a good definition for "non-perturbative QCD", but instead, lattice QED seems not to be a good definition for "non-perturbative QED"?

Intuitively, I would expect the lattice method to be a good method to define non-perturbative quantum theories, with its practicality depending on the feasibility to perform the calculations (or not).

But it seems something else is at play. Is it a problem of the lattice method, or is it that genuinely, non-perturbative QED is trivial? In other words, if we had another quantization method that gave a non-perturbative QED, would this method would also result in a trivial QED, or not necessarily?

 

[A. Neumaier]

Do you have any way to reconcile the idea that lattice QCD seems to be a good definition for "non-perturbative QCD", but instead, lattice QED seems not to be a good definition for "non-perturbative QED"?

QCD is asymptotically free, which guarantees a good lattice limit. In contrast, QED is not.

if we had another quantization method that gave a non-perturbative QED, would this method would also result in a trivial QED, or not necessarily?

The causal perturbation approach to QED is not susceptible to the usual triviality arguments, as it is manifestly covariant and works throughout without a cutoff. It starts off with nonperturbative axioms but then constructs QED only perturbatively. Partial resummation gives nontrivial partially nonperturbative results.

 

[atyy]

The causal perturbation approach to QED is not susceptible to the usual triviality arguments, as it is manifestly covariant and works throughout without a cutoff. It starts off with nonperturbative axioms but then constructs QED only perturbatively. Partial resummation gives nontrivial partially nonperturbative results.

The causal perturbation approach does not construct the theory, so it cannot be presented as a workable alternative to the lattice.

 

[A. Neumaier]

The causal perturbation approach does not construct the theory, so it cannot be presented as a workable alternative to the lattice.

Neither causal perturbation theory nor the lattice approach give a nonperturbative construction of QED; how to achieve the latter is a widely open problem.

However, causal perturbation theory constructs QED perturbatively, order by order, and produces highly accurate approximate covariant formulas for the S-matrix and the field operators that are in agreement with experiment.

On the other hand, the lattice approach to QED produces noncovariant approximate formulas that have never been shown to produce results agreeing with experiments, not even crudely. The numerical results produced are rather indicative of an approximately free theory, consistent with triviality.

Thus the lattice cannot be presented as a workable alternative to perturbation theory.

 

[Tendex]

The causal perturbation approach to QED is not susceptible to the usual triviality arguments, as it is manifestly covariant and works throughout without a cutoff. It starts off with nonperturbative axioms but then constructs QED only perturbatively. Partial resummation gives nontrivial partially nonperturbative results.

It is logically just as susceptible to possible triviality of the nonperturbative QED until the latter is actually achieved(if it exists) without triviality.

 

[A. Neumaier]

It is logically just as susceptible to possible triviality of the nonperturbative QED until the latter is actually achieved(if it exists) without triviality.

The point is that the standard triviality arguments - namely that a Landau pole must be traversed by the cutoff - breaks down. Thus, unlike in approaches with an explicit cutoff (such as in lattice methods), there is no longer a known mechanism for triggering triviality.

While this doesn't prove anything it removes any rational reason for believing in triviality beyond the very weak 'anything might happen as long as nothing has been proved'.

 

[vanhees71]

Isn't lattice quantization of electrodynamics then not just a not suitable regularization to define QED? So what? There are plenty of equivalent ways to define perturbative QED which is agreeing with observations. The modern understanding of QFT is anyway that it's an effective description for a limited range of energy scales. Whether or not there's a more fundamental description or not is not known yet.