Causal Perturbation Theory - Comments

[A. Neumaier]

in fact the first term is of order $e^2$. There are no one-vertex processes, because it isn't possible for all three particles (two fermions and a photon) to be on-shell.

This is true but irrelevant for causal perturbation theory.

First order means first order in the function $g$ in terms of which the expansion is made, not first order in coupling constants. Your argument does not apply since the parameterized S-matrix $S(g)$ is just a generating function, not an S-matrix in the physical sense. Only the adiabatic limit where $g\to 1$, has such an interpretation. I added a corresponding remark to my Insight article.

 

[maline]

First order means first order in the function g in terms of which the expansion is made, not first order in coupling constants.

$g$ is the test function that switches on the interaction, correct? So anything first-order in $g$ is also first-order in $e$.

Your argument does not apply since the parameterized S-matrix S(g) is just a generating function, not an S-matrix in the physical sense.

Oh, I think I see. You are saying that since $g(x)$ is not translation-invariant, energy and momentum are not conserved by $S(g)$, and so first-order processes are indeed possible. These terms will then vanish in the IR limit.

 

[A. Neumaier]

$g$ is the test function that switches on the interaction, correct? So anything first-order in $g$ is also first-order in $e$.

Oh, I think I see. You are saying that since $g(x)$ is not translation-invariant, energy and momentum are not conserved by $S(g)$, and so first-order processes are indeed possible. These terms will then vanish in the IR limit.

Yes. This is also the point where the attempt of a Lagrangian interpretation breaks down.

 

[maline]

since g(x) is not translation-invariant, energy and momentum are not conserved by S(g), and so first-order processes are indeed possible.

But this raises another issue: Per the axioms, $S(g)$ should not take us out of single-particle subspaces. But without 4-momentum conservation, won't a single fermion have an amplitude to spontaneously emit photons?

 

[Tendex]

But this raises another issue: Per the axioms, $S(g)$ should not take us out of single-particle subspaces. But without 4-momentum conservation, won't a single fermion have an amplitude to spontaneously emit photons?

That's only in the infinite volume limit g=1, the perturbative construction won't reach it.
Then again I feel the departure from the Lagrangian naive picture is obtained basically because CP theory exploits this perturbative artifact for its own benefit, kind like a sanitized version of Feynman's quick and dirty diagrams where in the latter since the renormalization is deferred to a later stage you get off-shell internal lines as artifacts of perturbation theory instead, to its mathematical detriment. It's all easily seen as a perturbative trade-off between 4-momentum conservation versus "on-shellness" where keeping the latter by using renormalized distributions is more mathematically elegant .

 

[maline]

That's only in the infinite volume limit g=1, the perturbative construction won't reach it.

What? I am talking about processes like $e \rightarrow e+ \gamma.$ When $g=1$ this cannot happen because of conservation laws, but otherwise it should occur already at first order in perturbation theory.

 

[Tendex]

What? I am talking about processes like $e \rightarrow e+ \gamma.$ When $g=1$ this cannot happen because of conservation laws, but otherwise it should occur already at first order in perturbation theory.

Sure, they occur. My comment was about you using axioms for $S(g)$ when you ought to use those for $g$ formal power series only. Then there is no issue that I see. If you are asking about spontaneous emission as a non-perturbative effect that's out of the scope of causal perturbative theory, and you don't want to use the off-shell argument from the traditional perturbative approach since avoiding it is mainly what brings us to CPT.
In fact, in the 50 years gone by from Epstein-Glaser formulation this causal approach has not led to a single clue about a non-perturbative theory to interacting fields.

 

[A. Neumaier]

But this raises another issue: Per the axioms, $S(g)$ should not take us out of single-particle subspaces. But without 4-momentum conservation, won't a single fermion have an amplitude to spontaneously emit photons?

