I Does a measurement setup determine the reality of spin measurement outcomes?

[A. Neumaier]

Yes, it is the same above regarding Euler Lagrange equation. They're physics, but that classification is a metaphysical one.

The classification of particle positions in Bohmian mechanics as 'real' or 'ontic' - i.e., the common interpretation of Bohmian mechanics - is also metaphysical, though Bohmian mechanics itself is, as an extension of the quantum mechanical formalism, (potentiallyy falsifiable) physics.

 

[PeterDonis]

Part 12 of this paper has an exposition

So basically, the criterion being given is that for something to be "real" it must be capable of being acted on by other "real" things, whereas the wave function is not acted on by anything. But this doesn't seem right, because the wave function is determined by Schrodinger's Equation, which includes the potential energy, and the potential energy is a function of the particle configuration. So I don't think it's correct to say that the wave function is not acted on by anything.

 

[DarMM]

So I don't think it's correct to say that the wave function is not acted on by anything.

I don't necessarily disagree, I was only stating what the "common" Bohmian position is.

 

[DarMM]

The classification of particle positions in Bohmian mechanics as 'real' or 'ontic' - i.e., the common interpretation of Bohmian mechanics - is also metaphysical, though Bohmian mechanics itself is, as an extension of the quantum mechanical formalism, (potentiallyy falsifiable) physics.

Agreed.

 

[kith]

We could just as well say that the EM field is "a relation obeyed by particles", since the only observations we have of the EM field are observations of particles whose motions are affected by the field. So what's the difference between the EM field and the wave function in Bohmian mechanics?

Without particles, there is no physical system which could be described by a wave function (in non-relativistic QM). There could still be an electric field because the Maxwell equations admit vacuum solutions. To me, this seems like an important difference and it justifies the view of the electromagnetic field as a physical system in its own right instead of a tool to describe particle interactions.

 

[A. Neumaier]

the wave function is determined by Schrodinger's Equation, which includes the potential energy, and the potential energy is a function of the particle configuration. So I don't think it's correct to say that the wave function is not acted on by anything.

But the coordinates are dummy variables, not Bohmian positions.

 

[PeterDonis]

Without particles, there is no physical system which could be described by a wave function (in non-relativistic QM). There could still be an electric field because the Maxwell equations admit vacuum solutions.

This seems to me to be a red herring. "Vacuum solutions" of the Maxwell equations just means there's no charge-current density "source" on the RHS. But there is no analogue to this "source" in Schrodinger's Equation anyway; the only "current density" is $\psi^* \psi$, which is always "there" anyway.

Also, talking about particles as the only "physical system which could be described by a wave function" already presupposes that the wave function itself is not "real". If the wave function itself is "real", there doesn't need to be any other physical system that it describes; it is a physical system all by itself.

 

[kith]

This seems to me to be a red herring. "Vacuum solutions" of the Maxwell equations just means there's no charge-current density "source" on the RHS. But there is no analogue to this "source" in Schrodinger's Equation anyway; the only "current density" is $\psi^* \psi$, which is always "there" anyway.

I need to better understand what you mean by the wave function being a physical system in order to comment on this. Right now it doesn't seem sensible to me to compare the equations like this.

Also, talking about particles as the only "physical system which could be described by a wave function" already presupposes that the wave function itself is not "real".

I just noted that in non-relativistic QM, the notion of particles (or more generally the notion of the physical system) comes before the notion of the wave function.

Similarly, in classical mechanics, the notion of particles comes before the notion of velocity. I wouldn't say that velocity being secondary to particles makes it not "real" but since "real" is notoriously ambiguous (in this thread alone, I count at least three different usages) I'd like to avoid this terminology completely.

If you are saying that I presuppose that the wave function is not a physical system similar to velocity not being a physical system in classical mechanics, I agree that I do this.

If the wave function itself is "real", there doesn't need to be any other physical system that it describes;[...]

Also in dBB -which the discussion seemed to be about- there needs to be a physical system which determines the observables. No such system, no wave function.

[...] it is a physical system all by itself.

