Insights Interview with Mathematician and Physicist Arnold Neumaier - Comments

[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)

 

[A. Neumaier]

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.

I don't understand how you can phrase in this way the theorem stated. You seem to say (i) below but the theorem seems to assert (ii) below.

(i) The space of possible Wilsonian effective field theories, viewed perturbatively, is identical with the collection of pQFTs formulated via causal perturbation theory.

(ii) The space of possible limits $\Lambda\to\infty$ of the Wilsonian flows is identical with the collection of pQFTs formulated via causal perturbation theory.

A Wilsonian effective theory has a finite $\Lambda$ and hence seems to me not to be one of the theories defined by causal perturbation theory. In any case, the Wilsonian flow is a flow on a collection of field theories, while causal perturbation theory does not say anything about flows on the space of renormalization parameters.

 

[Urs Schreiber]

I don't understand how you can phrase in this way the theorem stated. You seem to say (i) below but the theorem seems to assert (ii) below.

(i) The space of possible Wilsonian effective field theories, viewed perturbatively, is identical with the collection of pQFTs formulated via causal perturbation theory.

(ii) The space of possible limits $\Lambda\to\infty$ of the Wilsonian flows is identical with the collection of pQFTs formulated via causal perturbation theory.

A Wilsonian effective theory has a finite $\Lambda$ and hence seems to me not to be one of the theories defined by causal perturbation theory. In any case, the Wilsonian flow is a flow on a collection of field theories, while causal perturbation theory does not say anything about flows on the space of renormalization parameters.

I am pointing out that the following are two ways to converge to a fully renormalized pQFT according to the axioms of causal perturbation theory:

1. inductively in $k \in \mathbb{N}$ choose splittings/extensions of distributions in Epstein-Glaser renormalization as $k \to \infty$;
   
2. consecutively in $\Lambda \in [0,\infty)$ choose counterterms at UV-cutoff $\Lambda$ for $\Lambda \to \infty$.

In both cases we zoom in with a sequence of shrinking neighbourhods to a specific point in the space of renormalization schemes in causal perturbation theory. Only the nature and parameterization of these neighbourhoods differs. But the Wilsonian intuition, that as we keep going (either way) we see more and more details of the full theory, is the same in both cases.

BTW, that proof in DFKR 14, A.1 is really terse. I have spelled it out a little more: here.

 

[A. Neumaier]

1. inductively in $k \in \mathbb{N}$ choose splittings/extensions of distributions in Epstein-Glaser renormalization as $k \to \infty$;
   
2. consecutively in $\Lambda \in [0,\infty)$ choose counterterms at UV-cutoff $\Lambda$ for $\Lambda \to \infty$.

In both cases we zoom in with a sequence of shrinking neighbourhods to a specific point in the space of renormalization schemes in causal perturbation theory. Only the nature and parameterization of these neighbourhoods differs. But the Wilsonian intuition, that as we keep going (either way) we see more and more details of the full theory, is the same in both cases.

It seems to me that 2. involves a double limit since the cutoff is also applied order by order, and the limit at each order as the cutoff is removed gives the corresponding order on causal perturbation theory. Thus Wilson's approach is just an approximation to the causal approach, and at any fixed order one sees in the Wilsonian approach always fewer details than in the causal approach.

 

[A. Neumaier]

I wrote a first draft of a paper about my views on the interpretation of quantum mechanics, more precisely of what should constitute coherent foundations for quantum mechanics.

The draft includes among others my critique of Born's rule as fundamental truth and a description of my thermal interpretation of quantum mechanics.

Your comments are welcome.