What should be the mathematical way out of QFT's decades stagnation?

[Tendex]

TL;DR

The main approach in QFT uses Wightman axioms that define the quantum fields as smeared fields. This has the well-known drawback that multiplication of distributions is in general not well-defined and this kind of multiplication appears routinely in QFT calculations.

So which should be the best strategy to avoid this hurdle: maybe try and look for a different mathematical object as quantum fields? or rather try and avoid their multiplication in our ways to handle them to obtain predictions? any other ideas? perhaps leave it altogether as not very many phycisists and mathematicians care much for the whole thing anyway?

 

[vanhees71]

Well, I just use the standard renormalization methods. In perturbative calculations usually dim. reg. works nicely.

Conceptually a formulation to deal with the problem to make sense of multiplication of distributions is the "causal perturbation theory" aka Epstein-Glaser Approach. A nice textbook covering this for QED is

G. Scharf, Finite Quantum Electrodynamics, Springer (1995)

 

[Tendex]

Well, I just use the standard renormalization methods. In perturbative calculations usually dim. reg. works nicely.

Conceptually a formulation to deal with the problem to make sense of multiplication of distributions is the "causal perturbation theory" aka Epstein-Glaser Approach. A nice textbook covering this for QED is

G. Scharf, Finite Quantum Electrodynamics, Springer (1995)

Thanks for your answer, but the Wightman axioms are more general than the perturbative approach, and my questions are too.

 

[A. Neumaier]

Summary:: The main approach in QFT uses Wightman axioms that define the quantum fields as smeared fields. This has the well-known drawback that multiplication of distributions is in general not well-defined and this kind of multiplication appears routinely in QFT calculations.

So which should be the best strategy to avoid this hurdle: maybe try and look for a different mathematical object as quantum fields? or rather try and avoid their multiplication in our ways to handle them to obtain predictions? any other ideas? perhaps leave it altogether as not very many phycisists and mathematicians care much for the whole thing anyway?

It is known from the early days of relativistic quantum field theory that relativistic quantum fields necessarily involve operator-valued distributions - even in the free case. So avoiding this hurdle would mean avoiding relativistic quantum fields. Instead, the hurdle must be studied!

The way forward is to make oneself familiar with the rules for handling distributions and to apply them nonperturbatively, using causal perturbation theory as blueprint. For example,

• D. Buchholz and K. Fredenhagen, A C*-algebraic approach to interacting quantum field theories. Communications in Mathematical Physics 377 (2020), 947–969.

obtained in this way the nonperturbative field algebra of QED.

 

[Tendex]

It is known from the early days of relativistic quantum field theory that relativistic quantum fields necessarily involve operator-valued distributions - even in the free case. So avoiding this hurdle would mean avoiding relativistic quantum fields. Instead, the hurdle must be studied!

The way forward is to make oneself familiar with the rules for handling distributions and to apply them nonperturbatively, using causal perturbation theory as blueprint.

That blueprint doesn't seem a practical way forward as was discussed in another thread. Trying strategies that work perturbatively in the non-perturbative setting is like shooting a finite quantity of bullets in the dark to a moving target in a continuum of uncountably infinite possible positions.

 

[A. Neumaier]

That blueprint doesn't seem a practical way forward as was discussed in another thread.

But Buchholz and Fredenhagen achieved a significant nonperturbative progress in this way! Not yet the final thing but more than what could reasonably be hoped for 10 years ago. Difficult means not impossible!

 

[Tendex]

But Buchholz and Fredenhagen achieved a significant nonperturbative progress in this way! Not yet the final thing but more than what could reasonably be hoped for 10 years ago. Difficult means not impossible!

It is also not impossible that Schwartz distributions be demonstrably incompatible with quantum fields axiomatic properties. It seems there is no alternative plan for that eventuality.

 

[A. Neumaier]

It is also not impossible that Schwartz distributions be demonstrably incompatible with quantum fields axiomatic properties. It seems there is no alternative plan for that eventuality.