I have a hard time wrapping my head around this.

It makes a certain sense to me to regard the electromagnetic field as a tool for accounting for complicated direct particle interactions which gives it the same status as the wave function in Copenhagenish interpretations. That we usually don't do this but imagine the electromagnetic field as an independent entity has clear reasons: I can shoot a light pulse into empty space and have a mathematical description for what happens there in the absence of matter. This motivation is absent for the wave function. I cannot create a wave packet which propagates in the absence of matter (in NRQM). No matter, no wave function.

What does make a certain sense to me is to question the primacy of matter and say that the wave function and matter are intertwined. But I don't see any motivation to treat the wave function as a physical system all by itself.

 

[PeterDonis]

in dBB -which the discussion seemed to be about- there needs to be a physical system which determines the observables.

Yes, agreed.

No such system, no wave function.

I don't agree with this. The wave function is a separate thing from the Bohmian particle positions. It even has its own evolution equation (I know I said earlier that I thought that equation depends on the particle positions through the potential energy, but after @A. Neumaier's response to that I'm not so sure). So it's perfectly possible to look at the wave function in dBB as a separate entity, not dependent on particle positions.

 

[A. Neumaier]

@kith and @PeterDonis points of view can be reconciled by differentiating between the abstract physical system (n scalar particles in 3-space) and a concrete physical system (defined by its trajectory of states) providing an instance of it. The former is prior to the latter and determines the kinematics, but the latter is needed to make predictions.

 

[Demystifier]

Whatever terms we use, they are terms of philosophy (or perhaps "metaphysics"), not physics.

There is no physics without philosophy (meta-physics). For instance, when you systematically collect empirical data on nature and compare them with testable theoretical predictions, that's science. When you say that this is what science is, that's philosophy.

 

[kith]

I don't agree with this. The wave function is a separate thing from the Bohmian particle positions.

I don't see the disagreement with what I wrote. You compare the wave function with particle positions. So your statement seems to say that the wave function is conceptually on equal footing with a property of the system and not with the system itself. Or would you consider the particle position to be a physical system?

So it's perfectly possible to look at the wave function in dBB as a separate entity, not dependent on particle positions.

The Schrödinger equation indeed doesn't depend on the Bohmian particle positions. But this is not the important point for the analogy with the electric field. On the contrary, the Maxwell equations do depend on the position of the charged particles.

So this isn't an argument for considering the pilot wave a separate entity in a sense similar to the electric field (see my example in post #58). Without something to attribute a position to there is no wave function.

 

[martinbn]

I think the electric field itself can be confusing because the same term "electric field" is used for both the system and the mathematical description. May be a better analogy is a fluid and its velocity field. Is the wave function like the fluid or like the velocity field? To me it makes no sense to say that it is like the fluid.

 

[Demystifier]

I think the electric field itself can be confusing because the same term "electric field" is used for both the system and the mathematical description. May be a better analogy is a fluid and its velocity field. Is the wave function like the fluid or like the velocity field? To me it makes no sense to say that it is like the fluid.

That's an interesting analogy, but begs a question. If the wave function is analogous to the velocity field, then what is the thing that is analogous to the fluid?

 

[PeterDonis]

You compare the wave function with particle positions.

You can s/particle positions/particles/ in my post and it would still be correct. The point is that in Bohmian mechanics both the particles and the pilot wave (the wave function) are separate entities; the wave function is not assumed to only have meaning as a probability amplitude for various particle positions. It's an actual physical thing that pushes the particles around. Deriving the fact that the wave function also gives the probability amplitude for particle position measurements requires also assuming that the initial distribution of particle positions satisfies a particular constraint; it's not built into the wave function from the start.

Without something to attribute a position to there is no wave function.

I don't think this is correct in the Bohmian interpretation. See above.

 

[martinbn]

That's an interesting analogy, but begs a question. If the wave function is analogous to the velocity field, then what is the thing that is analogous to the fluid?

Whatever the quantum mechanical system is, say an electron.

 

[A. Neumaier]

Whatever the quantum mechanical system is, say an electron.

