Interview with Mathematician and Physicist Arnold Neumaier - Comments

[A. Neumaier]

Greg Bernhardt submitted a new PF Insights post

Interview with Mathematician and Physicist [URL='https://www.physicsforums.com/insights/author/a-neumaier/']Arnold Neumaier[/URL]

Continue reading the Original PF Insights Post.

Arnold will welcome science questions and comments only.

 

[dextercioby]

A warm Thank you to all 4 of you for making this read possible.

 

[bhobba]

I can't express my gratitude enough for this extremely insightful and informative interview.

Arnold has always been one of my favorite posters that I have learned an enormous amount from.

In particular his efforts to dispel, and explain in detail, myths like the reality of virtual particles is much appreciated, and IMHO very very important because of the huge amount of confusion about the issue, especially amongst beginners.

Thanks
Bill

 

[ftr]

Arnold, I am interested in your view on EPR. Also, the physical alternative to ^ virtual particles" seem to be a taboo. It seems to me that non locality is of essence.

 

[A. Neumaier]

Arnold, I am interested in your view on EPR.

See the following extended discussions:

https://www.physicsforums.com/threads/an-abstract-long-distance-correlation-experiment.852684/

https://www.physicsforums.com/threa...is-not-weird-unless-presented-as-such.850860/

https://www.physicsforums.com/threads/collapse-from-unitarity.860627/

the physical alternative to ^ virtual particles" seem to be a taboo. It seems to me that non locality is of essence.

The physical alternative is the interpretation of virtual particles as contributions to an infinite series of scattering amplitude approximations. There is nothing nonlocal in relativistic QFT. See my post on extended causality.

 

[DrChinese]

See my post on extended causality.

... Here are the definitions:

• Point causality: Properties of a point object depend only on its closed past cones, and can influence only its closed future cones.
• Extended causality: Joint properties of an extended object depend only on the union of the closed past cones of their constituent parts, and can influence only the union of the closed future cones of their constituent parts.
• Separable causality: Joint properties of an extended object consist of the combination of properties of their constituent points.

I believe that only extended causality is realized in Nature. It can probably be derived from relativistic quantum field theory. If this is true, there is nothing acausal in Nature. In any case, causality in this weaker, much more natural form is not ruled out by current experiments.

Nice definitions. I assume you consider an entangled system (of 2 or more particles) to be an "extended object", correct?

 

[A. Neumaier]

Nice definitions. I assume you consider an entangled system (of 2 or more particles) to be an "extended object", correct?

Yes.

Even a single particle is a slightly extended object. In quantum field theory, there are no point particles - only ''pointlike'' particles (which means particles obtained from point particles through renormalization - which makes them slightly extended).

On the other hand, entangled systems of 2 particles can be very extended objects, if the distance of the center of mass of the particles is large, as in long distance entanglement experiments. But these very extended objects are very fragile (metastable in the thermal interpretation, due to decoherence by the environment), and it takes a lot of experimental expertise to maintain them in an entangled state.

 

[thephystudent]

Thanks for the insight!

Do you believe gravity will turn out to be fundamentally an emergent force as Verlinde's picture (or other forces, e.g. for the coulomb force https://arxiv.org/abs/1704.04048) ?
Or would you consider such ideas rather meaningless by low falsiability?

 

[A. Neumaier]

That Coulomb's (nonrelativitic) force comes out of (relativistic) QED is well-known and no surprise. Getting gravity as emergent is in my opinion quite unlikely but not impossible. In fact there have been attempts to do so. (See this link and the surrounding discussion.)

 

[Urs Schreiber]

Scharf’s work also shows that there seems nothing wrong with canonically quantized gravity.

I feel that this statement deserves more qualification:

What Scharf argues is that from just the mathematics of perturbative QFT, there is no technical problem with a "non-renormalizable" Lagrangian: The non-renormalizability simply means (by definition) that infinitely many constants need to be chosen when renormalizing. While this may be undesireable, it is not mathematically inconsistent (unless one adopts some non-classical foundation of mathematics without, say, the axiom of choice; but this is not the issue that physicists are commonly concerned with). Simply make this choice, and that's it then.

