I Problems with Paper on QM Foundations

[Demystifier]

I am not advocating Bohmian mechanicsm in any way, but I do agree with this point.

This is an important point and a special case of the question of observer equivalence vs observer democracy. What may be fundamental, could be democracy - not equivalence. The equivalence may well be emergent only, is the sense of an evolutionary steady state. Allowing oneself to relax fundamental things, and instead focus on observer democracy may add explanatory value to why the effective equivalence is observed, but also why they aren't perfect. It's like a "noise" at the level of physical law, but it need not be a bad thing.

I even think that anyone that thinks about the logical arguements of which relativity builds, may find that the observer democracy condition is most certainly more obvious, but a weaker conditions than equivalence. The equivalence condition is handy, but may simply be wrong. (I obviously think it is, but one does not have to buy into something I think to at least see that it's a logical possibility that is perfectly rational and sound)

/Fredrik

What is observer democracy? That each observer has his own reality, not equivalent to the reality of other observers?

 

[Fra]

In other words, they have no status at all. A "calculational tool" can be things with no relation to the theory at all.

I think this is because we still do not understnad the theories deep enough. In a sense, we lack the physical understanding of the various calculations tools, because they tools are constructed in an extrinsic way, and they appear ad hoc; no matter how well corroborated. This gets more pressing one one tries to expand they theory an solve open questions. But I guess the ambition is that we could improve here.

But what if, we can translate the "calculations" down to the level of computations made by the observer/agent itself? Then the mathematics, should have a physical correspondence, which is automatically always "normalized" to the agents perpective. And unlike current theories, this renormalization must involves more variables than just spatial and energy scales, and it is not simply a statistical averaging to get macro variables.

This gets us a picture where the "theory" itself is not static, but itself evolving. Then one would have an ontology of agents, and a "inferred picture" of the environment which is regulated by the agents instrisic information processing capabilities. This would as i see it, unify the ontic and epistemological views. But "solving" the mathematical problem we get when formulating it like this, is not trivial, and it will in particular not take on the simple form of a differential equation, with a given state space and initial conditions, because forcing such a model would lead it seems to an infinitely complex model with a absurd fine tuning problem; that would render the theory imposible or any agent to actually implement. This "agent constraint" is what is the "natural regulator" when constructing the model.

/Fredrik

 

[Demystifier]

@Sunil What does it mean for $A^\mu$(or anything else) to be real?

Being real is a primitive notion, it cannot be defined in terms of something else. Somewhat like the notion of set in set theory. Which gives me an idea! The notion of set is not defined, but there are some axioms (ZF) that sets obey. Analogously, perhaps it's possible to make some list of axioms that any real object must satisfy, given a collection of measurable predictions. And then we can have various models that satisfy those axioms, which are nothing but various interpretations of the collection of measurable predictions. In particular, we can have standard and nonstandard interpretation of QM, just like we have standard and nonstandard interpretation of axioms for real numbers.

 

[Fra]

What is observer democracy? That each observer has his own reality, not equivalent to the reality of other observers?

As I always seen it the general constructing principle between not only relativity but also any gauge interaction, is that what any observer actually observes is just as valid description of reality as that of another one. This also includes that any inference of the apparent laws made by an observer, must be as valid as that of another observer. This is what i call the observer democracy.

Now, assuming that the various observers does not interact in a way that they significnatly influence each other or significantly modify the common environment, the natural next requirement is to expect that any observer should infer the SAME laws of physics, from their observations.

Then the invariants are formed from the equivalence class of all possible views. And one can talk about the equivalencec class or the invariants as the objective reality, all other things are gauged away as irrelevant observer choices.

This is a short description of key constructing principles of most of modern physics. And the arguments are hard to argue against, they are plausible and sound.

But if one allows for interaction of the observers, the game changes. We still have observer democracy, in the sense that no observer has priority of any other, but they may start to influence and interact with each other, and it's logically possible that two observers arrive at different views, where they can not even agree on their relations (which the equivalence implicit in symmetry transformations implies). In this case, instead of a timelesse symmetry transformation, we face a negotiation process where observers can literally and physically fight each other, and the result (after evolution time) may be an equilibiruym with a modified population of observers, that are simply TUNED for mutual equivalence.