But the system may be $N=10^{23}$ electrons

 

[martinbn]

But the system may be $N=10^{23}$ electrons

Yes, whatever it is. The point was that it wasnt the wave function.

 

[kith]

[...] the wave function is not assumed to only have meaning as a probability amplitude for various particle positions. It's an actual physical thing that pushes the particles around. Deriving the fact that the wave function also gives the probability amplitude for particle position measurements requires also assuming that the initial distribution of particle positions satisfies a particular constraint; it's not built into the wave function from the start.

I haven't written anything about probabilities and agree that they are not what gives meaning to the wave function in dBB. The wave function is as real as position and isn't secondary to it. But the two are not separate entities in the sense that you can have one without the other (see below).

Without something to attribute a position to there is no wave function.

I don't think this is correct in the Bohmian interpretation.

Then please give an example of a physical situation where the dBB description uses a wave function but no particle position. In the case of the electromagnetic field, I gave such an example of a light pulse traveling in a vacuum.

 

[PeterDonis]

Then please give an example of a physical situation where the dBB description uses a wave function but no particle position.

Of course I can't; dBB uses both to make predictions. But the claim of yours I was responding to was not "dBB uses both positions and the wave function to make predictions". It was "without something to attribute a position to there is no wave function", which I think is simply false for dBB since it attributes separate reality to the pilot wave.

 

[kith]

I'm not interested in discussing further whether the statement "without something to attribute a position to there is no wave function" is justified or not when we both can't think of a physical situation where the wave function is used in the absence of something to attribute a position to.

I'm still interested in discussing the commonalities and differences between the pilot wave and the electromagnetic field / @martinbn's fluid and how significant the differences are.

 

[Demystifier]

Whatever the quantum mechanical system is, say an electron.

But I can describe the fluid itself mathematically, without referring to the velocity field. For instance I can talk about its density and pressure. What can I say mathematically about the electron itself, without referring to its wave function?

 

[A. Neumaier]

But I can describe the fluid itself mathematically, without referring to the velocity field. For instance I can talk about its density and pressure. What can I say mathematically about the electron itself, without referring to its wave function?

In Bohmian mechanics, its position.

 

[Demystifier]

In Bohmian mechanics, its position.

Of course, but I wanted to see what martinbn will say.

 

[vanhees71]

This is interpretation dependent.

In the Copenhagen interpretation, the spin is not real before the measurement.

In Bohmian mechanics, the spin is determined before the measurement (by the beam in which it is)

In the thermal interpretation, the spin is a real number and the measurement outcome discretizes it, hence is only approximate.

I'd not use the word "real" here. It's misleading. According to QT in the orthodox interpretation (minimal interpretation, almost Copenhagen but without collapse and without "Heisenberg cut") observables are always "real" in the sense that you always have the oppotunity to measure them, independent of the state the system is prepared in. The state, however, determines whether the meaured observable has a determined value or not. If it has a determined value, you'll always obtain this certain value when you measure it. Otherwise, you get a random result when measuring this observable, and the state preparation implies the probabilities (or probability distribution in the case of continuous observables) for finding one of the possible values of the observable when measuring it.

I don't understand, what you mean by "the measurement outcome discretizes" the spin. Any spin component takes discrete values $m \in \{-s,-s+1,\ldots,s-1,s \}$. That's the possible values you get when measuring this spin component accurately (enough). If your detector resolution is not sufficient, you get some continuous distribution around these values of course, but that's due to the detector resolution and has nothing to do with the observable "spin component" per se.

 

[vanhees71]

Related to what @A. Neumaier said above, in the Copenhagen interpretation Spin is a phenomena that occurs in a classical system as a result of interaction with a microscopic system. It has no meaning outside of that interaction.

Well, it's not that simple unfortunately, at least not in the relativistic case. The macroscopic manifestation of spin is, e.g., ferromagnetism through the magnetic moments of the particles due to spin.

 

[vanhees71]

So what I am wondering about is if the spin is defined as a certain outcome of a certain measurement, or that it is regarded as something ontological real, in which case the measurement doesn't necessarily has to represent the value of it.

