A Is quantum theory a microscopic theory?

[A. Neumaier]

As a macroscopic object, it's not even possible to do so.

Surely, any single macroscopic object is a single quantum mechanical system. Thus it must be possible in principle to treat it as such, even though in practice one can do it only approximately. But approximations are ubiqitous anyway; even a single photon state can be ascertained only approximately.

 

[microsansfil]

Or would you also argue that the axioms of Euclidean geometry are also "nothing more than a metaphysical choice"?

Not the axioms of Euclidean geometry are just syntax ! They are neither true nor false. Theorems can be built, demonstrations can be made within the framework of the mathematical Proof theory. The link with semantics (Model theory) is through the completeness theorem.

/Patrick

 

[vanhees71]

Surely, any single macroscopic object is a single quantum mechanical system. Thus it must be possible in principle to treat it as such, even though in practice one can do it only approximately. But approximations are ubiqitous anyway; even a single photon state can be ascertained only approximately.

Sure, and that approximation leads to the classical behavior of the relevant observables of the macroscopic system.

 

[vanhees71]

Not the axioms of Euclidean geometry are just syntax ! They are neither true nor false. Theorems can be built, demonstrations can be made within the framework of the mathematical Proof theory. The link with semantics (Model theory) is through the completeness theorem.

/Patrick

Indeed, as any axiom system of mathematics also Euclidean geometry is an invention of human thought. Its application to the description of real-world phenomena is already physics, and it turns out that it is a pretty good description for the description of space for an inertial observer in the approximation that gravitational interactions can be neglected or treated in non-relativistic approximation(s).

 

[Demystifier]

Earlier you deny that perceptions are relevant to physics

I never denied that. I only denied that only perceptions are relevant to physics.

 

[Demystifier]

Moreover, the concept of existence is a concept from the field of metaphysics, but not from the field of physics.

I agree with the former but not with the latter. I don't think that there is a sharp borderline between physics and metaphysics.

 

[Auto-Didact]

Not the axioms of Euclidean geometry are just syntax ! They are neither true nor false. Theorems can be built, demonstrations can be made within the framework of the mathematical Proof theory. The link with semantics (Model theory) is through the completeness theorem.

/Patrick

Indeed, as Poincaré said 'fundamental principles are only conventions - adopted due to some convenience - and it is quite unreasonable to ask whether they are true or false as it is to ask whether the metric system is true or false.'

 

[DarMM]

As I said above there's nothing preventing you from treating the device quantum mechanically, thus it's not a problem with obtaining emergent classical behavior. It's a separate problem. It's that when you do model the device with QM you invoke a second device that is treated classically. You could treat this device with qm, but you invoke a third device and so on. This is sometimes known as the Von Neumann chain.

The presence of something not modeled with QM that selects a particular Boolean frame is always assumed in typical Copenhagen presentations of the theory.

This is not the case in classical theories, where the theory is not written with reference to a system lying outside the theory.

Thinking about this I think it needs to be supplemented by some realistic details about the Von Neumann chain.

For a device consisting of $N$ particles it can be shown that the second device required to measure it must contain particle number $N^{'}$ given by:
$$N^{'} > K\exp\left(\frac{CN^{r}}{qs}\right)$$
with $s$ a constant related to how the resolution of the second device scales with its size, $q$ a constant related to how fluctuations of results for the second device scale with its size, $r$ a constant depending on the type of material the first device is made from and $K$ and $C$ constant depending on the exact chemical make up of the first device.

$q$ and $s$ turn out to essentially be universal with values of $\frac{1}{2}$ and $\frac{1}{3}$ respectively. $r = \frac{2}{3}$ for fermionic matter. $C$ and $K$ are of $\mathcal{O}\left(1\right)$ for the metals most devices are made of.

