What is really that density matrix in QM?

[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.

 

[Auto-Didact]

Well, "shutup and calculate", is a better advice than getting lost in questions that cannot in any way objectively be settled.

Correction: has not been settled yet; it only needs to get settled once. You can only claim unsettleability of these arguments through mathematical proof of non-existence, which I as a constructivist almost see as an impossibility; obviously, there aren't any such proofs in the literature either.

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).

Mathematics is a form of science and mathematical physics is definitely science, despite what anyone claims; the minimal interpretation on the other hand is operationalism which replaces mathematics with heuristics. To use the words of Redhead: Operationalism is not sidestepping the need for philosophical analysis, but is itself just bad philosophy!

 

[PeterDonis]

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.

"shutup and calculate", is a better advice than getting lost in questions that cannot in any way objectively be settled.

It strikes me that this dispute cannot be objectively settled unless and until somebody either comes up with a proven theory of everything that is demonstrated to be mathematically consistent, or proves that one cannot exist. Neither of which seems likely to happen any time soon.

Given that, this subthread on foundations should end at this point.

 

[Auto-Didact]

As a closing remark, I paraphrase Kierkegaard: The history of physics can only be understood backwards; but it must be lived forwards.

 

[vanhees71]

Correction: has not been settled yet; it only needs to get settled once. You can only claim unsettleability of these arguments through mathematical proof of non-existence, which I as a constructivist almost see as an impossibility; obviously, there aren't any such proofs in the literature either.

Mathematics is a form of science and mathematical physics is definitely science, despite what anyone claims; the minimal interpretation on the other hand is operationalism which replaces mathematics with heuristics. To use the words of Redhead: Operationalism is not sidestepping the need for philosophical analysis, but is itself just bad philosophy!

Of course, mathematics is a structural science (not a natural science). What I talk about is fruitless philosophical debates about apparent problems, which are not objectively there, and cannot be settled other than finding new empirical evidence. There's not the slightest hint to where standard Q(F)T breaks down, which in some sense is a nuissance, because we lack guide from observations how to complete QFT to also describe gravity in a consistent way with all other interactions. For sure, philosophical debates won't solve this problem. We need empirical input and an ingenious physicists's and mathematician's idea, how to formulate it. Maybe it settles also some of the philosophical questions, but most likely the philosophers won't like the then hopefully revealed facts even less than the known ones and then claim to have new "problems".

 

[vanhees71]

It strikes me that this dispute cannot be objectively settled unless and until somebody either comes up with a proven theory of everything that is demonstrated to be mathematically consistent, or proves that one cannot exist. Neither of which seems likely to happen any time soon.

Given that, this subthread on foundations should end at this point.

The foundations have been settled in 1926, when Born gave his probabilistic interpretation of the quantum formalism. Quantum states are descibed by a statistical operator $\hat{\rho}$ and has the usual probabilisticmeaning, predicting the probabilities for the outcome of measurements on ensembles prepared in that state. With this you can close the thread ;-))).

 

[Lord Jestocost]

"Is it not good to know what follows from what, even if it is not really necessary FAPP? Suppose for example that quantum mechanics were found to resist precise formulation. Suppose that when formulation beyond FAPP is attempted, we find an unmovable finger obstinately pointing outside the subject, to the mind of the observer, to the Hindu scriptures, to God, or even only Gravitation? Would not that be very, very interesting?"

John Bell

 

[vanhees71]

But QT doesn't resist precise formulation. It's among the most precisely formulated theories ever. What Bell discredits as "FAPP" is just the physics content of QT, no more no less.

I'm never sure, whether Bell would have liked to see the predictions of QT concerning the violation of his inequality, based on local deterministic HV theories, would turn out to be wrong, i.e., whether he had wished to prove QT wrong and a return to the deterministic classical world view possible.

That's of course wishful thinking. Nature doesn't care about our philosophical prejudices, and thus Bell's work, which brought the philosophical speculations about possible problems with standard QT to the realm of objectively testable facts and thus indeed is among the most profound ideas concerning these foundations ever. Nowadays I think it's fair to say that the objective facts are revealed in a plethora of Bell tests in favor of standard QT, and local HV theories are simply counterfactual.

 

[PeterDonis]

The foundations have been settled in 1926

QT doesn't resist precise formulation. It's among the most precisely formulated theories ever

If you don't mind the term "measurement" not having a well-defined meaning. You can make these statements as your opinion, but I don't think you can assert them as simple facts.

With this you can close the thread ;-))).

And with that, it is closed.