I What is really that density matrix in QM?

[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.

 

[vanhees71]

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.

What do you mean by "not covariantly constructed"? We work with relativistic effective QFTs all the time, describing hadrons. They are all manifestly Poincare covariant. Of course it uses the usual "heuristics". We cannot wait until the mathematicians have found a way to make these heuristics rigoros. If we'd do so we'd still only describe free fields for the realistic case of (1+3) spacetime dimensions. Obviously the mathematicians haven't found a way to make QFTs describing real-world physics rigorous after more than 50 years of "axiomatic QFT" and its siblings.

 

[Lord Jestocost]

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 AA or BB" 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.

The distinction is a fundamental physical fact: A superposition remains a superposition!

“Decoherence is, formally, never complete. There always remain exponentially small non-diagonal terms in the reduced density matrix, reminding us that an initial pure state remains pure according to basic quantum mechanics.” (Roland Omnes, “Results and Problems in Decoherence Theory“)

“Let us go back however to less elevated questions. I did not yet mention that decoherence is a dynamical effect that is never perfectly exact. Entangled states of a measured quantum object and a measuring device are disentangled, but a tiny amount of entanglement (or superposition) always survives. The probability for observing a macroscopic interference effect between a dead and a live cat is never exactly zero, but extremely small and becoming exponentially smaller with larger values of time.” (Roland Omnes, “Decoherence And Ontology“)

 

[DarMM]

The distinction is a fundamental physical fact: A superposition remains a superposition!

I'm not disputing that.

“Decoherence is, formally, never complete. There always remain exponentially small non-diagonal terms in the reduced density matrix, reminding us that an initial pure state remains pure according to basic quantum mechanics.” (Roland Omnes, “Results and Problems in Decoherence Theory“)

Omnès argues in his books (Interpretation of Quantum Mechanics and Quantum Philosophy) that these exponentially small terms are actually zero since their only contribution is to self-adjoint operators that don't represent physical observables.

 

[DarMM]

What do you mean by "not covariantly constructed"? We work with relativistic effective QFTs all the time, describing hadrons. They are all manifestly Poincare covariant. Of course it uses the usual "heuristics". We cannot wait until the mathematicians have found a way to make these heuristics rigoros. If we'd do so we'd still only describe free fields for the realistic case of (1+3) spacetime dimensions. Obviously the mathematicians haven't found a way to make QFTs describing real-world physics rigorous after more than 50 years of "axiomatic QFT" and its siblings.

This is mostly due to analytically controlling infinite volume limits.

However the point is not to replace standard QFT, just to show that it makes sense. For instance the work on field theories in lower dimensions served to show that there is in fact a set of non-perturbative Schwinger functions and that renormalized perturbation theory as usually practiced is asymptotic to it. So it's more a case of showing the heuristic method is fine rather than finding a different "rigorous" method to replace it.

@A. Neumaier is referring to how effective field theories with a explicit $\Lambda$ cutoff that is not removed are not covariant. This is separate to the perturbative case where you introduce $\Lambda$, then renormalize and finally take $\Lambda \rightarrow \infty$.

 

[Auto-Didact]

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

The problem is that there is actually a subset of experts who agree with me, or more accurately phrased I agree with them; my assertions are merely a repeat of what is claimed in part of the constructive literature on the foundations of physics.

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.

It actually isn't wishful thinking: from the point of view of mathematics, BM is an existing mathematical object/theory, which naturally arises from a unique complex generalization of Hamilton-Jacobi theory, which moreover has several deep and unexpected connections with several other branches of pure and applied mathematics, which have no unambiguous relationship to textbook QM.

For comparison, it is unknown whether string theory, which is claimed to be an alternative possible foundation of physics, actually does exist as a mathematical theory at all; from a constructivist mathematics perspective I would say that string theory is a collection of frameworks still under construction, which may one day be demonstrated to be an existing mathematical theory.

Notice that I am not saying anything about the status of BM as a physical theory, which is far more complicated due to the pragmatically arisen classification - i.e. foundationally non-canonical classification - of 'empirically indistinguishable' interpretations of physical theories and so on; this is a very important distinction which needs to be made.

