I Nature Physics on quantum foundations

[gentzen]

I will elaborate it for Bosons, so that I can ignore the wavefunction for the equivalence relation.

Even for Fermions, I can just ignore the wavefunction for the equivalence relation. I realized this when the skeptic in me started to ponder over a "serious omission" in the given quotient construction:

If we interpret the trajectories as a function $(x_1, \ldots, x_n)(t) : \mathbb R \to \mathbb R^{3n}$ and consider "piecewise constant" permutations $\pi(t):\mathbb R \to S_n$, then $(x_{\pi(t)(1)}, \ldots, x_{\pi(t)(n)})(t)$ is only "piecewise continuous". So it is not a "strong" solution of the guiding equation. Weakening the continuity requirements is possible (and needed, because otherwise uniqueness of solution together with continuity allows identification of particles between different times), but it feels very much like a "patchwork construction".

Turns out my attempt to illustrate the character of "unnatural" constructions as "patchwork constructions" in the initial reply to A. Neumaier failed to identify the crucial points. Glueing together the endpoints of a closed interval to get a circle is an "unnatural" construction, but taking an open interval and identifying two small open intervals at both ends (pointwise) with each other to get a circle is a not an "unnatural" construction, despite allowing patchwork. (And the "quotient by a discrete subgroup" fails even worse to identify the crucial points, "discrete" is neither necessary nor sufficient, and "subgroup" instead of "group" was superfluous.)

I guess the point of "natural quotient" constructions is rather that at least locally, the real work should already be finished before taking the quotient, so that it doesn't make a difference for "local topological" constructions whether they are applied before or after taking the quotient.

Of course, it's much more efficient to simply not use any kind of trajectories as in the dBB interpretation. They do not provide anything physical to QT anyway. You may solve some philosophical quibble but introduce more complication without gaining any new insights from a scientific point of view.

Of course, it is completely unclear what "much more efficient" is supposed to mean in this context. The meaning of "anything physical" is clearer, but for me it is enough that dBB and quotient constructions provide "something mathematical".

The relationship between "mathematical constructions" and "philosophical quibbles" has always been a complicated one. Legend says that the Pythagoreans killed the one who discovered irrational numbers. And Zeno of Elea attacked the continuum on philosophical grounds. In both cases, the attacked concepts turned out to have hidden complexities and dangers, but the attacks themselves failed to clearly isolate those or show a way forward. Maybe philosophers are better at articulating hidden problems, than at helping to overcome them.

At least for me, reading philosophical texts is sometimes both fun and useful. SEP comes to mind, and also:

..., then I read An Interpretive Introduction to Quantum Field Theory cover to cover. It was easy to read, with a good mix of technical details, explanations, interpretations, and philosophical clarification.

… much of the interpretive work Teller undertakes is to understand the relationship and possible differences between quantum field-theory — i.e., QFT as quantization of classical fields — and quantum-field theory — i.e., a field theory of ‘quanta’ which lack radical individuation, or as Teller says, “primitive thisness.”

Teller made quite some effort to help his reader grasp how radically different truly "indistinguishable particles" are compared to our everyday experience. Looking at them in dBB on the other hand shows you how they require (anti-)symmetric wavefunctions and some form of discontinuity. As always, the discontinuity required in dBB is "too nonlocal" compared to what you actually need. And of course, lessons from dBB apply primarily to non-relativistic QM.

 

[RUTA]

There is something more to it though. Relativity is also different to our intuition, but far less physicists complain about it or work on the foundations/interpretations of it.

Here is why we don't have as much work on foundations/interpretations of relativity theory as we do on quantum mechanics per Zeilinger:

Physics in the 20th century is signified by the invention of the theories of special and general relativity and of quantum theory. Of these, both the special and the general theory of relativity are based on firm foundational principles, while quantum mechanics lacks such a principle to this day. By such a principle, I do not mean an axiomatic formalization of the mathematical foundations of quantum mechanics, but a foundational conceptual principle. In the case of the special theory, it is the Principle of Relativity, ... . In the case of the theory of general relativity, we have the Principle of Equivalences ... . Both foundational principles are very simple and intuitively clear. ...
I submit that it is because of the very existence of these fundamental principles and their general acceptance in the physics community that, at present, we do not have a significant debate on the interpretation of the theories of relativity. Indeed, the implications of relativity theory for our basic notions of space and time are broadly accepted.