Considering a measuring device in real life has around $N \approx 10^{27}$ particles this gives:
$$N^{'} > 10^{10^{18}}$$
far larger than the number of particles in the observable universe ($\approx 10^{80}$). Thus the second device would either be so massive as to collapse into a black hole even for our smallest devices, or need to be so large (bigger than superclusters) that special relativity would prevent it from working within the timescales it needs to analyse the first device.

Thus in reality the chain only has one step. The above calculations also mean that interference terms in our devices have no operational meaning as observables $\mathcal{A}$ that demonstrate them would require devices that cannot exist. This links into how in algebraic quantum field theory not every abstract operator is actually part of the local observable C*-algebra due to stress-energy constraints etc. Such interference observables $\mathcal{A}$ just don't exist physically.

It also probably means Wigner's friend isn't very meaningful for the actual classical world. (And this is ignoring how it requires macroscopic systems to basically be "frozen" to justify the pure state)

 

[A. Neumaier]

@DarMM: Where is this formula proved?

 

[DarMM]

@DarMM: Where is this formula proved?

I'm trying to order the original literature that gives this calculation to see where each of the constants is first calculated etc. However an essentially complete guide is given in:
Omnes, R., 1999. The Interpretation of Quantum Mechanics. 1st ed. Princeton: Princeton Univeristy Press.

Specifically Chapters 5-7. The formula itself is (7.32) in that book. Note that his derivation has gaps, e.g. he does sketch the calculation of $C$, but $K$ is simply quoted to be of the right order.

EDIT: However all other details are basically given, e.g. $q$ and $s$ are derived

 

[DarMM]

I should say this is part of why I have a hard time with results like Frauchiger-Renner.
The set up is basically:

Alice and Bob performed quantum measurements.

Wigner and Zeus then show up with superstructures built out of neutronium larger than the observable horizon. If we assume such a thing can even happen (was it built using material from outside the observable horizon?) somehow manage to circumvent the operational constraints of relativity (how?) they then perform a measurement on Alice and Bob. We can then show that if they use modal logic, generally thought to be invalid in QM anyway from earlier results like Hardy's paradox, we find a contradiction in quantum theory.

But really is this of any genuine interest?

 

[Auto-Didact]

Thus in reality the chain only has one step. The above calculations also mean that interference terms in our devices have no operational meaning as observables AA\mathcal{A} that demonstrate them would require devices that cannot exist. This links into how in algebraic quantum field theory not every abstract operator is actually part of local observable C*-algebra due to stress-energy constraints etc. Such interference observables AA\mathcal{A} just don't exist physically.

This raises one of the key issues: if all self-adjoint operators aren't actually observables, then there is a physically extraneous formal selection problem, i.e. the notion of observables is an unhelpful mathematical idealization which distracts more from the physics than that it is helpful.

 

[microsansfil]

I agree with the former but not with the latter. I don't think that there is a sharp borderline between physics and metaphysics.

Physics and metaphysics certainly complement each other in the construction of new theories. But in the use of current theories, I do not perceive their usefulness in the construction of models to carry out experiments. "shut up and calculate", works very well. Mastery of mathematics is much more useful.

Now I'm not saying that metaphysics is not useful to us. Have the courage to challenge his deepest metaphysical beliefs to question, for example, the scaffolding we use to build our theories. Metaphysics raises legitimate questions about the construction of our scientific knowledge.

/Patrick

 

[Auto-Didact]

Physics and metaphysics certainly complement each other in the construction of new theories. But in the use of current theories, I do not perceive their usefulness in the construction of models to carry out experiments. "shut up and calculate", works very well. Mastery of mathematics is much more useful.

Now I'm not saying that metaphysics is not useful to us. Have the courage to challenge his deepest metaphysical beliefs to question, for example, the scaffolding we use to build our theories. Metaphysics raises legitimate questions about the construction of our scientific knowledge.

/Patrick

Just as mathematics helps physicists in the construction of new physical theories, metaphysics gained from new physical theories communicated from physicists to mathematicians helps mathematicians in the construction of new mathematical theories.