With respect to BM serving as a solution to the measurement problem in the foundation of QM, and so be a new candidate foundations for physics itself - i.e. from the perspective of foundational research methodology - the most important thing that matters is that BM actually exists as a mathematical theory, either prior or after any experimental confirmation.

 

[Auto-Didact]

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.

I repeat the point to avoid confusion: from a foundations of physics perspective concerned with the evolution of the structure of physical theory, QM, despite its foundational problems, is conceptually much clearer than QFT and also has much less contingent elements in its formulations. No one doubts that QM exists as a mathematical theory, but there are constuctivists both within physics and mathematics who do doubt that QFT actually exists as a mathematical theory.

Physicists, being empirical scientists, can choose to be pragmatic and adopt an operationalist stance and/or justify their belief in QFT empirically as an instrumentally accurate framework and just ignore the whole point of mathematical existence altogether (exactly like string theorists do); from a foundational perspective however this is ultimately rationally unjustifiable exactly as the skeptical constructivist mathematicians have claimed.

 

[vanhees71]

Well, BM is to my knowledge not formulated for relativistic QFT yet (not even in the usual "heuristic" physicists' sense).

I also strongly disagree with @A. Neumaier that "QM is conceptually unclear since its inception". It's both mathematically and physically a very well understood physical theory. There's no problem whatsoever in its mathematical foundations, which were settled by von Neumann in his famous treatise, which is brillant on the mathematical part but a desaster concerning the physics part, nor in its physical interpretation, as far as the physics part is concerned, and the only part that bothers a physicist as a physicist is the purely instrumentalist purpose to describe observed phenomena and make predictions for observable phenomena. On both accounts QT (including also relatistic microcausal QFT) is an astonishing success, even on a quantitative level of high accuracy and in the comprehensibility of the covered phenomena in Nature. The only known gap is in our understanding of the gravitational interaction.

The apparent problems are rather "metaphysical" or "philosophical", namely in establishing its implications on the "ontology", i.e., what does it mean to have a "photon" or an "elementary particle", given the fact that not all possible observables can be determined by any state preparation (this is even true for the most simple observables, described by "qubit" states like spin 1/2 states or photon-polarization states), or what's the meaning of the very strong correlations described by entanglement that cannot in any way be explained by local deterministic theories?

From the physicists' point of view there's of course no issue with these EPR/Bell topics either since it's all compatible with relativistic microcausal QFT as soon as one is willing to except the probabilistic nature of nature and clearly distinguishes strong correlations between observables that are individually indetermined (for Bell states even maximally indetermined) from causal spooky interactions claimed by some flavors of the Copenhagen interpretation insisting on a collapse.

 

[A. Neumaier]

there are constuctivists both within physics and mathematics who do doubt that QFT actually exists as a mathematical theory.

Which constructivists do doubt that quantum Yang Mills theory actually exists as a mathematical theory?

 

[A. Neumaier]

I also strongly disagree with @A. Neumaier that "QM is conceptually unclear since its inception". It's both mathematically and physically a very well understood physical theory.

It is mathematically and physically very well understood as long as one treats interpretation question in a heuristic fashion. That's why many physicists, including you, don't bother. But conceptually, it is poorly understood, as the disagreement about foundations shows. That's why many physicists, including me, do bother.

 

[DarMM]

I repeat the point to avoid confusion: from a foundations of physics perspective concerned with the evolution of the structure of physical theory, QM, despite its foundational problems, is conceptually much clearer than QFT and also has much less contingent elements in its formulations. No one doubts that QM exists as a mathematical theory, but there are constuctivists both within physics and mathematics who do doubt that QFT actually exists as a mathematical theory

There are? I don't know many people who know the field who think QFT doesn't truly exist in the 3+1D case. There might have been some doubt years ago before Balaban, Magnen and Sénéor established the continuum limit exists, but I don't think there is any serious doubt now. Just a wall of estimates to work through.

I still find it odd to say we should disregard the physically more accurate theory (quantum field theory) because it hasn't given us another theory (quantum gravity) and instead focus on the physically less accurate theory (non-relativistic QM) because there is a theory equivalent to it (Bohmian Mechanics) that fits more with the research method of mathematicians like Grothendieck. I really don't follow.

 

[Auto-Didact]

There are? I don't know many people who know the field who think QFT doesn't truly exist in the 3+1D case. There might have been some doubt years ago before Balaban, Magnen and Sénéor established the continuum limit exists, but I don't think there is any serious doubt now. Just a wall of estimates to work through.

