# A The thermal interpretation and measurement

#### vanhees71

Gold Member
[Moderator's note: Spin off from previous thread since this is a separate topic from that one.]

But the thermal interpretation is not (yet?) a standard interpretation. I do agree that what you say would likely be true of an interpretation that solves the measurement problem (eg. maybe something like Bohmian Mechanics or the thermal interpretation, but that is also not a standard interpretation at this time).
It's hard to expect that it ever gets a standard interpretation, because it contradicts the very observations which lead to the discovery of modern quantum including Born's probability interpretation of the wave function (i.e., pure states): We do not find smeared distributions when we deal with single quanta (particles or photons) nor do we measure expectation values, if the measurement device is not constructed such that it does so. We rather measure what QT predicts: E.g., when measuring the position of an electron, prepared in some state, we find single pointlike hits (within the spatial resolution of the detector), with the single outcomes being random and distributed according to the probabilities predicted by QT (within the spatial resolution of the detector again).

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#### A. Neumaier

The standard interpretation is still one of the Copenhagen flavors, usually without the collapse postulate.

The standard interpretation has collapse. To see this, look at 20 different textbooks and make a measurement of the occurrence of the collapse in the postulates.
But the thermal interpretation is not (yet?) a standard interpretation.
It's hard to expect that it ever gets a standard interpretation, because it contradicts the very observations which lead to the discovery of modern quantum including Born's probability interpretation of the wave function (i.e., pure states): We do not find smeared distributions when we deal with single quanta (particles or photons) nor do we measure expectation values, if the measurement device is not constructed such that it does so. We rather measure what QT predicts: E.g., when measuring the position of an electron, prepared in some state, we find single pointlike hits (within the spatial resolution of the detector), with the single outcomes being random and distributed according to the probabilities predicted by QT (within the spatial resolution of the detector again).
You still misunderstand the thermal interpretation. It is consistent with all measurements that are consistent with QM since it makes partially weaker claims than the standard interpretation.

Instead of claiming that an eigenvalue is measured, the thermal interpretation claims only that the measurement result agrees with the q-expectation up to an error of the order of the associated uncertainty. There is always an eigenvalue deviating at most by the uncertainty, and the eigenvalues appearing with large probability deviate at most a few times the uncertainty. Thus a measured eigenvalue usually satisfies the claim of the thermal interpretation.

That Born's rule claims more, namely in its almost everywhere stated form that an exact eigenvalue is measured, does not constitute a disproof of the thermal interpretation.

On the contrary, experiment disproves this form of Born's rule whenever one measures an observable with irrational eigenvalues.

A relaxed Born's rule that claims to measure eigenvalues only approximately is better established
but is not even enough to justify the rule for calculating expectations, as it would give this rule only in some approximation.

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#### vanhees71

Gold Member
Measurements do in general NOT agree with the q-expectation values within some error, but they measure observables with an error determined by the measurement device and not by the quantum state of the measured system. That's an old misunderstanding, and it doesn't help to relabel it as if it were a revolutionary new "thermal interpretation".

Born's rule is the fundamental starting point for all descriptions of measurements using QT, including the POVM formalism. Of course, for a real-world experiment you have to analyze the statistical and systematical errors of the instrument used, but these are properties of the instrument, not of the measured objects!

#### A. Neumaier

they measure observables with an error determined by the measurement device
And this error is determined based on a convention what the true value to be approximated by the measurement is. Tradition and the thermal interpretation radically differ in this respect.

According to the thermal interpretation, measuring two spots with high accuracy is still a low accuracy measurement of the spin. It is like claiming to have measured the position of a car to mm accuracy by measuring the tip of the antenna, although the position of the car is defined to much less accuracy only.

#### A. Neumaier

There is no classical counterpart of spin. Spin is generically quantum, but that's semantics.
This is not true. Classical spin is given by the coadjoint representation of SO(3) on the Bloch sphere.
Its (geometric) quantization naturally produces the spin s representations for half-integral s.