I am just lifting the possibility, that this negotiation process, may be a KEY to understanding the laws and the hierarchy of interactions in nature. And in this view, one may need to see thgat elements of reality or ontologies may be subjective to the perspective, and at least in the transient sense, not necessarily related by perfect symmetry transformations as they don't exist on beforehand, they are the result of evolution.

/Fredrik

 

[Fra]

What is observer democracy? That each observer has his own reality, not equivalent to the reality of other observers?

Just to clarify. All observers are in the same world, there is only one universe in my view. (I'm not talking about MWI (which i never really got the real point of to be honest, it just seems weird and not solve any problems).

But each observer is it's own "inference-machinery" and has their own inferred, but real view of what the common world is like, that are inferred from real observations of what actually happens - from its perspetive.

But the differerent observers are not a priori assume to be related by symmetry transformations they can agree upon. It means as evolution forces agent-population changes, the symmetries are always approximate and not perfect, as they are evovling. There is also no non-physical "gods perspective" in thi, there are only the views of real physical material agents.

/Fredrik

 

[Sunil]

Nobody has yet made clear to me, what a clear definition of "ontology" is.

What's the problem? Ontology - that's the definition what, according to that

Physics is about well defined observable objective facts. There are no trajectories in phase space in quantum mechanics or quantum field theory.

Quantum theory is not a realist theory, but there exist realist interpretations, like Bohmian mechanics. These interpretations have trajectories in configuration space.

That's wrong. What's observed in the AB effect is a non-integrable phase factor, which depends however on a gauge-invariant quantity only, i.e., the flux of the magnetic field through a surface intersecting the solenoid,
$$\Phi_B=\int_{A} \mathrm{d}^2 \vec{f} \cdot \vec{B} = \int_{\partial A} \mathrm{d} \vec{x} \cdot \vec{A}.$$
It's an integral over the vector potential along a closed path, and this is gauge invariant. There are no gauge-dependent observables. This was a contractio in adjecto!

Do you read the text before you answer? Where I have claimed that there are gauge-dependent observables? The Wiki quote mentions the same thing as your formula.

The claim is different. You cannot compute this number using the $F_{\mu\nu}$ along the trajectory of the electrons.

If gauge symmetry is not exact, the model doesn't make physical sense. There can be approximate global symmetries like chiral symmetry of QCD in the light-quark sector, which make physical sense, but if a local gauge symmetry is broken in any way, it doesn't make any sense anymore.
This has nothing to do with Poincare invariance. ... but as soon as the underlying local gauge symmetry is broken, it doesn't make any sense anymore. If you know the BRST formalism or Gupta Bleuler for QED, then this should be immediately clear!

Again, do you read the text before you answer? Is it difficult to understand my "the Gupta-Bleuler formalism, resp. BRST quantization, which starts with a physically nonsensical indefinite Hilbert space for the fields, breaks down" which you have quoted.

The entire point of the BRST approach is to evaluate well-defined observables in a well-defined Hilbert space from a gauge theory, which by definition deals with a description with contains non-physical unobservable elements in the description.

Yes. Which obviously breaks down once the unobservable "calculational tools" become observable. Not because there is anything wrong with approximate gauge invariance, but because BRST uses an artificial indefinite metric for the $A^\mu$ which cannot make sense for observables.

 

[vanhees71]

I meant what's wrong is the claim "The Aharonov–Bohm effect shows that the local E and B fields do not contain full information about the electromagnetic field." Indeed in the AB experiment the particles don't have trajectories, and it's crucial that they don't have since the observable is a phase difference of the corresponding partial waves going the one or other way beyond the solenoid. The phase difference CAN be calculated with the observable field $\vec{B}$, and that's the important point. The observable (the shift of interference fringes when observing an ensemble of many particles) IS a gauge-invariant quantity although the "local formulation" needs the use of the gauge-dependent potential.

The BRST formalism does NOT use an ill-defined Hilbert space but constructs an adequate well-defined Hilbert space to quantize a gauge theory. Again: A gauge theory, for which gauge invariance is only "approximate" (a contradiction in itself) is not a well-defined theory at all. The entire BRST construction of the adequate Hilbert space, leading to a unitary S-matrix breaks down then.