I admit, most of my sources are a bit old and I am definitely a bit old fashioned as well. However, having theoretical experience in constructive non-pertubative methods, especially newer technical methods from applied mathematics which haven't seeped into the theoretical physics literature yet, I maintain that the constructivists are probably right w.r.t. the highly contingent nature of SR-based QFT which moreover literally requires a perturbative treatment making it a bad candidate to serve as the foundation of physics.

The argument against QFT as a foundation isn't so much an attack on QFT as a physical theory but on perturbation theory as a generically valid mathematical method. Perturbative renormalization group arguments were generally oversold by mathematicians and upon deeper analysis shown often to be very heuristic, especially during the 60s-80s, while research and results in non-perturbative methods only really took off in a more coherent sense afterwards.

I still find it odd to say we should disregard the physically more accurate theory (quantum field theory) because it hasn't given us another theory (quantum gravity) and instead focus on the physically less accurate theory (non-relativistic QM) because there is a theory equivalent to it (Bohmian Mechanics) that fits more with the research method of mathematicians like Grothendieck. I really don't follow.

Foundational research is far more focused on mathematical consistency of theories than on experimental accuracy, i.e. it is a matter of rationality and not of pure empiricism; mathematically speaking, any empirically vindicated physical theory regardless of its degree of accuracy can be generalized and then used to attempt to become the new foundational theory of physics and so knock the current foundations away; this is why from the perspective of theoretical methodology Einstein as a theoretician was allowed to completely disregard Newtonian foundations in his theorization, regardless of there being any experimental issues between Newtonian theory and Maxwell's theory.

This is why foundational research is so difficult: you have to be able to selectively disregard empirically validated theories in order to rewrite foundations, every route has to be tried and then the generalization towards mathematical reconstruction from first principles and reconceptualization evaluated and re-evaluated using creative insight and experience from structures from pure mathematics. Successfully being capable of navigating this minefield is very much an art form; it goes wothout saying that this is also how Newton invented calculus, Gauss invented differential geometry and so on.

From experience, as well as the history of physics and mathematics applied to the other sciences, we know that a scientific theory grounded in sophisticated mathematics has a much better chance of serving as the foundation for that science than an empirically highly accurate theory which is nevertheless ridden with mathematical inconsistencies; this may be a rare occurrence in physics, it is extremely typical in all other sciences.

 

[Lord Jestocost]

That's why many physicists, including me, do bother.

Bruce Rosenblum writes in "Quantum Enigma" (by Bruce Rosenblum and Fred Kuttner):

“I (Bruce) shared a taxi and conversation with John Bell in 1989 on the way to a small conference in Erice, Sicily, which focused on his work. At the conference, with wit, and in his Irish voice, Bell firmly emphasized the depth of the unsolved quantum enigma. In big, bold letters on the blackboard he introduced his famous abbreviation, FAPP, “for all practical purposes,” and warned against falling into the FAPPTRAP: accepting a merely FAPP solution for the enigma.”

 

[vanhees71]

It is mathematically and physically very well understood as long as one treats interpretation question in a heuristic fashion. That's why many physicists, including you, don't bother. But conceptually, it is poorly understood, as the disagreement about foundations shows. That's why many physicists, including me, do bother.

What do you mean by "heuristic fashion"? All a physical theory as a physical theory and only a physical theory has to provide is a description of the measured and observable phenomena. The fact that there are incompatible observables and that thus not all observables can take determined value is an observed fact you have to accept. Science is about observed facts and the prediction of possible yet unobserved predictions of new facts. If the latter turn out to be wrong, the theory is wrong. Otherwise it's a success, and so far QT is a great success.

It is, of course, hard to accept for our heuristics, based on our daily experience with classically behaving macroscopic objects, that there are incompatible observables and indeterminism, but it's a fact we must accept. If you have metaphysical and philosophical troubles with it, your metaphysics and philosophy have to be adapted not QT as a physical theory. As with relativity also QT provides a refined heuristics to describe the world according to the known well-reproducible facts. That's all I'm saying.