To paraphrase Feynman, mathematics is not physics and physics is not mathematics, they help each other: the symbols of physics have no semantic meaning to a mathematician, but to a physicist these same symbols convey an understanding of phenomena occurring in the real world; the physicist's mathematicized conceptual understanding is metaphysics.

 

[A. Neumaier]

This raises one of the key issues: if all self-adjoint operators aren't actually observables, then there is a physically extraneous formal selection problem, i.e. the notion of observables is an unhelpful mathematical idealization which distracts more from the physics than that it is helpful.

No. Only the name ''observable'' as a synonym for ''self-adjoint operator'' is questionable.

In finite-dimensional Hilbert spaces of not too high dimensions one can realize all Hermitian matrices as observables by giving a description for how to measure them. See, e.g., my paper

• U. Leonhardt and A. Neumaier, Explicit effective Hamiltonians for general linear quantum-optical networks, J. Optics B: Quantum Semiclass. Opt. 6 (2004), L1-L4.

Of course, the actual realization is limited by technological issues, but these change with time and should not be the subject of theoretical physics.

In infinite dimensions, the situation is similar though there are no rigorous results. But it is known how to represent by operators a number of key experimental quantities, so one can take these as building blocks and combine them in a similar way as done in finite dimensions. This gives a huge supply of operators for ''observables''.

 

[microsansfil]

the physicist's mathematicized conceptual understanding is metaphysics.

For the french philosopher Alain Badiou, mathematics is the ontology of the human being.

/Patrick

 

[Auto-Didact]

No. Only the name ''observable'' as a synonym for ''self-adjoint operator'' is questionable.

In finite-dimensional Hilbert spaces of not too high dimensions one can realize all Hermitian matrices as observables by giving a description for how to measure them. See, e.g., my paper

• U. Leonhardt and A. Neumaier, Explicit effective Hamiltonians for general linear quantum-optical networks, J. Optics B: Quantum Semiclass. Opt. 6 (2004), L1-L4.

Of course, the actual realization is limited by technological issues, but these change with time and should not be the subject of theoretical physics.

In infinite dimensions, the situation is similar though there are no rigorous results. But it is known how to represent by operators a number of key experimental quantities, so one can take these as building blocks and combine them in a similar way as done in finite dimensions. This gives a huge supply of operators for ''observables''.

I can see how that is an effective practical restriction, but as you say I don't see how that can be part of the subject of theoretical physics.

Hence I stand by my earlier point that for (theoretical) physics the notion of observables is ultimately an unhelpful notion, certainly not deserving of the primary role it is usually given in most treatments of textbook QM.

 

[Auto-Didact]

For the french philosopher Alain Badiou, mathematics is the ontology of the human being.

/Patrick

To quote Feynman, that would be wagging the dog by the tail.

 

[DarMM]

Relevant to the Von Neumann chain is not so much if a given self-adjoint operator/matrix in the observable algebra can be realized, but more so that the kind of observables discussed in the second level of the Von Neumann chain/Wigner's friend probably aren't actually elements of $\mathcal{A}\left(\mathcal{O}\right)$ for any spacetime region $\mathcal{O}$.

They're observables possible only from extrapolating the non-relativistic theory beyond its limit and to unrealistic scales, with no input as to the actual nature of spacetime and the constituency of real matter.

 

[A. Neumaier]

I can see how that is an effective practical restriction, but as you say I don't see how that can be part of the subject of theoretical physics.

Hence I stand by my earlier point that for (theoretical) physics the notion of observables is ultimately an unhelpful notion, certainly not deserving of the primary role it is usually given in most treatments of textbook QM.

Theory is always an idealization of practice. There is no need for theoretical physics to be different in this respect. Thus nothing needs to be changed except dropping the claim that every selfadjoint operator represents an observable. This has no effect at all on 99.9% of quantum theory.

 

[Auto-Didact]

Theory is always an idealization of practice. There is no need for theoretical physics to be different in this respect. Thus nothing needs to be changed except dropping the claim that every selfadjoint operator represents an observable. This has no effect at all on 99.9% of quantum theory.

