# What is the status of the existence problem for scalar QFT and QED?

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It is frequently claimed that in 4 dimensions, quantum field theories that are not asymptotically free (such as $\Phi^4$ theory or QED) cannot exist. How reliable are the arguments that try to substantiate this? In particular, to bring the discussion to a clear focus:

Is there a rigorous statement whose proof or disproof would settle the question of the existence of $\Phi^4$ theory with mathematical rigor?

asked Aug 2, 2014

## 1 Answer

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Let me formulate as an answer what I know about the question. I'd be happy to be corrected if something said here is inaccurate or incomplete. I assume throughout dimension $d=4$.

First of all, it seems to me that there is no definition of what $\Phi^4$ theory should mean in rigorous terms. Thus it is presently impossible to disproof the existence of $\Phi^4$ theory.

On the other hand, I think that there is consensus about how to judge a claimed existence proof of $\Phi^4$ theory: it would mean to have a Hilbert space and a family of operators rigorously defined on it, such that one can derive from it in a formal way (not necessarily rigorous, but valid on the level of the standards of the common textbooks on QFT) an asymptotic expansion of the S-matrix agreeing order by order with the textbook expansion.

The accepted mainstream view is that the existence of a Landau pole at very high energies in the approximate (low order perturbative or low resolution lattice) treatments of $\Phi^4$ theory, together with the most likely triviality of the continuum limit of the lattice version imply (in some unspecified roundabout way) the nonexistence of $\Phi^4$ theory. Useful papers representing this view are

Discussions of the existence problem (dubbed ''triviality of $\Phi^4$ theory) that take a less decisive stand are in

On the other hand, there is some evidence for existence of $\Phi^4$ theory or QED. None of these is accepted mainstream physics, but I haven't seen convincing reasons for dismissing these signs.

1. Klauder (well-known for work on coherent states in quantum optics) proposes alternative renormalization schemes that seem to work to some extent:

2. The so-called PT-symmetric QM formalism

gives a construction of a Hilbert space for the Lee model with a ghost mode, which exhibits a Landau pole in perturbation theory. There is also a perturbative QED version:

3. There are non-perturbative constructions for approximations of QED, where the Landau pole disappears, e.g.,

Note that 4D Yang-Mills nonabelian gauge theory, which is asymptotically free, also has a Landau pole in perturbation theory, and - worse than for QED and $\Phi_4^4$ - it is located at experimentally relevant energies! See https://arxiv.org/abs/1311.6116. But this is not taken as an indication of nonexistence of 4D Yang-Mills theory; instead (see the survey by Shifman, https://arxiv.org/abs/hep-ph/9802214) it is only taken to mean that at low energies, a nonperturbative treatment is called for.

answered Aug 2, 2014 by (14,019 points)
edited May 30, 2018

In the case of QED, if you take massive electrons with the fine-structure constant >1, the theory itself has a classical problem, which is that the field-mass in the field of the electron, long before it becomes relativistic, is greater than the rest mass. This creates a classical contradiction.

The calculation methods in the papers referenced do not fix the Landau pole problem at least in any way I could see from a quick superficial review, they are too formal to do anything more than rearrange the perturbation series. If you want to rearrange a series to get rid of the Landau pole, I recommend this simple trick: do a one loop running calculation, and discover that

$$de = Ce^2 d\Lambda$$

Then reinterpret the calculation for larger couplings by fixing one of the coupling factors at one particular value of the coupling $e_0$ and only let the other coupling be the running coupling, so that you come to the conclusion that

$$de = Ce e_0 d\Lambda$$

Since the running correction to e_0 is higher order in e/e_0, you can justify this rewriting at larger couplings by a formal rearrangement of the perturbation series. In this ridiculous rewriting, which is obviously not the physically correct thing to do, you see that e does not diverge at any $\Lambda$, it just grows exponentially. I could even do this nonsense:

$$de = C e_0^2 d\Lambda$$

in which case e only grows linearly with $\Lambda$. You can't exclude any of these behaviors at large coupling from the known dependence at small coupling, you really don't know anything about the beta function except an asymptotic expansion at small couplings, and by reshuffling the one-loop answer, you can fake any large-value growth from a given one loop calculation.

But, realistically, on physical grounds, the $e^2$ form Landau uses "looks correct", because you should use the current coupling at scale $\Lambda$ to get the next coupling, it "feels right". But since the difference can be rewritten as higher order, formally, and the true functional dependence only appears at large couplings, you don't really know in any rigorous way. The only point of this nonsense is that choosing all sorts of mathematical reshufflings can easily make the Landau pole go away.

But that doesn't mean that it does go away. The lack of rigor should not obfuscate the basic physical principle: the Landau pole is justified not just from the mathematics of the RG calculation to one loop, but also from the inconsistency of the classical physics at large fine structure constant.

I think it would be good to import the papers to the review section, because one can analyze the arguments in detail, and also try to resolve this ancient question, as the physical answer has been clear for 30 years already, and it's annoying that there is still doubt, a rigorous proof of triviality is very close in the case of 4d $\phi^4$, and should be possible even in QED.

