Infinities in QFT (and physics in general)

[Demystifier]

But I somehow have the impression that you are simply having fun with the reaction of mathematicians to claims questioning their traditions.

I admit, it's fun, but that's not all it is.

What I really want to say is this: the axioms of analytical mechanics, quantum mechanics, Bohmian mechanics, quantum field theory, ..., functional analysis, real analysis, ZF(C) set theory - they are all tools, not truths.

 

[martinbn]

@Demystifier You position is based on belief. You don't have any evedence that supports your stance.

 

[Demystifier]

@Demystifier You position is based on belief. You don't have any evedence that supports your stance.

Is the negation of my position a belief too? Is there any evidence that supports the negation of my stance?

 

[martinbn]

Is the negation of my position a belief too? Is there any evidence that supports the negation of my stance?

No one takes the negation of your stance as strongly as you do yours. No one said that the universe is exactly described by the continuous models. Everyone agrees that, unless proven otherwise, it is possible that all you need is finite sets. You, on the other hand, seem to exclude the possibility of the negation of your stance.

By the way it is completely irrelevant for this thread. Perhaps a new thread?

 

[A. Neumaier]

In analytic calculations I use real numbers just as you are. They really make calculations easier. But for me, that's the only true reason I use real numbers.

They also have a conceptual purpose. They make concepts well-defined, and independent of any whimsical approximation scheme. Without that, you have uncountably many equivalent discrete descriptions not involving real numbers, each defining a slightly different conceptual problem, and physical modeling becomes completely arbitrary.

They use real calculus due to tradition, that is to motivate numerical analysis as a way to approximate real analysis. But the numerical analysis itself does not rest on real numbers. It rests on numbers that can be represented by a finite number of digits in a decimal expansion.

Numerical analysis gives approximate answers to well-defined problems, often with well-defined error bounds, if not then at least with reasonable error estimates. A method for solving differential equations approximates their solution, independent of the myriad of ways one could replace the differential equation by a difference equation.

In a standard first course on QM, one is supposed to learn analytic techniques of calculation, rather than numerical ones.

Yes, since the concepts of physics are defined in these terms. The analytic techniques provide correct answers while numerical techniques only provide approximately correct answers.

If you work with discrete space and discrete time only, do you scrap all conservation laws? (But you even need one for Bohmian mechanics...)

You forgot to reply to this one. I haven't seen any conceptual definition of Bohmian mechanics without using real numbers to define what everything means.

 

[Demystifier]

I haven't seen any conceptual definition of Bohmian mechanics without using real numbers to define what everything means.

Sure, for conceptual definition of Bohmian mechanics, and for most of physics actually, real numbers are great. But for conceptual definition it is also great in QFT to write things like $\langle 0|\phi(x)\phi(y)|0\rangle$ or $\int{\cal D}\phi \, e^{iS[\phi]}$. Yet you know that in QFT it's very tricky to give those things a precise meaning. One approach is to take continuum seriously and deal with functional analysis. Another is to not take continuum seriously. Both approaches are legitimate, both have advantages and disadvantages. You prefer the former, I prefer the latter. You seem to argue that the latter is wrong a priori, I argue that it's not.

 

[Demystifier]

You, on the other hand, seem to exclude the possibility of the negation of your stance.

Actually, I don't. I appreciate any progress in understanding any kind of infinities, be it in axiomatic QFT, functional analysis or set theory. But my gut feeling (which, of course, can be wrong) is that approaches which do not insist that infinity should be taken very seriously have a better chance to succeed.

 

[martinbn]

Actually, I don't. I appreciate any progress in understanding any kind of infinities, be it in axiomatic QFT, functional analysis or set theory. But my gut feeling (which, of course, can be wrong) is that approaches which do not insist that infinity should be taken very seriously have a better chance to succeed.

It is not just infinities. It question the real numbers too. No infinities there.

 

[Demystifier]

It is not just infinities. It question the real numbers too. No infinities there.

Sets with infinite cardinality are also viewed as infinities in my dictionary.

 

[martinbn]

Sets with infinite cardinality are also viewed as infinities in my dictionary.

But you had no objections to the rational numbrs, nor the integers. They are also infinite sets!

 

[Demystifier]

But you had no objections to the rational numbrs, nor the integers.

Actually I did, but not explicitly. When I was talking about computations with a computer, I took for granted that a finite computer can represent only a finite set of different numbers.

 

[Demystifier]

By the way it is completely irrelevant for this thread.

