I Wilsonian viewpoint and wave function reality

[atyy]

Since http://www.physicsoverflow.org/a29578 to some (and indeed many) arbitrarily often differentiable function, vanhees71 is trivially right on this.

Yes. I was thinking that there is no specific theory he has in mind, thus the theory is not specified. Also, it is unknown whether the series are correlation functions of a relativistic quantum field theory.

But more generally, the technicalities are beside the point. We all agree there is no mathematically rigourous renormalization for physically relevant quantum field theories. The power of the Wilsonian viewpoint is that physicists feel the situation is like Newtonian calculus before Weierstrass or the early days of Fourier transforms - the theory is fine at the non-rigourous level and it matches experiments. Before Wilson, physicists did not feel the theory was ok at the non-rigourous level even though it matched experiements.

 

[A. Neumaier]

The power of the Wilsonian viewpoint is that physicists feel the situation is like Newtonian calculus before Weierstrass or the early days of Fourier transforms - the theory is fine at the non-rigourous level and it matches experiments. Before Wilson, physicists did not feel the theory was ok at the non-rigourous level even though it matched experiements.

Yes.

Causal perturbation theory is part of the next step - to make it more rigorous.I find it an improvement since no uncontrolled approximation is made. What is missing is the appropriate resummation technique that allows one to control the errors.

 

[Demystifier]

Why do you think Wilson did not taken Copenhagen seriously?

Because I don't see how the Wilsonian spirit can be compatible with Copenhagen spirit. But maybe that's just my lack of imagination.

 

[atyy]

Because I don't see how the Wilsonian spirit can be compatible with Copenhagen spirit. But maybe that's just my lack of imagination.

But is the Wilsonian spirit compatible with the Bohmian spirit? Naively, I think "no", because of Bell's theorem. In the Wilsonian picture, say in classical statistical mechanics or the "classical" intuition given by Bosonic path integrals, the coarse graining is local. But we know from Bell that there is no local reality, so the coarse graining always involves "wave function coarse graining" which seems to me not so Wilsonian in spirit.

 

[Demystifier]

@atyy you might find illuminating a quote from G.B. Folland, Quantum Field Theory: A Tourist Guide for Mathematicians
(the beginning of Chapter 6. Quantum Fields with Interactions):

"Everything we have done so far is mathematically respectable, although some of the results have been phrased in informal language. To make further progress, however, it is necessary to make a bargain with the devil. The devil offers us effective and conceptually meaningful techniques for calculating physically interesting quantities. In return, however, he requires us to compromise our mathematical souls by accepting the validity of certain approximation procedures and certain formal calculations without proof and — what is a good deal more disconcerting— by working with some putative mathematical objects that lack a rigorous definition. The situation is in some ways similar to the mathematical analysis of the eighteenth century, which developed without the support of a rigorous theory of limits and with the use of poorly defined infinitesimals." (my bolding)

Or let me put it in my own words. Just because a theory is not rigorous does not mean it doesn't make sense. Just because we don't fully understand something doesn't mean we don't understand it at all.

 

[atyy]

@Demystifier, yes what you extracted from Folland is what I said - but, but, but my big objection is that Folland nowhere mentions the Wilsonian conception. My impression of Folland's book is that he is stuck in the age of Feynman.

 

[Demystifier]

But is the Wilsonian spirit compatible with the Bohmian spirit? Naively, I think "no", because of Bell's theorem. In the Wilsonian picture, say in classical statistical mechanics or the "classical" intuition given by Bosonic path integrals, the coarse graining is local. But we know from Bell that there is no local reality, so the coarse graining always involves "wave function coarse graining" which seems to me not so Wilsonian in spirit.

You are mixing two different notions of locality: kinematic locality and dynamic locality. Bohmian mechanics is kinematically local (the degrees of freedom, namely particles, have precise positions), but dynamically non-local (particles influence each other by instantaneous action at a distance). Wilsonian picture is also kinematically local (local coarse graining of degrees of freedom). Bell theorem excludes dynamic locality, but not kinematic locality.

 

[atyy]

You are mixing two different notions of locality: kinematic locality and dynamic locality. Bohmian mechanics is kinematically local (the degrees of freedom, namely particles, have precise positions), but dynamically non-local (particles influence each other by instantaneous action at a distance). Wilsonian picture is also kinematically local (local coarse graining of degrees of freedom). Bell theorem excludes dynamic locality, but not kinematic locality.

But don't we have to coarse grain the wave function if we use Wilson's picture in Bohmian mechanics? I don't think one can think of the wave function as "kinematically local"?

 

[Demystifier]

@Demystifier, yes what you extracted from Folland is what I said - but, but, but my big objection is that Folland nowhere mentions the Wilsonian conception. My impression of Folland's book is that he is stuck in the age of Feynman.

