What is really that density matrix in QM?

[atyy]

That's an open question due to issues with infrared representations in QED.

Well, maybe QED doesn't exist - is this also expected to be a problem in say Yang Mills? Do we have pure states and unitary time evolution in Yang Mills?

 

[vanhees71]

here
View attachment 248039

/Patrick

Sure, that's why I said, if you use a complete set of compatible observables including $\hat{\rho}$. As any operator, a given $\hat{\rho}$ is of course unique. It doesn't depend on it's representation in terms of a complete orthnormal system (CONS), i.e.,
$$\hat{\rho} = \sum_{ij} |i \rangle \langle i|\hat{\rho}|j \rangle \langle j|.$$
If now you have a $\hat{\rho}$ with degnerate eigenvalues $p_j$, there are of course arbitrarily many CONS. Let $|i,\alpha \rangle$ be one such CONS, but in all of them you get
$$\hat{\rho}=\sum_{i} p_i \sum_{\alpha} |i,\alpha \rangle \langle i,\alpha |.$$
The inner sum is just the projector to the uniquely defined "degenerate" eigenspace to the degenerate eigenvalue $p_i$. These projectors are independent of the chosen degenerate CONS since obviously for another CONS $\widetilde{|i,\alpha \rangle}$
$$\hat{P}_i=\sum_{\alpha} |i,\alpha\rangle \langle i,\alpha| = \widetilde{|i,\alpha \rangle}\widetilde{\langle i,\alpha|}.$$

 

[DarMM]

Well, maybe QED doesn't exist - is this also expected to be a problem in say Yang Mills? Do we have pure states and unitary time evolution in Yang Mills?

Yes for Yang-Mills in flat spacetime we should have global pure states.

As for QED not existing I've always found the arguments for this very weak. It's based on the existence of a perturbative Landau pole, but the Gross-Neveu model has a Landau pole perturbatively, but still exists as a well-defined QFT in the non-perturbative sense.

However all of this would still only be in the idealisation of flat space. In curved space for any theory there simply isn't unitary time evolution simply because that's not how time evolution can be modeled for field theories in general, it is to be replaced with the notion of algebraic automorphisms.

 

[DarMM]

I should say I'm slowly in the process of gathering all this info in a coherent form. What nonperturbative QFT in curved spacetime is actually like renders many debates about QM pointless or seriously recasts the issues and I think it would be useful for others to know.

 

[atyy]

I should say I'm slowly in the process of gathering all this info in a coherent form. What nonperturbative QFT in curved spacetime is actually like renders many debates about QM pointless or seriously recasts the issues and I think it would be useful for others to know.

Well, at least it won't affect Bohmian Mechanics :)

 

[DarMM]

Well, at least it won't affect Bohmian Mechanics :)

Well that's because the Bohmians are objectively correct as we all secretly know in our heart but deny in public.

 

[A. Neumaier]

Well, maybe QED doesn't exist - is this also expected to be a problem in say Yang Mills? Do we have pure states and unitary time evolution in Yang Mills?

YangMills has a much worse infrared problem than QED.

 

[A. Neumaier]

In the case of no no pure states in flat spacetime, can we have unitary time evolution?

In principle, yes: Mixed states do not by themselves force a nonunitary evolution.

 

[charters]

I should say I'm slowly in the process of gathering all this info in a coherent form. What nonperturbative QFT in curved spacetime is actually like renders many debates about QM pointless or seriously recasts the issues and I think it would be useful for others to know.

This would be very interesting to read, but I would still be cautious about drawing foundational conclusions (conclusions universal to all quantum theory) from curved QFT, where curvature is purely classical. It isn't clear to me that curved QFT is a fully consistent idea or tells us more truth than flat QFT or isn't a red herring for foundational purposes.

For example Wald says the observables in curved QFT are just local fields and particles are not really valid ideas. But in quantum gravity, local fields are not gauge invariant and not observable. And in string theory, the background curvature itself is created/defined as a coherent state in the Fock space of graviton strings. So, if this facial contradiction stands, I think QG has to trump curved QFT.

 

[DarMM]

Well in a sense then anything could be a red herring when it comes to drawing conclusions. QFT in flat spacetime doesn't take curvature into account, String and Loop QG are partially worked out ideas that could be completely wrong.

