Interview with Mathematician and Physicist Arnold Neumaier - Comments

[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.