There is no need for an alternative plan as long as progress is made along the natural route. Progress in the last few years is clearly visible.

 

[mitchell porter]

I heard that the work of Fields medalist Martin Hairer might be relevant.

Also see section 6.2 of the official problem description for the Millennium Prize regarding Yang-Mills theory.

 

[Tendex]

There is no need for an alternative plan as long as progress is made along the natural route. Progress in the last few years is clearly visible.

Can you summarize specifically what recent progress you are referring to? The paper you mentioned developes a very old idea(1955) by Bogolubov so it seems more of the same or a variation on the same theme. What's qualitatively or in any way different or novel that can be considered a progress, in what way it takes us closer to the solution so that you can claim it was not expected as early as ten years ago ?

 

[A. Neumaier]

Can you summarize specifically what recent progress you are referring to? The paper you mentioned develops a very old idea(1955) by Bogolubov so it seems more of the same or a variation on the same theme. What's qualitatively or in any way different or novel that can be considered a progress, in what way it takes us closer to the solution so that you can claim it was not expected as early as ten years ago ?

Only since their 2020 paper (preprint from 2019) we know the field algebra of QED nonperturbatively. This algebra has lots of states, some of which explicitly constructed in the paper cited. What remains is to rigorously construct the states with the properties of interest. For example, the vacuum state gives rise to the few particle scattering matrix relevant for the interpretation of collision events in particle accelerators. Starting from the nonperturbative field algebra, this state can be approximately found by causal perturbation theory, which is all one needs to get the extraordinary accuracy of QED predictions.

Thus since the breakthrough in 2019, the situation in QED is essentially the same as that for the Navier-Stokes equations in 4 spacetime dimensions where the 150 year old equations are rigorously defined for more than 100 years, not only for 1 year. The construction of rigorous global solutions remained as elusive as that of the vacuum state of QED or Yang-Mills. On the other hand, people working in fluid dynamics know how to solve the equations numerically to get immensely useful predictions for fluid flow, such as optimal airfoils, etc.

But according to your argumentation, people should worry because it is not impossible that the existence of global solutions is demonstrably incompatible with the axiomatic properties of the Navier-Stokes equation. It seems there is no alternative plan for that eventuality. So one should ask for the best strategy to avoid this hurdle: maybe try and look for a different mathematical object as classical fields? Or ''perhaps leave it altogether as not very many physicists and mathematicians care much for the whole thing anyway?''

The truth is simply that hard problems are hard and their solution needs much more patience, time and effort that people like you are willing to allow.

Note that the Navier-Stokes problem is another Millennium problem, unsolved for more than 150 years. In contrast, the quantum Yang-Mills Millennium problem that you referred to is under investigation for less than 50 years. So we may possibly have to wait for another hundred years or so to see it solved in the rigorous sense required by the prize conditions - which are in terms of the Wightman axioms!

But I think much less time will be needed. Each of these problems will be solved by one of those prepared to study the question deep enough to find the missing insights. The Buchholz-Fredenhagen paper might well be the key for this...

 

[Tendex]

Only since their 2020 paper (preprint from 2019) we know the field algebra of QED - nonperturbatively. This algebra has lots of states, some of which explicitly constructed in the paper cited. What remains is to rigorously construct the states with the properties of interest.

It is a well known shortcoming of the C*-algebraic approach to QFT that the physical properties demanded on these states are not being found so it is difficult to connect the mathematical objects with the physics(even Haag admits that this approach "has given us a frame and a language not a theory.”. This mean that while mathematically might be a nice accomplishment to construct such field algebra for QED, it actually is too soon to know if this actually is real progress or not for the problem discussed.

Anyway, and since you addressed me personally in your rant after the quoted paragraphs, maybe I'm wrong but my perception is that you might be much too emotionally attached to this approach to be an impartial judge of its progress, as a general principle in science it is always the worst strategy to get too attached to a single pet approach or research line. It works better to keep an open-mind.

