Infinities in QFT (and physics in general)

[Demystifier]

Moderator's Note: Thread spun off from previous thread due to topic change.

Bell nonlocality is derived solely by proving that Schrödinger picture quantum mechanics in a finite-dimensional Hilbert space predicts violations of Bell inequalities. No quantum optics or quantum field theory is involved at all, not even relativity. Interacting relativistic QFT has not even a consistent particle picture at finite times. Hence there is a large gap between QFT and Bell nonlocality.

I see. Basically you are worried by stuff a mathematical physicist would study with fancy schmancy functional analysis. I'm not worried too much by such stuff because I don't think that actual infinities (actually infinite number of degrees of freedom, actually infinite dimensional Hilbert space) exist in the real world. Actual infinities are just idealizations that make some analytic calculations easier. When calculations with infinite sets become harder than those with big finite sets, then it's time to return to big finite sets.

But for those who take such infinite sets seriously and love stuff such as functional analysis and axiomatic QFT, I would mention the Reeh-Schlieder theorem. It seems that this theorem is a rigorous expression of a Bell-like nonlocality in axiomatic QFT.

 

[A. Neumaier]

I don't think that actual infinities (actually infinite number of degrees of freedom, actually infinite dimensional Hilbert space) exist in the real world.

How do you describe the motion of a nonrelativistic quantum particle in 3-space without an infinite-dimensional Hilbert space?

 

[Demystifier]

How do you describe the motion of a nonrelativistic quantum particle in 3-space without an infinite-dimensional Hilbert space?

E.g. by putting the 3-space on the lattice.

Or let me put it this way. When the Hilbert space is supposed to be infinite dimensional, how do you solve the Schrodinger equation on the computer?

 

[A. Neumaier]

E.g. by putting the 3-space on the lattice.

This is inadequate already for a free particle, and does not allow a discussion of scattering. Hence no chemical, nuclear, or elementary particle reactions. Thus it eliminates most of physics.

Or let me put it this way. When the Hilbert space is supposed to be infinite dimensional, how do you solve the Schrodinger equation on the computer?

Well, quantum chemists solve it all the time, but never using lattice discretizations.

 

[A. Neumaier]

But for those who take such infinite sets seriously and love stuff such as functional analysis and axiomatic QFT, I would mention the Reeh-Schlieder theorem. It seems that this theorem is a rigorous expression of a Bell-like nonlocality in axiomatic QFT.

Reeh-Schlieder gives entanglement. The latter is ubiquitous. Bell inequality violations and associated nonlocality issues are much more specific and therefore more rare (require special experiments to observe). They are unrelated to the Reeh-Schlieder theorem.

 

[Demystifier]

Well, quantum chemists solve it all the time, but never using lattice discretizations.

They certainly use some sort of discretization, because computer cannot work in the continuum. Numerical solutions of partial differential equations usually use a mesh, which is not much different from a lattice.

 

[Demystifier]

Bell inequality violations and associated nonlocality issues are much more specific and therefore more rare (require special experiments to observe).

Perhaps violation of Bell inequalities is rare, but Bell nonlocality is ubiquitous. See e.g. https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.72.1951

 

[A. Neumaier]

They certainly use some sort of discretization, because computer cannot work in the continuum. Numerical solutions of partial differential equations usually use a mesh, which is not much different from a lattice.

They use neither a mesh nor a lattice. Both would be extremely inefficient. Instead they use coherent states (Gaussians), which have nothing to do with lattices and need for their definition the standard structure of 3-space and an infinite-dimensional Hilbert space.

 

[A. Neumaier]

Perhaps violation of Bell inequalities is rare, but Bell nonlocality is ubiquitous. See e.g. https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.72.1951

What has this to do with the the Reeh-Schlieder theorem?

 

[Demystifier]

They use neither a mesh nor a lattice. Both would be extremely inefficient. Instead they use coherent states (Gaussians), which have nothing to do with lattices and need for their definition the standard structure of 3-space and an infinite-dimensional Hilbert space.

But then they use a finite number of coherent states, which, in practice, again makes the Hilbert space finite dimensional.

 

[A. Neumaier]

But then they use a finite number of coherent states, which, in practice, again makes the Hilbert space finite dimensional.

But these states are chosen adaptively, depending on the problem. Moreover, the approximation accuracy depends on the number of states chosen, and chemists add states until convergence is observed. This is possible only in an infinite-dimensional Hilbert space. And the basis varies from problem to problem, which shows how nonphysical the discretized setting is.

 

[A. Neumaier]

They certainly use some sort of discretization, because computer cannot work in the continuum.

By the same reasoning you must reject real numbers as unphysical, and work with rationals only. Then not even the diagonal of a square is physical anymore.

 

[PeterDonis]

By the same reasoning you must reject real numbers as unphysical, and work with rationals only. Then not even the diagonal of a square is physical anymore.

Well, square roots are still algebraic numbers, so you can still construct number fields that work like the rationals using them. But you could say the circumference of a circle isn't physical any more, since $\pi$ is transcendental.

