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Insights Interview with Mathematician and Physicist Arnold Neumaier - Comments

  1. Jan 3, 2018 #21

    A. Neumaier

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    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.
     
  2. Jan 3, 2018 #22

    A. Neumaier

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    Yes, this is quite similar to causal perturbation theory.
     
  3. Jan 3, 2018 #23

    A. Neumaier

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    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.
     
  4. Jan 3, 2018 #24

    Urs Schreiber

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    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.
     
  5. Jan 4, 2018 #25

    Urs Schreiber

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    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 any one 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.
     
  6. Jan 4, 2018 #26

    Urs Schreiber

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    I wrote:

    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!
     
  7. Jan 5, 2018 #27

    A. Neumaier

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    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.
     
    Last edited: Jan 5, 2018
  8. Jan 22, 2018 #28

    A. Neumaier

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    See also this thread from Physics Stack exchange, where solutions of an ''unrenormalizable'' QFT obtained by reparameterizing a renormalizable QFT are discussed.
     
    Last edited: Jan 22, 2018
  9. Jan 29, 2018 #29

    atyy

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    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.
     
  10. Jan 29, 2018 #30

    Urs Schreiber

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    I'd think this is only superficially so. In Epstein-Glaser-type causal perturbation theory (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.
     
  11. Jan 30, 2018 #31

    atyy

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    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.
     
  12. Jan 30, 2018 #32

    A. Neumaier

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    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.
    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.
     
    Last edited: Jan 30, 2018
  13. Jan 30, 2018 #33

    Urs Schreiber

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    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)
     
  14. Jan 30, 2018 #34

    A. Neumaier

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    Oh, there was a typo; I meant they don't contradict each other.
    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.
     
  15. Jan 30, 2018 #35

    Urs Schreiber

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    Ah, okay. :-)

    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)
     
    Last edited: Jan 30, 2018
  16. Jan 31, 2018 #36

    A. Neumaier

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    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.
     
    Last edited by a moderator: Jan 31, 2018
  17. Feb 2, 2018 #37

    Urs Schreiber

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    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.
     
  18. Feb 13, 2018 #38

    A. Neumaier

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    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.
     
    Last edited: Feb 13, 2018
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