I am aware that these measurements don't commute, but if one took the correlation between two measurements on the same spin to be real, but not de measurements themselves*, then we might have less of a problem for instance in considering that a two dimensional operator yields two real worlds in MWI, because the measurement outcomes are not real*.

Observables are generally defined as a quantified phenomenon that can be measured. This is true for all of physics, also in the realm, where classical physics is applicable as an approximation.

If you have two observables which are not compatible, i.e., if their representing self-adjoint operators do not commute, it's generally impossible to prepare states, where both observable take determined values. In your example: If you prepare a state, where the spin component $s_z$ of a particle is determined, spin components in other directions are necessarily indetermined.

It's also easy to understand, why this is the case in this example: To prepare a spin component you can use the Stern-Gerlach setup, i.e., you let the particle run through a magnetic field with a large homogeneous part in the $z$ direction and some gradient also in the $z$ direction (then, because of $\vec{\nabla} \cdot \vec{B}=0$ it necessarily has also a gradient in some other direction, but that you can almost neglect, as will become clear below). Then the spin rapidly precesses around the spin direction, and thus almost only the force in $z$ direction due to the field gradient is relevant for the motion of the particle. This leads to an almost perfect entanglement of the $z$-component of the particle's position with the spin component $s_z$, i.e., if you use an appropriate beam of particles this beam splits into $s$ partial beams, each of which contains particles with (almost) determined spin-$z$ components.

Now it's already clear even from these qualitative considerations only that in this way you can only determine the spin-$z$ component, while all others are necessarily pretty indetermined. So if you decide to determine the spin-$z$ component you cannot determine another spin component at the same time. To determine the other spin component you'd have to use a magnetic field in its direction rather than the $z$ direction. If you send a particle with determined $z$ direction, prepared using the corresponding Stern-Gerlach apparatus, through another Stern-Gerlach apparatus to determine, e.g., the spin-$x$ component this apparatus randomizes the spin-$z$ component again, i.e., you destroy the preparation of the spin-$z$ component necessarily if you want to determine the spin-$x$ component instead. For each individual particle with a determined spin-$z$ component you cannot say which value the spin-$x$ component you will get when running it through the Stern-Gerlach apparatus for the spin-$x$ prepration. All you know from the preparation in the state with determined spin-$z$ component are the probabilities to end up in any of the possible spin-$x$ states.

 

[vanhees71]

The common wave function that serves as a pilot wave for the particles.

That's another quibble I have with de Broglie Bohm. How do you define "spin" as a classical variable. In non-relativistic theory there may be a chance to define it from a classical model of a point particle with some intrinsic angular-momentum like direction (e.g., as a small rigid body) or simply only by the Bohmian trajectories of a particle with spin in an external magnetic field, leading in Stern-Gerlach like setups to an entanglement between spin component and position? In the latter case, it's however not clear to me, in which sense the spin is then defined as an observable in the Bohmian interpretation to begin with, and then of course it's even harder to decide whether, within the Bohm interpretation, the spin has some determined value or not.

 

[vanhees71]

The action has the dimension M·L²·T^-1. I don't think that the wave function has a dimension.

if the wave function is just a calculation tool for calculating measurement predictions, can it then be considered as an "ontic/beable entity"? A Platonist may answer yes, hence the ambiguity of these metaphysical notions of existence, ontology.

/Patrick

The wave function has a dimension. It's a probability density. In the position representation of a single particle the wave function has thus the dimension $L^{-3/2}$. I cannot say whether this has anything to do with philosophy or being ontic/beable or whatever, but that's a mathematical and physical fact ;-)) SCNR.

 

[vanhees71]

When Bohmians talk about ontology, they usually mean fundamental ontology. Temperature, in that sense, is not a fundamental ontology. Even kinetic energy of a classical nonrelativistic particle, given by $E=mv^2/2$, is not a fundamental ontology in classical mechanics. The only fundamental ontology in classical mechanics is the trajectory ${\bf x}(t)$, while everything else can be expressed in terms of that.

