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  What is the evidence against the triviality of pure QED and pure $\phi^4$?

+ 5 like - 0 dislike

This question sprouted from Arnold's comments of this post. I've seen more theoretical evidences hinting at the triviality of pure QED and pure $\phi^4$ in 4D spacetime than the opposite, I'll be more explicit:

(1)Perturbative expansions to a few loops order, for both QED and $\phi^4$. But I confess I myself have only verified the 1-loop results.

(2)Claims from lattice simulations. I never verified this for myself, and lattice $\phi^4$ numerics seem to be well cited, while the lattice QED numerics less so.

(3)Nonperturbative constructive results on $\phi^4$, a rigorous proof of triviality is said to be only a few lemmas away from completion (I remember this claim from an online lecture of Arthur Jaffe).

Now by "pure" I mean constructions that only make use of the infrared terms, for example if I remember correctly, evidence (3) can't exclude---even with the completion of the unproved lemmas---the possibility for the existence of a scalar field theory with additional $\phi^6, \phi^8, \phi \Box^2 \phi$ terms etc, such that it flows to a pure $\phi^4$ in the infrared.

What are the supporting evidences for the opposite claims?

asked Jul 31, 2015 in Theoretical Physics by Jia Yiyang (2,640 points) [ revision history ]

I'll collect my evidence; it may take a while...

@ArnoldNeumaier, sure take your time, it's bedtime here in my time zone...

I just noticed that some of what I know on this query is already in the thread ''What is the status of the existence problem for scalar QFT and QED? ''. However, in the mean time I learnt something more, so the new answers are still useful.

I am sorry to be ignorant, but what do you mean by "triviality of a theory"? Reducing the exact solutions to free ones or what?

@ColdCooler, it's just another way of saying Landau pole, except the perspective is a bit different by reversing the direction of renormalization group flow.

The difference $f(\infty)-\sum_{n=0}^{\infty}f^{(n)}x^n/n!, \; x\to\infty$ is exactly zero, but its "perturbation series" is non trivial and divergent. The Taylor series, if summed up selectively (not all its terms, but "the most divergent" ones), may result in something different from zero. That may be the origin for Landau-pole behavior of such "exact" sums.

1 Answer

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The Landau pole is most likely an artifact of perturbation theory, caused by the latter's lack of accounting for terms with an analytic dependence of the form $e^{const/\alpha}$, where (for QED) $\alpha$ is the fine structure constant. Evidence for this is the fact that the Landau pole in 1-loop perturbation theory for QED is cancelled exactly by a second, nonperturbative term that does not contribute perturbatively. As a result, the spectral function is well-defined and satisfies the causality conditions required by the Källen-Lehmann representation. See the paper by Bogolyubov, Logunov and Shirkov, The method of dispersion relations and perturbation theory, Soviet Physics JETP 37 (1960), 574-581. There is a highly cited review paper of the resulting analytic perturbation theory (concentrating on QCD, though) by Shirkov and Solovtsov, Ten years of the analytic perturbation theory in QCD. Theoretical and Mathematical Physics 150 (2007), 132-152. An interesting recent paper on the subject (for QCD) is https://arxiv.org/abs/1411.2554.

A 2017 paper by Djukanovic et al. shows that treating QED as an effective theory by adding an additional Pauli interaction term already removes the Landau pole at 1-loop order. 

For $\Phi^4$ theory, my positive evidence is somewhat limited, since this is somewhat aside from my main goal, to prove the existence of QED. 

My main argument for $\Phi^4$ theory is Section 8, ''IS DESTRUCTIVE FIELD THEORY POSSIBLE?'' in a paper by Gallavotti and Rivasseau from 1984, which discusses $\Phi^4$ theory in 4 dimensions and is very pessimistic that the existence of a rigorous $\Phi^4_4$ quantum field theory can be ruled out. The main reason given is that one needs arguments valid for all possible constructions, while the usual approach (leading to negative evidence) is to consider the simplest classes of approaches only. As far as I can tell no work at all has been done since then on their Super-strong Triviality Conjecture, for which they state ''we do not see at the moment any compelling reason to believe it at all''. (This 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

Callaway's 1988 comprehensive review article on triviality comments the first of these papers on p.290 as follows: 

Other attempts to construct a nontrivial $\phi^4$ theory are often a bit more abstract in nature. Rigorous discussions of triviality (see section 2) often require that a $\phi^4$ field theory is defined as an infinite-cutoff limit of a ferromagnetic lattice theory. It has been argued [4.31] that this is an assumption whose removal changes the nature of the problem dramatically. Indeed, no argument appears to prevent the existence of an interesting nontrivial ultraviolet limit of an antiferromagnetic lattice $\phi^4$ theory, even in d >4. This remains an interesting open problem.