OK, but even if he discussed the current status of the Wilsonian picture, the quoted part could refer to the Wilsonian picture as well.

 

[Demystifier]

But don't we have to coarse grain the wave function if we use Wilson's picture in Bohmian mechanics? I don't think one can think of the wave function as "kinematically local"?

Ah, now I see your point. Wilsonian coarse graining is kinematically local, wave function is not kinematically local, and yet wave function suffers Wilsonian coarse graining. How is that possible?

The answer is that the wave function is non-local in a very specific way that allows us to treat it by local methods. The wave function is not local, but it is multi-local. This means that it can be written as a sum of products of local wave functions.

 

[vanhees71]

It's amazing how you can come from a debate of the renormalization group and its interpretation to the interpretations of quantum theory. I think the entire renormalization theory is completely independent from the interpretation of quantum theory you follow. You only need the minimal interpretation to make sense out of it!

 

[Demystifier]

I think the entire renormalization theory is completely independent from the interpretation of quantum theory you follow.

I agree, but I think that such a statement is not obvious without an argument.

 

[vanhees71]

Well, I studied RG methods without ever thinking about interpretations, i.e., using the shutup-and-calculation interpretation ;-)).

 

[Demystifier]

Well, I studied RG methods without ever thinking about interpretations, i.e., using the shutup-and-calculation interpretation ;-)).

Just because you were not thinking about interpretations doesn't mean there is nothing to think about.

 

[atyy]

Ah, now I see your point. Wilsonian coarse graining is kinematically local, wave function is not kinematically local, and yet wave function suffers Wilsonian coarse graining. How is that possible?

The answer is that the wave function is non-local in a very specific way that allows us to treat it by local methods. The wave function is not local, but it is multi-local. This means that it can be written as a sum of products of local wave functions.

Yes, that must be it - one of the miracles of quantum mechanics.

 

[atyy]

Well, I studied RG methods without ever thinking about interpretations, i.e., using the shutup-and-calculation interpretation ;-)).

But your reply was the one reply that did not use shut-up-and-calculate - you said that in RG, you coarse grain degrees of freedom.

Also, I believe you said that quarks exist!

In contrast, Arnold Neumaier gave the shut-up-and-calculate answer - in RG one coarse grains correlation functions.

 

[vanhees71]

Yes, but there are more or less useful things to think about. I don't know, what the RG has to do with interpretational problems at all. I also don't know what you mean by "multi-local". A wave function is a state ket in the position representation and thus a field $\psi(t,\vec{x})$. You can say it describes a more or less well localized particle, depending on how sharply $|\psi|^2$ peaks around a certain value $\vec{x}_0$ of the position vector, but that's it.

 

[Demystifier]

Yes, that must be it - one of the miracles of quantum mechanics.

Yes. In fact, this multi-locality is the reason why QM is somehow on the borderline between local and non-local, i.e. why both local and non-local interpretations of QM exist.

 

[atyy]

Yes. In fact, this multi-locality is the reason why QM is somehow on the borderline between local and non-local, i.e. why both local and non-local interpretations of QM exist.

A further miracle is that in rigourous QFT, I think strict multilocality fails (Haag's theorem etc), but nonetheless in practice we have no problem with multilocality (eg. QED, using a lattice and tensoring the Hilbert spaces at each site).

 

[Demystifier]

I also don't know what you mean by "multi-local". A wave function is a state ket in the position representation and thus a field ψ(t,⃗x).

You have written down a single-particle wave function. I was talking about many-particle wave functions, the obvious thing one has in mind when talking about quantum non-locality.

 

[vanhees71]

What's coarse-grained is the resolution in time and space by introducing a cutoff in energy-momentum space. Then you introduce the counter terms and a typical energy-momentum scale where you look at scattering processes. This introduces effective renormalized parameters which are valid around this energy-momentum scales. You reorganize perturbation theory writing it in terms of the observable renormalized parameters. If the renormalized couplings are small the perturbative analysis makes (probably) sense.

Looking at processes at a much different energy-momentum scale, you encounter large logarithms, which have to be resummed, and this is very elegantly done using the RG equations. Of course, the perturbative approach usually fails when you try to apply it in regions of the scale, where the renormalized couplings become large. This happens in QCD if you want to make the scale small, due to asymptotic freedom, and then you have to switch to a different description than QCD. Nature tells us that then even the "relevant degrees of freedom" change from quarks and gluons to hadrons, and thus you try to invent effective hadronic theories, based on the symmetries (most importantly using the approximate chiral symmetry in the light-quark sector).

