So, if RQM isn't consistent; what makes us believe that QFT is?

[loop quantum gravity]

A conundrum indeed....


I still don't understand how you can use methods of semiclassical analysis in physics; what does it mean do they obey Heisenberg inequality or not? you can't have both, now can you?
It's OK from physics and engineering standpoint that the theories won't be necessarilly consistent mathematically, but from mathematical-logical standpoint it's sort of important you know...

 

[martinbn]

What is RQM? Relativistic quantum mechanics or relational quantum mechanics, or something else?

 

[gentzen]

It's OK from physics and engineering standpoint that the theories won't be necessarilly consistent mathematically, but from mathematical-logical standpoint it's sort of important you know...

Yeah, distributions, weak solutions, and all that stuff turned out to be interesting and mathematically relevant.

And there may be hidden mathematical lessons still buried in the dirty tricks required by QFT, say for example about multiplication of distributions as in:
How I Learned to Stop Worrying and Love QFT
or about ... how to put that in words ... let's say singular perturbed and singular parameterizations as in:
Renormalization without infinities - an elementary tutorial

 

[gentzen]

So, if RQM isn't consistent; what makes us believe that QFT is?

Relativistic quantum mechanics is not consistent, because it does not allow the creation or destruction of particles. And one can "show" relatively rigorously that this is required for compatibility with special relativity.

And if your point about QFT being potentially inconsistent is that point-particles could similarly be incompatible with special relativity, then this also contains some grains of truth. However, this sort of incompatibility also arises in other contexts, and still makes sense as a sort of approximation.

One example is that perfectly coherent light of a given wavelength is mathematically consistent, but perfectly incoherent light of a given wavelength does not exist mathematically. However, for direct write (i.e. maskless) laser lithography machines, this perfectly incoherent scenario is what one aims for. There are various engineering tricks how one can come very close to that scenario.

 

[loop quantum gravity]

Relativistic quantum mechanics is not consistent, because it does not allow the creation or destruction of particles. And one can "show" relatively rigorously that this is required for compatibility with special relativity.

And if your point about QFT being potentially inconsistent is that point-particles could similarly be incompatible with special relativity, then this also contains some grains of truth. However, this sort of incompatibility also arises in other contexts, and still makes sense as a sort of approximation.

One example is that perfectly coherent light of a given wavelength is mathematically consistent, but perfectly incoherent light of a given wavelength does not exist mathematically. However, for direct write (i.e. maskless) laser lithography machines, this perfectly incoherent scenario is what one aims for. There are various engineering tricks how one can come very close to that scenario.

Your second paragraph is what I am alluding.
I mean there's a pure millenium math problem on YM and mass gap; and my gut feeling is that if one found a suitable axiomatics he would have found it already. But to prove that there is no 3+1 axiomatics for QFT, i.e no-go theorem; it will take me decades to solve it. In the meantime, there's no tenure.... :-(
As I said it's ok for physics and engineering not to be as pedant as pure mathematicians.
As for semicalssical analysis, you didn't answer my question really, which is can we both have HUP and not have it at the same time? because that seems to be what we are doing there, which means that the particle both behaves classical and quantum in this regime? this is already contradictory, I mean [x,p]=0 or not, not both.
At least that's what I remember from WKB and all these perturbative and asymptotic methods.

We also have the Landau pole problem, I am not sure it's really resolved mathematically.
https://en.wikipedia.org/wiki/Landau_pole

But what do I understand, perhaps in quantum logic it makes sense...

 

[loop quantum gravity]

What is RQM? Relativistic quantum mechanics or relational quantum mechanics, or something else?

the former.

 

[martinbn]

it will take me decades to solve it.

Modest! That is why I don't work on the milenium problems, I don't want to spend the time.

 

[gentzen]

As for semicalssical analysis, you didn't answer my question really

I didn't even notice that there was a question.