I feel that the more popular Wilsonian perspective on this point is really pretty much the same statement, just phrased in different words: The more popular statement is that gravity makes sense as an effective field theory at any given cutoff energy, and that one needs to measure/fix further counterterms as one increases the energy scale.

So I feel there is actually widespread agreement on this point, just some difference in terminology. But the true issue is elsewhere: Namely the above statements apply to perturbation theory about a fixed gravitational background. The true issue of quantizing gravity is of course that the concept of background causality structure as used in Epstein-Glaser or Haag-Kastler does not apply to gravity, since for gravity the causal structure depends on the fields. For this simple reason it is clear that beyond perturbation theory, gravity definitely does not have "canonical quantization" if by this one means something fitting established axioms for QFT.

Instead, if quantum gravity really does exist as a quantum field theory (instead of, say, as a holographic dual of a quantum field theory), then this necessarily needs some slightly more flexible version of the Epstein-Glaser and/or Haag-Kastler-type axioms on causality: One needs to have a concept of causality that depends on the fields itself.

I am aware of one program aiming to address and solve this:

• Igor Khavkine "Characteristics, Conal Geometry and Causality in Locally Covariant Field Theory" (arXiv:1211.1914)

I think this deserves much more attention.

 

[A. Neumaier]

The non-renormalizability simply means (by definition) that infinitely many constants need to be chosen when renormalizing. While this may be undesireable, it is not mathematically inconsistent (unless one adopts some non-classical foundation of mathematics without, say, the axiom of choice; but this is not the issue that physicists are commonly concerned with).

This has nothing to do with the axiom of choice; the number of constants to choose is countably infinite only. The level of mathematical and physical consistency is precisely the same as when, in a context where people are used to working with polynomials defined by finitely many parameters, someone suggests to use instead power series. The complaint is completely unfounded that power series are not predictive since one needs infinitely many parameters to specify them. It is well-known how to specify infinitely many parameters by a finite formula for them!

I feel that the more popular Wilsonian perspective on this point is really pretty much the same statement, just phrased in different words: The more popular statement is that gravity makes sense as an effective field theory at any given cutoff energy, and that one needs to measure/fix further counterterms as one increases the energy scale.

In causal perturbation theory there is no cutoff, so Wilson's point of view is less relevant. Everything is constructed exactly; there is nothing effective. Gravity is no exception! Of course one can still make approximations to simplify a theory to an approximate effective low energy theory in the Wilson sense, but this is not intrinsic in the causal approach. (Nobody thinks of power series as being only effective approximations of something that fundamentally should be polynomials in a family of fundamental variables.)

The true issue of quantizing gravity is of course that the concept of background causality structure as used in Epstein-Glaser or Haag-Kastler does not apply to gravity, since for gravity the causal structure depends on the fields.

This is not really a problem. It is well-known that, just as massless spin 1 quantization produces gauge invariance, so massless spin 2 quantization produces diffeomorphism invariance. Hence as long as two backgrounds describe the same smooth manifold when the metric is ignored, they are for constructive purpose equivalent. Thus one may simply choose at each point a local coordinate system consisting of orthogonal geodesics, and you have a Minkowski parameterization in which you can quantize canonically. Locality and diffeomorphism invariance will make the construction behave correctly in every other frame.

 

[Urs Schreiber]

This has nothing to do

Careful with sweeping statements.

the number of constants to choose is countably infinite only.

Sure, but nevertheless, it needs a choice axiom, here the axiom of countable choice. Not that it matters for the real point of concern in physics, I just mentioned it for completeness.

The complaint is completely unfounded that power series are not predictive since one needs infinitely many parameters to specify them. It is well-known how to specify infinitely many parameters by a finite formula for them!

Don't confuse the way to make a choice with the space of choices. A formula is a way to write down a choice. But there are still many formulas.