 

[Sunil]

I meant what's wrong is the claim "The Aharonov–Bohm effect shows that the local E and B fields do not contain full information about the electromagnetic field." Indeed in the AB experiment the particles don't have trajectories, and it's crucial that they don't have since the observable is a phase difference of the corresponding partial waves going the one or other way beyond the solenoid. The phase difference CAN be calculated with the observable field $\vec{B}$, and that's the important point.

But the point was that this requires information about the $\vec{B}$ field inside the solenoid, where the electron is not. It is either on this trajectory or the other one - or nearby. Because of this point I have used the word "local".

The observable (the shift of interference fringes when observing an ensemble of many particles) IS a gauge-invariant quantity although the "local formulation" needs the use of the gauge-dependent potential.

No necessity to cry given that I have not questioned this.

The BRST formalism does NOT use an ill-defined Hilbert space but constructs an adequate well-defined Hilbert space to quantize a gauge theory.

Please read carefully what I write. Are you able to understand the difference between "physically nonsensical indefinite Hilbert space" and "ill-defined Hilbert space"? Of course, the indefinite Hilbert space is a well-defined mathematical object. It is physically nonsensical, because negative probabilities (however well-defined) are physically meaningless.

Again: A gauge theory, for which gauge invariance is only "approximate" (a contradiction in itself) is not a well-defined theory at all. The entire BRST construction of the adequate Hilbert space, leading to a unitary S-matrix breaks down then.

Again: There is no doubt that the BRST construction breaks down, no need to repeat this.

But this does not make a theory with approximate gauge invariance not well-defined. One has to use another approach to the quantization of such a theory, one which handles the $A^\mu(x)$ as observable quantum fields.

 

[vanhees71]

But the point was that this requires information about the $\vec{B}$ field inside the solenoid, where the electron is not. It is either on this trajectory or the other one - or nearby. Because of this point I have used the word "local".

It is indeed an astonishing topological feature of fields that you have information about them when knowing them only along a much smaller subset of their domain. One of the most simple cases are holomorphic functions, whose values you know everywhere in an area given the values along its boundary (Cauchy's theorem), but this indeed saves both, "locality", i.e., a local description of the interaction between charged particles and the electromagnetic field in terms of the gauge potential (and this must be so, because the em. field as a massless vector field must necessarily be described as a gauge field, as can be inferred from the representation theory of the proper orthochronous Poincare group a la Wigner 1939).

No necessity to cry given that I have not questioned this.

Please read carefully what I write. Are you able to understand the difference between "physically nonsensical indefinite Hilbert space" and "ill-defined Hilbert space"? Of course, the indefinite Hilbert space is a well-defined mathematical object. It is physically nonsensical, because negative probabilities (however well-defined) are physically meaningless.

I only wanted to emphasize that the BRST quantization formalism is necessary precisely because you need a well-defined Hilbert space and a unitary S-matrix to make physical sense of the entire construction.

Again: There is no doubt that the BRST construction breaks down, no need to repeat this.

But this does not make a theory with approximate gauge invariance not well-defined. One has to use another approach to the quantization of such a theory, one which handles the $A^\mu(x)$ as observable quantum fields.

Why are you then claiming again and again that a violation of gauge invariance. You are contradicting yourself with these two sentences, or can you quote a scientific paper where they successfully can interpret the gauge potentials as observable quantum fields?

 

[Sunil]

It is indeed an astonishing topological feature of fields that you have information about them when knowing them only along a much smaller subset of their domain. One of the most simple cases are holomorphic functions,

There are Hilbert spaces of holomorphic functions, but they have nothing to do with field theory.

I only wanted to emphasize that the BRST quantization formalism is necessary precisely because you need a well-defined Hilbert space and a unitary S-matrix to make physical sense of the entire construction.

And that's wrong. You can start from a vector field $A^\mu$ too.

