# Insights Introduction to Perturbative Quantum Field Theory - Comments (1 Viewer)

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#### A. Neumaier

I believe I did provide a pointer, to the section here , [...]
What you are referring to, and what remains unsolved in generality, is taking the adiabatic limit of the coupling constant.
The former only shows how to construct approximate observables at each order - This has nothing to do with the infrared limit. The latter is precisely the adiabatic limit. It is there (and only there) where the particle content of the theory (and hence issues such as confinement) would appear. For example, in QCD, the perturbative theory is in terms of quarks, but the infrared completed theory has no quarks (due to confinement) but only hadrons.

#### Urs Schreiber

Gold Member
Yes. And as you hint at, this is a totally unfounded argument. In an asymptotic power series in two variables there are an infinite number of terms to be chosen (at each order a growing number more), but nobody concludes that therefore power series at low order are not predictive. They are highly predictive as long as one is in the range of validity of the asymptotic expansion at this order (i.e., typically as long as the first neglected order contributes very little).
Right, the traditional lore highlights a would-be problem that does not actually arise because before it could, another problem kicks in (non-convergence of the perturbation series).

Classical books on diagrammatic techniques describe the construction of diagram series as if they were well defined. However, almost all important perturbation series are hopelessly divergent since they have zero radii of convergence. The first argument to this effect was given by Dyson

[...]

Even though Dyson’s argument is unquestionable, it was hushed up or decried for many years: the scientific community was not ready to face the problem of the hopeless divergency of perturbation series

[...]

The modern status of divergent series suggests that techniques for manipulating them should be included in a minimum syllabus for graduate students in theoretical physics. However, the theory of divergent series is almost unknown to physicists, because the corresponding parts of standard university courses in calculus date back to the mid-nineteenth century, when divergent series were virtually banished from mathematics.

#### Urs Schreiber

Gold Member
The former only shows how to construct approximate observables at each order - This has nothing to do with the infrared limit.
Sure, but why do you say "only"? This is the point that the perturbative interacting observables, as long as they have bounded spacetime support, may consistently be computed in perturbation theory without passing to the adiabatic limit. This says that the perturbation theory is well defined, irrespective of infrared divergencies.

In this sense it seems correct to me to write that "pAQFT deals with the IR-divergencies by organizing the observables into a local net". Or maybe instead of "deals with" it would be better to write "circumvents the problem of". (?)

#### vanhees71

Gold Member
The former only shows how to construct approximate observables at each order - This has nothing to do with the infrared limit. The latter is precisely the adiabatic limit. It is there (and only there) where the particle content of the theory (and hence issues such as confinement) would appear. For example, in QCD, the perturbative theory is in terms of quarks, but the infrared completed theory has no quarks (due to confinement) but only hadrons.
That's why low-energy QCD, if not using lattice-QCD simulations (within their range of applicability), is usually treated in terms of various effective field theories. For the light (+strange) quark domain one uses chiral symmetry (ranging from strict chiral perturbation theory for the ultra-low-energy limit to more or less "phenomenological" Lagrangians constrained by chiral symmetry). Another example is heavy-quark effective theory (also combined with chiral models if it comes to light-heavy systems like D-mesons).

The naive phenomenological physicists approach is indeed that such effective non-renormalizable theories use some low-loop orders of the effective theory with the corresponding low-energy constants, and this provides also predictive power. Often one has to resum ("unitarization"). Another quite popular non-perturbative approach is the renormalization-group approach ("Wetterich equation").

I guess, these more or less handwaving methods are not subject to the mathematically more rigorous approach discussed here, or can the here discussed approaches like pAQFT provide deeper insight to understand, why such methods are sometimes amazingly successful?