See, e.g., Section 2 of the paper

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#### vanhees71

Gold Member
And this error is determined based on a convention what the true value to be approximated by the measurement is. Tradition and the thermal interpretation radically differ in this respect.

According to the thermal interpretation, measuring two spots with high accuracy is still a low accuracy measurement of the spin. It is like claiming to have measured the position of a car to mm accuracy by measuring the tip of the antenna, although the position of the car is defined to much less accuracy only.
An error is not based on a convention but has to be measured itself. It's not always progress if a claim radically differs from tradition.

#### A. Neumaier

An error is not based on a convention but has to be measured itself.
To be able to talk about error you need a concept of the true value; the error is the difference. This true value is an eigenvalue in the traditional interpretations but the q-expectation in the thermal interpretation.
The thermal interpretation follows another tradition - the overwhelmingly useful tradition that the best estimate of the true value of something behaving randomly is its sample expectation. Thereby it restores the closeness of quantum thinking to classical thinking instead of making it radically different.

#### vanhees71

Gold Member
I can only repeat that this reveals that the "thermal interpretation" is in contradiction with observation. The very fact that QT has been discovered proves this. I've no clue how you can come to the idea that a measurement device always measures a quantum-theoretical expectation value. Again: What and how accurately a measurement device measures depends on the properties of this device and not on the preparation of the measured object. If you want to test the probabilistic predictions of QT you need a device which measures the observable under discussion with a much larger precision than the predicted standard deviation of the measured observable given the prepared state of the system. E.g., if you want to measure mass and width of a resonance in particle physics the mass resolution of the detector has to be much better than this resonance width.

#### A. Neumaier

Again: What and how accurately a measurement device measures depends on the properties of this device and not on the preparation of the measured object.
Again: To argue that the accurate position measurement of a spot is an accurate measurement of a spin depends on the convention what to call a measurement of a spin, and this convention is different in the thermal interpretation. Thus no contradiction to experiment arises, but only your misunderstanding of my claim.
E.g., if you want to measure mass and width of a resonance in particle physics the mass resolution of the detector has to be much better than this resonance width.
But we are discussing spin. You are telling me about a device for accurate position measurements.
I agree that it provides accurate position measurements.

But where is the proof that this accurate position measurement is an accurate spin measurement? There is no proof, only a claim based on convention. Conventions can be altered without altering the experimental record, and then provide a different interpretation

#### vanhees71

Gold Member
The value an observable takes is not mere convention. What's convention is the choice of the units. I referred to the SG experiment since you argued with it. There's a plethora of much more accurate measurements of the spin of e.g., the electron (or rather its magnetic moment) like the Zeeman effect, which led to the discovery of the electron spin through Goudsmit and Uhlenbek. There was even some famous confusion about the gyrofactor of 2 and the Thomas-Precession factor of also 2, and only after Thomas's clarification the explanation of Goundsmit and Uhlenbek was accepted. If the quantitative values were a mere convention, nobody had cared about this apparent problem.

In the SG experiment you can also determine with some accuracy the magnetic moment of the electron. Of course the necessary analysis needs the theory to make the quantitative connection between the involved quantities, i.e., the relation between the measured spatial deflection of the silver atoms, the (gradient of) the magnetic field etc. This, however, also doesn't imply that the quantitative results are just convention.

#### A. Neumaier

much more accurate measurements of the spin of e.g., the electron (or rather its magnetic moment)
A measurement of the magnetic moment of an electron is not a measurement of the electron spin but the measurement of a parameter related to an interaction term in its Hamiltonian which determines the angle between the beams generated by the Stern-Gerlach magnet.

#### vanhees71

Gold Member
Yes, and the value of this "parameter" is not mere convention but an definite outcome of a measurement. Measurement means comparison with defined units. Once the units are defined the numerical values being measured have a definite meaning and are not conventions!