There will be some problems with the implementation of the Lorenz gauge, but similar problems appear in condensed matter theories too if you have a continuity equation: If, say, the fundamental theory has exact particle conservation, and you want to have a field theory based on the density $\rho$ together with the continuity equation, it is not easy to reach that $\hat{\rho} \ge 0$ and that the conservation law is not even violated even by vacuum oscillations. Quantum condensed matter theory nonetheless works nicely.

Why are you then claiming again and again that a violation of gauge invariance. You are contradicting yourself with these two sentences, or can you quote a scientific paper where they successfully can interpret the gauge potentials as observable quantum fields?

Please quote me in a meaningful way. I do not claim "a violation of gauge invariance" but usually write complete sentences. Like the following: Theories with vector fields $A^\mu$ with some approximate gauge invariance are possible. They may be non-renormalizable, but this does not make them invalid as effective field theories. But these theories will certainly not use the Gupta-Bleuler resp. BRST approach, but start with a definite Hilbert space.

I only wanted to emphasize that the BRST quantization formalism is necessary precisely because you need a well-defined Hilbert space and a unitary S-matrix to make physical sense of the entire construction.

And that's simply wrong. You can make physical sense of vector fields if you start with a definite Hilbert space. Of course, you need exact gauge invariance to be able to factorize, and without factorization you cannot make physical sense of the whole construction build on the indefinite Hilbert space. But you are not at all obliged to start with some indefinite Hilbert space.

The original approach to QED did not use an indefinite Hilbert space. The references to the original approach:
Dirac P.A.M. (1927). The Quantum Theory of the Emission and Absorption of Radiation. Proc Roy Soc A114, 243-265
Fermi, E. (1932). Quantum Theory of Radiation. Rev Mod Phys 4(1), 87-132
But I would nonetheless recommend instead
Akhiezer, A.I., Berestetskii , V.B. (1965). Quantum Electrodynamics.
which give also the formulas for the original approach.

 

[vanhees71]

There are Hilbert spaces of holomorphic functions, but they have nothing to do with field theory.

You can take holomorphic functions as scalar fields (or their real and imaginary parts) in 2D Euclidean space. These are the "harmonic functions" with the properties described. It was only an example. Of course this mathematical phenomenon is more generally just Stokes's theorem for differential forms.

And that's wrong. You can start from a vector field $A^\mu$ too.

There will be some problems with the implementation of the Lorenz gauge, but similar problems appear in condensed matter theories too if you have a continuity equation: If, say, the fundamental theory has exact particle conservation, and you want to have a field theory based on the density $\rho$ together with the continuity equation, it is not easy to reach that $\hat{\rho} \ge 0$ and that the conservation law is not even violated even by vacuum oscillations. Quantum condensed matter theory nonetheless works nicely.

I'm talking about massless spin-1 fields in relativistic field theories. They are necessarily gauge fields, as can be derived from the representation theory of the Poincare group.

Please quote me in a meaningful way. I do not claim "a violation of gauge invariance" but usually write complete sentences. Like the following: Theories with vector fields $A^\mu$ with some approximate gauge invariance are possible. They may be non-renormalizable, but this does not make them invalid as effective field theories. But these theories will certainly not use the Gupta-Bleuler resp. BRST approach, but start with a definite Hilbert space.

You said repeatedly that local gauge symmetries can be approximate, but that's not true, because then they loose their physical meaning. This has nothing to do with (Dyson-) renormalizability or non-renormalizable effective theories.

And that's simply wrong. You can make physical sense of vector fields if you start with a definite Hilbert space. Of course, you need exact gauge invariance to be able to factorize, and without factorization you cannot make physical sense of the whole construction build on the indefinite Hilbert space. But you are not at all obliged to start with some indefinite Hilbert space.

No, massless vector fields must be necessarily quantized as gauge fields. For massive vector fields you are right.

The original approach to QED did not use an indefinite Hilbert space. The references to the original approach:
Dirac P.A.M. (1927). The Quantum Theory of the Emission and Absorption of Radiation. Proc Roy Soc A114, 243-265
Fermi, E. (1932). Quantum Theory of Radiation. Rev Mod Phys 4(1), 87-132
But I would nonetheless recommend instead
Akhiezer, A.I., Berestetskii , V.B. (1965). Quantum Electrodynamics.
which give also the formulas for the original approach.