In the thermal interpretation of QM, on the other hand, there is no fundamental ontology from which everything else can be expressed. All observables are on the same footing. I find it very weird, especially if I look at the classical limit.

Well, all this gibberish about ontic or not is just empty erudition. It reminds me of the debate whether forces are "ontic" or not, and one philosopher hit the other, who claimed forces are fictitious asking, whether he still believes what he just said ;-)).

Then it's also a bit a quibble, why you consider only position as ontic. Maybe you can get through with this quite radical view within classical Newtonian mechanics, but is it sufficient in connection with fields and quantum theory? There spin is obviously an additional fundamental degree of freedom. It depends a bit on how you define observables, but following the symmetry-principle arguments usually used to decide about the proper description of observables in QT, spin is a phenomenon appearing in addition to the classical point-particle observables that indeed can be traced back to the notion of position (and trajectories in position space) as the fundamental observable.

 

[DarMM]

Well, it's not that simple unfortunately, at least not in the relativistic case. The macroscopic manifestation of spin is, e.g., ferromagnetism through the magnetic moments of the particles due to spin.

How is that different from what I said?

 

[vanhees71]

I understood you statement to say that spin is not a phenomenon related to the object but emergent in its interaction with a macroscopic system. The point is that spin is an additional generic degree of freedom, which is generically quantum.

 

[martinbn]

But I can describe the fluid itself mathematically, without referring to the velocity field. For instance I can talk about its density and pressure. What can I say mathematically about the electron itself, without referring to its wave function?

I am not sure how this is related! The point was not that the velocity field is all that is needed for the mathematical description, add to it density, pressure, temperature etc. The point was to distinguish the thing that exists and interacts with other things on one hand and the mathematical notions that are used to model it on the other. The examples were obviously not exhaustive, they were there to make @kith 's point even more clear. Although I thought that he had said everything perfectly clearly.

 

[ftr]

The point was to distinguish the thing that exists and interacts with other things on one hand and the mathematical notions that are used to model it on the other.

Very good. How do you do that? There is no way, you will always describe the "things" mathematically by what they do and how they affect other "things" mathematically, Once you reach the theory that is exact (even in equivalent forms), then the "things" ARE the mathematical objects. This is the conclusion of MUH.

 

[vanhees71]

Mathematics is the language to talk about physics. As any language it's a description of real-world objects but not the same as the objects themselves. I've never seen a Hilbert space or a self-adjoint operator, a tensor field, or a time derivative, in any lab on the world. There are accelerators (preparation devices) and detectors (measurement devices) around but no abstract mathematical objects!

 

[ftr]

Well, you also have an electron in the lab, but you don't actually see it, do you?

 

[Demystifier]

That's another quibble I have with de Broglie Bohm. How do you define "spin" as a classical variable.

I don't.

In the latter case, it's however not clear to me, in which sense the spin is then defined as an observable in the Bohmian interpretation to begin with, and then of course it's even harder to decide whether, within the Bohm interpretation, the spin has some determined value or not.

You are using a wrong language. An observable is a hermitian operator, so spin is an observable in BM in exactly the same way as it is in standard QM. The right question, with right terminology, is whether the spin is a beable? (The terminology is explained in my paper which you said you read.) The answer is that spin is not a beable in BM. Only positions are beables. That's perfectly OK to explain the measurement of spin because, as you more or less agreed, all measurements can be reduced to measurements of position. All this is explained in the paper you said you read.

 

[vanhees71]

There are no hermitian operators in the lab. There's real-world equipment to prepare objects (like an "electron") and to measure its properties (mass, charge, magnetic moment, $g-2$, etc.).

It's of course true that, e.g., the SG apparatus uses a position preparation to measure spin (or more precisely the magnetic moment associated with spin), but still it's not clear to me, how to define spin in BM. It doesn't help to use buzz words like "beable". You have to give operational definitions together with the mathematical description. Spin, e.g., is no problem in standard QT whatsoever, but I thought the aim of BM is a kind of reestablishment of classical notions though (according to what we know today unavoidable) including some additional non-local dynamics. Neglecting spin, that's not a problem since you have a classical notion of the classical observable, which are after all maybe derivable by position only.