The fundamental relation between angular momentum (admittedly indeed only total angular momentum is observable, while the separation in orbital and spin angular momentum parts is only approximately possible in the non-relativistic limit, but let's discuss within non-relativistic QM first) is clear from again other experiments, e.g., the Einstein-de Haas experiment. It's a quite funny example for the fact that experimenters should not trust theorists too much, claiming they know, "what must come out". Indeed, de Haas found in measuring what's called the gyro-factor of the electron something not clearly to be 1 but something larger, but Einstein convinced him that it must be 1 according to the then only known model of magnetic moments at the time (around 1915, if I remember right), namely the Amperian idea of "molecular currents". The experiment was not that accurate, so de Haas got convinced he had to abandon his "wrong results". Though Einstein was wrong in this case, it clearly shows that indeed the quantitative outcomes of measurements of spin (angular momentum) and magnetic moments and its relation are not mere conventions but experimentally clearly measurable quantitative properties of matter. The mistake was soon corrected by other experimentalists. More accurate measurements show that indeed the main microscopic source of the magnetic field of conventional permenant magnets is the electron spin with a gyrofactor of about 2.

#### A. Neumaier

Yes, and the value of this "parameter" is not mere convention but an definite outcome of a measurement.
Of course. but it is not a measurement of spin, which is what is currently under discussion.
Already the units of spin and magnetic moment are different.

Moreover, the spin of both electron and proton is represented by the same operator vector $S=\frac12\hbar \sigma$, but they have very different magnetic moments.

#### A. Neumaier

In the SG experiment you can also determine with some accuracy the magnetic moment of the electron.
The determination of a magnetic moment by means of a Stern-Gerlach setting is completely independent of the notion of measuring spin. One has a Hamiltonian with an interaction term involving an unknown gyrofactor $\gamma$, one solves the Schrödinger equation for this Hamiltonian with initial state corresponding to a particle in a narrow beam and finds that the solution is a superposition of states located within two narrow beams whose angle $\alpha=a(\gamma)$ is determined by $\gamma$. This is unitary dynamics; no measurement (hence no controversial interpretation) is involved. Stern and Gerlach called this ''quantization of direction'' - not of spin!

The remainder is completely classical: Measure accurately the angle $\alpha$ between two beams and determine $\gamma$ by solving the equation $\alpha=a(\gamma)$. Neither Born's rule is involved nor any analysis in terms of eigenvalues of a component of the spin operator.

Thus the determination of a magnetic moment is completely independent of what is assumed about measuring a component of spin. Thus you cannot sensibly argue with it against the thermal interpretation, which assumes something different than tradition.

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#### PeterDonis

Mentor
Previous discussion of the TI and measurement, in the context of Stern-Gerlach, is here:

#### A. Neumaier

There is no classical counterpart of spin. Spin is generically quantum, but that's semantics.
This is not true. Classical spin is given by the coadjoint representation of SO(3) on the Bloch sphere.
Its (geometric) quantization naturally produces the spin s representations for half-integral s.

See, e.g., Section 2 of the paper
I replaced the link to one to a publicly available copy. A more general discussion of classical and quantum spin is in

#### vanhees71

Gold Member
I've looked at the introductory part. Of course, this is all well known, but the BMT equation is for homogeneous em. fields only. The classical equivalent of the Stern-Gerlach setup is not covered by this. The full theory is everything else than trivial. Here's a recent paper by some colleagues of mine:

https://arxiv.org/abs/1902.06513

#### A. Neumaier

I've looked at the introductory part. Of course, this is all well known, but the BMT equation is for homogeneous em. fields only. The classical equivalent of the Stern-Gerlach setup is not covered by this. The full theory is everything else than trivial. Here's a recent paper by some colleagues of mine:

https://arxiv.org/abs/1902.06513
In (4) (or, equivalently, in (7) and (10)) you can take the classical limit $\hbar\to0$ and find classical equations for a spinning particle. Thus spin is not intrinsically quantum, though it was discovered in a quantum context. Classical spin can even be modeled in genereal relativity on spin manifolds:

"The thermal interpretation and measurement"

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