Incidentally, this is what the string perturbation series gives: a formula for producing a certain choice of the infinitely many counterterms in (some extension) of perturbative gravity. To some extent one may think of perturbative string theory as parameterizing (part of) the space of choices in choosing renormalizaton parameters for gravity by 2d SCFTs. If these in turn arise as sigma models, this gives a way to parameterize these choices by differential geometry. It seems that the subspace of choices thus parameterized is still pretty large, though ("landscape"). Unfortunately, despite much discussion, little is known for sure about this.

This is not really a problem.

Not in perturbation theory, but that's clear.

 

[Urs Schreiber]

I expect that an appropriate 4-dimensional generalization of vertex algebras will play a role in giving rigorous foundations for operator product expansions.

Yes, maybe a key problem is to understand the relation between the Haag-Kastler axioms (local nets) for the algebras of quantum observables with similar axioms for the operator product expansion. These days there is a growing community trying to phrase QFT in terms of factorization algebras and since these generalize vertex operator algebras in 2d, they are to be thought of as formalizing OPEs in Euclidean (Wick rotated) field theory. Recently there is a suggestion on how to relate these to causal perturbation theory/pAQFT:

• Owen Gwilliam, Kasia Rejzner, "Comparing nets and factorization algebras of observables: the free scalar field" (arXiv.1711.06674)

but I suppose many things still remain to be understood.

 

[A. Neumaier]

Sure, but nevertheless, it needs a choice axiom, here the axiom of countable choice.

No. The parameters come with a definite grading and finitely many parameters for each grade, hence one can make the choice constructive (in many ways, e.g., by forcing all parameters of large grade to vanish).

Don't confuse the way to make a choice with the space of choices. A formula is a way to write down a choice. But there are still many formulas.

Of course. But making choices does not require a nonconstructive axiom. Each of the possible choices deserved to be called a possible theory of quantum gravity, so there are many testable constructive choices (and possibly some nonconstructive ones if one admits a corresponding axiom).

Making the right choice is, as with any theorem building, a matter of matching Nature through experiment. But of course, given our limited capabilities, there is not much to distinguish different proposed formulas for choosing the parameters; see C.P. Burgess, Quantum Gravity in Everyday Life: General Relativity as an Effective Field Theory. Only one new parameter (the coefficient of curvature^2) appears at one loop, where Newton's constant of gravitation becomes a running coupling constant with $$G(r) = G - 167/30\pi G^2/r^2 + ...$$ in terms of a renormalization length scale $r$, which is already below the current observability limit.

Numerically, the quantum corrections are so miniscule as to be unobservable within the solar system for the forseeable future. Clearly the quantum-gravitational correction is numerically extremely small when evaluated for garden-variety gravitational fields in the solar system, and would remain so right down to the event horizon even if the sun were a black hole. At face value it is only for separations comparable to the Planck length that quantum gravity effects become important. To the extent that these estimates carry over to quantum effects right down to the event horizon on curved black hole geometries (more about this below) this makes quantum corrections irrelevant for physics outside of the event horizon, unless the black hole mass is as small as the Planck mass

 

[Urs Schreiber]

In causal perturbation theory there is no cutoff, so Wilson's point of view is less relevant.

It's two different perspectives on the same phenomenon of pQFT. This is discussed in section 5.2 of Brunetti-Dütsch-Fredenhagen 09.

 

[A. Neumaier]

It's two different perspectives on the same phenomenon of pQFT. This is discussed in section 5.2 of Brunetti-Dütsch-Fredenhagen 09.

Yes. The Stueckelberg renormalization group is what is left from the Wilson renormalization semigroup when no approximation is permitted. In causal perturbation theory one only has the former. It describes a parameter redundancy of any fixed theory.

Approximating a theory by a simpler one to get better numerical access is a completely separate thing from constructing the theory in the first place. Causal perturbation theory clearly separates these issues, concentrating on the second.

 

[Urs Schreiber]

The parameters come with a definite grading and finitely many parameters for each grade,

The space of choices is of finite dimension in each degree, but it's is not a finite set in each degree. In general, given a function $p : S \to \mathbb{Z}$ one needs a choice principle to pick a section unless there is secretly some extra structure around. Maybe that's what you are claiming.

hence one can make the choice constructibly (in many ways, e.g., choosing all parameters of large grade as zero).