For example, the vacuum state gives rise to the few particle scattering matrix relevant for the interpretation of collision events in particle accelerators. Starting from the nonperturbative field algebra, this state can be approximately found by causal perturbation theory, which is all one needs to get the extraordinary accuracy of QED predictions.

This is no progress at all from perturbative usual ways if the right states can't be constructed.

 

[A. Neumaier]

It is a well known shortcoming of the C*-algebraic approach to QFT that the physical properties demanded on these states are not being found

Only if one requires them to be found rigorously in 4D. Finding them approximately, as in perturbation theory, is today easy - it is done for QED in causal perturbation theory. This is at the same level as one obtains S-matrices in quantum mechanics, where also perturbation theory is used.

Haag admits that this approach "has given us a frame and a language not a theory.”.

But this was long ago. There has been a lot of progress in algebraic QFT since his book. See, e.g., the 2019 book by Dütsch. It is fully rigorous and mainly contains theory, not just a frame and a language.

you might be much too emotionally attached to this approach to be an impartial judge of its progress

In making conjectures about the future there cannot be impartiality. But I don't earn any royalties from the publications on it, nor would I benefit in any other way beyond scientific curiosity. I am emotionally attached only because unlike you I see the potential in the approach. I studied in some detail all the alternatives, too, but none is as promising as the causal approach. Your suggestion of abandoning the whole field is surely the worst alternative, because it just means giving up on an important scientific challenge.

This is no progress at all from perturbative usual ways if the right states can't be constructed.

The progress consists in knowing precisely what the perturbative states are approximations of, in the same sense as one knows what the numerical solutions computed in fluid mechanics are approximations of. In both cases, one has no rigorous construction of the right states.

But before the B/F paper it wasn't even well-defined mathematically what a state in QED is.

 

[Tendex]

Only if one requires them to be found rigorously in 4D.

Sorry if I wasn't clear that this was the only dimension my comments refer to. Do you not consider the remote chance that there could be something special about 4D that makes all progress in different dimensions useless?

Your suggestion of abandoning the whole field is surely the worst alternative, because it just means giving up on an important scientific challenge.

I have nowhere suggested such thing. I've suggested to be open to alternatives, it's a different thing.

The progress consists in knowing precisely what the perturbative states are approximations of, in the same sense as one knows what the numerical solutions computed in fluid mechanics are approximations of. In both cases, one has no rigorous construction of the right states.

The analogy with fluid dynamics is not accurate enough, classical fields are very different from smeared fields both mathematically and physically and classical states still more so from the even higher level of abstraction of the field C*-algebras. It's not just the right states for solving the 4D equations that aren't rigorously constructed but the physical family of states out of which the solution might or not be found.

But before the B/F paper it wasn't even well-defined mathematically what a state in QED is.

It still isn't. A QED field algebra is not a 4D QED state. See above.

 

[A. Neumaier]

Do you not consider the remote chance that there could be something special about 4D that makes all progress in different dimensions useless?

No; there is nothing that would hint at that. The difficulty comes from the lack of good functional analytic techniques for obtaining the estimates needed for rigorous constructions in 4D, not from the distributional nature of quantum fields, a property shared by 2D and 3D QFTs. Moreover, the progress I had pointed out applies in 4D!

I've suggested to be open to alternatives,

... such as the following:

leave it altogether as not very many phycisists and mathematicians care much for the whole thing anyway?

Openness to powerless alternatives is a very poor option.

Most physicists and mathematicians do not care much for almost any particular problem; so your suggestion, taken to its logical conclusion, would mean to stop all physical and mathematical research. Instead, physicists and mathematicians choose to work on the problems they are interested in.

To succeed in solving hard problems it is enough that a handful of strong and dedicated researchers pursue these. It doesn't matter to them that there are people like you who think that easier alternatives are needed. There are enough easier problems for the less ambitious people.

classical fields are very different from smeared fields

No. Though the classical introductory textbook field equations have unsmeared solutions, many classical field equations on 4D Minkowski spacetime are only known to have global solutions in a Sobolev space, which need smearing to make them well-defined. Moreover, sometimes one can show that the physically correct solutions must be Young measure valued, which do not even allow an unsmeared interpretation for particular solutions of interest.