You are right. My axioms, inferred from the first (1989) edition of Scharf's QED book, were too strong; I corrected them by weakening the two stability axioms. The second (1995) edition silently corrected this mistake and only assume it (quite implicitly) in the adiabatic limit $g\to 1$, discussed there in Section 3.11 and later 4.1. I had not noticed that before since I didn't reread all the detail in the second edition. I added detailed references to the insight article indicating where the axioms are stated or used.

My comment was about you using axioms for $S(g)$ when you ought to use those for $g$ formal power series only.

The extraction of the nonperturbative axioms from Scharf was not done by Scharf but by me in the insight article. These axioms make sense nonperturbatively, even though the construction based on it is only perturbative. Presumably some resummation scheme may turn the latter into a provably full construction, though how to do this is presently unsolved.

In fact, in the 50 years gone by from Epstein-Glaser formulation this causal approach has not led to a single clue about a non-perturbative theory to interacting fields.

This cannot be held against it. Other approaches also did not lead to a single clue.

 

[vanhees71]

Of course, also axiomatic QFT cooks only with water and you have to take the proper limit of the regularization for the renormalized quantities as in the more sloppy standard approaches.

 

[A. Neumaier]

Of course, also axiomatic QFT cooks only with water and you have to take the proper limit of the regularization for the renormalized quantities as in the more sloppy standard approaches.

Not quite. There is no UV regularization in causal perturbation theory. The only water in it is the lack of convergence of the asymptotic series for $S(g)$. Thus without exact resummation one has no clue what the nonperturbative $S(g)$ would be.

 

[vanhees71]

What else is the "smearing" of the operators than a "regularization of the distributions"?

 

[maline]

In the article, you mention the hope that a "suitable summation scheme" will be found for Causal Perturbation Theory, thus proving the rigorous existence of $S(g)$. To me this hope seems unsupported and wildly optimistic. Remember that this is a power series in $g$, so we need is a summation that works all the way up to $g=1$. Furthermore, the summation must work for all possible energies of the incoming particles! This despite the fact that for large energies, even the lowest few term in the series will grow wildly rather than shrinking.

Also note that the summation scheme will probably have to be a relatively "tame" one like Borel summation. "Wilder" ideas like zeta function regularization are unlikely to give an $S(g)$ satisfying the axioms.

 

[A. Neumaier]

What else is the "smearing" of the operators than a "regularization of the distributions"?

In which sense could the occurrence of a test function in the definition $$\int g(x)\delta(x)dx=g(0)$$ of the Dirac delta distribution be a regularization of the latter?

The use of test functions is inherent in the definition of a distribution. Thus using test functions does not regularize the distribution in any meaningful sense. Nothing is smeared.

 

[vanhees71]

Sure, you call it not that, but that's what's behind it physically.

 

[A. Neumaier]

In the article, you mention the hope that a "suitable summation scheme" will be found for Causal Perturbation Theory, thus proving the rigorous existence of $S(g)$. To me this hope seems unsupported and wildly optimistic.

Probably only because you haven't thought enough about this matter.

Remember that this is a power series in $g$, so we need is a summation that works all the way up to $g=1$.

One has to sum a power series in a single variable $\tau$, introduced by replacing $g$ with $\tau g$. The resummed expression at $\tau=1$ will be a function of $g$ that can be analyzed for the adiabatic limit $g\to 1$ in the same way as one now analyzes this limit for the few loop contributions.

Furthermore, the summation must work for all possible energies of the incoming particles! This despite the fact that for large energies, even the lowest few term in the series will grow wildly rather than shrinking.

The whole point of resummation is that it includes important contributions from all energies. The size of the terms in the power series is completely irrelevant for the behavior of the resummed formulas.

Also note that the summation scheme will probably have to be a relatively "tame" one like Borel summation. "Wilder" ideas like zeta function regularization are unlikely to give an $S(g)$ satisfying the axioms.

Borel summation is not sufficient because of the appearance of renormalon contributions. The promising approach is via resurgent transseries, an approach much more powerful than Borel summation.