This sounds like you are claiming that there is a preferred section (zero section), when restricting to large elements. This doesn't seem to be the case to me, in general. But maybe I am missing something.

 

[Urs Schreiber]

Yes. The Stueckelberg renormalization group is what is left from the Wilson renormalization semigroup when no approximation is permitted. In causal perturbation theory one only has the former. It describes a parameter redundancy of any fixed theory.

Approximating a theory by a simpler one to get better numerical access is a completely separate thing from constructing the theory in the first place. Causal perturbation theory clearly separates these issues, concentrating on the second.

Not sure what this is debating, I suppose we agree on this. What I claimed is that the choice of renormalization constants in causal perturbation theory (in general infinite) corresponds to the choice of couplings/counterterms in the Wilsonian picture. It's two different perspectives on the same subject: perturbative QFT.

 

[vanhees71]

In causal perturbation theory there is no cutoff, so Wilson's point of view is less relevant. Everything is constructed exactly; there is nothing effective. Gravity is no exception! Of course one can still make approximations to simplify a theory to an approximate effective low energy theory in the Wilson sense, but this is not intrinsic in the causal approach. (Nobody thinks of power series as being only effective approximations of something that fundamentally should be polynomials in a family of fundamental variables.)

Well, conventional BPHZ also has no UV cutoff but a renormalization scale only.

Of course, a theory which is not Dyson renormalizable (like, e.g., chiral perturbation theory), needs necessarily a kind of "cut-off scale" since to make sense of the theory you have to read it as expansion in powers of energy-momentum. Since it's a low-energy theory you need to tell what's the "large scale" (in $\chi$PT it's $4 \pi f_{\pi} \simeq 1 \; \text{GeV}$).

 

[A. Neumaier]

The space of choices is of finite dimension in each degree, but it's is not a finite set in each degree. In general, given a function $p : S \to \mathbb{Z}$ one needs a choice principle to pick a section unless there is secretly some extra structure around. Maybe that's what you are claiming.

This sounds like you are claiming that there is a preferred section (zero section), when restricting to large elements. This doesn't seem to be the case to me, in general. But maybe I am missing something.

I am claiming that each particular choice made by a particular formula gives a valid solution. In particular, I gave the simplest constructive choice that leads to a valid solution, namely setting to zero enough higher order parameters in a given renormalization fixing scheme (that relates the choices available to values of higher order coefficients in the expansion of some observables, just like the standard renormalization scales are fixed by matching physical constants such as masses, charges, and the gravitational constant).This gives one constructive solution for each renormalization fixing scheme, each of them for the foreseeable future consistent with the experiments.

There are of course many other constructive solutions, but there is no way to experimentally distinguish between them in my lifetime. Thus my simple solution is adequate, and in view of Ockham's razor best. You may claim that the dependence on the renormalization fixing scheme is ugly, but this doesn't matter - Nature's choices don't have to be beautiful; only the general theory behind it should have this property.

Maybe there are additional principles (such as string theory) giving extra structure that would lead to other choices, but being experimentally indistinguishable in the foreseeable future, there is no compelling reason to prefer them unless working with them is considerably simpler than working with my simple recipe.

 

[A. Neumaier]

Not sure what this is debating, I suppose we agree on this. What I claimed is that the choice of renormalization constants in causal perturbation theory (in general infinite) corresponds to the choice of couplings/counterterms in the Wilsonian picture. It's two different perspectives on the same subject: perturbative QFT.

Wilson's perspective is intrinsically approximate; changing the scale changes the theory. This is independent of renormalized perturbation theory (whether causal or not), where the theory tells us that there is a vector space of parameters from which to choose one point that defines the theory. The vector space is finite-dimensional iff the theory is renormalizable.

In the latter case we pick a parameter vector by calculating its consequences for observable behavior and matching as many key quantities as the vector space of parameters has dimensions. This is deemed sufficient and needs no mathematical axiom of choice of any sort. Instead it involves experiments that restrict the parameter space to sufficiently narrow regions.