Another somewhat related question in my field (relativistic heavy-ion collisions) is the amazing agreement between relativistic viscous hdyrodynamics, derived from relativistic transport theory via the method of moments, Chapman-Enskog, and the like and full relativistic transport theory in a domain (of, e.g., Knudsen numbers around 1), where naively neither of these methods should work. On the other hand the finding of agreement suggest that two methods which are valid in opposite extreme cases (transport theory for dilute gases a la Boltzmann, where the particles scatter only rarely and otherwise are "asymptotically free" most of the time, i.e., large mean-free path vs. ideal hydrodynamics which is exact in the limit of vanishing mean-free path, i.e., the dynamics is slow compared to the typical (local) thermalization time).

#### A. Neumaier

Sure, but why do you say "only"? This is the point that the perturbative interacting observables, as long as they have bounded spacetime support, may consistently be computed in perturbation theory without passing to the adiabatic limit. This says that the perturbation theory is well defined, irrespective of infrared divergencies.

In this sense it seems correct to me to write that "pAQFT deals with the IR-divergencies by organizing the observables into a local net". Or maybe instead of "deals with" it would be better to write "circumvents the problem of". (?)
It avoids having to deal with it, just as standard renormalized perturbation theory does. The infrared divergences still show up (in both cases) when you try to calculate S-matrix elements. Indeed, the perturbatively constructed S-matrix elements cannot even have mathematical existence in case of QCD, because of confinement - there are no asymptotic quark states.

#### Urs Schreiber

Gold Member
It avoids having to deal with it, just as standard renormalized perturbation theory does.
Indeed this is standard renormalized perturbation theory, just done right.

Nothing in pAQFT is alternative to or speculation beyond traditional pQFT. It is traditional pQFT, but done cleanly. The observation that I have been highlighting, that the algebra of quantum observables localized in any compact spacetime region may be computed, up to canonical isomorphism, already with the adiabatically switched S-matrix supported on any neighbourhood of the causal closure of that spacetime region, is "just" the formal justification for why indeed it is possible to ignore the adiabatic limit in perturbation theory.

This is exactly like causal perturbation theory is "just" the formal justification for the standard informal construction of the perturbation series.

Anyway, we don't have a disagreement about the facts, maybe just about the wording.

#### A. Neumaier

Anyway, we don't have a disagreement about the facts, maybe just about the wording.
Yes. pAQFT removes cleanly all UV problems but none of the IR problems. The latter are resolved only by performing the adiabatic limit in causal perturbation theory - and there sit the constructive problems.

#### Urs Schreiber

Gold Member
Yes. pAQFT removes cleanly all UV problems but none of the IR problems.
The problem to be dealt with is that in the absence of the adiabatic limit, the perturbative S-matrix only exists in adiabatically switched form, which, taken at face value, does not make physical sense.

To make sense of causal perturbation theory in the absence of the adiabatic limit one needs to prove that the adiabatically switched S-matrix does, despite superficial appearance, serve to define the correct physical observables.

That proof is not completely trivial. It's result shows that the adiabatically switched S-matrix, while unable to define the global (IR) observables in the adiabatic limit, does, despite superficial appearance, induce the correct local net of localized physical perturbative observables. What is called pAQFT is just the name given to the result of this proof, the well-defined local net of perturbative observables obtained from unphysical switched S-matrices in absence of an adiabatic limit. This way pAQFT deals with the problem.

Without an argument like this you would have to make sense of the adiabatic limit in order to even define the perturbation theory. Which would essentially mean that you'd have to define the non-perturbative theory in order to define the perturbative theory. Which would be pointless.

I suppose the reason why we keep talking past each other is that you keep reading "deal with the IR problem" as "define the theory in the IR". But even before it gets to this ambitious and wide open goal, there is the problem of even defining the perturbation theory without taking the adiabatic limit. This second problem (which logically is the first one to consider) is what pAQFT solves.

#### vanhees71

Gold Member
But isn't the real solution of the IR problem in pQFT to use the correct asymptotic free states a la Kulish and Faddeev,

P. Kulish and L. Faddeev, Asymptotic conditions and infrared divergences in quantum electrodynamics, Theor. Math. Phys., 4 (1970), p. 745.
http://dx.doi.org/10.1007/BF01066485

and many other authors like Kibble?