 

[vanhees71]

Well, you also have an electron in the lab, but you don't actually see it, do you?

Sure, and electron is defined through what can be observed about it. It was found in gas discharge tubes and then its properties were carefully investigated. An electron is the collection of its unique properties, which are now all nicely formalized in a more or less simple mathematical scheme called the Standard Model of elementary particle physics, but the electron is not some quantized Dirac field living in a sloppily defined Fock space but it's an entitity measurable in the real world using all kinds of equipment in the lab.

 

[Demystifier]

There are no hermitian operators in the lab.

Of course. In the lab there are only perceptibles, which is also a term explained in the paper.

It's of course true that, e.g., the SG apparatus uses a position preparation to measure spin (or more precisely the magnetic moment associated with spin), but still it's not clear to me, how to define spin in BM. It doesn't help to use buzz words like "beable". You have to give operational definitions together with the mathematical description.

The operational definition of spin in BM is identical to the operational definition of spin in standard QM. You observe the position of the detector that clicked in the SG apparatus and from this you compute the spin from the standard formula.

 

[vanhees71]

Again this solidifies my view that there's no need for Bohmian mechanics to begin with. I don't need additional deterministic trajectories determined by a non-local pilot-wave concept to describe what's observed.

 

[Demystifier]

Again this solidifies my view that there's no need for Bohmian mechanics to begin with.

So effectively you reject Axiom 1 in my paper:
Axiom 1: All perceptibles are beables.

 

[A. Neumaier]

This is interpretation dependent.

In the Copenhagen interpretation, the spin is not real before the measurement.

I'd not use the word "real" here. It's misleading. According to QT in the orthodox interpretation (minimal interpretation, almost Copenhagen but without collapse and without "Heisenberg cut") observables are always "real" in the sense that you always have the oppotunity to measure them, independent of the state the system is prepared in.

I had meant the ''value of the spin'' (usually just called ''the spin''), and before the measurement.
This value does not exist. For me, real and existent are synonyms.

In the thermal interpretation, the spin is a real number and the measurement outcome discretizes it, hence is only approximate.

I don't understand, what you mean by "the measurement outcome discretizes" the spin. Any spin component takes discrete values $m \in \{-s,-s+1,\ldots,s-1,s \}$. That's the possible values you get when measuring this spin component accurately (enough). If your detector resolution is not sufficient, you get some continuous distribution around these values of course, but that's due to the detector resolution and has nothing to do with the observable "spin component" per se.

Yes, and this discrete response (a few well-separated blobs when $s$ is small) is what I refer to as discretizing. Note that in the thermal interpretation, the true value of a spin component is a real number, approximately found out by the measurement, with a discretization error of order $\hbar$, which is 1 in the units you are using.

The measurement by the Stern-Gerlach apparatus necessarily discretizes the outcome since the unitary dynamics of the magnet prepares the silver current in two discrete beams - already before measurement. The measurement therefore can do no more than find out which beam carries a particle, and hence gives an essentially binary response. Thus the continuous spin variable measured will produce only a single bit of information, and a large discretization error.

 

[A. Neumaier]

That's another quibble I have with de Broglie Bohm. How do you define "spin" as a classical variable.

There was already an answer in this thread:

From the Bohmian perspective it's indeed silly to call it measurement of spin. But Bohmians use such a silly language because that language is borrowed from standard QM (which is silly too, because standard QM says that spin doesn't exist before you measure it, so what does it mean to measure something which doesn't exist before measurement?). In other words Bohmians speak to "ordinary" physicists by saying something like this: The procedure that you call measurement of spin is really a measurement of position and I will tell you what is really going on when you think you measure spin.

 

[A. Neumaier]

I've never seen a Hilbert space or a self-adjoint operator, a tensor field, or a time derivative, in any lab on the world. There are accelerators (preparation devices) and detectors (measurement devices) around but no abstract mathematical objects!

