I Wilsonian viewpoint and wave function reality

[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!

 

[stevendaryl]

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!

I can't tell who wrote the above paragraph, but that example doesn't refute Einstein. Einstein was completely in sympathy with that point of view. He believed that the perfect correlations in EPR type experiments could be explained by hidden variables, and that a measurement simply revealed the pre-existing (though unknown) value of those variables. Bell showed that he was wrong---EPR type correlations cannot be explained by hidden variables unless they are nonlocal.

 

[atyy]

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.

But if there is a cut-off, there is no need for causal perturbation theory, since there are no UV divergences.

 

[A. Neumaier]

But if there is a cut-off, there is no need for causal perturbation theory, since there are no UV divergences.

There is no need, but there is also no harm. It may be easier to see in this way how the covariant causal perturbation theory is related to the Hilbert spaces whose ''limit'' (which is no longer a Fock space) would have to be constructed in a fully constructive version. Perhaps this will be the way how it is done one day.

 

[atyy]

There is no need, but there is also no harm. It may be easier to see in this way how the covariant causal perturbation theory is related to the Hilbert spaces whose ''limit'' (which is no longer a Fock space) would have to be constructed in a fully constructive version. Perhaps this will be the way how it is done one day.

I agree, but I still don't quite see why if we use a cutoff that you prefer the momentum cutoff over the lattice. In both cases, using the Wilsonian viewpoint we need to show that the low energy corase grained observables reproduce the traditional recipe plus experimentally negligible corrections. But I don't believe there is a rigourous version of Wilson's viewpoint for QED, whether one uses a momentum or a lattice cutoff.

 

[Tollendal]

Dear stevendaryl,

I was trying to say that Einstein never accepted the non locality of the Universe, that my naive example pretended illustrate. Of course, the entanglement results from this non locality - is a fact dayly demonstrated.

 

[A. Neumaier]

why if we use a cutoff that you prefer the momentum cutoff over the lattice.

There are three reasons;

1. The value of the momentum cutoff doesn't make a difference to the computational complexity, as it is just a parameter in the final formulas. On the other hand, changing the lattice size makes a tremendous difference to the work involved. As a consequence one gets for QED numerical lattice results only to a few digits of accuracy but numerical perturbation results to 10 digits of accuracy if desired, with comparable amount of work.

2. In case of the asymptotic series, it has been proved rigorously in the 1950s that, order by order, the limit exists in which the cutoff is removed, and one obtains a covariant result, interpretable in terms of the traditional relativistic scattering variables and highly accurate when truncated to low order. On the other hand, there is not a single convergence result for the lattices, only numerical studies of poor accuracy and without any associated covariance.

3. Renormalizability proofs are associated with the perturbative setting only; in the lattice case there are no such proofs for realtivistic QFTs, only plausibility arguments by analogy to soldi state physics.

Thus the lattice case has far weaker theoretical and practical properties.

I would be interested in a lattice study of the anomalous magnetic moment of the electron - haven't seen any.

 

[vanhees71]

I can't tell who wrote the above paragraph, but that example doesn't refute Einstein. Einstein was completely in sympathy with that point of view. He believed that the perfect correlations in EPR type experiments could be explained by hidden variables, and that a measurement simply revealed the pre-existing (though unknown) value of those variables. Bell showed that he was wrong---EPR type correlations cannot be explained by hidden variables unless they are nonlocal.

Indeed, that's my point! Einstein understood quantum theory completely. There's no doubt about that, and that's why I quoted this paper of 1948 which is much clearer than the now famous EPR work of 1948. There is indeed no action at a distance. It cannot be by construction of local relativistic microcausal QFT. The "problem" for Einstein was not causality (which is however indeed a problem in that flavors of interpretations that assume a collapse outside of the validity of quantum dynamics which leads to an instantaneous influence of a measurement process at far distances) but inseparability, and indeed there's no violation of causality by making a local measurement of the polarization of a singl photon at place A of a polarization entangled photon pair, with the 2nd photon registered a far distance apart at location B. The correlation of the polarizations of the photons was always there from the moment on they were created, and nothing changes instantaneously at B when the polarization of the photon is measured at A. Of course, the single-photon polarizations for the measurement were maximally random, but the correlation was there, and before the measurement the two photons are inseparable due to the entanglement. The problem thus is not quantum theory itself, when you take the probabilistic meaning of the state seriously and refer only to descriptions of ensembles (minimal statistical interpretation). Whether or not you consider this as a complete description of nature or not is your personal taste (Einstein didn't, and that's why he looked for a unified classical field theory for the last 30 years of his life), but so far we have nothing better.