It's not just the right states for solving the 4D equations that aren't rigorously constructed but the physical family of states out of which the solution might or not be found.

As in quantum mechanics, the family of all states defines precisely what a solution is. This makes everything well-specified. This is fully analogous to the requirement in classical PDEs of belonging to some Sobolev space or some bigger space with less regularity.

The solutions of interest are then specified by adding appropriate symmetry requirements, boundary conditions, and/or regularity conditions, again as in quantum mechanics and in the case of PDEs. To select the vacuum state (satisfying the Wightman axioms) one just needs to specify Poincare invariance, a regularity condition at infinity (tempered distributions), and irreducibility (which can be enforced afterwards). Constructing these rigorously is therefore a task conceptually completely analogous just like constructing a global solution of the Navier-Stokes equations with specified boundary conditions.

 

[Tendex]

No. Though the classical introductory textbook field equations have unsmeared solutions, many classical field equations on 4D Minkowski spacetime are only known to have global solutions in a Sobolev space, which need smearing to make them well-defined. Moreover, sometimes one can show that the physically correct solutions must be Young measure valued, which do not even allow an unsmeared interpretation for particular solutions of interest.

As in quantum mechanics, the family of all states defines precisely what a solution is. This makes everything well-specified. This is fully analogous to the requirement in classical PDEs of belonging to some Sobolev space or some bigger space with less regularity.

The solutions of interest are then specified by adding appropriate symmetry requirements, boundary conditions, and/or regularity conditions, again as in quantum mechanics and in the case of PDEs. To select the vacuum state (satisfying the Wightman axioms) one just needs to specify Poincare invariance and a regularity condition at infinity (termpered distributions), irreducibility (which can be enforced afterwards). Constructing these rigorously is therefore a task conceptually completely analogous just like constructing a global solution of the Navier-Stokes equations with specified boundary conditions.

I'm afraid you are not understanding that by smeared fields in RQFT I meant Operator-Valued(or in this case also algebra-valued) distributions already in the linear regime, not just weak solutions of nonlinear equations, in Sobolev spaces or Young measures where real-valued distributions and linear functionals may appear. We seem to be talking about different smearings.

Quantized relativistic fields are not the same mathematically as classical fields, which was what I claimed, regardless of your insistence on the Navier-Stokes equations, and neither is mathematically analogous quantizing fields to quantizing non-relativistic single particles. But the weaker analogy that you use as answer about solutions of PDEs and constructions QFTs is merely a wish, not a mathematically proved fact as you appear to claim using it so liberally.

 

[A. Neumaier]

Quantized relativistic fields are not the same mathematically as classical fields.

I didn't claim this. But classical fields are real-valued or vector-valued distributions or (in elasticity) matrix-valued distributions. (The latter are already operator-valued, though only operators on $R^2$.)

It is extremely natural that quantization promotes these to operator valued distributions. Something different would not be a quantum theory anymore.

I'm afraid you are not understanding that by smeared fields in RQFT I meant Operator-Valued (or in this case also algebra-valued) distributions already in the linear regime.

But the linear regime (linear quantum fields in a classical external field in flat or curved background) is well-understood and rigorously constructed, so there is no incentive to contemplate an alternative.

Difficulties appear only in the nonlinear case. There you have the same kind of difficulties already in the simpler classical setting. It is not to be expected that the quantum version should be easier than the classical version!

 

[Tendex]

classical fields are real-valued or vector-valued distributions or (in elasticity) matrix-valued distributions.

I guess this is a typo and you meant functions, not distributions.

 

[A. Neumaier]

I guess this is a typo and you meant functions, not distributions.

No.

Classical fields are in the nicest generic cases (i.e., excluding solutions of differential equatiosn with analytic coefficients) elements of Sobolev spaces, hence not functions but only equivalence classes of functions agreeing except for a set of measure zero. Even for linear PDEs in compact domains (the simplest generic case), the existence of continuous solutions (which distinguish a unique member of the equivalence class) requires the presence of an additional Sobolev inequality.