In the infinite-dimensional case the situation is similar, except that we would need to match infinitely many key quantities, which we do not (and will never) have. This just means that we are not able to tell which precise choice Nature is using.

But we don't know this anyway for any physical theory - even in QED, the best theory we ever had, we know Nature's choice only to 12 digits of accuracy or so. Nevertheless, QED is very predictive.

Infinite dimensions do not harm predictability elsewhere in physics. In fluid dynamics, the solutions of interest belong to an infinite-dimensional space. But we are always satisfied with finite-dimensional approximations - the industry pays a lot for finite element simulations because its results are very useful in spite of their approximate nature. Thus there is nothing bad in not knowing the infinite-dimensional details as long as we have good enough finite-dimensional approximations.

 

[A. Neumaier]

Well, conventional BPHZ also has no UV cutoff but a renormalization scale only.

Yes, this is quite similar to causal perturbation theory.

 

[A. Neumaier]

Recently there is a suggestion on how to relate these to causal perturbation theory/pAQFT:

• Owen Gwilliam, Kasia Rejzner, "Comparing nets and factorization algebras of observables: the free scalar field" (arXiv.1711.06674)

This is nice in that it illustrates the concepts on free scalar fields, so that one can understand them without all the technicalities that come later with the renormalization. I don't have yet a good feeling for factorization algebras, though.

 

[Urs Schreiber]

I'd eventually enjoy a more fine-grained technical discussion of some of these matters, to clear out the issues. But for the moment I'll leave it at that.

By the way, not only may we view the the string perturbation series as a way to choose these infinitely many renormalization parameters for gravity by way of other data, but the same is also true for "asymptotic safety". Here it's the postulate of being on a finite-dimensional subspace in the space of couplings that amounts to the choice.

 

[Urs Schreiber]

I gave the simplest constructive choice that leads to a valid solution, namely setting to zero enough higher order parameters in a given renormalization fixing scheme

The space of choices of renormalization parameters at each order is not a vector space, but an affine space. There is no invariant meaning of "setting to zero" these parameters, unless one has already chosen an origin in these affine spaces. The latter may be addressed as a choice of renormalization scheme, but this just gives another name to the choice to be made, it still does not give a canonical choice.

You know this, but here is pointers to the details for those readers who have not see this:

In the original Epstein-Glaser 73 the choice at order $\omega$ happens on p. 27, where it says "we choose a fixed auxiliary function $w \in \mathcal{S}(\mathbb{R}^n)$ such that...". With the choice of this function they build one solution to the renormalization problem at this order (for them a splitting of distributions) which they call $(T^+, T^-)$. With this "origin" chosen, every other solution of the renormalization at that order is labeled by a vector space of renormalization constants $c_\alpha$ (on their p. 28, after "The most general solution"). It might superficially seem the as if we could renormalize canonically by declaring "choose all $c_\alpha$ to be zero". But this is an illusion, the choice is now in the "scheme" $w$ relative to which the $c_\alpha$ are given.

In the modern reformulation of Epstein-Glaser's work in terms of extensions of distributions in Brunetti-Fredenhagen 00 the analogous step happens on p. 24 in or below equation (38), where at order $\omega$ bump functions $\mathfrak{w}_\alpha$ are chosen. The theorem 5.3 below that states then that with this choice, the space of renormalization constants at that order is given by coefficients relative to these choices $\mathfrak{w}_\alpha$.

One may succintly summarize this statement by saying that the space of renormalization parameters at each order, while not having a preferred element (in particular not being a vector space with a zero-element singled out) is a torsor over a vector space, meaning that after anyone point is chosen, then the remaining points form a vector space relative to this point. That more succinct formulation of theorem 5.3 in Brunetti-Fredenhagen 00 is made for instance as corollary 2.6 on p.5 of Bahns-Wrochna 12.

