I Wilsonian viewpoint and wave function reality

[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.

 

[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).