The point is more so that most foundational work uses a very idealised form of NRQM and thus misses many subtleties from QFT or even just a realistic application of NRQM. Most of the results I will discuss will be generic to QFT regardless of background.

 

[charters]

The point is more so that most foundational work uses a very idealised form of NRQM and thus misses many subtleties from QFT or even just a realistic application of NRQM. Most of the results I will discuss will be generic to QFT regardless of background.

Yes that's very fair. Look forward to it.

 

[Auto-Didact]

I should say I'm slowly in the process of gathering all this info in a coherent form. What nonperturbative QFT in curved spacetime is actually like renders many debates about QM pointless or seriously recasts the issues and I think it would be useful for others to know.

This issue was already addressed in Cao, 1999. Because QFT (based in SR) is explicitly an EFT and nothing more, it turns out automatically to actually be less interesting from a foundational perspective, i.e. the conceptual issues facing QFT seem at every twist and turn to be completely contingent on the idealized structures which have no clear relation to physical structures in general, nor any actually known strong mathematical basis of consensus coming directly from the practice of pure mathematics; this is the heavy price one pays for advocating operationalism, which w.r.t. QT of course has the minimal interpretation as its poster boy.

 

[DarMM]

explicitly an EFT and nothing more

What is his argument for this point?

less interesting

Less interesting than NRQM?

 

[atyy]

In principle, yes: Mixed states do not by themselves force a nonunitary evolution.

But from DarMM's post #70, it seems that even in flat spacetime for the Type III algebras with no pure states, the question of unitary evolution is open?

 

[Auto-Didact]

Well in a sense then anything could be a red herring when it comes to drawing conclusions. QFT in flat spacetime doesn't take curvature into account, String and Loop QG are partially worked out ideas that could be completely wrong.

The point is more so that most foundational work uses a very idealised form of NRQM and thus misses many subtleties from QFT or even just a realistic application of NRQM. Most of the results I will discuss will be generic to QFT regardless of background.

This is exactly why foundational physics methodology has - as Smolin and Hardy have argued - to be attempted as a deferential team effort which is per capita not too strongly focussed on overt specialization in possibly irrelevant technicalities, definitely not with any single dominant research programme dominating a field but instead with many possibly intercompatible frameworks, where the researchers should have no lasting ties to any single programme but instead rotate periodically. This type of research methodology has proven itself very effective in many other academic fields and human endeavors, but it is practically as far as one can imagine from typical physics training today.

 

[DarMM]

But from DarMM's post #70, it seems that even in flat spacetime for the Type III algebras with no pure states, the question of unitary evolution is open?

It is open. I know that Buchholz thinks there are global pure states and unitary time evolution.

 

[Auto-Didact]

What is his argument for this point?

The book is a collection of arguments made by Weinberg, Nelson, Shankar, Redhead, Teller et al.

From what I remember, there are three responses in the mathematical physics literature to QFT failing as a foundation of physics: 1) the EFT attitude, 2) string theory and 3) mathematics, i.e. the axiomatic, constructive and algebraic approaches. (NB: string theory is generally not even seen as a possible solution for it is conceptually very close to QFT, so close in fact that there is doubt that they actually differ conceptually in a non-trivial sense.)

The argument that QFT is an EFT and nothing more is simply the Wilsonian point of view, which itself is embracing a possibly infinite amount of stable limited domains of validity each with their own separate dynamics and ontology; logically this actually does away with reductionism itself.

Less interesting than NRQM?

Yes, less interesting than the standard problems of NRQM precisely because the conceptual issue in the QFT case is so muddied by foundationally irrelevant contingencies such that we end up with all these meta-problems, whereas in NRQM the issues in contrast are (or have become) quite clear and are therefore explicitly logically and mathematically solvable as demonstrable by the existence of BM and spontaneous collapse models.

 

[atyy]

It is open. I know that Buchholz thinks there are global pure states and unitary time evolution.

What is the status of energy conservation in these subsets of spacetime and the global spacetime? If energy conservation holds, could one have unitary evolution just writing (by analogy) $\dot{\rho} = -i[H,\rho]$ ?

 

[Auto-Didact]

The constructive approach which tries to find a nonpertubative formulation of renormalization is actually from the historical perspective, the most canonical mathematical approach; if quantum gravity exists this will be the mathematics needed to form the conceptual framework of the physical theory.

