Field theories in 4 dimensions

[kelly0303]

Hello! I know this is a very general question (and I am really a very beginner in the field) so I am sorry if it is dumb, but here it goes. In Schwarz book on QFT, at the end of Section 14.4 (path integrals chapter) he says: "We do not know if QED exists, or if scalar $\phi^4$ exists, or even if asymptotically free or conformal field theories exists. In fact, we do not know if any field theory exists, in a mathematically precise way, in four dimensions". Can someone explain (preferably in layman terms) what does he mean? I heard so many times that QED is the most well tested physical theory with the best predictions and so on. So if QED is not mathematically consistent (is this what he means?), are all these amazing results (experimental confirmations) just a mix of good intuition and luck i.e. can it be that at a point they might just fail? Also in the very beginning (before the experimental confirmation) what motivated physicists to continue working on QED, if they were not even sure if their work was mathematically consistent, let allow describing the reality? Thank you!

 

[haushofer]

Others are more qualified to give you a decent answer, but maybe you like the following paper as a comfort:

https://arxiv.org/abs/1201.2714

 

[vanhees71]

Another great book in this direction is

A. Duncan, The conceptual framework of quantum field theory, Oxford University Press, Oxford (2012).

 

[DarMM]

In the Euclidean path integral approach we normally write something like:
$$\int{\mathcal{D}\phi\:e^{-S\left[\phi\right]}\:\mathcal{O}\left(\phi\right)}$$
where $\phi$ refers to any generic collection of fields and $\mathcal{O}\left(\phi\right)$ is some observable.

In general this is split up as:
$$\int{\mathcal{D}\phi\:e^{-S_{0}\left[\phi\right]}e^{-S_{I}\left[\phi\right]}\:\mathcal{O}\left(\phi\right)}$$
where $S_0$ is the free action and $S_I$ is the interaction part of the action.

Since integration with respect to:
$$\mathcal{D}\phi\:e^{-S_{0}\left[\phi\right]}$$
is a "solved problem" essentially just an infinite dimensional version of Gaussian integration, we tackle integration in general by expanding the second exponential:
\begin{align*}\int{\mathcal{D}\phi\:e^{-S_{0}\left[\phi\right]}e^{-S_{I}\left[\phi\right]}\:\mathcal{O}\left(\phi\right)} & = \int{\mathcal{D}\phi\:e^{-S_{0}\left[\phi\right]}\:\mathcal{O}\left(\phi\right)}\\
& + \int{\mathcal{D}\phi\:e^{-S_{0}\left[\phi\right]}\:\left(-S_{I}\left[\phi\right]\right)\:\mathcal{O}\left(\phi\right)} + \cdots
\end{align*}
Where each term is a Gaussian integral. Some of these integral diverge since they involve multiplying distributions at the same point. These divergences have to be repaired via an ad-hoc formalism called perturbative renormalization.

Now there were a few problems with this naive way of handling these integrals. Firstly even the free part of the integral:
$$\mathcal{D}\phi\:e^{-S_{0}\left[\phi\right]}$$
makes no sense as $\mathcal{D}\phi$ can be proven not to exist. However we can prove that there is a Gaussian measure on the space of fields, often denoted $d\mu_{C}$ which is the continuum limit of the lattice equivalent of
$$\mathcal{D}\phi\:e^{-S_{0}\left[\phi\right]}$$

Note we don't need the lattice formalism to prove it exists, a result called Minlos theorem can be used to prove its existence directly in the continuum.

So that's integration with respect to free action handled. Since the perturbative series is a series of integrations with respect to the free measure and renormalization can make them finite we also know that we can define the perturbative series as used in practice by physicists.

However this perturbative series diverges so we cannot sum it up to obtain the exact answer for the interacting theory or prove such an exact answer exists. For that we need to demonstrate that integration with respect to full actions makes sense, i.e. non-perturbatively without expanding the interacting action's exponential.

Now we face another problem. It can be proven (Haag/Nelson's theorem) that for interacting theory measures $d\nu$:
$$d\nu \neq d\mu_{C}\:e^{-S_{I}\left[\phi\right]}$$
In other words they're not really a free measure multiplied by an exponential with the interacting action. They are simply some other measure.

We would hope that we could try making some kind of approximated version of them by making the degrees of freedom finite, i.e. using ultraviolet and infrared cutoffs, and then removing the cutoff to construct the true measure. This does work, there are various ways of cutting the integrals off, though some only appear in the constructive field theory literature especially the infrared cutoffs. We cutoff the naive measure given above:
$$d\mu_{C}\:e^{-S_{I}\left[\phi\right]}$$
and carefully correct $S_{I}$ so that the limit where the cutoffs are removed is well defined.

To date we have been able to show that in the limit as the cutoffs are removed that measures in $d = 2,3$ do in fact exist. We have not been able to show this is the case for realistic $d = 4$ theories.

 

[vanhees71]

Well, the pragmatic way is to just use a quantization volume and periodic boundary conditions as a regulator and then taking the limit to infinite volume for the adequate quantities. Haag's theorem is an interesting mathematical fact, which however is of little relevance for the application of the (admittedly mathematically not well-defined) QFT in (1+3) dimensions in terms of perturbation theory or (for QCD) lattice calculations, where the lattice is just another way to regularize our sloppily defined ignorance of mathematical finesses regarding operator valued distributions or the as sloppily defined path integrals over field configurations.

 

[DarMM]

Haag's theorem is an interesting mathematical fact, which however is of little relevance for the application of the (admittedly mathematically not well-defined) QFT in (1+3) dimensions in terms of perturbation theory

This is because the perturbative series calculated in Fock space (canonical) or against the Gaussian measure (path integral) is still asymptotic to the true values. Haag/Nelson's theorem only means the usual method of deriving the perturbative series presented in textbooks is invalid. It doesn't cast any mathematical doubt on perturbation theory itself.

 

[vanhees71]

Well, I'd say the presented derivation in textbooks is not mathematically rigorous, but at the end the physicist get it right, because the math forces them to get it right: The problem formalized in Haag's theorem is circumvented by regularizing the issue by introducing a finite "quantization volume" and impose the periodic boundary conditions adequate to describe scattering events (for "cavity QED" you'd impose rigid boundary conditions to discribe an idealized cavity with perfectly reflecting walls of inifinite electric conductivity).

 

[DarMM]

I'd say the presented derivation in textbooks is not mathematically rigorous, but at the end the physicist get it right

That's basically what I'm saying. The derivation doesn't hold but the end result is correct. The perturbative series is asymptotic to the true integral.