Wilsonian viewpoint and wave function reality

[atyy]

If you take Copenhagen seriously and conceive that degrees of freedom are not real, then Wilson coarse graining is also not real. It is just a calculation tool. A very useful tool. Which, for someone who takes Copenhagen seriously, should be enough.

Is the tool "physical"? Yes, if you are persistent in taking Copenhagen seriously. Bohr said that the task of physics is not to find out how nature is, but what we can say about nature. So if Wilson coarse graining helps you to say something about nature, then, according to Copenhagen, it's physical.

Is it consistent with the Wilsonian spirit? Probably not, for Wilson himself was probably not someone who was taking Copenhagen very seriously. But people can use the same tools even when they have different spirits. (For instance, using a computer not for computing but for discussions on the forum is not in the original spirit of the idea of computer.)

Yes, that's really what I'm asking about - is Copenhagen consistent with the Wilsonian spirit?

It is the spirit that is important, since the spirit is the main reason why physicists no longer believe renormalization to be a conceptual problem, even if they cannot execute Wilsonian renormalization in a mathematically sound way (dimensional regularization!). The chief value of Wilsonian thinking is not calculational, but conceptual or "spiritual" or "moral", as physicists say.

Why do you think Wilson did not taken Copenhagen seriously?

 

[Jimster41]

But if one chooses to conceive Wilson as coarse graining degrees of freedom, yet conceive the degrees of freedom as not real, then why would we accept Wilson's explanation as a "physical" explanation for why renormalization works?

I thought the re-normalization step on a lattice corresponded to a self-similarity step of a fractal (recursive system). I kind of thought that was the idea.

And that therefore re-normalization doesn't discard degrees of freedom. It's just that under exponentiation recursive systems go back and forth between appearing to have lots of degrees of freedom and appearing to be self similar (the same thing over and over - fewer degrees of freedom).

And If I understood Schroeder at all multi-fractals might explain how such a process could be noisy and impure, create what seem to be permanent degrees of freedom and never create perfect self-similarity - why we always see a mix of repetition and novelty. It's because there is really more than one recursive system or structure involved. It's multi-fractal.

Not trying to suggest a theory. I just thought that's what the existing theories re-normalization on lattices were playing with. Why else the repetitive lattice?

 

[vanhees71]

Yes, that's really what I'm asking about - is Copenhagen consistent with the Wilsonian spirit?

It is the spirit that is important, since the spirit is the main reason why physicists no longer believe renormalization to be a conceptual problem, even if they cannot execute Wilsonian renormalization in a mathematically sound way (dimensional regularization!). The chief value of Wilsonian thinking is not calculational, but conceptual or "spiritual" or "moral", as physicists say.

Why do you think Wilson did not taken Copenhagen seriously?

What do you mean by that physicists can't do the renormalization? The RG equations are in use for more than 50 years now, and particularly the functiona RG approach in finite-temperature QFT is very popular again today in nuclear and also condensed-matter theory!

 

[atyy]

What do you mean by that physicists can't do the renormalization? The RG equations are in use for more than 50 years now, and particularly the functiona RG approach in finite-temperature QFT is very popular again today in nuclear and also condensed-matter theory!

As an example, even in classical physics, physicists cannot do renormalization - does the epsilon expansion really make any sense? There is still a lot of work for mathematicians to do. But in classical physics, the physical picture of Wilson (Landau, Kadanoff) is so good that although we may have qualms about the mathematics, we believe that mathematicians will ultimately succeed.

 

[vanhees71]

What specifically do you think doesn't make sense? It makes very much sense in perturbation theory and certain non-perturbative approximations. In QED it makes so much sense that it's among the most accurate descriptions of properties of particles and their interactions we have in physics. I simply don't see your point. Of course, there's no mathematical rigorous description of interaction QFTs, but the Wilson RG methods rehabilitated QFT as effective theories, explaining why you can ignore high energy-momentum scales to describe the low-energy-momentum physics relevant for your system under consideration.