In the standard treatment one uses arguments a la Bloch&Nordsieck, Kinoshita&Lee&Nauenberg and soft-photon/gluon resummation to resolve the IR problems. It's of course far from being rigorous.

I've also no clue, how you can define proper S-matrix elements without adiabatic switching (in both the remote past and the remote present). Forgetting this leads to pretty confusing fights in the literature. See, e.g.,

F. Michler, H. van Hees, D. D. Dietrich, S. Leupold, and C. Greiner, Off-equilibrium photon production during the chiral phase transition, Annals Phys., 336 (2013), p. 331–393.
http://dx.doi.org/10.1016/j.aop.2013.05.021
http://arxiv.org/abs/1310.5019

All this is, of coarse, far from being mathematically rigorous, but maybe it's possible to make it rigorous in the sense of pAQFT?

#### A. Neumaier

induce the correct local net of localized physical perturbative observables.
but this has nothing to do with the infrared (i.e., low energy) behavior, so you shouldn't use the term IR in this connection.

The basic conflict in QCD (or quantum Yang-Mills) is that there are no physical quark fields although there are perturbative quark fields.

In QED, the conflict is less obvious but you may look at Weinberg's Volume 1, Chapter 13 for a discussion of IR effects in QED. These effects appear although the renormalized perturbative asymptotic series is already completely well-defined! The reason is that at a given energy the number of massless particles produced is unbounded, and to get physical results one must integrate over all these soft photon degrees of freedom. This is most correctly (but still in a mathematically nonrigorous way) handled by using coherent state techniques, as in the references given by Handrik van Hees.

#### Urs Schreiber

Gold Member
but this has nothing to do with the infrared (i.e., low energy) behavior
Exactly, and so one needs to prove that this may indeed be ignored in the perturbation theory. It is commonly said that causal perturbation theory disentangles the UV from the IR effects, but this only becomes completely true once one proves that the adibatically switched S-matrix produces correct physical observables even without taking its adiabatic limit.

I feel like we have exchanged this same point a couple of times now. And we still don't disagree about any facts, the only disagreement you have seems to be against the words by which I referred to the issue of proving that causal perturbation theory makes physical sense without taking the adiabatic limit. I called this "deal with the IR divergences". You seem to be saying that "deal with the IR divergences" sounds to you like "define the theory in the IR". Maybe a resolution would be if I changed the wording to "deal with the decoupling of the IR divergences"?

I am open for suggestions of the rewording, if it gets us past this impasse. You have so many interesting things to say, it is a pity that we seem to be stuck on a factual non-issue.

#### Urs Schreiber

Gold Member
By the way, the next article in the series is ready, but it is being delayed by some formatting problems.

I have prepared my code for the next article in the "Instiki"-markup language, on an nLab page here
My plan had been to simply port this code here to Physics Forums. Unfortunately, this turns out to be impractical, due to numerous syntax changes that would need to be made.

With Greg we are looking for a solution now. A technically simple solution would be to simply include that webpage inside an "iframe" within the PF-Insights article. But maybe this won't be well received with the readership here? If anyone with experience in such matters has a suggestion, please let me know.

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#### A. Neumaier

Maybe a resolution would be if I changed the wording to "deal with the decoupling of the IR divergences"?
Saying something like ''cleanly decouples the fully resolved UV issues from the (in causal perturbation theory still unresolved) IR issues'' would be fine with me.

You have so many interesting things to say, it is a pity that we seem to be stuck on a factual non-issue.
Seemingly being stuck is also a factual non-issue. As you can see from my contributions, even when I discuss terminology, I enrich it with interesting information for other readers....

#### vanhees71

Gold Member
That's interesting. I always thought the IR divergences of standard PT are easier cured than the UV problems. It's just the soft-photon/gluon (or whatever is soft in some model with massless quanta) resummation, and then there's "theorems" like Bloch/Nordsieck and/or Kinoshita/Lee/Nauenberg:

https://en.wikipedia.org/wiki/Kinoshita-Lee-Nauenberg_theorem

What are the issues that you call them "still unresolved" in pAQFT?