The problem however is again that the mathematical experience required for achieving the constructive goal is not the kind of mathematics that physicists tend to be familiar with, at least not how they conceptualize it; in stark contrast, most physicists seem to only have a very meager and very weakly generalizae grasp of the branches of mathematics required. Those more in the know tend to be applied or pure mathematicians, e.g. Tao, Villani and Klainerman; see here.

This is partly because the mathematics is actually relatively new, conceptually scattered across multiple fields and that the literature lacks consensus on terminology due to different applied specialists across the sciences reinventing the same mathematical objects without communicating the results to each other; all of this messiness had prevented direct standardization and hence has very much delayed the push into the curriculums even until this day (NB: I poll physics graduates yearly to see if things have changed).

 

[Auto-Didact]

What is the status of energy conservation in these subsets of spacetime and the global spacetime? If energy conservation holds, could one have unitary evolution just writing (by analogy) $\dot{\rho} = -i[H,\rho]$ ?

In general, there typically aren't any conservation laws of particular interest in open systems, with instead other kinds of equations being more interesting to characterize and understand the open system e.g. relative entropy. John Baez and his (former) students have written a lot on these topics.

 

[A. Neumaier]

What is the status of energy conservation in these subsets of spacetime and the global spacetime? If energy conservation holds, could one have unitary evolution just writing (by analogy) $\dot{\rho} = -i[H,\rho]$ ?

If the Wightman axioms hold on the vacuum sector, this is automatically valid.

 

[atyy]

If the Wightman axioms hold on the vacuum sector, this is automatically valid.

So is @DarMM saying the Wightman axioms may fail to hold in the vacuum sector in rigorous QED?

 

[A. Neumaier]

So is @DarMM saying the Wightman axioms may fail to hold in the vacuum sector in rigorous QED?

Not as I understand him. He talked about the question of whether there are global pure states, which is independent of unitary evolution.

 

[DarMM]

The argument that QFT is an EFT and nothing more is simply the Wilsonian point of view

The argument that QFT necessarily must be an EFT seems weak to me since we know that Yang-Mills theories have a well defined continuum limit.

The problem however is again that the mathematical experience required for achieving the constructive goal is not the kind of mathematics that physicists tend to be familiar with, at least not how they conceptualize it; in stark contrast, most physicists seem to only have a very meager and very weakly generalizae grasp of the branches of mathematics required. Those more in the know tend to be applied or pure mathematicians, e.g. Tao, Villani and Klainerman

Most of the major names in Constructive Field Theory were theoretical physicists or theoretical chemists by training, not mathematicians.

it turns out automatically to actually be less interesting from a foundational perspective, i.e. the conceptual issues facing QFT seem at every twist and turn to be completely contingent on the idealized structures which have no clear relation to physical structures in general

I don't understand this, especially the automatically less interesting part. Could I have an example?

Yes, less interesting than the standard problems of NRQM precisely because the conceptual issue in the QFT case is so muddied by foundationally irrelevant contingencies such that we end up with all these meta-problems, whereas in NRQM the issues in contrast are (or have become) quite clear and are therefore explicitly logically and mathematically solvable as demonstrable by the existence of BM and spontaneous collapse models

I think you are mixing things up here. There are mathematical issues in QFT, this is separate to what QFT has to say about the issues discussed in the foundations of Quantum Theory.

So for example there are technical issues with the infinite volume limit in Yang-Mills, but a technicality like this isn't really related to or takes away from points such as that QFT causes the difference between proper and improper mixtures to dissolve which has an effect on foundational debates in QM.

I don't think because there are open questions about the mathematics of QFT this renders the proven modifications QFT makes to issues in QM irrelevant.

You're talking in very vague generalities here, could you give specific examples of what you mean?

the conceptual issues facing QFT seem at every twist and turn to be completely contingent on the idealized structures which have no clear relation to physical structures in general, nor any actually known strong mathematical basis of consensus coming directly from the practice of pure mathematics

Again I'm not really sure what is being referenced here. Could you give an example, what are these "idealized structures which have no clear relation to physical structures in general"?

 

[vanhees71]

Sorry for my ignorance, but what's the "difference between proper and improper mixtures", and what is the issue concerning foundational debates in QM.