 

[A. Neumaier]

Well, square roots are still algebraic numbers, so you can still construct number fields that work like the rationals using them. But you could say the circumference of a circle isn't physical any more, since $\pi$ is transcendental.

Even for square roots, computational physicists and quantum chemists use computations with floating point arithmetic, which consists of rational numbers only. The use of exact arithmetic with algebraic numbers is restricted to a minority of mathematicians and computer scientists.

 

[Demystifier]

Moreover, the approximation accuracy depends on the number of states chosen, and chemists add states until convergence is observed. This is possible only in an infinite-dimensional Hilbert space.

No, it's possible only in a finite dimensional Hilbert space. When the dimension becomes too big, the computer program cracks.

 

[Demystifier]

By the same reasoning you must reject real numbers as unphysical, and work with rationals only. Then not even the diagonal of a square is physical anymore.

Exactly!

Of course, you can work with reals as an approximation if that makes the calculations simpler. But physical theories in continuum are a map, not a territory.

 

[Demystifier]

What has this to do with the the Reeh-Schlieder theorem?

Both demonstrate that nonlocality associated with entanglement is ubiquitous.

Another important point! Bell nonlocality is essentially a property of free entangled particles, interactions between particles are not essential at all. On the other hand, free QFT is understood completely and rigorously. The Hilbert space of free QFT can be reduced to the direct sum of Hilbert spaces with fixed numbers of particles. So there is really no problem to understand Bell nonlocality in QFT rigorously.

 

[gentzen]

Seeing this discussion between A. Neumaier and Demystifier about the necessity of infinite Hilbert spaces and real numbers is fascinating for me. It seems to confirm an observation I recently suggested as a possible explanation for the different prevalences of Mathematical fictionalism vs. physical fictionalism:

Like many other mathematicians, I believe in a principle of 'conservation of difficulty'. This allows me to believe that mathematics stays useful, even if it would be fictional. I believe that often the main difficulties of a real world problem will still be present in a fictional mathematical model.
...
From my experience with physicists (...), their trust in 'conservation of difficulty' is often less pronounced. As a consequence, physical fictionalism has a hard time

A. Neumaier's explanations why infinite Hilbert spaces are relevant in practice exhibit the places where the 'conservation of difficulty' actually shows up. But they fail to convince Demystifier, because he does not even acknowledge the relevance of 'conservation of difficulty' to the question of whether infinite Hilbert spaces are necessary or not. (In that same answer, I later defend Bohmian mechanics as an illustration of how 'conservation of difficulty' works in practice. Would be interesting to see whether Demystifier accepts the relevance of 'conservation of difficulty' for that example.)

And it is not as if A. Neumaier would try to defend a non-standard position here, like defending the necessity of nonseparable Hilbert spaces. Even in "Foundations of quantum physics III. Measurement," he only explained where nonseparable Hilbert spaces can arise, but didn't try to defend their necessity for doing practical physical calculations:

The exact state of the interacting system is now a complicated state in a renormalized quantum field Hilbert space^* that no one so far was able to characterize; it is only known (Haag’s theorem) that it cannot be the asymptotic Fock space describing the noninteracting particles.

^* Because of superselection sectors, this Hilbert space is generally nonseparable, a direct sum of the Hilbert spaces corresponding to the different superselection sectors.

 

[Demystifier]

Would be interesting to see whether Demystifier accepts the relevance of 'conservation of difficulty' for that example.

Bohmian trajectories can be computed on a computer, so real numbers are not necessary for Bohmian mechanics. If Gisin claims the opposite, he's wrong.

Concerning the general problem of continuum in physics, an obligatory read is
https://arxiv.org/abs/1609.01421

 

[gentzen]

Bohmian trajectories can be computed on a computer, so real numbers are not necessary for Bohmian mechanics. If Gisin claims the opposite, he's wrong.

Oh, I forgot that Gisin's attack on Bohmian mechanics was also included in that post. The illustration I had in mind was the nonlocality of Bohmian mechanics, which initially seemed like a weekness of Bohmian mechanics, but in the end turned out to be one of its biggest successes.

However, I have to admit that as far as I can tell, I might be the original author of Gisin's attack:

There is also a philosophical side, which is a total different story. A real number can contain an infinite amount of information, but nature probably doesn't contain an infinite amount of information in a finite volume. So nature has to use QM (or something similar) to blur details enough such they no longer contain an infinite amount of information. But how does QM achieve this? The many world interpretation seems to go exactly into the opposite direction, and Bohmian mechanics is even worse than many worlds with respect to the amount of information required for its ontology.

But what I had in mind was more related to a paradox in interpretation of probability than to an attack on using real numbers to describe reality. The paradox is how mathematics forces us to give precise values for probabilities, even for events which cannot be repeated arbitrarily often (not even in principle). And if I would state my uncertainty about the exact probability, it would once again be an exact number, so in a certain sense even more ridiculous. The interpretation of probability used by the thermal interpretation is an appropriate resolution of that paradox for me, and I am really glad that the thermal interpretation is significantly older than my quoted text above.