Hence for a general Lagrangian there is no formula for choosing the renormalization parameters at each order. It is in very special situations only that we may give a formula for choosing the infinitely many renormalization parameters. Three prominent such situations are the following:

1) if the theory is "renormalizable" in that it so happens that after some finite order the space of choices of parameters contain a unique single point. In that case we may make a finite number of choices and then the remaining choices are fixed.

2) If we assume the existence of a "UV-critical hypersurface" (e.g. Nink-Reuter 12, p. 2), which comes down to postulating a finite dimensional submanifold in the infinite dimensional space of renormalization parameters and postulating/assuming that we make a choice on this submanifold. Major extra assumptions here. If they indeed happen to be met, then the space of choices is drastically shrunk.

3) We assume a UV-completion by a string perturbation series. This only works for field theories which are not in the "swampland" (Vafa 05). It transforms the space of choices of renormalization parameters into the space of choices of full 2d SCFTS of central charge 15, the latter also known as the "perturbative landscape". Even though this space received a lot of press, it seems that way too little is known about it to say much at the moment. But that's another discussion.

There might be more, but the above three seem to be the most prominent ones.

 

[Urs Schreiber]

I wrote:

In the original Epstein-Glaser 73 the choice at order $\omega$ happens on p. 27, where it says "we choose a fixed auxiliary function $w \in \mathcal{S}(\mathbb{R}^n)$ such that...".

Hm, I guess Arnold will argue that we can construct choices for these auxiliary functions. There won't be a canonical choice but at least constructions exist and we don't need to appeal to non-constructive choice principles. Okay, I suppose I agree then!

 

[A. Neumaier]

Hm, I guess Arnold will argue that we can construct choices for these auxiliary functions. There won't be a canonical choice but at least constructions exist and we don't need to appeal to non-constructive choice principles. Okay, I suppose I agree then!

Yes.

More specifically, there is no significant difference between choosing from a finite number of finite-dimensional affine spaces in the renormalizable case and choosing from a countable number of finite-dimensional affine spaces in the renormalizable case. The same techniques that apply in the first case to pick a finite sequence of physical parameters (a few dozen in the case of the standard model) that determine a single point in each of these spaces can be used in the second case to pick an infinite sequence of physical parameters that determine a single point in each of these spaces. Here a parameter is deemed physical if it could be in principle obtained from sufficiently accurate statistics on collision events or other in principle measurable information.

Any specific such infinite sequence provides a well-defined nonrenormalizable perturbative quantum field theory. Thus there is no question of being able to make the choices in very specific ways. As in the renormalizable case, experiments just restrict the parameter region in which the theory is compatible with experiment. Typically, this region constrains the first few parameters a lot and the later ones much less.This is precisely the same situation as when we have to estimate the coefficients of a power series of a function $f(x)$ from a finite number of inaccurate function values given together with statistical error bounds.

 

[A. Neumaier]

The non-renormalizability simply means (by definition) that infinitely many constants need to be chosen when renormalizing. While this may be undesirable, it is not mathematically inconsistent

See also this thread from Physics Stack exchange, where solutions of an ''unrenormalizable'' QFT obtained by reparameterizing a renormalizable QFT are discussed.

 

[atyy]

I feel that this statement deserves more qualification:

What Scharf argues is that from just the mathematics of perturbative QFT, there is no technical problem with a "non-renormalizable" Lagrangian: The non-renormalizability simply means (by definition) that infinitely many constants need to be chosen when renormalizing. While this may be undesireable, it is not mathematically inconsistent (unless one adopts some non-classical foundation of mathematics without, say, the axiom of choice; but this is not the issue that physicists are commonly concerned with). Simply make this choice, and that's it then.

I feel that the more popular Wilsonian perspective on this point is really pretty much the same statement, just phrased in different words: The more popular statement is that gravity makes sense as an effective field theory at any given cutoff energy, and that one needs to measure/fix further counterterms as one increases the energy scale.

So I feel there is actually widespread agreement on this point, just some difference in terminology.

That's an interesting comparison. But maybe this aspect of the Wilsonian viewpoint is different. In the Wilsonian viewpoint, we don't need to know the theory at infinitely high energies, whereas I don't think Scharf's work makes sense unless a theory exists at infinitely high energies.