 

[atyy]

There are three reasons;

1. The value of the momentum cutoff doesn't make a difference to the computational complexity, as it is just a parameter in the final formulas. On the other hand, changing the lattice size makes a tremendous difference to the work involved. As a consequence one gets for QED numerical lattice results only to a few digits of accuracy but numerical perturbation results to 10 digits of accuracy if desired, with comparable amount of work.

2. In case of the asymptotic series, it has been proved rigorously in the 1950s that, order by order, the limit exists in which the cutoff is removed, and one obtains a covariant result, interpretable in terms of the traditional relativistic scattering variables and highly accurate when truncated to low order. On the other hand, there is not a single convergence result for the lattices, only numerical studies of poor accuracy and without any associated covariance.

3. Renormalizability proofs are associated with the perturbative setting only; in the lattice case there are no such proofs for realtivistic QFTs, only plausibility arguments by analogy to soldi state physics.

Thus the lattice case has far weaker theoretical and practical properties.

I would be interested in a lattice study of the anomalous magnetic moment of the electron - haven't seen any.

OK, I see what you mean. But maybe one day lattice can get there. Then the main difference is that you are much more hopeful that a UV complete QED will be found, whereas I suspect it doesn't exist, so causal perturbation theory is a red herring.

I can kinda believe the momentum cutoff gives a well-defined quantum theory, but would like to know more before accepting that idea. Do you have a reference? Can gauge invariance really be preserved using a momentum cutoff?

 

[A. Neumaier]

I can kinda believe the momentum cutoff gives a well-defined quantum theory, but would like to know more before accepting that idea. Do you have a reference?

It is the usual starting point - how can it need a reference?

Can gauge invariance really be preserved using a momentum cutoff?

Why do you insist on exact gauge invariance if you are willing to violate exact Poincare invariance? it will be valid like the latter in the limit of removing the cutoff.

Gauge invariance is tied to masslessness of the photon. But this is unprovable - though we have excellent, very tiny upper bounds on the mass. Note that QED with massive photons is still renormalizable, and indeed the infrared problems are often (though imperfectly) addressed by assuming during the calculations that the photon has a tiny mass, put to zero at the very end of the calculations.

 

[atyy]

It is the usual starting point - how can it need a reference?

Why do you insist on exact gauge invariance if you are willing to violate exact Poincare invariance? it will be valid like the latter in the limit of removing the cutoff.

Gauge invariance is tied to masslessness of the photon. But this is unprovable - though we have excellent, very tiny upper bounds on the mass. Note that QED with massive photons is still renormalizable, and indeed the infrared problems are often (though imperfectly) addressed by assuming during the calculations that the photon has a tiny mass, put to zero at the very end of the calculations.

Well, I can provide a reference supporting the lattice approach:

http://arxiv.org/abs/hep-th/0603155
"But the lattice approach is very important also for more fundamental reasons: it is the only known constructive approach to a non-perturbative definition of gauge field theories, which are the basis of the Standard Model."

Edit: Google Erhard Seiler - he's associated with the Chopra Foundation?! Christof Koch too?

 

[A. Neumaier]

the lattice approach is very important also for more fundamental reasons: it is the only known constructive approach

You should not forget that after they discuss this approach in more detail they close to say at the end of Section 6:

On the constructive side, the success with gauge theories in four dimensions has been much more modest, even though some impressive mathematical work towards control of the continuum limit has been done by Balaban

There are many approaches to relativistic quantum field theory in 4 dimensions , none so far leading to a construction. So the statement by Fredenhagen, Rehren, and Seiler that you quoted is an article of faith, not a fact.

Fact is that all very high accuracy predictions (which only exist in QED) are done starting from the covariant formulation.