 

[maline]

One has to sum a power series in a single variable $\tau$, introduced by replacing $g$ with $\tau g$. The resummed expression at $\tau=1$ will be a function of $g$ that can be analyzed for the adiabatic limit $\g\to 1$ in the same way as one now analyzes this limit for the few loop contributions.

The whole point of resummation is that it includes important contributions from all energies. The size of the terms in the power series is completely irrelevant for the behavior of the resummed formulas.

Borel summation is not sufficient because of the appearance of renormalon contributions. The promising approach is via resurgent transseries, an approach much more powerful than Borel summation.

Thank you for this, I will need to try to absorb this material before responding.

 

[A. Neumaier]

Sure, you call it not that, but that's what's behind it physically.

No. In physics, regularization always involves changing a problem to a nearby less singular problem and restoring the original problem later by taking a limit. Nothing like this happens in causal perturbation theory in the UV, i.e., regarding the treatment of the distributions. Note that the adiabatic limit $g\to 1$ is not needed for the construction of the local field operators and hence for the perturbative construction of the quantum field theory in terms of formally local operators in a Hilbert space.

On the perturbative level, the adiabatic limit is the only limit appearing in the causal approach, needed for the recovery of the IR regime, including the physical S-matrix. The lack of convergence of the perturbative series also has nothing to do with regulaization.

 

[vanhees71]

I don't think that we discuss the lack of convergence of the perturbative series. That's another issue. It's an asymptotic series not a convergent one already in simply QM problems.

In my understanding, the problem with not taking the said adiabatic limit however seems to be that you loose Poincare invariance then. In that sense also this approach to renormalization is just another type of regularization with the necessity of the limit to be taken at the end in order to have a Poincare invariant/covariant scheme.

 

[A. Neumaier]

In my understanding, the problem with not taking the said adiabatic limit however seems to be that you lose Poincare invariance then.

The field operators defined by functional differentiation with respect to he test fuctions $g$ satisfy causal commutation rules and transform in a Poincare covariant way. For this to work, $S(g)$ is just a formal object without any pretense of being an S-matrix. No adiabatic limit is involved here.

In that sense also this approach to renormalization is just another type of regularization with the necessity of the limit to be taken at the end in order to have a Poincare invariant/covariant scheme.

The adiabatic limit is only needed to recover the Poincare invariant S-matrix. However, this limit is a long distance (low energy) IR limit.

On the other hand, conventional regulaization schemes such as dimensional regularization or procedures with a cutoff regularize instead the short distance (high energy) UV behavior. The latter would correspond to requiring somewhere in causal peturbation theory a limit where $g$ approaches a delta function. But such a limit is never even contemplated in the literature on the causal approach.

 

[Tendex]

I actually see the analogy as looking at the CPT treatment of distributions kind like an IR regularization, not taking the usual UV regularization limit to $g$ in CPT which it obviously doesn't.

 

[vanhees71]

The field operators defined by functional differentiation with respect to he test fuctions $g$ satisfy causal commutation rules and transform in a Poincare covariant way. For this to work, $S(g)$ is just a formal object without any pretense of being an S-matrix. No adiabatic limit is involved here.

The adiabatic limit is only needed to recover the Poincare invariant S-matrix. However, this limit is a long distance (low energy) IR limit.

On the other hand, conventional regulaization schemes such as dimensional regularization or procedures with a cutoff regularize instead the short distance (high energy) UV behavior. The latter would correspond to requiring somewhere in causal peturbation theory a limit where $g$ approaches a delta function. But such a limit is never even contemplated in the literature on the causal approach.

Now I'm puzzled. I thought the entire business of the causal PT approach is the usual UV regularization. I guess, I have to study this approach in more detail to understand what's behind it.

 

[A. Neumaier]

Now I'm puzzled. I thought the entire business of the causal PT approach is the usual UV regularization. I guess, I have to study this approach in more detail to understand what's behind it.

The causal approach achieves UV renormalization without any regularization. But to be able to work with free fields it regularizes the physical S-matrix in the IR by means of test functions with compact support (rather than arbitrary smooth ones), which amounts to switching off the interaction at large distances. The adiabatic limit restores the long distance interactions.