But I am actually quite happy that I didn't notice it. Asking three unrelated questions at the same time is bound to create confusion. And if you additionally prod anybody who tried to answer one of those questions to also answer the other questions, very soon everybody will have learned their lesson.

The right thing to do in such cases seems to me to suggest to you to only ask one question at a time. And try to be clear about what you are actually asking.

which is can we both have HUP and not have it at the same time? because that seems to be what we are doing there, which means that the particle both behaves classical and quantum in this regime? this is already contradictory, I mean [x,p]=0 or not, not both.
At least that's what I remember from WKB and all these perturbative and asymptotic methods.

Wow, once again three different only vaguely related concepts in a single sentence: "WKB and all these perturbative and asymptotic methods". But why not, if it is the relation between these three concepts which confuses you. So if you want, ask a separate question.

We also have the Landau pole problem, I am not sure it's really resolved mathematically.
https://en.wikipedia.org/wiki/Landau_pole

OK, this seems related to this QFT question. But maybe somebody else wants to answer it.

But what do I understand, perhaps in quantum logic it makes sense...

What do you mean by "quantum logic"? Is it ... or ...? Or ...?

 

[Morbert]

Relativistic quantum mechanics is not consistent, because it does not allow the creation or destruction of particles.

Hmm, a literature search brings up stuff like this: https://arxiv.org/abs/nucl-th/0308061

 

[Morbert]

PS A post from @A. Neumaier a while back:

Graduate Post in thread 'Consistency of Relativistic QM'

Jan 20, 2026

Hi All

While going through my book on QFT for mathematicians (What Is a Quantum Field Theory?
by Michel Talagrand) I came across the following:

'Quantum Mechanics, in its standard formulation, is not compatible with Special Relativity,
and it is very difficult to reconcile these two theories. Early attempts in this direction go
under the name of “Relativistic Quantum Mechanics”. This theory, which is the subject of
numerous textbooks, runs into severe inconsistencies, and we will not try to describe it.'

Talagrand is a mathematician who tried to understand standard relativistic...

• A. Neumaier

• 
• 

 

[loop quantum gravity]

Modest! That is why I don't work on the milenium problems, I don't want to spend the time.

I didn't even start it.
I am first trying to understand the references that are in Jaffa's and Witten's proposal, I stopped somewhere in one of the papers by Konrad Schrader and another Professor. (I think they passed away).
I guess if I ever find something tangible for a proof that such a theory doesn't exist mathematically, I would be the only one who would understand if it's correct or not. (not that modest I know...
).

 

[loop quantum gravity]

I didn't even notice that there was a question.

But I am actually quite happy that I didn't notice it. Asking three unrelated questions at the same time is bound to create confusion. And if you additionally prod anybody who tried to answer one of those questions to also answer the other questions, very soon everybody will have learned their lesson.

The right thing to do in such cases seems to me to suggest to you to only ask one question at a time. And try to be clear about what you are actually asking.


Wow, once again three different only vaguely related concepts in a single sentence: "WKB and all these perturbative and asymptotic methods". But why not, if it is the relation between these three concepts which confuses you. So if you want, ask a separate question.


OK, this seems related to this QFT question. But maybe somebody else wants to answer it.


What do you mean by "quantum logic"? Is it ... or ...? Or ...?

There's a book that some professor called Cohen with that name.
I mean before measurement we are in a superposition of complementary states; which classically cannot be achieved since they contradict each other. I bet you know simple classical propositional calculus. So 0 and 1 are just like T and F; and in superposition before you measure anything that thing is in those two contradicting states (if you adhere to classical logic you can infer anything from a contradicition).

Perhaps the consiousness of the observer dictates what he will observe somehow; I don't know how and it's quite a terrifying thought.

BTW, before reading the books on QFT I never heard of the term 'ghosts' in physics context, it seems also to appear in fluid mechanics literature (at least in one book I had been reading).

I can't say that I don't like physics; but sometimes it feels illogical and very perplexing; I mean there are some problems that don't spell out all that is needed to be given in the problem (I mean those physical assumptions that for some are intuitive, but obviously nothing is intuitive in quantum theory).