While this is no direct positive evidence it is doubly negated evidence, which may count as positive, too. Klauder claims that an approach based on affine coherent states should work in any dimension, but his arguments are not rigorous.

See also my answer in the thread ''What is the status of the existence problem for scalar QFT and QED?'', which contains additional information.

answered Jul 31, 2015 by Arnold Neumaier (14,547 points) [ revision history ]
edited May 29, 2018 by Arnold Neumaier
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Thanks, I'll read the reference later.

I remember you mentioned something like "Wightman axioms imply QED is nontrivial, given....", what's the precise statement?

The exact statement is: If QED exists at low energies (which you accepted), it must exist at all energies. For existence means that the uncharged sector satisfies the Wightman axioms. From the Wightman axioms together with the massless photon field one can deduce that the spectrum of the momentum vector (generator of Poincare translations) is unbounded, and all (smeared) creation operators for any timelike momentum $p$ exist.

Wilsonian type renormalization group arguments do not apply to QED in causal perturbation theory, since there is no cutoff to be taken to infinity. A Landau pole can therefore at worst mean that there is no scaling limit, i.e., no limit of QED where the electron mass vanishes.

I'll review other positive evidence after returning form my holiday.

@ArnoldNeumaier, ok thanks, and enjoy the holiday!


whatever the final fundamental theory is, it must somehow approach something that contains the interacting QED at low energies

Whatever the final fundamental theory is (if one exists at all, which is by no means certain), it contains an effective vacuum state and effective fields describing the electromagnetic field $F_{\mu\nu}$ and the electron field $\Psi$, and a Poincare group acting on it. These fields generate a field algebra whose gauge invariant part satisfied the Wightman axioms and hence provides a model for QED at all energies, which coincides with standard renormalized QED at low energies. Possibly the standard renormalized QED field equations get (formally nonrenormalizable) higher order correction terms that can be ignored at low energies but not at higher ones. It is this version of QED that exists beyond reasonable doubt, due to experimental evidence and very general symmetry considerations. (Note that in 2D there are formally nonrenormalizable theories that nevertheless exist rigorously.)

However, beyond that, I think it is likely that this version of QED actually satisfies the renormalized QED field equations exactly.

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@Dilaton, unless you take the philosophical stand point of "There's no theory of everything at all", or you must have missed the entire point Arnold and I have been discussing. We are discussing the possibilities of the fundamental theory being a QFT,  while what you have been saying are correct but irrelevant stuff. 

@JiaYiyang: Referring to a fundamental theory only guarantees existence of QED possibly with infinitely many higher order terms. However, these additional terms then encode (part of) all other content of the fundamental theory. On the other hand, one would expect that any consistent theory containing QED would give one of these variants of QED, so that there should be infinitely many of them (unless there is only a unique theoretical possibility for the fundamental theory, which I doubt very much). This means that if one wants to construct QED without the fundamental theory one should have much freedom in choosing the higher order terms. Indeed, standard renormalization theory suggests that one can choose freely infinitely many parameters. But one needs to construct only one of these to have a good theory.

On the other hand, power counting arguments say that in perturbation theory, one gets a valid simplified theory form any theory with infinitely many terms by simply deleting all nonrenormalizable terms. I expect this statement to be valid rigorously. Thus if one of the QED variants incorporating some physics beyond QED exists, I expect that standard QED also exists.

As I had already mentioned, in causal perturbation theory a Landau pole doesn't force nonexistence. The Landau pole only forbids constructions that are based on an energy cutoff that must be moved to infinity (or a short distance cutoff that must be moved to zero) - beyond the pole. This forbids lattice approximations. Indeed the only positive triviality results are for (particular classes of) the latter only. But all other approximations that keep from the start the correct short-distance structure are unaffected by Landau pole arguments.

I do not really know what (if anything) to expect as consequences of a Landau pole in causal perturbation theory - One day I'll have to do corresponding calculations.

Isn't effort better spent on more pressing and rewarding issues like quantum gravity? 

I believe that the issues with quantum gravity will sort out themselves once the existence of QED is positively settled, and not before. I have many indications that suggest that the main problems to be overcome in both cases lie in the inadequateness of traditional renormalized perturbation theory rather than in the nonexistence of the canonical quantizations of electrodynamics or gravity.

The same techniques that will construct QED will also (though probably with considerably more work) construct the standard model, quantum gravity, and their unification.

This is why I think constructing QED is by far the most important open problem in theoretical physics. It surely is the oldest one, hence it has for mathematical physics the same status that the Riemann hypothesis has for mathematics. And it appears to me tractable enough that I study in detail all techniques that promise some contribution to this problem.

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