The relation with the thread is indirect, as seen in my post #72 https://www.physicsforums.com/threads/why-is-qft-local.997102/post-6478915

 

[A. Neumaier]

One approach is to take continuum seriously and deal with functional analysis. Another is to not take continuum seriously. Both approaches are legitimate, both have advantages and disadvantages. You prefer the former, I prefer the latter. You seem to argue that the latter is wrong a priori, I argue that it's not.

The latter does not give an unambiguous logical definition, since ambiguitiy is introduced through the multitude of possible approximations. Thus it is conceptually fuzzy.

Sets with infinite cardinality are also viewed as infinities in my dictionary.

So you have already problems with texts, which form an infinite set.

 

[Demystifier]

Thus it is conceptually fuzzy.

Fuzzy is better than inefficient.

So you have already problems with texts, which form an infinite set.

The set of all texts that was ever written and will ever be written by humans and human made machines is finite. With non-written texts I don't have problems at all.

 

[A. Neumaier]

Fuzzy is better than inefficient.

But clarity is much more efficient efficient than fuzziness. This is demonstrated well by the fact that all quantum physics textbooks base their concepts on real numbers and infinite-dimensional Hilbert spaces, and no physicist (not even you yourself) defines concepts using your fuzzy, strictly finitist point of view.

The set of all texts that was ever written and will ever be written by humans and human made machines is finite.

This assumes that the universe has a finite lifetime, which is questionable.

 

[Demystifier]

But clarity is much more efficient efficient than fuzziness. This is demonstrated well by the fact that all quantum physics textbooks base their concepts on real numbers and infinite-dimensional Hilbert spaces, and no physicist (not even you yourself) defines concepts using your fuzzy, strictly finitist point of view.

I agree with that. When real numbers, infinite-dimensional Hilbert spaces etc give rise to clarity, I am happy to use them. But my point is that sometimes they don't seem to give rise to clarity, an example being attempts to make interacting 3+1 dimensional QFT rigorous. In such cases different strategies towards clarity may be more efficient.

Or to relate all this to the topic of this thread. If Bell non-locality is well understood in QM, but not so well understood in QFT, then my practical philosophy is to reformulate QFT such that it looks more like QM. It seems that the easiest way to do this is to replace some (not all!) QFT infinities with appropriate finite objects. A lattice may be one very specific way to do it, but not necessarily the best way.

 

[A. Neumaier]

If Bell non-locality is well understood in QM, but not so well understood in QFT, then my practical philosophy is to reformulate QFT such that it looks more like QM.

This just sweeps the difficulties under the carpet instead of solving them - giving the deceptive illusion of understanding when there is none.

If physicists had always done this we'd not have the conceptual highlights of Hamiltonian mechanics or quantum optics.

 

[Demystifier]

This just sweeps the difficulties under the carpet instead of solving them - giving the deceptive illusion of understanding when there is none.

Maybe! In general, how to distinguish illusion of understanding from true understanding?

If physicists had always done this we'd not have the conceptual highlights of Hamiltonian mechanics or quantum optics.

Could you be more specific about those examples? What was the specific problems that could have been swept under the carpet, but were not with development of Hamiltonian mechanics and quantum optics?

 

[A. Neumaier]

how to distinguish illusion of understanding from true understanding?

The latter is achieved if almost everyone working on the topic agrees with you.

Changing the problem is never true understanding.

 

[Demystifier]

Changing the problem is never true understanding.

Wasn't renormalization in QFT a sort of a change of problem? Didn't it produce a lot of understanding, e.g. in computation of g-2? Isn't lattice QCD also a change of problem? Didn't it also contribute to computation of g-2 just a few days ago?
https://www.nature.com/articles/s41...AQd_8PAc9FEooX7yL9urOjqbhXLL7HEG-zpNxocqUgWy4

 

[Demystifier]

Changing the problem is never true understanding.

And there is more. Didn't Einstein change the problem of aether when he postulated that there is no aether? Didn't Planck changed the problem of UV catastrophe when he postulated discreteness in the form $E=nh\nu$ out of nowhere?

 

[A. Neumaier]

Wasn't renormalization in QFT a sort of a change of problem? Didn't it produce a lot of understanding, e.g. in computation of g-2? Isn't lattice QCD also a change of problem?

No. It was changing the status of computations that lead to manifest and meaningless infinities from ill-defined to perturbatively well-defined, through a better understanding of what in the original foundations was meaningful and what was an artifact of naive manipulation.