 

[Fra]

Here is why we don't have as much work on foundations/interpretations of relativity theory as we do on quantum mechanics per Zeilinger:

I agree about the situation. But do we agree on the conclusion from it?

I think for example "reason" for WHY there is a observer invariant upper limit on communication speed is an important one to answer. Taking it as an axiom or emprical observation is I think not satisfactory. When deriving SR from axioms details of EM field or light has no distinguished role. Just the existence of an invariant common max speed(in 3D space) is enough. What is it with the construction or emergence of space that explains this? If we can answer that (and i think we shouldn try) then i think we will get many clues towards unifying GR and QM.

But i still agree its a value to (like you try to) to point to potential principal explanations of qm as well. But it does not give me peace of mind.

/Fredrik

 

[Fra]

After all space is just an "index" of events that gives them relational structure. How is this index built and defined from the observer that distinguishes the events. If we start bt thinking of "classical pointers" don't we already with the word "classical" imply not only "macroscopic" but also the embedding in classical 3D space? How can one imagine "classical observer" prior to spacetime?

/Fredrik

 

[RUTA]

I agree about the situation. But do we agree on the conclusion from it?

I think for example "reason" for WHY there is a observer invariant upper limit on communication speed is an important one to answer. Taking it as an axiom or emprical observation is I think not satisfactory. When deriving SR from axioms details of EM field or light has no distinguished role. Just the existence of an invariant common max speed(in 3D space) is enough. What is it with the construction or emergence of space that explains this? If we can answer that (and i think we shouldn try) then i think we will get many clues towards unifying GR and QM.

But i still agree its a value to (like you try to) to point to potential principal explanations of qm as well. But it does not give me peace of mind.

/Fredrik

In principle explanation, the empirically-discovered fact is not fundamental, it must be justified by a compelling fundamental principle. For example, in special relativity we explain length contraction as follows:

Relativity principle $\rightarrow$ Justifies light postulate $\rightarrow$ Dictating length contraction.

So, the "invariant upper limit on communication speed" (light postulate) is not the fundamental explanans in special relativity, the relativity principle is. As Norton notes (https://sites.pitt.edu/~jdnorton/papers/companion.pdf):

Until this electrodynamics emerged, special relativity could not arise; once it had emerged, special relativity could not be stopped.

Wouldn't it be nice if we had a principle explanation of entanglement that was just as compelling?

 

[vanhees71]

From the relativity principle (i.e., the equivalence of all inertial frames and the assumption of their existence) alone + further symmetry assumptions (homogeneity of time, euclidicity of space for all inertial observers) you can derive that there are two spacetime models: Galilei-Newton spacetime (with the Galilei group as its symmetry group) and Einstein-Minkowski spacetime (with the Poincare group as its symmetry group). To decide, which one describes nature is empirical, i.e., you cannot separate fundamentals from empirics. All fundamental laws must be grounded in solid empirical evidence.

I don't see, where there should be a general difference between the foundations of quantum theory and the description of space and time. The only difference is the lack of determinism in quantum theory and the possibility of classical, deterministic physics within the spacetime models, which have been successful to build the foundation of both classical and quantum physics.

The reluctance to accept "irreducible randomness" in the (observed!) behavior of Nature seems to me to be just a psychological phenomenon without any objective empirical foundation. At least there's only emprical evidence for this irreducible randomness, and none against it.

 

[martinbn]

From the relativity principle (i.e., the equivalence of all inertial frames and the assumption of their existence) alone + further symmetry assumptions (homogeneity of time, euclidicity of space for all inertial observers) you can derive that there are two spacetime models: Galilei-Newton spacetime (with the Galilei group as its symmetry group) and Einstein-Minkowski spacetime (with the Poincare group as its symmetry group). To decide, which one describes nature is empirical, i.e., you cannot separate fundamentals from empirics. All fundamental laws must be grounded in solid empirical evidence.