If I understand it right, a "proper mixture" is a statistical operator coming from a thought experiment used to introduce the idea of mixed states to begin with. The argument goes like this. Suppose Alice prepares some pure states, e.g., a stream of electron-spin states each prepared either with $\sigma_z = 1/2$ or $\sigma_x=1/2$ and she randomly chooses with probability $p_1$ the former and with probability $p_2$ the latter and sends this stream of electrons to Bob, who measures the spin in some direction. So how should Bob describe this "proper mixture" of states. As it turns out, the answer is the statistical operator
$$\hat{\rho}=p_1 \hat{P}_{z+} + p_2 \hat{P}_{x-}, \quad \hat{P}_{j,\sigma}=|\sigma_j=\sigma \rangle \langle \sigma_j=\sigma|.$$
An "improper mixture" is usually called a "reduced state", i.e., you have some composite system prepared in some state (usually one assumes a pure state) like two electrons in the spin-singlet state
$$|\psi \rangle = \frac{1}{\sqrt{2}} (|\sigma_z=1/2,\sigma_z=-1/2 \rangle-|\sigma_z=-1/2,\sigma_z=1/2 \rangle$$
and you want to know the spin state of only one of these two electrons. The answer is the partial trace
$$\hat{\rho}_2=\mathrm{Tr}_1 |\psi \rangle \langle \psi|=\frac{1}{2} \hat{1}.$$
`This are of course quite different procedures to construct the state of the system for the given situation, but at the hand you have a statistical operator, and there's no fundamental difference in their meaning, providing just the probabilities for the outcome of measurements on the so prepared electron spins. There's no way to distinguish between how this state preparation came about. In the second example there's no way to figure out that the unpolarized electrons described by $\hat{\rho}_2$ are part of a pair of electrons prepared in the spin-singlet state. To determine this state you need to perform experiments on the spin of both electrons. There's no way to figure this out by measuring only spin components of the one electron. So the description of the 2nd electrons state is complete in the sense that you cannot know more than what's given by the statistical operators associated with the probabilities for outcomes of further measurements on the accordingly prepared ensemble.

But where in all this is a fundamental problem?

 

[DarMM]

Sorry for my ignorance, but what's the "difference between proper and improper mixtures", and what is the issue concerning foundational debates in QM.

Some people say there is an interpretive difference and thus one can't read the lack of interference for macroscopic observables in a classical probabilistic way and so decoherence can't be said to leave a device in a state you can read as "pointer state $A$ or $B$" because decoherence only leaves you with an improper mixture.

Basically they say only a proper mixture can be read in a classical probabilistic "or" sense, an improper one cannot because the system as a whole is entangled.

I don't see the distinction as being valid myself, but that is what they say.

 

[vanhees71]

Ah well, one more "pseudo problem" ;-)). This is as if you'd claim that it makes a difference in the (local) thermal-equilibrium state of my cup of coffee whether it cooled down to its present temperature or whether I reheated it up from a old cup of coffee (which of course I'd never do). Also there you cannot decide from just measuring the temperature of my coffee, what really has happened...

Probabilities are probabilities are probabilities...

 

[Auto-Didact]

The argument that QFT necessarily must be an EFT seems weak to me since we know that Yang-Mills theories have a well defined continuum limit.

You are misapprehending the argument by neglecting the conditional premise: Given that QFT is to serve as a foundational theory for physics as a science, then there are only three types of answers that are offered in the literature, i.e. physicists claim either

1. EFT through perturbative renormalization arguments with cutoffs, which using the customary rigor necessary for theoretical physics is seen as a sufficient argument, but using the customary rigor necessary for mathematical physics is seen as an insufficient argument; i.e. from a foundational standpoint, the argument is de facto an insufficient argument because perturbative renormalization is an approximative methodology whose validity is contingent upon making specific assumptions about the structure being renormalized; in the case of QFT making these assumptions is completely unjustifiable and therefore this line of argument does not actually meet the stringent requirements for direct demonstration within the practice of mathematics itself i.e. proof by construction. Within mathematics, there is an entire field devoted to directly solving such problems using non-perturbative methods, where familiarity with these methods exposes perturbation theory for the joke that it is.
2. String theory, which fails pragmatically - independent of any mathematical consistency and existence claims - due to string theory being conceptually in the same class of mathematical structures as QFT, and fails formally due to string theory not actually having a rigorously proven mathematical existence.
3. Mathematics, which is separated into the non-constructive approaches (e.g. the axiomatic approach of Wightman et al. and the C*-algebraic approach using the GNS construction) and the constructive approach, i.e. directly solving the issue from first principles by directly constructing/finding the unique nonperturbative answer by using sophisticated mathematics (from the practice of pure and applied mathematics) and/or inventing this new mathematics in the process in order to do so.