Isn't lattice QCD also a change of problem?

No; it is a method of approximately solving QCD. To do it well one needs the continuum limit - a single lattice calculation is never trusted. See, e.g., point 4 of the recent post of André W-L in the context of muon g-2 figuring in the Nature article you cited. Thus lattices are useful for reliable physics prediction only in the context of the continuum theory.

 

[A. Neumaier]

And there is more. Didn't Einstein change the problem of aether when he postulated that there is no aether? Didn't Planck changed the problem of UV catastrophe when he postulated discreteness in the form $E=nh\nu$ out of nowhere?

No. They replaced a questionable hypothesis by a much stronger one. While Nikolic-physics removes from physics all strong concepts (which need infinity).

 

[Nullstein]

This conversation seems very unproductive to me. Of course, in the end, calculations are made on a computer with finite precision, but physics is about understanding the laws of nature and that's hardly possible without continnuum mathematics. Just think about what a great insight special relativity was. A simple gedanken experiment based on intuitive principles leads to the length contraction formula $L' = L\sqrt{1-\frac{v^2}{c^2}}$. However, if there were no square roots in physics, there would be no sane way to argue in favor of special relativity and we would still not understand the origin of the Lorentz transformations. Continuum mathematics leads to simple and compelling insights about nature. There are no analogous gedanken experiments without continuum mathematics and thus, while we could write down formulas and put them on the computer, we would be unable to understand their origin. They would just be way less convincing.

 

[Demystifier]

but physics is about understanding the laws of nature and that's hardly possible without continnuum mathematics.

I certainly agree that continuum mathematics helps a lot in understanding physics. But at the same time, in many cases it also creates serious difficulties: https://arxiv.org/abs/1609.01421

 

[Nullstein]

That's true, but the appearance of difficulties doesn't refute continuum mathematics. We can only make things as simple as possible, but not simpler. If difficulties arise, they must certainly be fixed, but just dumping continuum mathematics if singularities appear is quite excessive and introduces more problems than it solves. And it is often the case that singularities arise exactly because we didn't worry enough about continuum mathematics. For instance, the inifinities in QFT are due to using perturbation theory in circumstances where perturbation theory is not applicable and the perturbation series don't converge (see e.g. Dyson's argument). By paying more attention to correctly taking the continuum limit, the singularities can be resolved and we obtain perfectly reasonable theories (so far at least in 1+1 or 2+1 dimensions). And while the problem is still difficult in 3+1 dimensions, the beauty and simplicity of the results in lower dimensions strongly suggest that this is the correct way to go. And these considerations even led to the insight, that there is new physics in the non-perturbative regime that is invisible to perturbation theory. So the initial difficulties really seem to be an argument in favor of continuum mathematics instead of one against it.

 

[stevendaryl]

There is a sense in which it is clearly true that we don't need the continuum. Any observations we make as long as they are finite precision (which they always are) can be simulated by a sufficiently complicated deterministic finite automaton.

But conceptually, if the universe really is discrete, there is the puzzle as to why it appears continuous. Assuming that it's not a computer program that was specifically designed to fool us...

So I personally would not find it satisfying to see an argument that things could be discrete. I would want an argument that there is a plausible (non ad-hoc) discrete model that you could prove gave rise to the illusion of continuous spacetime.

 

[Demystifier]

But conceptually, if the universe really is discrete, there is the puzzle as to why it appears continuous.

Because the minimal distance is too small to be seen by our present experimental techniques. For instance, many theories argue that discreteness might start to appear at the Planck scale.

 

[stevendaryl]

Because the minimal distance is too small to be seen by our present experimental techniques. For instance, many theories argue that discreteness might start to appear at the Planck scale.

That's a slightly different issue. If the discrete model is the discrete counterpart to 4D spacetime, then at sufficiently large length scales, it might look continuous. But what is the reason that a discrete model would happen to look like the discrete counterpart to 4D spacetime, other than if you are trying to simulate the latter with the former?

 

[Nullstein]

The central issue is: All plausible, intuitive and beautiful arguments that have been succesfully used to derive our best physical theories, such as e.g. symmetry principles, and that really make the difference between physics and stamp collection, heavily rely on continuum mathematics.

Sure, we could discretice our theories, but we would lose all deep insights that we had gained and it would convert physics into mere stamp collection. Unless we can come up with even more plausible, intuitive and beautiful arguments for discrete theories, we shouldn't go that route.