I don't see, where there should be a general difference between the foundations of quantum theory and the description of space and time. The only difference is the lack of determinism in quantum theory and the possibility of classical, deterministic physics within the spacetime models, which have been successful to build the foundation of both classical and quantum physics.

The reluctance to accept "irreducible randomness" in the (observed!) behavior of Nature seems to me to be just a psychological phenomenon without any objective empirical foundation. At least there's only emprical evidence for this irreducible randomness, and none against it.

So what is the fundamental principle of QM that is similar to the invariance of the speed ot light?

 

[vanhees71]

It's the notion of quantum states represented by statistical operators on a Hilbert space with their probabilistic interpretation.

 

[RUTA]

So what is the fundamental principle of QM that is similar to the invariance of the speed ot light?

There are different ways to render a principle account of QM. Bohr did it using the quantum postulate (discontinuous jumps between stationary states) and the correspondence principle (quantum transitions correspond to harmonics of classical motion). [Heisenberg used these to generate his matrix formulation of QM.] More recently (starting in 90's) we have information-theoretic reconstructions of QM, e.g., Rovelli based his on non-commutativity, Bub based his on non-Boolean algebraic structure, and Hardy based his on continuity of reversible transformations between pure states for the qubit. That last one leads to Brukner and Zeilinger's fundamental principle of Information Invariance & Continuity which is the equivalent of the light postulate for SR. In information-theoretic form it's not as transparent as the light postulate, but when instantiated physically it means the measurement of Planck's constant is invariant between inertial reference frames related by spatial rotations or translations. So, the invariant measurement of the speed of light between inertial reference frames related by boosts leads to SR and the invariant measurement of Planck's constant between inertial reference frames related by spatial rotations and translations leads to QM. The relativity principle is the compelling fundamental principle justifying these empirically-discovered facts in both cases.

 

[vanhees71]

Of course, QT must be compatible with the spacetime model you use and that's why you come to unitary ray representations of the Galilei or Poincare group for the Newtonian and special-relativistic spacetime model, and $\hbar$ is a scalar parameter, because it's just an arbitrary conversion factor to define the SI units.

 

[RUTA]

Of course, QT must be compatible with the spacetime model you use and that's why you come to unitary ray representations of the Galilei or Poincare group for the Newtonian and special-relativistic spacetime model, and $\hbar$ is a scalar parameter, because it's just an arbitrary conversion factor to define the SI units.

Planck's constant h appears in Planck's radiation law, the Planck-Einstein relationship, and Einstein's photoelectric equation, for example. In that sense, it is a fundamental constant of Nature. As Weinberg points out, measuring an electron's spin "is a measurement of a universal constant of Nature, h." Thus, Information Invariance & Continuity applied to the electron spin measurement (a qubit) means everyone must measure the same value of h, regardless of their relative Stern-Gerlach magnet orientations. Reads just like the light postulate and is justified the same way, too.

 

[Fra]

In principle explanation, the empirically-discovered fact is not fundamental, it must be justified by a compelling fundamental principle. For example, in special relativity we explain length contraction as follows:

Relativity principle $\rightarrow$ Justifies light postulate $\rightarrow$ Dictating length contraction.

So, the "invariant upper limit on communication speed" (light postulate) is not the fundamental explanans in special relativity, the relativity principle is. As Norton notes (https://sites.pitt.edu/~jdnorton/papers/companion.pdf):

Wouldn't it be nice if we had a principle explanation of entanglement that was just as compelling?

I see I was vauge as often, I agree the relativity principles is somehow more paramount, but my main point was that the conservative notion of "relativity principles" relates specifically to spacetime frames. But how are these frames and spaces justified in the first place - without accepting the very things we aim to explain?
As I think the heart of the relativity principle (which I always take to be the observer equivalence(or preferably democracy), noting that an observer is more than just a preferred coordinate frame) is much more profound than is the notion of "space".

What that in mind, I think the conventinal meaning of "relativity principle" as PRESUMING the notion of 4D spacetime is not sufficiently compelling when trying to incorporate also gravity. This very distinction is what IMO is why QG seems to slippery.