But for psychological reasons such a change can have quite drastic effects for theoreticians trying to construct theories beyond QM. This is analogous to the difference between orbits being almost circles and actually being circles; this may seem like a negligible difference in terms of precision, but conceptually it is a worldview shattering fundamental change.

 

[DarMM]

Thus nothing needs to be changed except dropping the claim that every selfadjoint operator represents an observable

You're correct here of course, but I think @Auto-Didact is getting at the fact that some no-go theorems or thought experiments are based on literally every self-adjoint operator being an observable. As Jürg Fröhlich mentioned in his recent paper the idealizations in the thought experiments of some quantum foundations papers might be more a hindrance than a help.

 

[A. Neumaier]

the idealizations in the thought experiments of some quantum foundations papers might be more a hindrance than a help.

There are other idealizations that have a more severe effect on quantum foundations. One is that Born's rule cannot be fundamental as it implies a nonzero probability for almost instantaneous effects of local preparations at positions light years away. This can be seen without Bell's inequality just by looking at Hegerfeldt's theorem.

 

[vanhees71]

That's NOT what Born's rule implies. It's implied by non-relativistic physics, where this is not a problem at all since within Newtonian physics interactions at a distance are part of the theory. Relativistic microcausal QFTs are constructed such that this cannot be happen since local operators representing observables, particularly the Hamilton density, commute at space-like separation of their arguments.

 

[A. Neumaier]

That's NOT what Born's rule implies. It's implied by non-relativistic physics, where this is not a problem at all since within Newtonian physics interactions at a distance are part of the theory. Relativistic microcausal QFTs are constructed such that this cannot be happen since local operators representing observables, particularly the Hamilton density, commute at space-like separation of their arguments.

Commutation is about operators, not about probabilities, which are introduced solely through Born's rule. Thus ''this cannot happen'' is no argument, unless you can show why causal commutation rules imply constraints on Born's rule.

Note that Hegerfeldt's theorem covers both the relativistic and the nonrelativistic case:

• G.C. Hegerfeldt, Instantaneous spreading and Einstein causality in quantum theory, Annalen der Physik 7 (1998), 716--725.

 

[vanhees71]

I was referring to your claim that there are "effects of local preparations at positions light years away". This is a claim on the dynamics. The paper of Hegerfeldt is no surprise. It's known since the first attempts to generalize Schrödinger's successful formulation of wave mechanics to the relativistic realm. It's well known that this fails for exactly the reason we discuss here. That's why in the 21st century relativistic QT is introduced right away as relativistic (microcausal) QFT (see e.g., Peskin/Schroeder).

 

[A. Neumaier]

I was referring to your claim that there are "effects of local preparations at positions light years away". This is a claim on the dynamics.

Yes. But it is a claim for probabilities for measurement results, governed by Born's rule for the dynamically propagated wave function, and Hegerfeldt's paper is about that.

On the other hand, the theoretical apparatus of relativistic QFT is about q-expectation values of products of field operators (n-point functions), and causal results are valid only for these. Since expectation values say nothing at all about low probability events, the causal commutation relations have no implications for very low probability "effects of local preparations at positions light years away".

 

[A. Neumaier]

From

• G.C. Hegerfeldt, Instantaneous spreading and Einstein causality in quantum theory, Annalen der Physik 7 (1998), 716--725.

a few sentences (see p.7):

The Hamiltonian can be quite general, only boundedness from below is required, and this ensures either instantaneous spreading or confinement in a fixed bounded region for all times. [...]
This example [of the Dirac equation] is instructive since it shows the importance of the positive energy condition. The Dirac equation contains positive and negative energy states, and therefore we conclude from our results that positive-energy solutions of the Dirac equation always have infinite support to begin with! This is phrased as a mathematical result for instance in the book of Thaller [17].