Fact is also the gauge theories that have been constructed rigorously (in lower dimensions, e.g., QED_2 = the Schwinger model and 2-dimensional Yang-Mills) were constructed through the covariant approach.

As you well know, Balaban abandoned his work trying to construct 4D Yang-Mills theory through a lattice approach, and nobody took it up again. The continuum limit which should provide O(4) invariance seems too difficult to be tackled. Even then, one obtains a Euclidean field theory, not a Minkowski one, and needs the Osterwalder-Schrader theorem (which assumes exact O(4) invariance and reflection positivity) to get to the physical theory - and the resulting physical theory is exactly Poincare invariant. I am not awae of an approximate lattice version of the Osterwalder-Schrader theorem that would provide a physical, not quite covariant theory. All this shows that exact poincare invariance is fundamental. It is even used to classify the particle content of a theory - on a finite lattice there is no S-matrix and no notion of (asymptotic) particle states.

 

[atyy]

You should not forget that after they discuss this approach in more detail they close to say at the end of Section 6:There are many approaches to relativistic quantum field theory in 4 dimensions , none so far leading to a construction. So the statement by Fredenhagen, Rehren, and Seiler that you quoted is an article of faith, not a fact.

Fact is that all very high accuracy predictions (which only exist in QED) are done starting from the covariant formulation.

Fact is also the gauge theories that have been constructed rigorously (in lower dimensions, e.g., QED_2 = the Schwinger model and 2-dimensional Yang-Mills) were constructed through the covariant approach.

As you well know, Balaban abandoned his work trying to construct 4D Yang-Mills theory through a lattice approach, and nobody took it up again. The continuum limit which should provide O(4) invariance seems too difficult to be tackled. Even then, one obtains a Euclidean field theory, not a Minkowski one, and needs the Osterwalder-Schrader theorem (which assumes exact O(4) invariance and reflection positivity) to get to the physical theory - and the resulting physical theory is exactly Poincare invariant. I am not awae of an approximate lattice version of the Osterwalder-Schrader theorem that would provide a physical, not quite covariant theory. All this shows that exact poincare invariance is fundamental. It is even used to classify the particle content of a theory - on a finite lattice there is no S-matrix and no notion of (asymptotic) particle states.

Well, we probably have more a difference of taste than a technical disagreement. I should say that perhaps Balaban abandoned his work not because his approach is not good, but because when he moved house one summer, the movers lost all his notes on Yang Mills ... (not sure if this is true, heard this anecdote from a talk by Jaffe, and I might not be retelling it quite correctly)

 

[A. Neumaier]

Well, we probably have more a difference of taste than a technical disagreement. I should say that perhaps Balaban abandoned his work not because his approach is not good, but because when he moved house one summer, the movers lost all his notes on Yang Mills ... (not sure if this is true, heard this anecdote from a talk by Jaffe, and I might not be retelling it quite correctly)

You gave here a link to the lecture where Jaffe mentioned this.

But the real point is that an approach is taken up by others if they find it promising enough, and this hasn't happened in almost 30 years passed since the bulk of the work appeared. Recently Dimock wrote a few papers giving a streamlined exposition of part of Balaban's work - apparently nothing else happened. Having lost his notes could have been a good excuse for Balaban to quit working on the topic without having to admit that it seemed a dead end.

No one found the matter promising enough to put a PhD student on it, where losing notes is not a criterion since they are expected to create new notes based on their understanding on what was published (which was a lot).

You would be of the right age to choose it as your PhD topic - may I challenge you?

 

[atyy]

You would be of the right age to choose it as your PhD topic - may I challenge you?

Ha, ha, I am pleased to know I still seem young to some people. I am way past the usual age to do a PhD. Being a lattice fan, and more a non-rigourist, if I were to make any progress with this hobby, I would try the chiral fermion problem :P

BTW, thanks for the pointer to the exposition by Dimock!

 

[Demystifier]

I would try the chiral fermion problem

Since you are fan of it, can you recommend a good and not too technical review of the chiral fermion problem, which in simple terms explains what exactly this problem is? Or even better, could you write (perhaps as an insight entry on this forum) a non-technical review by yourself?