 

[A. Neumaier]

A further miracle is that in rigourous QFT, I think strict multilocality fails (Haag's theorem etc), but nonetheless in practice we have no problem with multilocality (eg. QED, using a lattice and tensoring the Hilbert spaces at each site).

None of the high quality predictions of QED were achieved with the lattice approximation. All were computed with the (within the level of rigor of theoretical physics) fully local renormalized continuum version in carefully chosen approximations (NRQED).

 

[Demystifier]

A further miracle is that in rigourous QFT, I think strict multilocality fails (Haag's theorem etc), but nonetheless in practice we have no problem with multilocality (eg. QED, using a lattice and tensoring the Hilbert spaces at each site).

Yes, but I would propose to modify the language. I would interpret Haag's theorem as no-go theorem, so what we call rigorous QFT is not rigorous QFT (due to the no-go theorem), while lattice QFT is a rigorous QFT.

 

[vanhees71]

Also many-particle states are neither local nor non-local. This doesn't make sense at all. The only information contained in it concerning "localization" is in the Born rule, and also there you can have a more or less localized many-body system, depending on the state.

I think, what's again mixed up here is the difference between non-locality, non-separability, and long-ranged correlations. The latter two are closely related and described by entanglement. This mixing up different notions made Einstein pretty unhappy with the famous EPR paper later, and he wrote another paper alone to clarify this point. It's, unfortunately, in German:

A. Einstein, Dialectica 2, 320 (1948)

 

[Demystifier]

Also many-particle states are neither local nor non-local. This doesn't make sense at all. The only information contained in it concerning "localization" is in the Born rule, and also there you can have a more or less localized many-body system, depending on the state.

I think, what's again mixed up here is the difference between non-locality, non-separability, and long-ranged correlations. The latter two are closely related and described by entanglement. This mixing up different notions made Einstein pretty unhappy with the famous EPR paper later, and he wrote another paper alone to clarify this point. It's, unfortunately, in German:

A. Einstein, Dialectica 2, 320 (1948)

OK, so you agree that many-body wave functions are, in general, non-separable. Right?

 

[atyy]

Yes, but I would propose to modify the language. I would interpret Haag's theorem as no-go theorem, so what we call rigorous QFT is not rigorous QFT (due to the no-go theorem), while lattice QFT is a rigorous QFT.

I think the language is ok, because there are rigourous relativistic QFTs constructed in infinite volume and in which the lattice spacing is taken to zero. The resolution to Haag's theorem just means that although the argument and "interaction picture" used by physicists is not strictly correct in infinite volume, the mathematicians are in fact able to get around Haag's theorem by using different arguments (they construct using a version of Wilson (but I think they came up with it independently), and things like Osterwalder-Schrader).

 

[atyy]

None of the high quality predictions of QED were achieved with the lattice approximation. All were computed with the (within the level of rigor of theoretical physics) fully local renormalized continuum version in carefully chosen approximations (NRQED).

Yes, but that was before Wilson. In the Wilsonian picture, one takes the cut-off to be finite, corresponding to say the lattice. At present, the lattice is the only rigourous construction of a theory that we believe to be quantum mechanical and also give the correct predictions. Of course, this is only "believed", but it is believed because of Wilson.

In the Wilsonian picture, there is no need for causal perturbation theory, since there are no UV divergences if one starts from the lattice.

 

[Demystifier]

I think the language is ok, because there are rigourous relativistic QFTs constructed in infinite volume and in which the lattice spacing is taken to zero.

I don't think it's true. I think all so called "rigorous" interacting QFT's need some sort of regularization of UV divergences, not much different from a non-zero lattice spacing.

 

[atyy]

I don't think it's true. I think all so called "rigorous" interacting QFT's need some sort of regularization of UV divergences, not much different from a non-zero lattice spacing.

I'm pretty sure it's correct, one can really take the lattice spacing to zero (at any rate, IIRC from other threads Haag's concerns infrared), eg. section 6.2 of http://www.claymath.org/sites/default/files/yangmills.pdf.

 

[A. Neumaier]

Yes, but that was before Wilson.

No. The most recent $>10$ digit accuracy agreement of a QED prediction dates from 2014, and is done using standard 1948 renormalized Lorentz-covariant perturbation theory, neither using lattices nor Wilson's ideas.

At present, the lattice is the only rigourous construction of a theory that we believe to be quantum mechanical and also give the correct predictions. [...] there is no need for causal perturbation theory, since there are no UV divergences if one starts from the lattice.

There are no UV divergences since these only appear in the limit where the lattice spacing tends to zero and the full theory with the full symmetry group is recovered. The lattice is a rigorous construction of an approximation with UV and IR cutoff - but such rigorous constructions without a lattice were already known in 1948. The unsolved quest for rigor is only in controlling the limits where the cutoffs are removed!