This sounds like you'd agree to the quote from Asher Peres in another thread!

There are also no observables described by linear operators, but only positive operators describing the response of the detectors to the preparation devices.

 

[ftr]

Sure, and electron is defined through what can be observed about it. It was found in gas discharge tubes and then its properties were carefully investigated. An electron is the collection of its unique properties, which are now all nicely formalized in a more or less simple mathematical scheme called the Standard Model of elementary particle physics, but the electron is not some quantized Dirac field living in a sloppily defined Fock space but it's an entitity measurable in the real world using all kinds of equipment in the lab.

Suppose no humans ever existed would you say that the concept of a circle is non existent, I hope you wouldn't. Let's say humans came about and wrote an equation for a circle, would you make any real distinction between the two, i.e. the concept and the equation, I hope you wouldn't. Just think of the electron in the same way, after all you admit that only its properties which are mathematical describe it.

 

[A. Neumaier]

Suppose no humans ever existed would you say that the concept of a circle is non existent, I hope you wouldn't. Let's say humans came about and wrote an equation for a circle, would you make any real distinction between the two, i.e. the concept and the equation, I hope you wouldn't. Just think of the electron in the same way, after all you admit that only its properties which are mathematical describe it.

A circle and its equation is like your income and the corresponding bank account statement.

 

[ftr]

A corle and its equation is like your income and the corresponding bank account statement.

That is not true for the circle is it, both the equation( what does its jotting represent!) of the circle and the conceptual ethereal are the same, both mathematical in nature. The equations explain what matter properties and their interactions are which are mathematical. That makes sense because we only know of two things that exist, matter and mathematics(relations between numbers) and so science seem to strongly suggest that the former is really nothing but the later. And there is nothing else to explain what matter is, you can't explain matter with matter or matter with some unknown that will be ultra ultra metaphysics.

 

[vanhees71]

So effectively you reject Axiom 1 in my paper:
Axiom 1: All perceptibles are beables.

Can you translate this into understandable language?

 

[vanhees71]

I had meant the ''value of the spin'' (usually just called ''the spin''), and before the measurement.
This value does not exist. For me, real and existent are synonyms.Yes, and this discrete response (a few well-separated blobs when $s$ is small) is what I refer to as discretizing. Note that in the thermal interpretation, the true value of a spin component is a real number, approximately found out by the measurement, with a discretization error of order $\hbar$, which is 1 in the units you are using.

The measurement by the Stern-Gerlach apparatus necessarily discretizes the outcome since the unitary dynamics of the magnet prepares the silver current in two discrete beams - already before measurement. The measurement therefore can do no more than find out which beam carries a particle, and hence gives an essentially binary response. Thus the continuous spin variable measured will produce only a single bit of information, and a large discretization error.

In my understanding of the quantum formalism there are observables, i.e., measurable quantities. You can always measure these quantities, no matter in which state the system is prepared in. Whether or not a specific observable takes a determined value is completely described by the state. It takes a determined value if and only if with 100% probability you find one and only one value when measuring the observable. Otherwise it does not take a determined value, and the outcome of the measurement is irreducibly random. The only thing the state preparation tells you are the probabilities for the outcome of measurements of this observable.

A spin component is discrete. If measured accurately, you get one of the eigenvalues of the corresponding operator, i.e., it's in the discrete set $\{-s,-s+1,\ldots,s-1,s \}$. The SGA, if properly set up, entangles (almost perfectly) the position of the particle with the spin component in the direction given by the magnetic field. An unpolarized beam splits in $(2s+1)$ discrete partial beams because the measured spin component is discrete. That's why you can use the SGA as (almost perfect) von Neumann filter measurement of the corresponding spin component. It's not the expectation value of the spin component what's measured, as you seem to claim in your "thermal interpretation" but the resolution of the SGA is sufficient to accurately resolve the discrete values of the spin components. The measurement of the magnetic moment of the electron is among the most accurate fundamental quantities ever meausured (though of course not with a simple SGA of course).