 

[gentzen]

There's a book that some professor called Cohen with that name.

"An Introduction to Hilbert Space and Quantum Logic" by David W. Cohen
Seems to be a slim little book, which introduces the math of QM, connects it to Brikhoff/von Neumann quantum logic, then goes on with even more math for QM like spectral theorem, and then also discusses questions arising in the context of interpretation of QM a bit.

Perhaps the consiousness of the observer dictates what he will observe somehow; I don't know how and it's quite a terrifying thought.

I guess Cohen won't claim stuff like that. And I would say it is totally unrelated to the issues with QFT. Even for QM, consciousness in the "mysterious" sense plays no role. The observer only plays a role like in classical thermodynamics or fluid mechanics, where details of a state or of noise are abstracted away, because they are inaccessible to the observer and "also" irrelevant in a certain sense. Here, "inaccessible" can refer both to what can be observed, and to what can be influenced or controlled.

I can't say that I don't like physics; but sometimes it feels illogical and very perplexing; I mean there are some problems that don't spell out all that is needed to be given in the problem (I mean those physical assumptions that for some are intuitive, but obviously nothing is intuitive in quantum theory).

OK, not sure what to make of this. It somehow feels to me like you are convinced that your questions don't have answers anyway, or at least nobody here will be able to answer them.
Therefore, you see no problem in mixing them all together into one big unanswerable question.

I would try to not mix the mathematical troubles of QFT with the discussions about the interpretation of QM.

 

[A. Neumaier]

Relativistic quantum mechanics is not consistent, because it does not allow the creation or destruction of particles. And one can "show" relatively rigorously that this is required for compatibility with special relativity.

Hmm, a literature search brings up stuff like this: https://arxiv.org/abs/nucl-th/0308061

Indeed, relativistic multiparticle quantum mechanics is consistent (and useful) but has, compared to relativistic quantum field theory, a much more limited domain of applicability. A good survey article is

• B.D. Keister and W.N. Polyzou, Relativistic Hamiltonian dynamics in nuclear and particle physics, Adv. Nuclear Physics 20 (1991), 226--479.

 

[A. Neumaier]

obviously nothing is intuitive in quantum theory

Try my book on algebraic quantum physics, it might convince you of the opposite!

 

[loop quantum gravity]

Try my book on algebraic quantum physics, it might convince you of the opposite!

Hi @A. Neumaier I mean in quantum logic as it's written in the wiki page we don't have the distribuiton rule that connects between conjunction and disjounction, which in classical logic is called distirubitve law, i.e: $p\wedge(q\vee r) = (p\wedge q)\vee (p\wedge r)$, and also by replacing the logical connectives, these two equalities are assumed in classical logic (they can be inferred or proved from their truth tables); but in quantum logic we don't have these two equalities. As the example they give in the wikipedia page on quantum logic, that I read years ago. There's also a book by Jan Halmhalter called Quantum Measure Theory that I believe is discussed there, I haven't finished reading it though.

 

[A. Neumaier]

Hi @A. Neumaier I mean in quantum logic as it's written in the wiki page we don't have the distribuiton rule that connects between conjunction and disjounction, which in classical logic is called distirubitve law, i.e: $p\wedge(q\vee r) = (p\wedge q)\vee (p\wedge r)$, and also by replacing the logical connectives, these two equalities are assumed in classical logic (they can be inferred or proved from their truth tables); but in quantum logic we don't have these two equalities. As the example they give in the wikipedia page on quantum logic, that I read years ago. There's also a book by Jan Halmhalter called Quantum Measure Theory that I believe is discussed there, I haven't finished reading it though.

All arguments in textbooks and the research literature are based on classical logic. Nothing at all in quantum theory is deduced by means of quantum logic. Quantum logic has not even a notion of implication. That it is called a logic is based purely on some superficial mathematical similarities with real logic.