Of course, in the early days they completely fixed the gauge before quantizing. That's another equivalent way to quantize the em. fieeld, which is even preferrable if you learn the subject for the first time. It's only disadvantage is that it is not manifestly covariant, which makes calculations of higher-order perturbative corrections a nightmare.

Nevertheless also there gauge invariance is needed to make physical sense of the theory. There's no way out: The math of the Poincare group tells you that a massless vector field must be a gauge field.

 

[Sunil]

I'm talking about massless spin-1 fields in relativistic field theories. They are necessarily gauge fields, as can be derived from the representation theory of the Poincare group.

Given that I do not claim that the result of the limit $m\to 0$ of massive vector fields will be something different in its observable predictions, I see no reason to care about such results. If that result tells us that both methods give, in the result, the same theory, fine. If you name them gauge fields or not does not matter.

No, massless vector fields must be necessarily quantized as gauge fields. For massive vector fields you are right.
...
You said repeatedly that local gauge symmetries can be approximate, but that's not true, because then they loose their physical meaning.

And here I simply disagree. AFAIU this is a misunderstanding based on the memories of time when people thought that non-renormalizable theories make no physical sense.

Take a naive discretization of a chiral lattice theory. It will not have exact gauge invariance on the lattice, but will be a well-defined theory. A finite number of degrees of freedom on a finite lattice (say on a large cube) and no infinities. Quantized in a straightforward canonical way. What will be the large distance limit?

Of course, in the early days they completely fixed the gauge before quantizing. That's another equivalent way to quantize the em. fieeld, which is even preferrable if you learn the subject for the first time. It's only disadvantage is that it is not manifestly covariant, which makes calculations of higher-order perturbative corrections a nightmare.

Ok, that's already much better.

I think that I have already said that I don't care about the mathematical tricks for approximate computations. You use "dimensional regularization" with physically completely meaningsless "dimensions" $4-\varepsilon$ in your renormalization? Fine, as long as you don't sell this as being something physically meaningful. You use an indefinite Hilbert space for computing your scattering coefficients? Fine, as long as you don't sell this as being something physically meaningful.

If this particular trick needs gauge invariance, it follows that you cannot apply this trick if there is no such exact gauge invariance, as for a naive chiral gauge field on the lattice. But this is a problem for computations, not for the meaningful definition of the theory itself.

There's no way out: The math of the Poincare group tells you that a massless vector field must be a gauge field.

And what makes the difference for me if I have a well defined theory with massive gauge fields with observable potentials and take the limit $m\to 0$? The result will be a gauge theory? Fine. What's the problem?
(BTW, the math of the Poincare group may be also irrelevant on the fundamental lattice level.)

 

[vanhees71]

Again. This has nothing to do with Dyson renormalizability or effective field theories with an infinite number of parameters ("low-energy coupling constants"). A gauge theory, for which the local gauge symmetry is broken has no physical interpretation, because it leads to acausality and a non-unitary S-matrix.

The limit $m \rightarrow 0$ for massive vector fields is non-trivial. You can describe the massive vector field as a Proca field. Then the limit is not well defined (as you can already see looking at the free propagator of this field, containing a piece $\propto p_{\mu} p_{\nu}/m^2$. Another possibility is to describe it as an Abelian gauge field, which is the Stueckelberg approach. Then the limit $m \rightarrow 0$ can be taken and leads to the usual gauge theory for a massless vector field. Again, whenever you deal with massless vector fields you end up with a gauge theory!

 

[Sunil]

You have not got the point that I'm not afraid of ending up with a gauge theory? My point is that this does not require that the $A^\mu$ should be handled as completely unphysical fields living in an indefinite Hilbert space of the Gupta-Bleuler resp. BRST approach.

One possibility would be the one you have accepted with your "in the early days they completely fixed the gauge before quantizing". Another one, canonical quantization on a lattice of a chiral gauge field which does not have exact lattice gauge symmetry on the lattice. It is nonetheless well-defined, unitary, and has some large distance continuous limit.