 

[Auto-Didact]

What do you mean by "heuristic fashion"? All a physical theory as a physical theory and only a physical theory has to provide is a description of the measured and observable phenomena. The fact that there are incompatible observables and that thus not all observables can take determined value is an observed fact you have to accept. Science is about observed facts and the prediction of possible yet unobserved predictions of new facts. If the latter turn out to be wrong, the theory is wrong. Otherwise it's a success, and so far QT is a great success.

If the focus in the practice of physics is on getting or matching experiment, then heuristics are acceptable; this is true for experimentalists and applied physicists, as well as those theorists whose theoretical focus are in close proximity to their contemporary state of experimental physics. In other words, heuristic arguments are only acceptable for non-fundamental research; most sciences aren't mathematical in their foundations so they do not have to directly address this problem.

Physics however is mathematical, not only in its daily practice, but even way down in its foundations, i.e. the subject of physics is about physical laws which are as far as we can tell properties or aspects of nature which have or can be given a mathematical form. Because of this, the foundations of physics necessarily requires the same level of rigour as that required in the practice of mathematics, therefore heuristic arguments are clearly foundationally unacceptable.

Changing the meaning of what a science is - i.e. insistence on the new philosophy of science about observables invented by the pioneers of QM - is fully a heuristic argument which was made in the 20th century only to forget about foundational issues and pursue new available experiments; this is the correct approach if there are new available experiments to explain, which was the case for much of the 20th century until the completed construction of the Standard Model during the 70s.

Trying to reform the foundations based on heuristic arguments such as the operational success of QFT is not even wrong; all that such often made suggestions demonstrate is an immense ignorance of what foundations research is among those making the suggestion, i.e. most contemporary physicists. This is actually something which is to be expected because there aren't any practicing physicists who are still alive and were already practicing before, during and after the last completed foundational change i.e. the SR/GR revolutions; to make matters worse the practice of mathematics is still divorced from that of physics.

The foundational revolution for QT was never completed, but its completion just ignored for heuristic reasons. In other words, the heuristic argument was a good argument for physics until the 70s. Thereafter however the heuristic argument becomes the wrong approach, precisely because there aren't any more experiments to explain, yet there have remained glaring mathematical inconsistencies in the foundations of the still ongoing revolution, which moreover get exacerbated when trying to theoretically merge fundamental theories.

In physics, w.r.t. QT we are today obviously in this theory crisis situation at the moment, meaning both theoreticians and mathematicians are needed to resolve the problems in the foundations of QT. Any theorist who doesn't see this has actually stopped pursuing the theoretical practice of fundamental physics, but instead has for practical reasons chosen to disregard mathematics as the ultimate method for engaging in fundamental physics and instead settle for heuristics i.e. for philosophy instead.

This heuristic attitude among physicists actually seems to be imparted during the training of students (shut up and calculate), which contemporary physicists of course learned from their own teachers, because it was adequate for that period in history of physics when there were experiments to analyze; unluckily, this attitude has become educational dogma which doesn't change even when change is needed.

Today we don't have the excuse of unfinished experiment anymore and we have through the professionalized educational system managed to remove practically all aspiring foundational physicists from physics, leaving us unable to complete the QT revolution even if we desperately want to finish it. This mental inertia among physicists is incidentally also why I chose to leave physics after getting my degree and instead just continue to pursue theoretical and mathematical physics from outside the academic establishment, next to medical practice and applied mathematical research.

 

[vanhees71]

Well, "shutup and calculate", is a better advice than getting lost in questions that cannot in any way objectively be settled. The "interpretational issues of QT" are just a matter of personal belief but not natural science. The only thing decidable by objective science is the minimal interpretation, which just tells you to take the probabilistic nature of QT (generalized Born's rule) as a fundamental property of nature. Every assumption in addition (like Bohmian trajectories in non-relativistic QM, parallel universes a la MWI, etc.) is just not part of science, though maybe sometimes of some intellectual interest or amusement (as some esoterics is really funny, as long as it doesn't hurt anybody).

It's something else with speculative (but reasonable) alterations of the minimally interpreted QT, like the spontaneous-collapse theory a la GRW, which lead to (at least in principle) the prediction of at least in principle empirically testable deviations from the predictions of standard QT. Here the only question is whether the scientific community is willing to invest the money to really do the experiments needed to test it, which of course needs an idea of a feasible experiment first.