Why does (on the heuristic level you argue) the nonexistence of the classical limit imply nonexistence of the quantum theory? After all, $\hbar$ is a constant, so the classical limit is a nonphysical mathematical abstraction.

@ArnoldNeumaier: The reason I argue this is because if QED were consistent at one value of alpha, you don't expect a breakdown at larger values, because the coupling gradually runs up at shorter and shorter distances, so you would expect it is consistent at all alpha. Taking alpha sufficiently large (or, equivalently up to choice of units, making a supermassive electron, so that the coupling running freezes at some large value), you have an object which has a classical approximation should be valid, but it can't be valid.

Consider alpha=100, then the self-field of an electron at mass m has, classically, locally, more energy in the field in the region 100 times the Compton wavelength than it's mass! To say that the theory still makes sense, you are saying that somehow the classical description either doesn't work, or else, there is some classical negative energy density very close to the electron, to make the total mass come out right. This negative energy is the reason for the runaway solution classical paradoxes in classical QED renormalization, like in Dirac.

If you are arguing that QED is consistent at alpha=100 then you must show why a massive classical particle description should somehow break down. It's not mathematics, I know, it's physics, but it's a solid argument.

Hmm. The superheavy electron should still be a fermion, hence does not satisfy the equations from classical QED. The classical limit of QED would be a Vlasov-like classical field theory, and I haven't seen any paper that discusses its properties, so how can you know whether it is inconsistent (as the old classical electrodynamics)?

I am talking about the classical limit of a single heavy Fermion state, not of many particles. The statistics (and spin) are completely irrelevant for the classical nonrelativistic dynamics.

The concept of a single fermion state makes sense for energies significantly less than the mass, there are superpositions of these which make nonrelativistic states which are localized roughly like a delta function, so long as the delta-function is wider than the Compton wavelength. These states obey a classical Schrodinger equation, and approximately obey classical mechanics in the usual WKB limiting way.

Around a classical localized single-electron state, there is an approximate classical field, and the classical field energy of this classical field is greater than the rest mass already in the region larger than the Compton wavelength. This is a real paradox, because you can't fix up the energy using positive classical contributions at smaller distances.

There is no "Vlasov like" anything, this is not a field theory limit, but a classical particle dynamics limit.

There is nothing wrong with the argument, but making it a rigorous mathematically is a pain in the ass, because you need to talk about approximately classical states of a quantum particle. But it might still be possible to make the argument rigorous by using the property that the energy should be an integral of a local Hamiltonian density which is locally positive, and using only the classical Gauss's law to bound the field and then the field energy for a best-possible-localized single electron state to greater than the mass. It's not something I know how to do, or invest effort in doing, simply because the heuristic argument is already completely persuasive to me without rigor.

Ron, there is another nonsense when a self-action is taken seriously. You start from a Dirac (classical) equation for an electron with a self-action and as a result you obtain an electrically neutral magnetic monopole (see also here and here).

This paper is doing the type of classical nonsense that Arnold Neumaier is warning about--- taking the Dirac field to be a classical real-valued field. That's not what I was talking about.

The results in that paper come from treating the Dirac field with wrong statistics. To have a classical real-number field limit, you need a bosonic field. When Dirac used his equation to find real-valued solutions, he didn't use it to produce a classical "Dirac real valued field", he used it to describe the wavefunction of single-electron states. The equation happens to be the same formally at the linear level, but there is no nonlinearity in a wavefunction description, the interaction with the electromagnetic field is by perturbations, by producing new electrons in new waves. Getting a simple wavefunction equation is what motivated Dirac, even though this is not a good motivation.

This paper should be imported to the review section. It has a ridiculous unphysical conclusion which can be explained very simply. The reason he "unavoidably" gets a monopole solution is because he is giving classical solutions  the "assumptions of spherical symmetry". Since Dirac spinors rotate by a half-angle, if you try to make a spherically symmetric solution to a Dirac field, you end up with a singularity. The singularity can be explained as follows--- you start with some spinor values at a position along the positive x-axis, and you rotate the spinor as you rotate the coordinates, to make a spherically symmetric solution. When you rotate by 180 degrees to get to the negative x-axis either clockwise or counterclockwise, you end up with opposite sign Dirac-field values, because a 360 degree rotation is not the identity on the Dirac field. So it makes no sense to ask for a rotationally invariant solution to a Dirac field, you need at least a line-singularity where the field jumps.

This "cotangent term" he then misinterprets as the singularity of a magnetic monopole. The physics of this paper is all jumbled up. It perhaps, however, has historical value, because considerations of this sort might have historically motivated Dirac to that incredible original leap of 1930 or 1931, where he produce the proper description of monopole field in two sheets. The two-sheet description does indeed naturally emerge from considering the unphysical question of what a rotationally invariant solution to the real-valued Dirac field equation looks like.

We know today that the proper interpretation of classical Dirac field is as a Grassman number field, not as a real valued field, so the questions in this paper are not interesting.

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