/Fredrik

 

[bhobba]

So, the "invariant upper limit on communication speed" (light postulate) is not the fundamental explanans in special relativity, the relativity principle is.

All it is, is the fixing of a constant that naturally occurs in the theory. It could be infinity, and intuitively you think it would be. But physics is an experimental science, and its value is the speed of light for all sorts of reasons, both theoretical and experimental. Theoretically, if you leave it as just an undetermined constant C then you can use it to derive Maxwell's equations which have solid experimental support for C being the speed of light:
http://cse.secs.oakland.edu/haskell/Special Relativity and Maxwells Equations.pdf

Thanks
Bill

 

[RUTA]

All it is, is the fixing of a constant that naturally occurs in the theory. It could be infinity, and intuitively you think it would be. But physics is an experimental science, and its value is the speed of light for all sorts of reasons, both theoretical and experimental. Theoretically, if you leave it as just an undetermined constant C then you can use it to derive Maxwell's equations which have solid experimental support for C being the speed of light:
http://cse.secs.oakland.edu/haskell/Special Relativity and Maxwells Equations.pdf

Thanks
Bill

Indeed, c was infinity until Maxwell's equations. Once you had a finite value from Maxwell's equations, the relativity principle demands everyone measure the same value for it, regardless of their inertial reference frame. That gives you SR.

 

[RUTA]

I see I was vauge as often, I agree the relativity principles is somehow more paramount, but my main point was that the conservative notion of "relativity principles" relates specifically to spacetime frames. But how are these frames and spaces justified in the first place - without accepting the very things we aim to explain?
As I think the heart of the relativity principle (which I always take to be the observer equivalence(or preferably democracy), noting that an observer is more than just a preferred coordinate frame) is much more profound than is the notion of "space".

What that in mind, I think the conventinal meaning of "relativity principle" as PRESUMING the notion of 4D spacetime is not sufficiently compelling when trying to incorporate also gravity. This very distinction is what IMO is why QG seems to slippery.

/Fredrik

You might be interested in this article then https://www.nature.com/articles/s41467-018-08155-0.pdf

 

[A. Neumaier]

there's only emprical evidence for this irreducible randomness, and none against it.

There is empirical evidence for randomness, but not for its irreducibility. The latter cannot be empirical since it is an intrisically theoretical question!

 

[vanhees71]

Well, yes, if you believe in a non-local deterministic HV, you are right, and of course one cannot exclude it by the simple fact that nobody has been able to provide a convincing one (in the relativistic case; for non-relativistic QM you can take de Broglie-Bohm as an example).

 

[bhobba]

Well, yes, if you believe in a non-local deterministic HV,

It sort of reminds one of the differences between LET and SR. There is no way to tell the difference experimentally. Still, virtually everyone recognises SR as more elegant, more straightforward and more penetrating to the essence of what is going on - namely, the symmetries of the POR implying up to a constant to be determined by experiment (it turns out to be the speed of light), the transformation between inertial frames. It is like some wit remarked. If I push some antique pen, an educated person would say it was the force I applied that moved it. But if I said no - when I pushed, it was not the force that moved it but the shade of Newton I annoyed by pushing on his favourite pen. You can't prove me wrong - but it is obvious it is a superfluous, unnecessary, even silly assumption and reject it. A deeper analysis boils down to a philosophical discussion of 'simplicity' which we do not do here. It is just noted virtually everyone recognises the force explanation is more straightforward.

Thanks
Bill

 

[Demystifier]

nobody has been able to provide a convincing one

Define "convincing"!

 

[vanhees71]

It sort of reminds one of the differences between LET and SR. There is no way to tell the difference experimentally. Still, virtually everyone recognises SR as more elegant, more straightforward and more penetrating to the essence of what is going on - namely, the symmetries of the POR implying up to a constant to be determined by experiment (it turns out to be the speed of light), the transformation between inertial frames. It is like some wit remarked. If I push some antique pen, an educated person would say it was the force I applied that moved it. But if I said no - when I pushed, it was not the force that moved it but the shade of Newton I annoyed by pushing on his favourite pen. You can't prove me wrong - but it is obvious it is a superfluous, unnecessary, even silly assumption and reject it. A deeper analysis boils down to a philosophical discussion of 'simplicity' which we do not do here. It is just noted virtually everyone recognises the force explanation is more straightforward.

