Philosophy of quantum field theory

[Sunil]

But properly arguing about rigorous matters of any kind requires technical details (here about rigorous quantum field theory, which means AQFT) rather than foundational issues or philosophical principles. The only nontechnical argument you gave is that you think there is little reason for a rigorous QFT involving gravity and hence for a rigorous UV limit of physical relevance. But on the nontechnical level there is as little reason against a rigorous QFT involving gravity; so a discussion on this level is moot.

Sorry, but if the question is about the possibility of existence of some technical construction, in a situation where it is not known to exist, proper arguments do not necessarily require technical details. The classical argument that if such a thing would exist it would have, with high probability, already been found is non-technical. The clarification where the burden of proof is located - namely on the side of those who argue for the existence - and the consequence that the "zero hypothesis" in this case is that it does not exist is also completely non-technical but nonetheless valid.

Instead, technical arguments for the non-existence have - justly - already a bad reputation. These would be impossibility theorems. But the list of misguided impossibility theorems is long. The most well-known one is von Neumann's impossibility proof for hidden variables. Actually, history repeats itself by a large number of $\psi$-ontology theorems, starting with the PBR theorem, in a situation where Caticha has already given a nice realist $\psi$-epistemic interpretation with his "entropic dynamics".

Once one does not require rigor, there is at least overwhelming evidence that QCD has a valid UV limit, independent of the unsolved questions about quantum gravity.

A simple question where I have never seen an answer: What would be the difference between a theory with free particles from the start and the UV limit of an asymptotically free theory? If the interaction goes to zero in the limit, the limit itself would be useless.

Literature:
Leifer, M.S. (2014). Is the Quantum State Real? An Extended Review of $\psi$-ontology Theorems. Quanta 3(1), 67-155, arxiv:1409.1570
Caticha, A. (2011). Entropic Dynamics, Time and Quantum Theory, J. Phys. A 44 , 225303, arxiv:1005.2357

 

[A. Neumaier]

The classical argument that if such a thing would exist it would have, with high probability, already been found is non-technical.

It is non-technical but logically faulty, hence irrelevant.

Everything that is difficult to find but has not yet been found would not exist with high probability, according to this argument. Moreover, the probabilities involved are extremely subjective, and do not mean anything except for the subject uttering them.

I gave the example of Kepler's conjecture where it took several hundred years to settle it positively. This doesn"t mean that the conjecture was almost certainly false before it was proved. Instead, those informed always considered it most likely to be true (for technical reasons)! The Riemann hypothesis is also regarded as true by many even though in spite of the attempt of a number of the best mathematicians, no proof has been found in the last 150 years.

 

[A. Neumaier]

What would be the difference between a theory with free particles from the start and the UV limit of an asymptotically free theory? If the interaction goes to zero in the limit, the limit itself would be useless.

You mix up two different limits.

An asymptotically free quantum field theory is not free except in the limit of infinitely tiny distances. But only the theory at finite distances has physical relevance. Particles are not free at all in an asymptotically free theory! In QCD, the fundamental particles, the quarks, are even confined to short distances.

The observable particles are those that are asymptotically free in a very different asymptotic limit - the one where time goes to plus or minus infinity.

 

[Sunil]

It is non-technical but logically faulty, hence irrelevant.

Everything that is difficult to find but has not yet been found would not exist with high probability, according to this argument. Moreover, the probabilities involved are extremely subjective, and do not mean anything except for the subject uttering them.

"Logically faulty" is itself faulty once applied to an argument which does not even claim that it is decisive, and which was, moreover, reduced here to a short phrase, not more than a reference to this argument, instead of the argument itself.

In fact, most of the arguments people like to name "logically faulty" are completely adequate if understood not at certain logical conclusions but as part of plausible reasoning. So if from A follows B, it is "logically faulty" to conclude from B that A, but if we have B this makes the hypothesis nonetheless A more plausible. If this increase in plausibility is a serious one or not depends on the circumstances, that means, on the details. To summarize, in domains where one cannot expect certain proofs, but only plausible reasoning, one can misrepresent as "logically faulty" almost everything.

What would be the details in this case? First of all, the amount of intellectual energy which has been already spend trying to solve the problem. In this case, we have a whole subdomain of mathematical physics, AQFT, working essentially only on this problem, and a large prize for those who solve it. Then, observation of what has been reached, and why the methods used have not been sufficient yet. This requires some knowledge of what has been reached, which I have not. But what I have heard is that the successful examples are all superrenormalizable. In this case, I would see no base for hope.

This doesn"t mean that the conjecture was almost certainly false before it was proved. Instead, those informed always considered it most likely to be true (for technical reasons)! The Riemann hypothesis is also regarded as true by many even though in spite of the attempt of a number of the best mathematicians, no proof has been found in the last 150 years.

The question what is most plausibly true if the problem itself is not solved is, of course, something which depends on the problem. Last but not least, the hypothesis that A is true is nothing but a reformulation of the hypothesis that not A is true, solving one solves automatically the other too, but the answers are opposite. So, one needs at least some information to identify what plays the role of the null hypothesis.

But in this case, it is quite simple. We look for a theory of physics with certain properties. But, in fact, we don't look for a completely arbitrary one, but, for other reasons, for a simple one. If a research program with the final aim to construct a simple theory has failed to construct yet even a horribly complex one, and simple theories are in general easier to find and construct (if one is not restricted by fitting observations, as in this case) the answer is the natural one: To give up the research program because it has no chance to reach the final aim.

 

[A. Neumaier]

In fact, most of the arguments people like to name "logically faulty" are completely adequate if understood not at certain logical conclusions but as part of plausible reasoning.

So it is logically faulty but in your eyes plausible. Very well. I and many others look with different eyes.

What would be the details in this case? First of all, the amount of intellectual energy which has been already spend trying to solve the problem. [...] In this case, I would see no base for hope.

By the same logically faulty but in your eyes plausible argument there is no base of hope for solving any of the Clay Millenium Problems - not only the 7th that you address. But those responsible for them posed them because they expect them to be solved!

the answer is the natural one: To give up the research program because it has no chance to reach the final aim.

So you should give up! Indeed, with what you find plausible you'll never solve one of the millennium problems.

Others, with their own plausibilities, see hope and do not give up. Some will succeed in due time.

 

[Sunil]

This would require first a proof of the unfeasibility of the standard mathematical continuum way as commented by Neumaier, and if that happened it is not clear what kind of theory that situation could allow to construct, at least with the current concept of the mathematical continuum of the reals.

I disagree. If a theory with a cutoff is a well-defined theory (say, a lattice theory), it does not matter at all if a limit lattice distance to zero exists or not. The mathematical concept of a continuum is irrelevant here, the lattice theories use real numbers too.

So it is logically faulty but in your eyes plausible. Very well. I and many others look with different eyes.

Don't distort what I write. Only a misinterpretation of it as a rigorous statement would be logically faulty. In the straightforward interpretation it assigns only some low probability and is in no way logically faulty.

By the same logically faulty but in your eyes plausible argument there is no base of hope for solving any of the Clay Millenium Problems - not only the 7th that you address. But those responsible for them posed them because they expect them to be solved!

Wrong. I have clarified that the power of the argument depends on what one adds. And I have added some considerations which cannot be applied to the other Millenium problems.

So you should give up! Indeed, with what you find plausible you'll never solve one of the millennium problems.

I cannot give up something I have never started. Those problems are simply not interesting for me.

Others, with their own plausibilities, see hope and do not give up. Some will succeed in due time.

No problem, they can do whatever they like. Most will give up in due time.