Mathematical Quantum Field Theory - Interacting Quantum Fields - Comments

[Urs Schreiber]

Greg Bernhardt submitted a new PF Insights post

Mathematical Quantum Field Theory - Interacting Quantum Fields

Continue reading the Original PF Insights Post.

 

[Urs Schreiber]

A few months back we had had some discussion here on orders of $\hbar$ in Feynman diagrams, maybe somebody remembers it and has the link. There was a request back then to turn the reply into a PF-Insights. The corresponding discusssion is now prop. 15.67.

(By the way, does anyone see why equation (253) does not render? Maybe I am too tired now to spot it, I'll try again tomorrow.)

 

[Duong]

I checked that the maths code is fine in itself (by copy-pasting into a latex environment and rendering.)

 

[David Corfield]

The error message is supposed to indicate the problem, but I can't see what it's objecting to:

<math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
<merror>
<mtext>\label{OnRegularObservablesQuantumMasterWardIdentityViaTimeOrdered}&#xA0;
&#xA0;&#xA0;\{-S',(-)\}&#xA0;\circ&#xA0;\mathcal{R}^{-1}&#xA0;
&#xA0;&#xA0;\;=\;&#xA0;
&#xA0;&#xA0;\mathcal{R}^{-1}&#xA0;
&#xA0;&#xA0;\left(\left\{&#xA0;-(S'&#xA0;+&#xA0;g&#xA0;S_{int})&#xA0;\,,\,&#xA0;(-)&#xA0;\right\}_{\mathcal{T}}&#xA0;-i&#xA0;\hbar&#xA0;&#xA0;&#xA0;\Delta_{BV}\right)</mtext>
</merror>
</math>

 

[Duong]

At the end of remark 15.18 you wrote that the phenomenon of quark confinment is "invisible to perturbative quantum chromodynamics in its free field vacuum state, due to infrared divergences."

As I understand, quark confinement occurs already at the energy scale low enough for the coupling constant goes large enough; at this scale perturbation theory gives us divergent series, and we simply don't know what to do and so can't make prediction. I would feel ok if you blame this failure to the lack of a good interacting vacuum, but I don't get why IR divergence has its part in here as IR divergence = caring about arbitrarily low energy scale?

 

[Urs Schreiber]

At the end of remark 15.18 you wrote that the phenomenon of quark confinment is "invisible to perturbative quantum chromodynamics in its free field vacuum state, due to infrared divergences."

As I understand, quark confinement occurs already at the energy scale low enough for the coupling constant goes large enough; at this scale perturbation theory gives us divergent series, and we simply don't know what to do and so can't make prediction. I would feel ok if you blame this failure to the lack of a good interacting vacuum, but I don't get why IR divergence has its part in here as IR divergence = caring about arbitrarily low energy scale?

The remark you refer to is side remark that would deserve much more discussion to do the topic justice. But briefly, the statement is that QCD happens to be strongly coupled at low energies (corresponding to large wavelengths), which is the downside of its "asymptotic freedom", saying that, conversely, it is weakly coupled at high energies (small wavelengths). The low-energy bound states of QCD (protons, neutrons) are invisible to the QCD perturbation series. In this sense it is an IR problem due to an incorrect perturbative vacuum. I should also say that the arguments that a good interacting vacuum would help to address the issue are plausible, but remain conjectural as far as actual mathematics goes, as far as I am aware.

Apart from pointing it out, this issue is not discussed in the lectures. Generally, it remains a fairly white spot on the map of physics.

Notice that, maybe counter-intuitively, this issue is completely unrelated to the convergence of the perturbation series: The perturbation series never converges for non-trivial theories!, neither for large nor for small non-zero coupling (see here for more on the general non-convergence of the perturbation series).

 

[Duong]

I see. My confusion is only due to the words "IR divergence", which I think must necessarily refer to taking the analytic adiabatic limit (which I also understand as not needed.)

Maybe you told me about this, but I forgot some important points:

• When the coupling is small, do we have a good justification to truncate the series (even for theories that work, e.g. QED)?
• As I understand, for a fixed physical process, an S-matrix scheme gives an expansion series in the couplings with coefficients being fixed numbers. We then do experiment to find out the values of the couplings. Then how do different S-matrix schemes, giving different expansion series now with necessarily same couplings values, all fit with experiments? i.e. how do different renormalization schemes all work when being compared with experiment?

 

[Urs Schreiber]

When the coupling is small, do we have a good justification to truncate the series (even for theories that work, e.g. QED)?

This is a bit of a black magic.

One thing that people do is called "optimal truncation" or "superasymptotics". Underlying this is a simple rule-of-thumb: Given an asymptotic series, find its smallest contribution, then truncate at this term, i.e. form the finite sum of terms up to this smallest one. At least for some classes of asymptotic series one may prove that the resulting value is the closest to the "actual value" of any function with that asymptotic series. This black magic is called optimal truncation or superasymptotics, see here for pointers. Of course if one does not know the "actual value" (which is the situation of interest), then one still does not know how close one is to it, only that one is as close to it as possible by truncating the given asymptotic series.

If you are seriously interested in making sense of the asymptotic perturbation series, the keyword to go for is "resurgence of the trans-series" (here). I sketched some of this when we talked in Hamburg. It seems that people are slooowly making progress with this idea.

 

[Urs Schreiber]

As I understand, for a fixed physical process, an S-matrix scheme gives an expansion series in the couplings with coefficients being fixed numbers. We then do experiment to find out the values of the couplings. Then how do different S-matrix schemes, giving different expansion series now with necessarily same couplings values, all fit with experiments? i.e. how do different renormalization schemes all work when being compared with experiment?

Oh, they don't all fit experiment. Only one of them is supposed to be the right one in any given situation. But the point is that the space of renormalization constants is a countable sequence of affine spaces (see theorem 16.14) and picking any one S-matrix serves to pick one point in each of these affine spaces. This makes the affine spaces become vector spaces and allows to express any other (notably the physically correct) S-matrix now as a sequence of actual numbers (the renormalization constants) relative to the fixed "renormalization scheme".

This is an important subtlety, clarified by Epstein-Glaser back in 1973, which many practicing physicists do not appreciate: Without a choice of renormalization scheme, which is like a choice of coordinates on the space of renormalization parameters, there is no sense in which a renormalization parameter (a "quantum correction") is large or small.

 

[A. Neumaier]

then one still does not know how close one is to it,

Well, the error is typically of the order of the last term used or the first neglected term, though this is only a rule of thumb.