Causal Perturbation Theory - Comments

[Tendex]

But then it's indeed equivalent to the standard techniques for evaluating proper vertex functions within a given renormalization scheme.

In practice, that's what it looks like to me, just more mathematically sophisticated in that it perturbatively avoids the interaction picture and gets rid of the UV divergences in a more elegant way if you like. I found this video that talks about this equivalence stressing the renormalization and distributional issues.

 

[A. Neumaier]

In practice, that's what it looks like to me, just more mathematically sophisticated in that it perturbatively avoids the interaction picture and gets rid of the UV divergences in a more elegant way if you like. I found this video that talks about this equivalence stressing the renormalization and distributional issues.

The moral of this primarily historical and conceptually not demanding lecture on the causal approach is given in minute 47:34:

There are compelling reasons to adopt an effective field interpretation of QFT, but providing the only available solution to the UV problems of the theory is not one of them.

(since causal perturbation theory settles these in a more convincing manner)

 

[strangerep]

I found this video

What a weird, empty talk. It's "conceptually not demanding" because it's almost totally content-free. That's an hour that would have been better spent studying Scharf's textbook.

 

[A. Neumaier]

What a weird, empty talk. It's "conceptually not demanding" because it's almost totally content-free. That's an hour that would have been better spent studying Scharf's textbook.

Well, it is not quite empty. It explains how nonlinear operations with distributions force naively infinite constants and properly done lead to undetermined coefficients, already in simpler situations than quantum field theory. Thus it explains why the well-known distributional nature of quantum fields (even free ones) must run into difficulties and the need for renormalization.

By the way, the author is a philosopher with a PhD in physics.

 

[strangerep]

Well, it is not quite empty. It explains how nonlinear operations with distributions force naively infinite constants [...]

I suspect you see those explanations in the talk because you've already been thinking about the subject for ages.

By the way, the author is a philosopher with a PhD in physics.

Yes, I was aware. To me, that fact explained his tendency to waffle on for quite a while without saying very much.

 

[Tendex]

The talk was given in the context of a Philosophy of physics meeting and the audience was not just theoretical physicists but also philosophers so that is the level and "style" it was geared towards. I agree with Neumaier that some important points about distributions and renormalization were clearly made.

I must say, though, that I much prefer the directly non-perturbative way to deal with these renormalization need related concepts, namely the Källén-Lehmann spectral representation and the explanations in the talk enticed me because they were general enough to relate to free fields, distributions and the constraints imposed in non-perturbative interacting fields and not just to the particular perturbative strategy of Epstein-Glaser that must ultimately be justified by the former to exist.

 

[maline]

Epstein-Glaser?

I don't know very much on these topics, so please enlighten me if I'm wrong. But if I understand correctly, Epstein-Glaser and the like merely describe in precise terms how renormalization is to be done, without making any attempt to justify the procedure "from first principles". They don't describe what it is that we are trying to calculate using these prescriptions. They provide an answer, but not the question!

Indeed, this will be true of any formalism that begins with a Lagrangian. Renormalization does not allow us to calculate the observable values of masses and coupling constants from the Lagrangian, meaning that predicting physical values requires some additional input. A properly specified theory would include "fundamental" parameters that eventually fix the measured values. If all we have is a Lagrangian, we simply do not know what we are calculating.

Another aspect of the same thing (I believe) is that Epstein-Glaser is fundamentally perturbative - that is, the power series is the only output; there is no function that the series is intended to approximate!

So I'm not sure these methods bring us any closer to explaining what we actually mean by QFT interaction terms...

 

[A. Neumaier]

this will be true of any formalism that begins with a Lagrangian.

Causal perturbation theory does not work with Lagrangians!

So I'm not sure these methods bring us any closer to explaining what we actually mean by QFT interaction terms...

In the causal approach, the meaning is precisely given by the axioms for the parameterized S-matrix. The construction is at present perturbative only. Missing are only suitable summation schemes for which one can prove that their result satisfies the axioms nonperturbatively. This is a nontrivial and unsolved step but not something that looks completely hopeless.

 

[A. Neumaier]

In your insights article you write: "To define particular interacting local quantum field theories such as QED, one just has to require a particular form for the first order approximation of S(g). In cases where no bound states exist, which includes QED, this form is that of the traditional nonquadratic term in the action, but it has a different meaning."
Exactly in what way is the meaning different from the one in the traditional action? Does the approximation follow an local action principle or not?
It looks to me like it just uses a renormalized Lagrangian instead of the usual bare one since it changes the moment when the renormalizations is performed to a previous step instead of the usual latter one. But the local action is still there in the background, just more rigurously renormalized from the start.

One can force causal perturbation theory into a Lagrangian framework then it looks like this.

But nothing in causal perturbation theory ever makes any use of Lagrangian formalism or Lagrangian intuition. No action principle is visible in causal perturbation theory; it is not even clear how one should formulate the notion of an action!

Instead, causal perturbation theory starts with a collection of well-informed axioms for the parameterized S-matrix (something not at all figuring in the Lagrangian approach) and exploits the relations that follow from a formal expansion of the solution of these equations around a free quantum field theory. The latter need not be defined by a Lagrangian either but can be constructed directly from irreducible representations of the Poincare group, as in Weinberg"s book (where Lagrangians are introduced much later than free fields).

 

[maline]

To define particular interacting local quantum field theories such as QED, one just has to require a particular form for the first order approximation of S(g). In cases where no bound states exist, which includes QED, this form is that of the traditional nonquadratic term in the action, but it has a different meaning.

This statement needs to be corrected. It indicates that the S-matrix for QED includes a term of first order in $e$, when 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.

To see this, assume WLOG that the photon is outgoing, and consider the energy in the rest frame of the incoming fermion.

 

[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.