#### A. Neumaier

What are the issues that you call them "still unresolved" in pAQFT?
The IR problem in QED is well understood only in the absence of nuclei (i.e., if only external fields are present beyond photons, electrons and positrons). If there are nuclei (whether assumed pointlike or with appropriate assumed form factors doesn't matter much) there are many bound states, and their treatment is very poorly understood.

Symptomatic for the state of affairs is the remark in Weinberg's QFT book, Vol.1, p.560: ''It must be said that the theory of relativistic effects and radiative corrections in bound states is not yet in entirely satisfactory state.'' This is a very euphemistic description of what in reality is a complete and ill-understood mess.

In QCD all low energy phenomena involve bound states - due to confinement, and these problems permeate everything.

The Lee-Nauenberg theorem is flawed when analyzed carefully:
https://arxiv.org/abs/hep-ph/0511314

#### A. Neumaier

By the way, the next article in the series is ready, but it is being delayed by some formatting problems.

I have prepared my code for the next article in the "Instiki"-markup language, on an nLab page here
I need to digest the concept of a smooth set employed in your setting. Are there relations to the Conceptual Differential Calculus of Wolfgang Bertram? (This exists in a number of variants, one of them being in https://arxiv.org/abs/1503.04623 .) Since this is somewhat off-topic here, I asked a corresponding questions at PhysicsOverflow; please reply there.

My plan had been to simply port this code here to Physics Forums. Unfortunately, this turns out to be impractical, due to numerous syntax changes that would need to be made.

With Greg we are looking for a solution now. A technically simple solution would be to simply include that webpage inside an "iframe" within the PF-Insights article. But maybe this won't be well received with the readership here? If anyone with experience in such matters has a suggestion, please let me know.
I always converted by hand, though it takes a considerable amount of time.

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#### DrDu

Interesting article!
You say that pQFT is a perturbational expansion not only in coupling constant but also in Plancks constant. The latter point is not immediately clear to me.

#### vanhees71

Gold Member
It is since the number of loops counts the powers of $\hbar$. This is clear from the path-integral formalism since you can understand the Dyson series also as saddle-point approximation of the path integral. See, e.g., Sect. 4.6.6 in

https://th.physik.uni-frankfurt.de/~hees/publ/lect.pdf

#### Demystifier

2018 Award
Happily, no experiment occurs in an infinite laboratory, so IR divergences are a mere calculation inconvenience (it is not very practical to perform analytic calculations with big but finite IR cutoffs), not a genuine physical problem.

#### Urs Schreiber

Gold Member
You say that pQFT is a perturbational expansion not only in coupling constant but also in Planck's constant. The latter point is not immediately clear to me.
Here is how to see it:

The explicit $\hbar$-dependence of the perturbative S-matrix is

$$S(g_{sw} L_{int} + j_{sw} A) = T \exp\left( \tfrac{1}{i \hbar} \left( g_{sw} L_{int} + j_{sw} A \right) \right) \,,$$

where $T(-)$ denotes time-ordered products. The generating function

$$Z_{g_{sw}L_{int}}(j_{sw} A) \;:=\; S(g_{sw}L_{int})^{-1} \star S(g_{sw}L_{int} + j_{sw} A)$$

involves the star product of the free theory (the normal-ordered product of the Wick algebra). This is a formal deformation quantization of the Peierls-Poisson bracket, and therefore the commutator in this algebra is a formal power series in $\hbar$ that, however, has no constant term in $\hbar$ (but starts out with $\hbar$ times the Poisson bracket, followed by possibly higher order terms in $\hbar$):

$$[L_{int},A] \;=\; \hbar(\cdots) \,.$$

Now by Bogoliubov's formula the quantum observables are the derivatives of the generating function

$$\hat A := \tfrac{1}{i \hbar} \frac{d}{d \epsilon} Z_{g_{sw}L_{int}}(\epsilon j A)\vert_{\epsilon = 0}$$

Schematically the derivative of the generating function is of the form

\begin{aligned} \hat A & := \tfrac{1}{i \hbar} \frac{d}{d \epsilon} Z_{g_{sw}L_{int}}(\epsilon j A)\vert_{\epsilon = 0} \\ & = \exp\left( \tfrac{1}{i \hbar}[g_{sw}L_{int}, -] \right) (j A) \end{aligned} \,.