 

[Urs Schreiber]

But maybe this aspect of the Wilsonian viewpoint is different. In the Wilsonian viewpoint, we don't need to know the theory at infinitely high energies, whereas I don't think Scharf's work makes sense unless a theory exists at infinitely high energies.

I'd think this is only superficially so. In Epstein-Glaser-type [URL='https://www.physicsforums.com/insights/causal-perturbation-theory/']causal perturbation theory[/URL] (which is what Scharf's textbooks work out, but Scharf is not the originator of these ideas) one has in front of oneself the entire (possibly infinite) sequence of choices of renormalization contants, but one also has complete control over the space of choices and hence one has directly available the concept "all those pQFTs whose first $n$ renormalization constants have the following fixed values, with the rest being arbitrary". This is exactly the concept of knowing the theory up to that order.

 

[atyy]

I'd think this is only superficially so. In Epstein-Glaser-type [URL='https://www.physicsforums.com/insights/causal-perturbation-theory/']causal perturbation theory[/URL] (which is what Scharf's textbooks work out, but Scharf is not the originator of these ideas) one has in front of oneself the entire (possibly infinite) sequence of choices of renormalization contants, but one also has complete control over the space of choices and hence one has directly available the concept "all those pQFTs whose first $n$ renormalization constants have the following fixed values, with the rest being arbitrary". This is exactly the concept of knowing the theory up to that order.

But in the Epstein-Glaser theory, no theory is constructed, ie. the power series are formal series, and it is unclear how to sum them. In contrast, if we use a lattice theory as the starting point for Wilson, then that starting point is at least a well defined quantum theory.

 

[A. Neumaier]

That's an interesting comparison. But maybe this aspect of the Wilsonian viewpoint is different. In the Wilsonian viewpoint, we don't need to know the theory at infinitely high energies, whereas I don't think Scharf's work makes sense unless a theory exists at infinitely high energies.

I'd think this is only superficially so. In Epstein-Glaser-type [URL='https://www.physicsforums.com/insights/causal-perturbation-theory/']causal perturbation theory[/URL] (which is what Scharf's textbooks work out, but Scharf is not the originator of these ideas) one has in front of oneself the entire (possibly infinite) sequence of choices of renormalization contants, but one also has complete control over the space of choices and hence one has directly available the concept "all those pQFTs whose first $n$ renormalization constants have the following fixed values, with the rest being arbitrary". This is exactly the concept of knowing the theory up to that order.

The two point of views do not contradict each other. Causal perturbation theory has no effective cutoff and is at fixed loop order defined at all energies. In this sense it exists and gives results that compare (in case of QED) exceedingly well with experiment. The only question is how accurate the fixed order results are at energies relevant for experiments. Here numerical results indicate that one never needs to go to more than 4 loops.

But in the Epstein-Glaser theory, no theory is constructed, ie. the power series are formal series, and it is unclear how to sum them. In contrast, if we use a lattice theory as the starting point for Wilson, then that starting point is at least a well defined quantum theory.

Formal power series can be approximately summed by many methods, including Pade approximation, Borel summation, and extensions of the latter to resurgent transseries. The result is always Poincare invariant and hence in agreement with the principles of relativity; unitarity is guaranteed to the order given, which is usually enough. Thus for practical purposes one has a well-defined theory. only those striving for rigor need more.

On the other hand, lattice methods don't respect the principles of relativity (not even approximately) unless they are extrapolated to the limit of vanishing lattice spacing and infinite volume. In this extrapolation, all mathematical problems reappear that were swept under the carpet through the discretization. The extrapolation limit of lattice QFT - which contains the real physics - is, to the extend we know, as little well-defined as the limit of the formal power series in causal perturbation theory.

Moreover, concerning the quality of the approximation, lattice QED is extremely poor when compared with few loops QED, and thus cannot compete in quality. The situation is slightly better for lattice QCD, but there both approaches have up to now fairly poor accuracy (5 percent or so), compared with the 12 relative digits of perturbative QED calculations.