This is fully analogous to truncating short range potentials in quantum mechanics in order to be able to use free particles at large negative and positive times to of obtain an S-matrix without any limit. In quantum mechanics, the adiabatic limit restores the original potential. The mathematically proper treatment has to introduce a Hilbert space of asymptotic states and a Möller operator that transforms from infinite time to finite time. This makes the whole procedure less intuitive and requires more machinery from functional analysis, described rigorously in the 4 mathematical physics volumes of Reed and Simon.

 

[Tendex]

Now I'm puzzled. I thought the entire business of the causal PT approach is the usual UV regularization. I guess, I have to study this approach in more detail to understand what's behind it.

There are no UV divergences in CPT since it uses renormalized distributions, so no UV regularization is needed. The causal approach with $S(g)$ is based on switching the interaction in a finite spacetime region featured by the function $g(x)$ of which $S(g)$ is the functional. The physical S-matrix corresponds to the limit when the spacetime volume goes to infinity. So the adiabatic limit is an IR limit.

 

[Tendex]

There is no regularization in causal perturbation theory.

But to be able to work with free fields it regularizes the physical S-matrix in the IR by means of test functions with compact support (rather than arbitrary smooth ones), which amounts to switching off the interaction at large distances.

I guess in the first sentence you meant UV regularization then. I thought vanhees was all along talking about the obvious IR regularization in CPT. In any case there is some kind of regularization always involved.

 

[A. Neumaier]

I guess in the first sentence you meant UV regularization then.

Yes, corrected. In the above context, I was referring to regularization in the UV sense, like vanhees7. I became more precise when it was clear that misunderstandings resulted.

 

[vanhees71]

The causal approach achieves UV renormalization without any regularization. But to be able to work with free fields it regularizes the physical S-matrix in the IR by means of test functions with compact support (rather than arbitrary smooth ones), which amounts to switching off the interaction at large distances. The adiabatic limit restores the long distance interactions.

This is fully analogous to truncating short range potentials in quantum mechanics in order to be able to use free particles at large negative and positive times to of obtain an S-matrix without any limit. In quantum mechanics, the adiabatic limit restores the original potential. The mathematically proper treatment has to introduce a Hilbert space of asymptotic states and a Möller operator that transforms from infinite time to finite time. This makes the whole procedure less intuitive and requires more machinery from functional analysis, described rigorously in the 4 mathematical physics volumes of Reed and Simon.

But renormalization has nothing to do with regularization. Regularization in the usual approach is just to get finite quantities to be able to calculate the "unrenormalized quantities" before you renormalize them, i.e., to express the unobservable "infinite constants of the theory" in terms of "measuarable finite ones".

BPHZ-like schemes directly define the renormalized proper vertex functions in a given scheme without previous regularization.

As I said, I guess I've simply not really understood, how this special scheme of causal PT works to regularize/renormalize the UV divergences, and it must somehow handle this first, before one can address the IR/collinear divergences, which only occur in theories with massless fields.

 

[A. Neumaier]

Regularization in the usual approach is just to get finite quantities to be able to calculate the "unrenormalized quantities" before you renormalize them, i.e., to express the unobservable "infinite constants of the theory" in terms of "measurable finite ones". [...] I've simply not really understood, how this special scheme of causal PT works to regularize/renormalize the UV divergences,

Such "infinite constants of the theory" or ''UV divergences'' nowhere arise in the causal approach, hence no regularization is needed. Even the word ''re''normalization is a misnomer in this approach, since from the start only physical parameters appear.

The only remnant of the traditional renormalization approach is due to subtracted dispersion relations, which introduces at each order some constants. But these are fixed immediately by relations that lead to a unique distribution splitting.

 

[vanhees71]

So what's the strategy to get rid of the usual infinities and where comes renormalization in in the scheme of causal perturbation theory. The point is that you need renormalization no matter whether you have infinities or not. If you don't have infinities of course you don't need regularization.

