Wilsonian viewpoint and wave function reality

[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!

 

[atyy]

No. The most recent $>10$ digit accuracy agreement dates from 2014, and is done using standard 1948 renormalized Lorentz-covariant perturbation theory, neither using lattices nor Wilson's ideas.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 controling the limits where the cutoffs are removed!

Throughout you are assuming that relativistic QED exists - there is no proof of such a thing.

 

[A. Neumaier]

Throughout you are assuming that relativistic QED exists - there is no proof of such a thing.

No. It is only assumed that the perturbative expansion of QED exists, which is known rigorously since 1948. It is this expansion which provides the 10 digit accuracy when compared with experiment.

 

[Demystifier]

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.

In this paper they also use a UV regularization. Let me quote:
"Because of the local singularity of the nonlinear field, one must first cut off the interaction. The simplest method is to truncate the Fourier expansion of the field ..."
At the end of calculation they consider the continuum limit and show that the limit exists, but my only claim was that, at least as an intermediate step, a regularization cannot be avoided.

 

[atyy]

In this paper they also use a UV regularization. Let me quote:
"Because of the local singularity of the nonlinear field, one must first cut off the interaction. The simplest method is to truncate the Fourier expansion of the field ..."
At the end of calculation they consider the continuum limit and show that the limit exists, but my only claim was that, at least as an intermediate step, a regularization cannot be avoided.

I agree.

 

[atyy]

No. It is only assumed that the perturbative expansion of QED exists, which is known rigorously since 1948. It is this expansion which provides the 10 digit accuracy when compared with experiment.

But it doesn't make any sense without Wilson.

 

[Demystifier]

I don't know, what the RG has to do with interpretational problems at all.

One thing that relates them is ontology. If one is using an ontological interpretation, then it is reasonable to ask whether an RG transformation changes ontology.

 

[Demystifier]

But it doesn't make any sense without Wilson.

It depends on what do you mean by "make sense". Does a calculation algorithm giving right predictions make sense?

Or to use Follands analogy, does infinitesimal calculus as defined by Newton and Leibniz (before Weierstrass or Robinson) make sense?

 

[atyy]

It depends on what do you mean by "make sense". Does a calculation algorithm giving right predictions make sense?

Or to use Follands analogy, does infinitesimal calculus as defined by Newton and Leibniz (before Weierstrass or Robinson) make sense?

By make sense, I mean define a quantum theory with well defined Hilbert space etc that gives finite predictions. With lattice QED and Wilson, we can understand QED as being the low energy limit of a well-defined quantum theory.

Calculus made sense before Weierstrass if one believes that velocity = distance X time, and that things should be "nice" at our low resolution, even if space and time were discrete, which is a forerunner to the Wilsonian argument.

 

[A. Neumaier]

But it doesn't make any sense without Wilson.

It makes perfect sense if derived via causal perturbation theory. Not a single infinity, not a single cutoff, and not a single nonphysical parameter appears.

By make sense, I mean define a quantum theory with well defined Hilbert space etc that gives finite predictions. With lattice QED and Wilson, we can understand QED as being the low energy limit of a well-defined quantum theory.

The QED limit of the lattice approximation of QED is not well-defined at all. And at fixed IR and UV cutoff lattice QED lacks all relevant invariance properties.

So to me, causal perturbation theory makes much more sense, is much better understood, and gives a much better definition of QED than the completely uncontrolled lattice approximations. (Indeed, the only way to verify if a future construction of a QFT ''is'' QED is to verify that the asymptotic expansion of its S-matrix reduces to that constructed by causal perturbation theory. There is no such statement for lattice QED.)

But of course, ''making sense'', ''understanding'' and ''better'' are as subjective as the various interpretations of QM...

 

[atyy]

It makes perfect sense if derived via causal perturbation theory. Not a single infinity, not a single cutoff, and not a single nonphysical parameter appears.

The QED limit of the lattice approximation of QED is not well-defined at all. And at fixed IR and UV cutoff lattice QED lacks all relevant invariance properties.

So to me, causal perturbation theory makes much more sense, is much better understood, and gives a much better definition of QED than the completely uncontrolled lattice approximations. (Indeed, the only way to verify if a future construction of a QFT ''is'' QED is to verify that the asymptotic expansion of its S-matrix reduces to that constructed by causal perturbation theory. There is no such statement for lattice QED.)

But of course, ''making sense'', ''understanding'' and ''better'' are as subjective as the various interpretations of QM...

I prefer the Wilsonian spirit, in which we take lattice QED with finite spacing in finite volume to define the theory. Then argue non-rigourously that the the standard perturbative expansions are very good approximations to low energy coarse grained correlation functions. I feel this is better because it makes it physically clear that the expansions are only low energy coarse grained approximations, and that we do not need to take the cutoff to infinity.