Thanks
Bill

LET is just another interpretation of standard SR. In contradistinction to that there's not even a convincing non-local deterministic reinterpretation of relativistic QT to be discussed about. With convincing I mean a non-local theory that obeys the causality structure of relativistic spacetime, i.e., that there cannot be causal connections between space-like separated events.

 

[bhobba]

Define "convincing"!

That's like 'simplicity' in my post - a philosophical question. There is no right or wrong answer - just what most seem to find simple or convincing.

Thanks
Bill

 

[vanhees71]

For me convincing means to find a non-local (field?) theory that is Poincare covariant and obeying the causality constraint, i.e., that there are no causal connections between space-like separated events.

 

[Demystifier]

For me convincing means to find a non-local (field?) theory that is Poincare covariant and obeying the causality constraint, i.e., that there are no causal connections between space-like separated events.

In what sense would such hypothetical theory be "non-local", if there are no causal connections between space-like separated events?

 

[vanhees71]

That's the question! If you want a realistic HV theory in accordance with the measured facts (violation of Bell's inequality) it must be non-local, and the question is whether there is a non-local realistic theory that is Poincare covariant and causal. I think, it's very difficult to find such a theory, but of course I'm not aware of any "no-go theorem" proving that such a theory is impossible.

 

[Demystifier]

That's the question! If you want a realistic HV theory in accordance with the measured facts (violation of Bell's inequality) it must be non-local, and the question is whether there is a non-local realistic theory that is Poincare covariant and causal. I think, it's very difficult to find such a theory, but of course I'm not aware of any "no-go theorem" proving that such a theory is impossible.

The problem is not that it is difficult to find such a theory. The problem is that it is not clear what it even means that the theory is "non-local", if the causal connections between space-like separated events are forbidden. In my view, the kind of theory you would find "convincing" is self-contradictory by definition, because "non-local" in the Bell theorem means existence of causal connections between space-like separated events. I'm sure you would not find convincing something which is self-contradictory, so for you "non-local" must mean something else, but I don't know what.

 

[vanhees71]

Then you say that there is no realistic HV theory at all, and the case is closed. I tend to agree with that since I also don't know, how to construct a non-local theory that obeys the causality constraint. I also don't see any necessity to look for such a model since local relativistic QFT describes all known matter with high accuracy correctly. Of course this Standard Model is incomplete, because it doesn't describe the gravitational interaction.

In field theory it would perhaps be described by a Lagrangian that is not a polynomial of the fields and their derivatives at one spacetime argument and which is not a gauge theory, where you can have such constructions without violating the locality/microcausality of observables (e.g., QED in Coulomb gauge).

As I said, I don't have an idea, whether such a model exists but also don't know whether there's a no-go theorem, which definitely excludes such a possibility.

 

[Demystifier]

Then you say that there is no realistic HV theory at all,

No I don't.

In field theory it would perhaps ... not a gauge theory, where you can have such constructions without violating the locality/microcausality of observables (e.g., QED in Coulomb gauge).

A Bohmian theory of that kind has been constructed, see e.g. my https://arxiv.org/abs/2205.05986
and references therein. The locality/microcausality is not violated for observables, but is violated for beables.

As I said, I don't have an idea, whether such a model exists but also don't know whether there's a no-go theorem, which definitely excludes such a possibility.

As I just said, a model exists.

 

[A. Neumaier]

The problem is that it is not clear what it even means that the theory is "non-local", if the causal connections between space-like separated events are forbidden.

It just means violation of Bell type inequalities, since this is the only reason why people say that QM is nonlocal. There is no problem with this meaning.

 

[vanhees71]

I said convincing! I never understood what "beables" should be, if not "observables". I want of course a physical argument and theory and not some philosophical gibberish. Also in your abstract you already say that it violates Lorentz (and thus also Poincare) covariance. So that's not convincing for me either on a technical level.

 

[A. Neumaier]

"non-local" in the Bell theorem means existence of causal connections between space-like separated events.

Not in general, only assuming hidden variables with certain properties...