I'm quite happy if these examples all lead to the same gauge theory. In this case, the gauge potentials would be normal physical fields living in physical definite Hilbert space. Even if the operators measuring their values would not be observables because of gauge symmetry in the continuous limit, they would be handled with the same mathematics as usual observables. And there would be the BRST approach which would be preferable for computations because some manifestly covariant integrals are easier to compute.

 

[vanhees71]

You have not got the point that I'm not afraid of ending up with a gauge theory? My point is that this does not require that the $A^\mu$ should be handled as completely unphysical fields living in an indefinite Hilbert space of the Gupta-Bleuler resp. BRST approach.

I try one last time: There is no indefinite Hilbert space. The entire point of the covariant operator quantization of gauge theories (BRST) is that there is no such thing!

One possibility would be the one you have accepted with your "in the early days they completely fixed the gauge before quantizing". Another one, canonical quantization on a lattice of a chiral gauge field which does not have exact lattice gauge symmetry on the lattice. It is nonetheless well-defined, unitary, and has some large distance continuous limit.

If you quantize on a lattice it's by construction clear that there's nothing observable which is not gauge invariant.

I'm quite happy if these examples all lead to the same gauge theory. In this case, the gauge potentials would be normal physical fields living in physical definite Hilbert space. Even if the operators measuring their values would not be observables because of gauge symmetry in the continuous limit, they would be handled with the same mathematics as usual observables. And there would be the BRST approach which would be preferable for computations because some manifestly covariant integrals are easier to compute.

The values of the gauge fields cannot be observables, because the operators do not obey the microcausality condition.

 

[Sunil]

I try one last time: There is no indefinite Hilbert space. The entire point of the covariant operator quantization of gauge theories (BRST) is that there is no such thing!

Ok, then I give up.

"The scalar photons are treated by using an indefinite metric"
Gupta, S.N. (1950). Theory of longitudinal photons in quantum electrodynamics, Proc Phys Soc A 63(7), 681-691

"Gupta has introduced an alternative method of quantization for the Maxwell field which differs from the usual one in that the scalar part of the field is quantized by means of the indefinite metric of Dirac. It is shown that this method can be extended into a general and consistent theory, including the case of interaction with electrons."
Bleuler, K. (1950). Eine neue Methode zur Behandlung der longitudinalen und skalaren Photonen. Helvetica Physica Acta 23(V), 567-586

Of course, as I have written already many many times, at the end of the construction some definite Hilbert space is constructed. But this does not make the Hilbert space where the potential operators $A^\mu$ live definite.

If you quantize on a lattice it's by construction clear that there's nothing observable which is not gauge invariant.

No. There are gauge-invariant lattice theories, namely Wilson lattice gauge fields. But it works only for vector gauge fields. If you use a naive lattice approximation for chiral gauge fields then there will be no gauge invariance in the lattice theory. But this does not forbid you to define the lattice theory in a reasonable canonical way (without indefinite Hilbert spaces).

The values of the gauge fields cannot be observables, because the operators do not obey the microcausality condition.

In the Lorenz gauge the equations are Lorentz-covariant and classically Einstein-causality holds. So there is no base to assume that the corresponding quantum operators will not fulfill the microcausality condition.

Then, there is no base for assuming that a lattice regularization fulfills relativistic symmetry requirements. A lattice regularization is the most straightforward way to regularize quantum gravity at Planck scale and to obtain a regularized theory with a finite number of degrees of freedom which one can quantize canonically. To expect from such a lattice regularization relativistic symmetry would be strange.

 

[A. Neumaier]

In the Lorenz gauge the equations are Lorentz-covariant and classically Einstein-causality holds.

No. Without gauge fixing, the coupled Maxwell-Klein Gordon equations (the nearest classically to QED) are not hyperbolic, hence do not satisfy Einstein-causality.

 

[Sunil]

No. Without gauge fixing, the coupled Maxwell-Klein Gordon equations (the nearest classically to QED) are not hyperbolic, hence do not satisfy Einstein-causality.

$\square A^\mu = j^\mu$ not hyperbolic?

Then, what has your "without gauge fixing" to do with my "in the Lorenz gauge"? Are you about the remaining gauge freedom after the Lorenz gauge?

 

[A. Neumaier]

$\square A^\mu = j^\mu$ not hyperbolic?