In other words, the only actual argument which can be acceptable from a foundational point of view is the mathematical constructive approach. The only way to argue against this is to take an operationalist stance, i.e. just blatantly ignore the problem and pretending everything is alright because experiments can be done; while this may be an acceptable strategy in the rest of science, it has never been acceptable in the practice of fundamental physics precisely because an answer can in principle be given at the level of rigor of mathematical physics in stark contrast to almost all other sciences.

Anyone arguing that perturbative renormalization with cutoffs is fully sufficient such that QFT can function as a foundation of physics is either unfamiliar with the limited validity thereof which can all be adressed using non-perturbative techniques and just bluffing or taking it on faith in experts who probably are more or less familiar with the difficulties of the non-perturbative methods but just are deliberately bluffing by being sloppy at sophisticated mathematics i.e. behaving as a cavalier physicist w.r.t. a physical theory as a mathematical object, instead of behaving as a careful mathematician.

Most of the major names in Constructive Field Theory were theoretical physicists or theoretical chemists by training, not mathematicians.

One needn't actually be a mathematician by training in order to be able to intuit or to identify and construct sophisticated mathematics. There are numerous examples in history, e.g. Faraday was mathematically illiterate but invented classical field theory, Cardano was a physician who invented complex numbers in his spare-time as a hobbyist, Hubble was a lawyer disillusioned with law and more interested in astronomy, even Witten was a historian before he found his way to mathematics and physics!

I don't understand this, especially the automatically less interesting part. Could I have an example?

In foundational research methodology an argument is less cogent if despite serious attempts at clarification it is still more vague than any other argument which can or has been made conceptually clear.

From experience it is known that less cogent arguments which remained so for indefinitely long periods of time usually turned out to be that the clarification attempts either were actually impossible or unachieved because the correct choice of some branch of mathematics for that particular idea hadn't been made, found or even invented yet.

Therefore less cogent arguments which repeatedly defy clarification indicate that the former or latter is occurring directly making the argument less interesting foundationally; notice that this has nothing whatsoever to do with if an argument is correct or not, merely whether it is properly and justifiably arguable or not from a standpoint of a high degree of rigour.

I think you are mixing things up here. There are mathematical issues in QFT, this is separate to what QFT has to say about the issues discussed in the foundations of Quantum Theory.

So for example there are technical issues with the infinite volume limit in Yang-Mills, but a technicality like this isn't really related to or takes away from points such as that QFT causes the difference between proper and improper mixtures to dissolve which has an effect on foundational debates in QM.

This is where the disagreement is. In foundational research, the existence of technical issues can not ever be used as an excuse to neglect remaining foundational issues; doing this is non lege artis practice of foundational research.

The foundational question for physics is not whether there are foundational issues which can be "resolved" in the foundations of QT by embracing QFT but instead whether contemporary QFT as is is itself sufficient as a foundation for physics; conveniently labeling the pre-existing foundational issues of QFT as technical issues is just begging the question, missing the entire point of foundational research.

Foundational answers always require direct resolution i.e. proof by construction; "answering" issues by dissolving issues while shifting the burden of proof from one set of issues to some other set of issues is making a category error i.e. nothing short of fallacious reasoning and therefore foundationally unacceptable unless it is accompanied by a direct constructive proof.

I don't think because there are open questions about the mathematics of QFT this renders the proven modifications QFT makes to issues in QM irrelevant.

That is certainly a pragmatic stance which one can choose to take, but simply not an acceptable foundational stance to take as I laid out above; only a fully constructive proof based on a first principles argument can show otherwise i.e. an actual construction of a quantum gravity theory which shows that QFT is conceptually adequate as a foundation of physics in this respect; all the known constructive evidence so far points to the contrary.

You're talking in very vague generalities here, could you give specific examples of what you mean?