However, each element of a Sobolev space determines a unique distribution. Indeed, the theory of linear partial differential equations features distributions very prominently; see, e.g., the series of books by Hörmander. The techniques used today in causal perturbation theory are based on wave front sets characterizing certain properties of distributions, and the properties of wave front sets were explored by people working on classical PDEs long before they were exploited in QFT.

 

[Tendex]

No.

Classical fields are in the nicest generic cases (i.e., excluding solutions of differential equatiosn with analytic coefficients) elements of Sobolev spaces, hence not functions but only equivalence classes of functions agreeing except for a set of measure zero. Even for linear PDEs in compact domains (the simplest generic case), the existence of continuous solutions (which distinguish a unique member of the equivalence class) requires the presence of an additional Sobolev inequality.

I thought you were referring to the usual classical field functions that we all know and appretiate. I don't actually mind such generality. I guess the relevant phrase was:

quantization promotes these to operator valued distributions

This promotion is the key difference I was referring to all along.

 

[A. Neumaier]

I thought you were referring to the usual classical field functions that we all know and appreciate. I don't actually mind such generality.

Well, solutions of PDEs behave in general worse than the textbook examples would suggest. One needs distributions to describe their properties in general. Therefore distributions are the natural rigorous setting for classical fields.

This promotion is the key difference I was referring to all along.

What else should classical fields be promoted to upon quantization, according to this key difference?
There are no natural alternatives!

Therefore operator-valued distributions and hence Wightman fields (or the more general Yaffe fields, which do not require temperedness of the distributions) are the natural mathematical objects in terms of which a rigorous relativistic QFT should be described.

 

[vanhees71]

But in classical field theory the distributions are due to approximations stemming from simplifications to get some analytical solutions (like "point charges" in electrodynamics, fields from a current-conducting infinite cylindrical wire, etc.) .

 

[Tendex]

What else should classical fields be promoted to upon quantization, according to this key difference?
There are no natural alternatives!

Sure, I was pointing out the difference that made your conceptual analogy not quite complete.

 

[A. Neumaier]

But in classical field theory the distributions are due to approximations stemming from simplifications to get some analytical solutions (like "point charges" in electrodynamics, fields from a current-conducting infinite cylindrical wire, etc.) .

Not always. There are other kinds of distributions that are relevant in physics, for example in the study of microstructures. There the PDEs arise from nonconvex actions, and their solutions are complicated oscillating distributions described by Young measures. See, for example

• Kružík, M., Mielke, A., & Roubíček, T. Modelling of microstructure and its evolution in shape-memory-alloy single-crystals, in particular in CuAlNi. Meccanica, 40 (2005), 389-418.

This is typical for many multiscale problems in physics.

 

[A. Neumaier]

Sure, I was pointing out the difference that made your conceptual analogy not quite complete.

The unsolved problems in the classical case of rigorously proving existence of global solutions for PDEs where distributions matter are analogous to the unsolved problems in the quantum case of rigorously proving existence for operator valued distributions. In both cases, the problem is posed in completely rigorous terms and physically useful approximation methods flourish, but the rigorous construction has been elusive so far.

It is only an analogy because there are obvious differences. But the analogy is strong enough to justify aiming at solving the problem rather than giving up or changing the foundations to something in a vague nowhere land, as suggested by you in post #1.

 

[Tendex]

the analogy is strong enough to justify aiming at solving the problem rather than giving up or changing the foundations to something in a vague nowhere land, as suggested by you in post #1.

I think you took my last sentence in the OP to seriously in this thread. I suppose a bit of sarcasm is not well suited in a science forum. Thanks for your inputs anyway.

 

[PeterDonis]

I think you took my last sentence in the OP to seriously in this thread. I suppose a bit of sarcasm is not well suited in a science forum.

When you title a thread the way you titled this one, you lose the right to claim sarcasm regarding remarks like the last sentence in your OP.

This thread is closed.