 

[A. Neumaier]

No, it's possible only in a finite dimensional Hilbert space. When the dimension becomes too big, the computer program cracks.

Your proposed Nikolic-physics is really poor in content. It cannot make sense of the free particle, the harmonic oscillator, the notion of atoms or molecules or chemical reactions. In fact it can do almost nothing.

I'd like to suggest that you give a first course on quantum mechanics covering the standard stuff to see how little of it can be stated in finite-dimensional Hilbert spaces,

Exactly!

Of course, you can work with reals as an approximation if that makes the calculations simpler. But physical theories in continuum are a map, not a territory.

... let alone without using real numbers conceptually to define the terms of interest.

 

[A. Neumaier]

Both demonstrate that nonlocality associated with entanglement is ubiquitous.

Another important point! Bell nonlocality is essentially a property of free entangled particles, interactions between particles are not essential at all. On the other hand, free QFT is understood completely and rigorously. The Hilbert space of free QFT can be reduced to the direct sum of Hilbert spaces with fixed numbers of particles. So there is really no problem to understand Bell nonlocality in QFT rigorously.

Reeh-Schlieder demonstrates entanglement, not Bell nonlocality. For the latter you need particles moving along trajectories - otherwise the whole argument of Bell breaks down! Goldstein's paper assumes a 2-particle picture and says nothing at all about fields.

 

[A. Neumaier]

Bohmian trajectories can be computed on a computer, so real numbers are not necessary for Bohmian mechanics.

How are they defined without the concept of real numbers? You cannot use differential equations without real calculus!

 

[Demystifier]

How are they defined without the concept of real numbers? You cannot use differential equations without real calculus!

I suggest you to read some introduction to numerical analysis. Roughly it's like ordinary analysis, except with finite $\Delta x$ instead of infinitesimal $d x$. And there are no $\varepsilon$'s and $\delta$'s.

 

[A. Neumaier]

I suggest you to read some introduction to numerical analysis. Roughly it's like ordinary analysis, except with finite $\Delta x$ instead of infinitesimal $d x$. And there are no $\varepsilon$'s and $\delta$'s.

Every introduction to numerical analysis (including the book I wrote on this topic) assumes real calculus when discussing differential equations.

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

Or how do you prove that energy is conserved for a particle in a time-independent external field?

Any attempt to give a full exposition of physics without using real numbers and continuity is doomed to failure. Not a single physics textbook does it. Claiming that physics does not need real numbers is simply ridiculous.

 

[Demystifier]

Or how do you prove that energy is conserved for a particle in a time-independent external field?

There is no experimental proof that energy is exactly conserved. Experimentally, it's conserved only up to some small error. Therefore, it is not necessary that energy is exactly conserved in theory. To reproduce at least approximate energy conservation consistent with experimental bounds, discrete theory without real numbers is sufficient.

The real difference between you and me is that you are a mathematical Platonist, while I am a conventionalist: https://en.wikipedia.org/wiki/Philosophy_of_mathematics#Conventionalism
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. In situations in which real numbers (or other kinds of infinities) create more trouble than utility, I have no problems with abandoning them and applying a different approach, whatever works best, including finitism. Kronecker famously said "God created the natural numbers, all else is the work of man." I mostly agree, except that I would add that man created even the natural numbers. What man created, man can change.

 

[Demystifier]

Every introduction to numerical analysis (including the book I wrote on this topic) assumes real calculus when discussing differential equations.

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.

 

[Demystifier]

Claiming that physics does not need real numbers is simply ridiculous.

It depends on what do you mean by "need". Does human physicist needs a pen and paper? Yes she does. Does human physicist needs her brain? Yes she does. But she needs them in a different sense. The latter is absolutely necessary, while the former is very very useful but not absolutely necessary. The need for real numbers is of the former kind. I can imagine an advanced civilization with advanced theoretical physics which does not use real numbers at all.

 

[gentzen]

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.

But if real numbers are used as the source of randomness, then just blindly computing with floating point numbers out of convention and convenience is simply not enough. In his preprint Indeterminism in Physics and Intuitionistic Mathematics from Nov. 2020, Gisin pointed out that

scientists working on weather and climate physics explicitly use finite-truncated numbers and stochastic remainders [40]
[40] T.N. Palmer, Stochastic Weather and Climate Models, Nature Reviews— Physics 1, 463-471 (2019)

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

 

[Demystifier]

I'd like to suggest that you give a first course on quantum mechanics covering the standard stuff to see how little of it can be stated in finite-dimensional Hilbert spaces

In a standard first course on QM, one is supposed to learn analytic techniques of calculation, rather than numerical ones. In such a context, infinities of all kinds, e.g. those appearing in $\int_{-\infty}^{\infty} dx e^{ikx}$, are a very useful tool. But some of those students will one day become applied physicists whose main tool will be the computer performing numerical computations. They will need to learn how to restate all these infinities in terms of big finite numbers.

 

[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.