Again I'm not really sure what is being referenced here. Could you give an example, what are these "idealized structures which have no clear relation to physical structures in general"?

You aren't misunderstanding me, I am literally saying what you think I can not possibly be saying: I do not believe that QED or any other similar QFT actually exists mathematically and I am saying that the perturbative renormalization arguments are wrong because they have not actually understood renormalization correctly i.e. constructively.

I am doing this deliberately precisely because of experience and familiarity with a wide array of obscure non-perturbative methods in a range of different technical and practical situations - some of which were analogous to the Wilsonian arguments which actually turned out to be wrong upon deeper inspection for any of a myriad of reasons. Moreover, I am taking such an extreme stance because I am a strong advocate of constructivist mathematics as the sole proper research methodology to finding conceptually cogent answers within the foundations of physics.

So to reiterate, my criticism is aimed at all perturbative renormalization arguments and even to algebraic QFT in general, which isn't so much a physical theory about nature but instead an operationalization of statistical methodology parading as a "new kind of physics" based on an operationalist philosophy which is less concerned with ontology and more concerned with completely unwarranted reification of unjustifiable limits such as flat space limits and artificially imposing background dependent vacuum states purely for axiomatic consistency reasons in orderto give an illusion of rigour i.e. putting makeup on a pig; I put contemporary QFT at same level of validity for a foundation of physics as Ptolemaic epicycles is as a foundation for celestial mechanics.

 

[DarMM]

This is where the disagreement is. In foundational research, the existence of technical issues can not ever be used as an excuse to neglect remaining foundational issues

I'm not saying to neglect them, I'm saying they seem independent. For example control over the infinite volume limit of Yang-Mills doesn't make any difference to the algebraic properties of QFT relevant to issues you see in the Foundations of QM.

only a fully constructive proof based on a first principles argument can show otherwise i.e. an actual construction of a quantum gravity theory which shows that QFT is conceptually adequate as a foundation of physics in this respect; all the known constructive evidence so far points to the contrary.

This is hard for me to parse, there has to be a quantum theory of gravity for you to consider modifications QFT makes to QM as relevant?

But QM itself doesn't give a quantum theory of gravity, so why are we considering it over QFT?

I do not believe that QED or any other similar QFT actually exists mathematically and I am saying that the perturbative renormalization arguments are wrong because they have not actually understood renormalization correctly i.e. constructively.

But we've found several QFTs that exist mathematically, including gauge theories.

 

[Auto-Didact]

I'm not saying to neglect them, I'm saying they seem independent. For example control over the infinite volume limit of Yang-Mills doesn't make any difference to the algebraic properties of QFT relevant to issues you see in the Foundations of QM.

They are independent, fully independent in the sense that QFT is not to serve as a concomitant ad hoc solution bed for problems in the Foundations of QM - which is itself a foundation of physics - but directly as a foundation of physics itself.

If QFT is chosen to be used to solve problems in the Foundations of QM, then there is no guarantee whatsoever that these solutions given by QFT do not precisely arise purely because of questionable technical issues related to the mathematical structure of QFT itself which might fully break down in any deeper formulation of QFT (such as a GR based version of QFT); following such a non lege artis approach to foundational research almost always leads to the introduction of conceptual meta-problems.

Such foundational meta-problems are always contingent and pragmatic but foundationally speaking typically nonsensical, e.g. such as the highly contingent claim that the photon wave function doesn't exist because there is no position representation for photon states, while from constructive mathematics such an object can be almost trivially directly constructed using a mathematically pedestrian generalization. It is impossible to claim a priori that this mathematical generalization isn't an aspect of a more correct theory to which textbook quantum theory is only a limiting case.

Another example of such an issue often claimed to be "resolved" by taking seriously a vacuous meta-problem is the invalid claim that AdS/CFT can be applied to dS as elaborated by Smolin in some of his 2016 - 2018 papers.

This is hard for me to parse, there has to be a quantum theory of gravity for you to consider modifications QFT makes to QM as relevant?

But QM itself doesn't give a quantum theory of gravity, so why are we considering it over QFT?

Final conclusions in foundational research are time independent: either a quantum gravity theory exists or it doesn't; whether we have discovered it yet mathematically or not is frankly speaking irrelevant but if it doesn't we can never discover it.