The reason why I never was much interested in the book by Scharf was that my (maybe too superficial) glance over it I had indeed the impression that it's not more than the use of subtracted dispersion relations. This is of course also a way to renormalize in the standard approach, as shown in Landau Lifshitz vol. IV. I'm only not so sure, whether it's practical for higher than one-loop calculations.

 

[A. Neumaier]

So what's the strategy to get rid of the usual infinities

The strategy is to never introduce them. The distributions used have the mathematically correct singularities, and these distributions are manipulated in a mathematically well-defined way. Thus infinities cannot appear by design.

where comes renormalization in in the scheme of causal perturbation theory.

Only in the fact that the final results agree with the results of conventional renormalization schemes. The starting point (i.e., the axioms and the first order ansatz) does not refer to anything that would need renormalization.

The point is that you need renormalization no matter whether you have infinities or not.

This is a wrong, unsupported claim. One needs it only if one starts with the ill-defined Dyson series.

I'm only not so sure, whether it's practical for higher than one-loop calculations.

How many loops are you using for your QCD calculations?

 

[A. Neumaier]

The recent book

• Michael Dütsch, From Classical Field Theory to Perturbative Quantum Field Theory, Birkhäuser 2019

treats causal perturbation theory in a different way than Scharf, using off-shell deformation quantization rather than Fock space as the starting point. From the preface:

the aim of this book is to give a logically satisfactory route from the fundamental principles to the concrete applications of pQFT, which is well intelligible for students in mathematical physics on the master or Ph.D. level. This book is mainly written for the latter; it is made to be used as basis for an introduction to pQFT in a graduate-level course.
[...]
This formalism is also well suited for practical computations, as is explained in Sect. 3.5 (“Techniques to renormalize in practice”) and by many examples and exercises.
[...]
The observables are constructed as formal power series in the coupling constant and in $\hbar$.
[...]
This book yields a perturbative construction of the net of algebras of observables (“perturbative algebraic QFT”, Sect. 3.7), this net satisfies the Haag–Kastler axioms [93] of algebraic QFT, expect that there is no suitable norm available on these formal power series.

In contrast to Scharf, he often uses renormalization language. However, he also writes (p.165, his italics):

However, we emphasize: Epstein–Glaser renormalization is well defined without any regularization or divergent counter terms. We introduce these devices only as a method for practical computation of the extension of distributions (see Sect. 3.5.2 about analytic regularization), or to be able to mimic Wilson’s renormalization group (see Sect. 3.9).

 

[vanhees71]

The strategy is to never introduce them. The distributions used have the mathematically correct singularities, and these distributions are manipulated in a mathematically well-defined way. Thus infinities cannot appear by design.

Only in the fact that the final results agree with the results of conventional renormalization schemes. The starting point (i.e., the axioms and the first order ansatz) does not refer to anything that would need renormalization.

This is a wrong, unsupported claim. One needs it only if one starts with the ill-defined Dyson series.

How many loops are you using for your QCD calculations?

Well, I guess I'm only wasting your time and I should rather make another attempt to read Scharf's book again, but why is it wrong that you need renormalization in perturbation theory?

In the conventional theory you need to choose a renormalization scheme and the proper vertex functions depend on this choice. The S-matrix elements are independent of the choice (at the order you've calculated them), which is the content of the renormalization group equations. So is causal perturbation theory just a special choice of a renormalization scheme and where in this scheme are the renormalization-group equations hidden?

 

[A. Neumaier]

How many loops are you using for your QCD calculations?

why is it wrong that you need renormalization in perturbation theory?

In the conventional theory you need to choose a renormalization scheme and the proper vertex functions depend on this choice. The S-matrix elements are independent of the choice (at the order you've calculated them), which is the content of the renormalization group equations. So is causal perturbation theory just a special choice of a renormalization scheme and where in this scheme are the renormalization-group equations hidden?