Thus a single QED electron prepared in an arbitrary state has - according to the Born rule, taken at face value - a nonzero probability of being immediately detected arbitrarily far away.

 

[Auto-Didact]

You're correct here of course, but I think @Auto-Didact is getting at the fact that some no-go theorems or thought experiments are based on literally every self-adjoint operator being an observable. As Jürg Fröhlich mentioned in his recent paper the idealizations in the thought experiments of some quantum foundations papers might be more a hindrance than a help.

See, this is why I love physicsforums; this is literally a discussion we have had here quite recently purely through honest discussion with multiple participants, who only have partially overlapping viewpoints constantly engaging each other from multiple sides, which is now also reflected in the literature. Some of the things I have learned here from by reading and engaging in discussions open to criticism from all, are in some cases so novel I cannot even find it in the literature, let alone in books.

Moreover, both the respect and criticism I experience w.r.t some recurring 'allies' and 'opponents' here is to me of quite a similar nature as the deferential and deontic attitudes I have gained professionally, by attending weekly meetings for critical scientific appraisal of specific issues and medical-ethical analysis; these are both proven philosophical methods which can be utilized in order to come to a consensus for what the appropriate course of action is in wickedly vague and complicated situations together with fellow clinicians, who all moreover tend to have conflicting opinions on how to proceed yet still are willing to try working together in good faith.

I dare say that physicsforums by exhaustively discussing the foundations of QM issues, has in the last years actually become a beacon for direct access to reliable knowledge on this domain, given that one goes through a representative portion of these threads and that the descriptions and terminology continue to be used reliably within these subforums; this seems to be quite a unique feat among scientific communities on the internet or elsewhere as far as I am aware. Who said that these philosophical discussions do not contribute to the foundations of physics?

 

[vanhees71]

Yes. But it is a claim for probabilities for measurement results, governed by Born's rule for the dynamically propagated wave function, and Hegerfeldt's paper is about that.

On the other hand, the theoretical apparatus of relativistic QFT is about q-expectation values of products of field operators (n-point functions), and causal results are valid only for these. Since expectation values say nothing at all about low probability events, the causal commutation relations have no implications for very low probability "effects of local preparations at positions light years away".

But you cannot describe this in terms of a non-local single-particle Hamiltonian, $\hat{H}=\sqrt{\hat{p}^2+m^2}$. This has been abandoned for a long time by now for exactly the reason, it leads to the acausal behavior, we discuss here. Relativistic QM isn't even consistent with relativistic causality constraints for free particles, if treated in this way!

Maybe I don't understand what you mean by "local preparations light years away", because if they are assumed to be "local", how then can they be causally connected at the same time? It's a contradiction in adjecto!

 

[A. Neumaier]

But you cannot describe this in terms of a non-local single-particle Hamiltonian, $\hat{H}=\sqrt{\hat{p}^2+m^2}$. This has been abandoned for a long time by now for exactly the reason, it leads to the acausal behavior, we discuss here. Relativistic QM isn't even consistent with relativistic causality constraints for free particles, if treated in this way!

But the free QED electron can be described in terms of a more complicated non-local single-particle Hamiltonian!

The single electron sector of renormalized QED including infrared dressing is invariant under Poincare transformations, since there is no scattering. Its Hilbert space is the Hilbert space of a Poincare invariant infraparticle, and the time shift generator defines the Hamiltonian. The infraparticle structure is discussed in posts #30 and #31 of another thread. The Hamiltonian is that of a quasifree particle with a reducible representation of the Poincare group given by a mass spectrum with a branch point at the physical electron mass, where the continuous mass spectrum has a sharp peak. The details of the mass density are not completely known but the basic structure is in the reference of #31 of the other thread. The resolvent $(E-H)^{-1}$ is the renormalized electron propagator, as given by the Kallen-Lehmann formula associated with this mass density.

 

[PeterDonis]

Thread closed for moderation.

 

[PeterDonis]

The thread has run its course and will remain closed.