The coupled equations must be hyperbolic. Without coupling you only have a free field.

 

[Sunil]

The coupled equations must be hyperbolic. Without coupling you only have a free field.

The point being? I have included a source term. I would ask you to show me which part of the coupled system makes it non-hyperbolic. Is the equation for the matter fields not hyperbolic?

 

[A. Neumaier]

The point being? I have included a source term. I would ask you to show me which part of the coupled system makes it non-hyperbolic. Is the equation for the matter fields not hyperbolic?

I think it is not the coupling to the scalar field (as I first thought) but the constraint $\partial_\mu A^\mu =0$ that spoils hyperbolicity. I don't remember the details and don't have the time now to check; thus maybe I am wrong.

 

[Sunil]

I think it is not the coupling to the scalar field (as I first thought) but the constraint $\partial_\mu A^\mu =0$ that spoils hyperbolicity.

Ok, this has at least some plausibility. To introduce conservation laws into a field theory is known to be problematic.

I know (memory, without good sources, sorry) that this creates some problems in condensed matter theory too. Take a particle theory with exact conservation law. Then you can use, in a large distance approximation, a classical field theory with fields $\rho(x,t)$ and the corresponding momentum $\pi^i(x,t) = \rho(x,t) v^i(x,t)$. Try to define a quantum field theory for these four fields such that it is the large distance limit of the fundamental theory. How to reach that $\hat{\rho}(x,t) \ge 0$ given that particle number in the fundamental theory is always positive? How to reach that $\partial_t \hat{\rho}(x,t) + \partial_i \hat{\pi}^i(x,t) = 0$ exactly, without even vacuum oscillations around zero, given that particle number is exactly conserved in the fundamental theory? AFAIR there are no good answers for this.

So it seems quite plausible that there are similarly no good answers how to preserve Einstein-covariance of a classical theory with such an exact conservation law during quantization.

 

[Demystifier]

I know (memory, without good sources, sorry)

Now @PeterDonis will come at you.

 

[zonde]

I was on you-tube and saw a video from Oxford on QM foundations. I didn't agree with it, but that is not an issue - I disagree with a lot of interpretational stuff. The video mentioned a paper they thought essential reading:
https://www.mathematik.uni-muenchen.de/~bohmmech/BohmHome/files/three_measurement_problems.pdf

Really nice idea in this paper. If you want to understand what are the problems with QM you have to investigate the areas where QM is weak and that is individual events.

Is it me, or has the author not shown the appropriate care? In particular, they claim theories that violate 1a are hidden variable theories. I thought - what - how does that follow. It may simply mean nature is fundamentally probabilistic. Or, in other words, the three assumptions are not inconsistent.

Specifically, I do not think the following is logically justified:
'And since we are interested in individual cats and detectors and electrons since it is a plain physical fact that some individual cats are alive and some dead, some individual detectors point to "UP" and some to "DOWN", a complete physics, which is able at least to describe and represent these physical facts, must have more to it than ensemble wave-functions.'

My response is - that might be your idea of what compete physics is, but it might be best not to assume that is everyone's idea of complete physics. Einstein, of course, thought QM incomplete, but I am not sure that is necessarily why.

Hmm, to me it seems rather clear just by reading 1A statement: "1.A The wave-function of a system is complete, i.e. the wave-function specifies (directly or indirectly) all of the physical properties of a system."
Complete/incomplete means whether wave-function is or is not best possible description of individual system. And if it's not then additional variables (HV) can improve description of individual system.
It's just a definition of a term. And it is related to history of QM so it does not appear out of the blue.

Anyways, I thought of slightly modified three statements after reading Maudlin's paper.
Given single preparation of quantum state and measurement of the state that gives two easily distinguishable groups of events, we can say that considering each event and taking into account that there are two possible outcomes at least one of the options is true:
1. Measurement event revealed something physical.
2. Measurement event created something physical.
3. Measurement event is not physical.

 

[stevendaryl]

The ensemble interpretation, analogous to the frequentist interpretation of probability (which has the issue that the law of large numbers is only valid for infinite 'experiments'), is a 'practical' interpretation that can't answer fundamental matters. However, it is used all the time in applications.