 

[Tollendal]

Dear Vanhees71,

If the factor psi signifies a probability, as Max Born demonstrated, when the measurement is made all indefinition desappears. It seems to me there is no mystery whatsoever in this (as the Copenhagen interpretation implies).

 

[Jimster41]

I found a cool and recent paper on spin lattice quantum percolation modeled with multifractals. They do seem to equate re-normalization with self-similarity. But I am pretty confused about the implications of re-normalization vis-a-vis the complaint about throwing away degrees of freedom and the implications of self-similarity on those degrees of freedom.

Partly it stems from an inability to really grasp what a fractional dimension represents. I can't understand that even when talking about a single fractal sysem - much less a multfractal system. But I can imagine that there are important differences if the multifractal case.http://arxiv.org/pdf/1405.1985v3.pdf

Quantum percolation transition in 3d: density of states, finite size scaling and multifractality
Laszlo Ujfalusi, Imre Varga
(Submitted on 8 May 2014 (v1), last revised 23 Oct 2014 (this version, v3))
The phase diagram of the metal-insulator transition in a three dimensional quantum percolation problem is investigated numerically based on the multifractal analysis of the eigenstates. The large scale numerical simulation has been performed on systems with linear sizes up to L=140. The multifractal dimensions, exponents Dq and αq, have been determined in the range of 0≤q≤1. Our results confirm that this problem belongs to the same universality class as the three dimensional Anderson model, the critical exponent of the localization length was found to be ν=1.622±0.035. The mulifractal function, f(α), appears to be universal, however, the exponents Dq and αq produced anomalous variations along the phase boundary, pQc(E).

 

[vanhees71]

Dear Vanhees71,

If the factor psi signifies a probability, as Max Born demonstrated, when the measurement is made all indefinition desappears. It seems to me there is no mystery whatsoever in this (as the Copenhagen interpretation implies).

I couldn't agree more!

 

[atyy]

What specifically do you think doesn't make sense? It makes very much sense in perturbation theory and certain non-perturbative approximations. In QED it makes so much sense that it's among the most accurate descriptions of properties of particles and their interactions we have in physics. I simply don't see your point. Of course, there's no mathematical rigorous description of interaction QFTs, but the Wilson RG methods rehabilitated QFT as effective theories, explaining why you can ignore high energy-momentum scales to describe the low-energy-momentum physics relevant for your system under consideration.

How does a fractional dimension make sense?

It is wrong to use the accurate predictions to justify the lack of sense. There is no need for Wilson at all if we accept insensible calculations that happen to match experiments closely. The point of Wilson is that he gave a physically sensible picture of renormalization, so that even if we cannot exactly carry it out, we believe the present wrong calculations involving fractional dimensions are close enough in spirit to the right calculations.

 

[vanhees71]

Dimensional regularization is a mathematical tool to organize the evaluation of Feynman diagrams in perturbation theory. It's nothing essential. You can also renormalize without any regularization using the BPHZ technique of subtraction on the level of the integrands. It's just a bit less convenient than dim. reg. Perhaps you should read some good book about renormalization...

 

[atyy]

Dimensional regularization is a mathematical tool to organize the evaluation of Feynman diagrams in perturbation theory. It's nothing essential. You can also renormalize without any regularization using the BPHZ technique of subtraction on the level of the integrands. It's just a bit less convenient than dim. reg. Perhaps you should read some good book about renormalization...

In fact these only construct formal power series. They are not physical.

 

[vanhees71]

A series is not physical but it describes physical quantities, namely S-matrix elements.

BTW you can also define the perturbative series without ever having trouble with ill-defined (divergent) integrals. See, e.g.,

Finite Quantum Electrodynamics, the Causal Approach, Springer (1995)

 

[atyy]

A series is not physical but it describes physical quantities, namely S-matrix elements.