Unless an explicit proof can be given that such a theory doesn't exist, every failure in construction along the way is just incremental progress. In this sense foundational research is almost diametrically opposite to regular science research and highly reminiscent of cutting edge mathematical research.

But we've found several QFTs that exist mathematically, including gauge theories.

These aren't non-perturbative constructivist models which arose naturally as applications of candidate structures from a unique combination of different strands of traditional pure mathematics, but highly artificially/non-authentic formulated framework, concerned not with authentic discovery but with formal axiomatics; algebraic QFT is a bit better but in my opinion to prematurely cavalier w.r.t. many issues.

In fact, neither a proof for the existence of fixed points in four dimensional spacetime for the Standard Model has been achieved nor has the constructive existence of Yang-Mills theory in $\mathbb{R}^4$ been proven, which is of course one of the Millenium Problems.

 

[DarMM]

Final conclusions in foundational research are time independent: either a quantum gravity theory exists or it doesn't; whether we have discovered it yet mathematically or not is frankly speaking irrelevant but if it doesn't we can never discover it.

I'm not talking about whether QG can be discovered, I'm asking why we can't consider QFT prior to its discovery. Even more so though I'm asking why QM can be considered but QFT cannot considering neither give you QG.

These aren't non-perturbative constructivist models which arose naturally as applications of candidate structures from a unique combination of different strands of traditional pure mathematics, but highly artificially/non-authentic formulated framework, concerned not with authentic discovery but with formal axiomatics; algebraic QFT is a bit better but in my opinion to prematurely cavalier w.r.t. many issues

I know constructive and algebraic QFT pretty well and I literally have no idea what this means. Can you explain?

 

[Auto-Didact]

I'm not talking about whether QG can be discovered, I'm asking why we can't consider QFT prior to its discovery. Even more so though I'm asking why QM can be considered but QFT cannot considering neither give you QG.

One can, but as I said all results will be contingent arguments. Foundational research is only concerned with necessary arguments.

For QM as a foundation of physics the situation is different because QM is conceptually clearer, i.e. the underlyjng mathematical issues can be given a complete constructive mathematical resolution, e.g. Bohmian mechanics.

I know constructive and algebraic QFT pretty well and I literally have no idea what this means. Can you explain?

Constructive QFT is a very small offshoot of constructive mathematics. Constructive mathematics is the traditional style of practicing pure mathematics (and today also arises in applied mathematics) which completely eschews formalism and recognizes that axiomatics and logicism have absolutely nothing whatsoever of substance to offer to the practice of mathematics except for rigour, but rigour is actually only a secondary concern and not actually involved in the primary concern of pure mathematics which is the creation/discovery of new mathematics. Poincaré, Hadamard, Strogatz, Frenkel and many others have written extensively on this topic. The problem is again that all of these writings are disjointed and disparate and not yet carefully and neatly pieced together into an easily absorbable format which requires little to no experience or understanding on the part of the reader, instead requiring the reader to be historically aware, well read and capable of thematic analysis and other qualitative research methodologies.

In fact, all unity of mathematics programmes and ideas, such as Wigner's unreasonable effectiveness argument, Dyson's argument on missed opportunities, the Langlands Program and this thread about PDE theory by Klainerman, all share the same underlying concept, namely that creation/discovery of mathematics is a highly constructive approach not so much concerned with following the known rules, but creatively breaking them and so going beyond what is known. This is in stark contrast to formal axiomatics which is completely uncreative but instead purely logical; formalism and axiomatics are only a priori cleaning up of messy creative mathematics, they have nothing to do with creativity and therefore nothing to do with constructive mathematics; this is what I mean when I say a piece of mathematics is artificial or inauthentic.

Grothendieck gives a wonderful description of his own experience which I think illustrates the constructive spirit best:

In those critical years I learned how to be alone. [But even] this formulation doesn’t really capture my meaning. I didn’t, in any literal sense, learn to be alone, for the simple reason that this knowledge had never been unlearned during my childhood. It is a basic capacity in all of us from the day of our birth. However these three years of work in isolation [1945-1948], when I was thrown onto my own resources, following guidelines which I myself had spontaneously invented, instilled in me a strong degree of confidence, unassuming yet enduring in my ability to do mathematics, which owes nothing to any consensus or to the fashions which pass as law. By this I mean to say: to reach out in my own way to the things I wished to learn, rather than relying on the notions of the consensus, overt or tacit, coming from a more or less extended clan of which I found myself a member, or which for any other reason laid claim to be taken as an authority. This silent consensus had informed me both at the lycee and at the university, that one shouldn’t bother worrying about what was really meant when using a term like” volume” which was “obviously self-evident”, “generally known,” ”in problematic” etc … it is in this gesture of ”going beyond to be in oneself rather than the pawn of a consensus, the refusal to stay within a rigid circle that others have drawn around one-it is in this solitary act that one finds true creativity. All others things follow as a matter of course.