I don’t understand how the ensemble interpretation of QM is an interpretation.

In classical statistical mechanics, the meaning of an ensemble of systems is a collection of systems that are macroscopically identical but microscopically different.

But in quantum mechanics, an electron that is in a superposition of spin-up in the z-direction and spin-down in the z- direction cannot be interpreted in that way. If you have an ensemble of electrons in that state, it cannot be interpreted as some fraction of them being spin-up and some fraction being spin-down.

Maybe after a measurement of the spin in the z-direction, you can interpret it that way. But that seems to me equivalent to a “collapse” interpretation of measurement.

 

[A. Neumaier]

But in quantum mechanics, an electron that is in a superposition of spin-up in the z-direction and spin-down in the z- direction cannot be interpreted in that way. If you have an ensemble of electrons in that state, it cannot be interpreted as some fraction of them being spin-up and some fraction being spin-down.

In the ensemble interpretation, a superposition is supposed to describe an ensemble of electrons in the superposition, not an ensemble of electrons in one of two particular states.

 

[PeterDonis]

In classical statistical mechanics, the meaning of an ensemble of systems is a collection of systems that are macroscopically identical but microscopically different.

But in quantum mechanics, an electron that is in a superposition of spin-up in the z-direction and spin-down in the z- direction cannot be interpreted in that way. If you have an ensemble of electrons in that state, it cannot be interpreted as some fraction of them being spin-up and some fraction being spin-down.

In the QM ensemble interpretation as compared with classical statistical mechanics, "macroscopically the same" corresponds to "all prepared by the same process", and "microscopically different" corresponds to "will not all give the same result when measured in the same way". Whether or not this is a good enough correspondence to justify the term "ensemble interpretation" is a matter of choice of words; the interpretation itself is clear about what it's saying, and does not claim to be in exact correspondence with the classical statistical mechanics definition of an ensemble.

 

[zonde]

Anyways, I thought of slightly modified three statements after reading Maudlin's paper.
Given single preparation of quantum state and measurement of the state that gives two easily distinguishable groups of events, we can say that considering each event and taking into account that there are two possible outcomes at least one of the options is true:
1. Measurement event revealed something physical.
2. Measurement event created something physical.
3. Measurement event is not physical.

Speaking about option 3. which basically is about MWI. In Maudlin's paper objections to MWI are related to Born's rule. I would say that there is more serious philosophical problem.
MWI denies that my perception of certain consistent observations as objective physical facts is justified.

I will explain. I make an observation, say I look at a track in Wilson cloud chamber. Now I ask: am I seeing things or is this objective physical fact that there is a track in that cloud chamber?
So I ask somebody else to look at this track and ask him if he is seeing the same. But even if he agrees with me he could be influenced by unconscious bias say because he tends to consider more seriously the option that I am right. Or he just lies because it benefits him somehow.
So I ask him to do the experiment according to given instructions and to write down what he is seeing. Then to avoid possibility that my memory is cheating me I make a record myself and after he has made the experiment we compare our records.
If we do that and the records are consistent with each other I can say with great confidence that I am not seeing things and I should consider the track in cloud chamber as a objective physical fact.

So, what does MWI say about all that? It says I can't relay on all these things as a method of getting objective physical facts. Measurement records are not facts at all, they are just illusions of facts because some QM aspect of me is aligned with some QM aspect of measurement record. And communication with other person gives me consistent experience because some QM aspect of me is aligned with certain QM aspect of other person including this QM aspect of all the communication media between us.

So what I can do instead? Well, I should pretend that measurement records are objective facts and repeat to myself the magic word "decoherence" while remembering at the back of my mind that deep down it's a fake certainty.

How this approach is different from superdeterminism?

 

[zonde]

Speaking about option 1. it is actually quite simple - it contradicts observations.
It is obvious with quite simple three polarizers experiment:

 

[Morbert]

Speaking about option 1. it is actually quite simple - it contradicts observations.
It is obvious with quite simple three polarizers experiment:

For posterity:

Consistent Histories as described by Robert Griffiths would assert option 1, and would do so without recourse to hidden variables or similar extension to QM.

When I get a chance I can lay out how CH would approach this experiment.