BTW you can also define the perturbative series without ever having trouble with ill-defined (divergent) integrals. See, e.g.,

Finite Quantum Electrodynamics, the Causal Approach, Springer (1995)

The series is formal power series because it is not convergent, and there is no construction of the theory to which it is an approximation.

 

[vanhees71]

It's an asymptotic series, most probably with divergence radius 0. So what?

 

[atyy]

It's an asymptotic series, most probably with divergence radius 0. So what?

For it to be asymptotic series, the thing that it is approximating must exist. In other words, the theory must be constructed. Does a construction of QED exist?

 

[vanhees71]

I don't understand what you mean by "constructed". You have to check whether the series converges in the sense of asymptotic series of not. If it doesn't you are in trouble and have to try other descriptions than perturbation theory to make sense of it (some resummation, e.g.).

 

[atyy]

I don't understand what you mean by "constructed". You have to check whether the series converges in the sense of asymptotic series of not. If it doesn't you are in trouble and have to try other descriptions than perturbation theory to make sense of it (some resummation, e.g.).

"Constructed" in the mathematical sense, eg. has Yang-Mills (UV complete in finite volume) been constructed? As of this date, the answer is no.

Similarly, has QED (UV complete in finite volume) been constructed? As of this date, the answer is no.

So I don't believe your claim that the series is asymptotic is justified.

 

[vanhees71]

I'm sorry, but I'm not familiar with what "constructed" means. I'm not an expert in axiomatic QFT. The only thing I know is that so far there is no rigorous mathematical definition of a realistic interacting QFT in (1+3) dimensions.

I'm not aware, however, that there is a practical problem with QED. The Lamb shift calculations are done up to 4 or 5 loop order without any indication for "divergence".

In QCD a famous (or better infamous) example for the failure of purely perturbative methods is the evaluation of the equation of state, which is complete up to the order possible for technical reasons, and this series is highly "divergent" in standard perturbation theory. One way out is hard-thermal-loop resummed perturbation theory. A nice summary can be found in the following talk:

http://www.helsinki.fi/~rummukai/talks/trento06.pdf

 

[atyy]

I'm sorry, but I'm not familiar with what "constructed" means. I'm not an expert in axiomatic QFT. The only thing I know is that so far there is no rigorous mathematical definition of a realistic interacting QFT in (1+3) dimensions.

I'm not aware, however, that there is a practical problem with QED. The Lamb shift calculations are done up to 4 or 5 loop order without any indication for "divergence".

In QCD a famous (or better infamous) example for the failure of purely perturbative methods is the evaluation of the equation of state, which is complete up to the order possible for technical reasons, and this series is highly "divergent" in standard perturbation theory. One way out is hard-thermal-loop resummed perturbation theory. A nice summary can be found in the following talk:

http://www.helsinki.fi/~rummukai/talks/trento06.pdf

As an example, one way to get Stirling's approximation for the factorial is via divergent, but asymptotic series. In this case we do know that the factorial exists rigourously, ie. it has been constructed.

 

[vanhees71]

Ok, then we have no constructed interacting QFT in (1+3) dimensions, as far as I know.

 

[Jimster41]

How does a fractional dimension make sense?

It is wrong to use the accurate predictions to justify the lack of sense. There is no need for Wilson at all if we accept insensible calculations that happen to match experiments closely. The point of Wilson is that he gave a physically sensible picture of renormalization, so that even if we cannot exactly carry it out, we believe the present wrong calculations involving fractional dimensions are close enough in spirit to the right calculations.

Do you mean, "how could a fractional dimension be physical"?
Hasn't a good bit of sensible theory around such a thing been developed at this point?
... though I agree the idea does sort of defy physical intuition - at least mine anyway.

https://en.wikipedia.org/wiki/Fractal_dimension

 

[Jimster41]

How does a fractional dimension make sense?

It is wrong to use the accurate predictions to justify the lack of sense. There is no need for Wilson at all if we accept insensible calculations that happen to match experiments closely. The point of Wilson is that he gave a physically sensible picture of renormalization, so that even if we cannot exactly carry it out, we believe the present wrong calculations involving fractional dimensions are close enough in spirit to the right calculations.