The parameterization of the S-matrix of QED in terms of the physical mass and charge fixes the first order term in $S(g)$ and hence everything, so there is nothing to be renormalized.

But there is some freedom in the construction. It can be used to introduce a redundant parameter at the cost of introducing running coupling constants and more complex formulas. Since the physical electron charge corresponds to a running charge at zero energy, the parameterization of the S-matrix in terms of the physical mass and charge corresponds to a conventional renormalization at zero photon mass.

Scharf writes in the 1995 edition:

(p.260:) It should be remembereded that the vacuum polarization tensor, for other reasons, is normalized by the conditions (3.6.34, 35). Then, it gives no contribution to charge normalization, too. If one assumes a different normalization of $\Pi(k)$, then the coupling constant in $T_1(x)$ and the physical charge are no longer equal. This is the starting point for the renormalization group. This subject will be discussed in Sect. 4.8.

(p.271:) The subject of this section is called renormalizability in other textbooks. The reader will agree that the prefix "re" is of no use here. By renormalization we always mean finite renormalization of an already normalized T-distribution, as discussed in Sect. 3.13, for example.

The redundant parameter would have no effect in the nonperturbative solution. But since the expansion point is different, it leads to different results at each order of perturbation theory. These perturbative results are then related by finite renormalizations in terms of a Stückelberg-Petermann renormalization group. It expresses the charge appearing in the coupling constant - now no longer the experimental charge but running with the energy scale - in terms of the physical mass and charge.

Thus renormalization is finite and optional. Maybe this is special to QED since the free physical parameters have a direct physical meaning.

 

[Tendex]

I'm also new to this approach and haven't got around to getting a copy of Scharf's book yet, so I might be misunderstanding concepts. Acording to your comment in #62 about the moral of the video lecture I linked, it seemed to me that the CPT approach is in some sense opposite to the effective field renormalization group approach that is oriented to the traditional perturbative approach with Feynman propagators plagued with UV divergences, so it looks to me that trying to recover renormalization group equations in it goes against the spirit of CPT, is this so?

 

[A. Neumaier]

I'm also new to this approach and haven't got around to getting a copy of Scharf's book yet, so I might be misunderstanding concepts. Acording to your comment in #62 about the moral of the video lecture I linked, it seemed to me that the CPT approach is in some sense opposite to the effective field renormalization group approach that is oriented to the traditional perturbative approach with Feynman propagators plagued with UV divergences, so it looks to me that trying to recover renormalization group equations in it goes against the spirit of CPT, is this so?

Not really. There are two very different renormalization groups which should not be mixed up. The first one by Wilson is important in nonequilibrium thermodynamics and for condensed but approximate descriptions in terms of composite fields. The second, older one by Stückelberg is the most important one in local quantum field theory and is not related to effective fields but to overparameterization.

• The Wilson renormalization group (actually only a semigroup, but the name has stuck) is based on removing high energy degrees of freedom by repeated infinitesimal coarse graining. It loses information and hence leads to approximate effective field theories and the Wetterich renormalization group equation.
• The Stückelberg-Petermann renormalization group (a true group) expresses the running coupling constant through the Callan-Symanzik renormalization group equation. This group is due to the existence of a redundant mass/energy parameter and has nothing to do with effective fields, as it does not change the contents of the theory, only the perturbative expansion.

The Stückelberg-Petermann renormalization group already appears in the quantum mechanics of an anharmonic oscillator when one wants to relate the perturbation series obtained by perturbing around Hamiltonians describing harmonic oscillators with different frequency. The frequency chosen is arbitrary and hence nonphysical; it is the analogue of the renormalization scale in QFT.

 

[Tendex]

Ok, so in #62 you(and the lecturer) just meant that effective theory in the sense of Stückelberg-Petermann renormalization group was not as nice dealing with perturbative UV divergences as CPT?

Also I believe in particle physics they sometimes mix the philosophy of the Wilsonian RG approach with the perturbative RG in their quest for machines with ever higher energies.