 

[Urs Schreiber]

The two point of views contradict each other.

They'd better not, if both are about the same subject, pQFT.

There are various ways to parameterize the ("re"-.)normalization choices in causal perturbation theory, and one is by Wilsonian flow of cutoff. This is explained in section 5.2 of

• Romeo Brunetti, Michael Dütsch, Klaus Fredenhagen,
  "Perturbative Algebraic Quantum Field Theory and the Renormalization Groups",
  Adv. Theor. Math. Physics 13 (2009), 1541-1599
  (arXiv:0901.2038)

 

[A. Neumaier]

They'd better not

Oh, there was a typo; I meant they don't contradict each other.

if both are about the same subject, pQFT.

But they aren't. I think atyy's point was that Wilson's conceptual view is in principle nonperturbative. The noncontradiction stems from the fact that both lead to valid and time-proved approximations of QFT.

 

[Urs Schreiber]

Oh, there was a typo

Ah, okay. :-)

I think atyy's point was that Wilson's conceptual view is in principle nonperturbative.

To make progress in the discussion we should leave the non-perturbative aspect aside for the moment, and first of all find agreement for pQFT, where we know what we are talking about.

What I keep insisting is that Wilsonian effective field theory flow with cutoff-dependent counterterms is an equivalent way to parameterize the ("re"-)normalization freedom in rigorous pQFT formulated via causal perturbation theory.

Namely the theorem by Dütsch-Fredenhagen et. al. which appears with a proof as theorem A.1 in

• Michael Dütsch, Klaus Fredenhagen, Kai Keller, Katarzyna Rejzner,
  "Dimensional Regularization in Position Space, and a Forest Formula for Epstein-Glaser Renormalization",
  J. Math. Phy. 55(12), 122303 (2014)
  (arXiv:1311.5424)

says the following:

Given a gauge-fixed free field vacuum around which to perturb, and choosing any UV-regularization of the Feynman propagator $\Delta_F$ by non-singular distributions $\{\Delta_{F,\Lambda}\}_{\Lambda \in [0,\infty)}$, in that

$$ \Delta_F = \underset{\Lambda \to \infty}{\lim} \Delta_{F,\Lambda}$$

and writing

$$
\mathcal{S}_\Lambda(O) := 1 + \frac{1}{i \hbar} + \frac{1}{2} \frac{1}{(i \hbar)^2} O \star_{F,\Lambda} O + \frac{1}{6} \frac{1}{(i \hbar)^3} O \star_{F,\Lambda} O \star_{F,\Lambda} O
+ \cdots
$$

for the corresponding regularized S-matrix at scale $\Lambda$ (built from the star product that is induced by $\Delta_{F,\Lambda}$) then:

1. There exists a choice of regularization-scale-dependent vertex redefinitions $\{\mathcal{Z}_\Lambda\}_{\Lambda \in [0,\infty)}$ (sending local interactions to local interactions), hence of "counterterms" such that the limit
   $\mathcal{S}_\infty := \underset{\Lambda \to \infty}{\lim} \mathcal{S}_\Lambda \circ \mathcal{Z}_\Lambda$
   exists and is an S-matrix scheme in the sense of causal perturbation theory (this def., hence is Epstein-Glaser ("re"-)normalized);
2. every Epstein-Glaser ("re"-)normalized S-matrix scheme $\mathcal{S}$ arises this way;
3. the corresponding Wilsonian effective field theory at scale $\Lambda$ is that with effective (inter)action given by
   $S_{eff,\Lambda} = \mathcal{S}_\Lambda^{-1} \circ \mathcal{S}_\infty(S_{int})$.

This exhibits the choice of scale-dependent effective actions of Wilsonian effective field theory as an alternative way to parameterize the ("re"-)normalization choice in causal perturbation theory.

See also

• Michael Dütsch,
  "Connection between the renormalization groups of Stückelberg-Petermann and Wilson",
  Confluentes Mathematici, Vol. 4, No. 1 (2012) 12400014
  (arXiv:1012.5604)