Surely a cantor dust can be imagined physically - as a non-finite process (I know that's probably not the right technical term) - but I mean it just keeps going. It's an infinite series, without limit (is that what you are getting at?) - there are no cases where a bound can be drawn around it's "state" because it doesn't have one in the classical sense.

It seems possible (to me at least) that if everything we see and are is built on a Cantor Dust-like process - QM is an accurate description of reality including the part where the "state" of the fundamental process is always partly indeterminate, really. And it really means the horizon - whatever it is, supports only a Cantor Dust-like (or generally fractal recursive) process. In fact that is why there is a "horizon" between observed and unobserved in the first place.

In which case Copenhagen is right (you can't really ask an answerable question about full state evolution at the horizon - nothing ever finally gets there). And maybe Wilson is precise and physical because multi-fractal processes do create seemingly complete structures even though they never complete.

So...maybe "God created the fractals" and we invented everything else.

 

[A. Neumaier]

You can also renormalize without any regularization using the BPHZ technique of subtraction on the level of the integrands. It's just a bit less convenient than dim. reg. Perhaps you should read some good book about renormalization...

you can also define the perturbative series without ever having trouble with ill-defined (divergent) integrals. See, e.g.,
Finite Quantum Electrodynamics, the Causal Approach, Springer (1995)

See my Insight Article on this. Thus renormalization does not depend on fractional dimensions - the latter are just very convenient since they preserve all symmetries.

In fact these only construct formal power series. They are not physical.

Anything done in interacting 4D relativistic quantum field theory only constructs approximations (crude lattice regularizations or asymptotic series) to theories that have not yet been constructed. Nevertheless, QED, which is such a theory, is among all physical theory the one that was checked experimentally to the highest accuracy (up to 12 significant digits). Thus having constructed only approximations (and asymptotic series are infinite families of such approximations) doesn't make the latter unphysical. Only the intermediate terms leading to the final results are unphysical. Note that lattice approximations have the same problem - and even worse since not even an asymptotic expansion of the approximation error made is known.

You have to check whether the series converges in the sense of asymptotic series of not.

This doesn't make sense - there is no convergence in the sense of asymptotic series. The problem is that every power series is an asymptotic series to a huge number of aribitrarily often differentiable functions, taking arbitrary at any given fixed nonzero value of the argument. For if $f(x)$ is such a function then $g(x)=f(x)+\sum_{k=1:N} c_ke^{-(k/x)^2}$ is for any choice of the $c_k$ another such function, with the same asymptotic expansion, and the free constants can be matched to arbitrary function values. Thus knowing the asymptotic series is still very, very far from knowing anything significant about the function itself. One needs (proofs for, or assumptions of) uniform estimates for the error in order to pin down (at least to some degree) the function itself. The implicit (in 4D unproved) assumption in asymptotic QFT calculations is that the error is of the order of the first neglected term, which seems at least to hold very well for QED.

So I don't believe your claim that the series is asymptotic is justified.

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

 

[vanhees71]

The implicit (in 4D unproved) assumption in asymptotic QFT calculations is that the error is of the order of the first neglected term, which seems at least to hold very well for QED.

It was this sense I meant when I said "convergent in the sense of an asymptotic series".

 

[A. Neumaier]

You have to check whether the series converges in the sense of asymptotic series of not.

The implicit (in 4D unproved) assumption in asymptotic QFT calculations is that the error is of the order of the first neglected term, which seems at least to hold very well for QED.

It was this sense I meant when I said "convergent in the sense of an asymptotic series".

Even if the first statement was meant in the second sense it cannot be checked without proving error bounds, and hence couldn't be checked for QED.

I think what current practice amounts to is: One checks whether the first few terms produce an answer consistent with experiments, and when this is the case one counts it as a success and believes that the error is small enough.

 

[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.