(The precise expression is given by the "retarded products", see (Rejzner 16, prop. 6.1).)
By the above, the exponent $\tfrac{1}{\hbar} [L_{int},-]$ here yields a formal power series in $\hbar$, and hence so does the full exponential.

Here is how this relates to loop order in the Feynman perturbation series:

Each Feynman diagram $\Gamma$ is a finite labeled graph, and the order in $\hbar$ to which this graph contributes is

$$\hbar^{ E(\Gamma) - V(\Gamma) }$$

where

1. $V(\Gamma) \in \mathbb{N}$ is the number of vertices of the graph
2. $E(\Gamma) \in \mathbb{N}$ is the number of edges in the graph.
This comes about (see at S-matrix -- Feynman diagrams and Renormalization for details) because

1) the explicit $\hbar$-dependence of the S-matrix is

$$S\left(\tfrac{g}{\hbar} L_{int} \right) = \underset{k \in \mathbb{N}}{\sum} \frac{g^k}{\hbar^k k!} T( \underset{k \, \text{factors}}{\underbrace{L_{int} \cdots L_{int}}} )$$

2) the further $\hbar$-dependence of the time-ordered product $T(\cdots)$ is

$$T(L_{int} L_{int}) = prod \circ \exp\left( \hbar \int \omega_{F}(x,y) \frac{\delta}{\delta \phi(x)} \otimes \frac{\delta}{\delta \phi(y)} \right) ( L_{int} \otimes L_{int} ) \,,$$

where $\omega_F$ denotes the Feynman propagator and $\phi(x)$ the (generic) field observable at point $x$ (where we are notationally suppressing the internal degrees of freedom of the fields for simplicity, writing them as scalar fields, because this is all that affects the counting of the $\hbar$ powers).

The resulting terms of the S-matrix series are thus labeled by

1. the number of factors of the interaction $L_{int}$, these are the vertices of the corresponding Feynman diagram and hence each contibute with $\hbar^{-1}$

2. the number of integrals over the Feynman propagator $\omega_F$, which correspond to the edges of the Feynman diagram, and each contribute with $\hbar^1$.

Now the formula for the Euler characteristic of planar graphs says that the number of regions in a plane that are encircled by edges, the faces, here thought of as the number of "loops", is

$$L(\Gamma) = 1 + E(\Gamma) - V(\Gamma) \,.$$

Hence a planar Feynman diagram $\Gamma$ contributes with

$$\hbar^{L(\Gamma)-1} \,.$$

So far this is the discussion for internal edges. An actual scattering matrix element is of the form

$$\langle \psi_{out} \vert S\left(\tfrac{g}{\hbar} L_{int} \right) \vert \psi_{in} \rangle \,,$$

where

$$\vert \psi_{in}\rangle \propto \tfrac{1}{\sqrt{\hbar^{n_{in}}}} \phi^\dagger(k_1) \cdots \phi^\dagger(k_{n_{in}}) \vert vac \rangle$$

is a state of $n_{in}$ free field quanta and similarly

$$\vert \psi_{out}\rangle \propto \tfrac{1}{\sqrt{\hbar^{n_{out}}}} \phi^\dagger(k_1) \cdots \phi^\dagger(k_{n_{out}}) \vert vac \rangle$$

is a state of $n_{out}$ field quanta. The normalization of these states, in view of the commutation relation $[\phi(k), \phi^\dagger(q)] \propto \hbar$, yields the given powers of $\hbar$.

This means that an actual scattering amplitude given by a Feynman diagram $\Gamma$ with $E_{ext}(\Gamma)$ external vertices scales as

$$\hbar^{L(\Gamma) - 1 + E_{ext}(\Gamma)/2 } \,.$$

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