Since then I’ve had the chance in the world of mathematics that bid me welcome, to meet quite a number of people, both among my “elders” and among young people in my general age group who were more brilliant, much more ‘gifted’ than I was. I admired the facility with which they picked up, as if at play, new ideas, juggling them as if familiar with them from the cradle–while for myself I felt clumsy, even oafish, wandering painfully up an arduous track, like a dumb ox faced with an amorphous mountain of things I had to learn (so I was assured) things I felt incapable of understanding the essentials or following through to the end. Indeed, there was little about me that identified the kind of bright student who wins at prestigious competitions or assimilates almost by sleight of hand, the most forbidding subjects.

In fact, most of these comrades who I gauged to be more brilliant than I have gone on to become distinguished mathematicians. Still from the perspective or thirty or thirty five years, I can state that their imprint upon the mathematics of our time has not been very profound. They’ve done all things, often beautiful things in a context that was already set out before them, which they had no inclination to disturb. Without being aware of it, they’ve remained prisoners of those invisible and despotic circles which delimit the universe of a certain milieu in a given era. To have broken these bounds they would have to rediscover in themselves that capability which was their birthright, as it was mine: The capacity to be alone.

Strogatz also rejoins in this attitude in his latest book saying:

As Archimedes says, “It is easier to supply the proof when we have previously acquired, by the method, some knowledge of the questions than it is to find it without any previous knowledge.” In other words, by noodling around, playing with the Method, he gets a feel for the territory. And that guides him to a watertight proof.
This is such an honest account of what it’s like to do creative mathematics. Mathematicians don’t come up with the proofs first. First comes intuition. Rigor comes later. This essential role of intuition and imagination is often left out of high-school geometry courses, but it is essential to all creative mathematics.

 

[zekise]

... (The two systems become entangled with respect to each other.) Two systems could be entangled together without a third system being involved, but if it were, then they would all be entangled together.

Coherence is simply the state before decoherence, when phase relationships are known. Once coherence is lost it is pretty much lost forever.

Hopefully I have answered all your points?

Thanks Michael. I am not sure what you mean by phases or how entropy figures into decoherence.

Is it correct to say that if system A is coherent to system B, then A appears as a superposition to B. And if system A decoheres wrt system B, then an entanglement has been established between the two systems. If so, if TAB is a measure of entanglement between A & B and SAB is a measure of superposition, then TAB.SAB = 1?

Notation: coherence (ǂ) and decoherence (≡). Are the following correct?:

If A ≡ B and A ≡ C then B ≡ C.
If A ≡ B and A ǂ C then B ǂ C.

Since ǂ is commutative, when a photon P is emitted from a laser diode towards a screen (call it environment E), and before it collides with the screen, then P ǂ E. Therefore, it must be the case that E is coherent to P. We can figure the wave function for P, which is typically in respect to the environment E. Can we also figure the wave function for E in respect to P? I hope I am making sense.

 

[A. Neumaier]

I do not believe that QED or any other similar QFT actually exists mathematically

But this is a pure belief while you phrase your assertions as if they were facts.

There is no evidence at all that QED does not exist as a mathematically sound theory. EFT is just a bag of heuristics (none of the EFTs are covariantly constructed) that pushes the unresolved issues under
the carpet.

 

[A. Neumaier]

QM is conceptually clearer, i.e. the underlyjng mathematical issues can be given a complete constructive mathematical resolution, e.g. Bohmian mechanics.

This is wishful thinking.

The meaning of QM is conceptually unclear since its inception. Bohmian mechanics doesn'gt fix the unclearness but contradicts much of the available (canonical) structure of QM. If it had clarifies things there were virtually full agreement about the foundations of quantum mechanics.