I dislike the arguments behind causal perturbation theory, because it seems to solve the UV divergence problem, but in fact leaves it untouched, since no UV complete theory is constructed. If a UV complete theory exists, the causal perturbation theory is not needed, one can just construct the old fashioned Feynman series in the nonsensical subtracting infinities way, and directly prove (since one has the UV complete theory) that the nonsensically constructed series is asymptotic and give the error bounds.

Edit: You will probably argue the Wilson plus lattice viewpoint is unsatisfactory, since it doesn't explain why the invariance properties emerge at low energies. So the additional bit of philosophy that goes with the lattive viewpoint is that relativity can be emergent at low energies, eg. massless relativistic Dirac fermions in graphene.

As I understand it, the major argument against the lattice viewpoint is that there is no consensus lattice construction of chiral fermions interacting with non-Abelian gauge fields. So the lattice viewpoint at the moment is restricted to say QED. But given that there is no rigourous relativistic QFT in 3+1D, the lattice viewpoint is ahead, since it can rigourously construct at least a candidate theory. Furthermore, some attempts at constructing Yang-Mills in 3+1D rigourously do start from the lattice (eg. Balaban).

 

[A. Neumaier]

the lattice viewpoint is ahead, since it can rigourously construct at least a candidate theory.

we take lattice QED with finite spacing in finite volume to define the theory.

This does not define QED but a huge infinite family of mutually inequivalent theories, one for each possible lattice and lattice spacing. Each one makes different predictions, most of them very poor ones.

Moreover, in lattice QED one must already work very hard to get 3 digits of relative accuracy. It requires more than astronomical resources to get 10 digits. Moreover, all practically computable lattice predictions are done in Euclidean (imaginary-time) QFT, and one has to resort to a nonexistent discrete analogue of the Osterwalder-Schrader theorem to get a real-time version.

All this even holds for scalar QED where no fermion doubling problem exists.

 

[A. Neumaier]

But given that there is no rigourous relativistic QFT in 3+1D

QED with a fixed large momentum cutoff is a well-defined, rigorous nearly relativistic QFT whose S-matrix elements have an asymptotic series that converges coefficient-wise to that of causal perturbation theory, say.

Thus the results obtained by truncating the covariant perturbation series at 5 loops (needed for the 10 digits) are provably equivalent (within computational accuracy) to those of a rigorously defined nearly relativistic QFT.

This is far better than what one has for the lattice approximations, which are completely uncontrolled.

 

[atyy]

QED with a fixed large momentum cutoff is a well-defined, rigorous nearly relativistic QFT whose S-matrix elements have an asymptotic series that converges coefficient-wise to that of causal perturbation theory, say.

Thus the results obtained by truncating the covariant perturbation series at 5 loops (needed for the 10 digits) are provably equivalent (within computational accuracy) to those of a rigorously defined nearly relativistic QFT.

This is far better than what one has for the lattice approximations, which are completely uncontrolled.

I don't know enough to know if this is correct (eg. how is the Hilbert space defined, is time evolution unitary etc?), but this is certainly in the Wilsonian and lattice spirit. It is not in the causal perturbation spirit, since there is a cut-off.

 

[A. Neumaier]

I don't know enough to know if this is correct (eg. how is the Hilbert space defined, is time evolution unitary etc?)

The Hilbert space is standard Fock space with momentum states up to the cutoff energy. The Hamiltonian is derived from the action (written in momentum space and with the cut-off). Hence the dynamics is unitary. It is well-known that for QED the resulting renormalized perturbation series is (in the limit of infinite cutoff) independent of the details of the regularization, hence agrees with that of any of the established procedures, including dimensional renormalization (the best computational heuristics) and causal perturbation theory (the covariant and manifestly finite derivation of the perturbation series). Note that causal perturbation theory never claimed results different from the traditional ones, only a mathematically agreeable procedure to arrive at them.

this is certainly in the Wilsonian and lattice spirit. It is not in the causal perturbation spirit, since there is a cut-off.

It is certainly not in the lattice spirit, since this is an uncontrolled approximation.

Note that one could do causal perturbation theory with a cutoff and then obtain exactly the approximate perturbation series in a completely analogous way. But this is needed only for people like you who want to see an explicit family of Hilbert spaces and don't trust the perturbation series otherwise. So nobody working in the field is interested in writing it out explicitly.

The real mathematical obstacles only show up when one tries to justify limits. And these difficulties seem at present unsurmountable both in the covariant approaches and in the lattice approaches. So in this respect none has an advantage over the other.

 

[Tollendal]

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)

Dear Atvy,

Suppose I take a pair of gloves and put each of them in a box. I keep one with me and give the other to you, who then take a rocket to the Moon and there open it. Instantaneously you know what glove remained with me - there is no "communication" between us, as the situation was deffined the moment I closed the boxes. I imagine that's an example the Universe is non local, an empyrical constatation we must accept as a datum from reality.
Einstein didn't understand it. It seems to me his "ghostly action at distance" is nonsense!