I Many measurements are not covered by Born's rule

[A. Neumaier]

Why do you have what you call a "Thermodynamic Interpretation"?

My ''thermal interpretation'' is an interpretation of quantum mechanics in general, and the measurement problem in particular. I developed this since none of the present interpretations gives an interpretation fully compatible with the actual practice of using quantum mechanics - where many things - such as spectral lines, Z-boson masses, or electric fields) are measured that are impossible to capture with the Born interpretation underlying all previous interpretations.

Well, all these quantities are measured with the corresponding measurement devices like spectrometers, particle detectors (for the Z-boson mass and width you measure dilepton spectra in various ways), etc. you find in the physics labs around the world. In physics in fact quantities are defined by giving appropriate (equivalence classes of) measurement protocols to quantitatively observe them. That's why they are called observables after all. Also there is nothing more needed concerning the application of the quantum-theoretical formalism (e.g., formulated as the representation of an observable algebra on Hilbert space, based on various symmetry principles which themselves are discovered by observation of conservation laws) than Born's rule, i.e., the minimal interpretation.

1. Within the framework of a Hilbert space for an atom one cannot find an observable in the sense of ''self-adjoint Hermitian operator'' that would describe the measurement of the frequency of a spectral line of the atom. For the latter is given by the differences of two eigenvalues of the Hamiltonian, not by an eigenvalue itself, as Born's rule would require.

2. The measurement of a Z-boson is not a simple, essentially instantaneous act as the Born rule would require but a complex inference from a host of measurements on decay products. That the Born rule is used implicitly to derive the rules for working with an S-matrix does not make it applicable to the measurement of a Z-boson itself.

3. Electric field operators $E(x)$ are not Hermitian operators to which eigenvalues could be associated but oerator-valued distributions. Thus Born's rule cannot even be applied in principle. The measurement of an electric field already involves an averaging process and really measures an expectation, not an eigenstate.

Where you need something like a "thermal interpretation" is when it comes to understand the overwhelming success of classical physics (including classical relativistic and non-relativistic mechanics, electrodynamics, and thermodynamics) to describe macroscopic systems. Here you need some coarse graining to describe macroscopic effective (relevant) degrees of freedom as (spatio-temporal) averages over many microscopic degrees of freedom.

And this is already needed to explain why we can measure electric fields.

 

[vanhees71]

1. Within the framework of a Hilbert space for an atom one cannot find an observable in the sense of ''Hermitian operator'' that would describe the measurement of the frequency of a spectral line of the atom. For the latter is given by the differences of two eigenvalues of the Hamiltonian, not by an eigenvalue itself, as Born's rule would require.

What you measure is the energy of the emitted photon, and that's represented by a self-adjoint operator in QED. Of course, to get a clear spectral line you have to measure a lot of photons to get a clear spectral line. As any probabilistic measurement you "collect a lot of statistics" to get a sharp line.

2. The measurement of a Z-boson is not a simple, essentially instantaneous act as the Born rule would require but a complex inference from a host of measurements on decay products. That the Born rule is used implicitly to derive the rules for working with an S-matrix does not make it applicable to the measurement of a Z-boson itself.

The Z boson was discovered in the UA1 experiment by Rubbia et al in reactions like $p+\bar{p} \rightarrow \mathrm{e}^+ + \mathrm{e}^- \quad \text{or} \quad \mu^++\mu^-$:

http://cerncourier.com/cws/article/cern/28849

It's a nice example for measuring a cross section in particle physics or, in more theoretical words, the corresponding S-matrix elements. Although for sure not "simple" technology wise, the principle is very simple: You make a lot of scattering experiments and collect statistics for the reaction channel of interest. After all the cross section is a typical probabilistic quantity as defined by the formalism of QFT. As you say, it directly uses the definition of S-matrix elements, which of course is based on the standard (minimal) interpretation of QFT.

3. Electric field operators $E(x)$ are not Hermitian operators to which eigenvalues could be associated but oerator-valued distributions. Thus Born's rule cannot even be applied in principle. The measurement of an electric field already involves an averaging process and really measures an expectation, not an eigenstate.And this is already needed to explain why we can measure electric fields.

Sure, but where is a contradiction here to the standard minimal interpretation of QT? QT tells you how to calculate the pertinent expectation values. For an electromagnetic field, it's in both quantum and classical field theory a time average of the field-energy density ("intensity"): $\left \langle \frac{1}{2}(\vec{E}^2+\vec{B}^2) \right \rangle$.

How to evaluate averages in the QT formalism for given (pure or mixed) states is derived from Born's rule. There's no additional assumption going into that. So I don't see, where you need more interpretation than the minimal one to analyse/describe any of the above examples within the QT formalism than the minimal interpretation (aka Born's rule).

 

[A. Neumaier]

What you measure is the energy of the emitted photon, and that's represented by a self-adjoint operator in QED.

No. What one measures is the presence of photons in a particular region of a spectral resolution defining the peak. Their energy is usually not measured - at least not in simple measurements of spectral lines in a prism-generated spectrum, which surely counts as a measurement.

to get a clear spectral line you have to measure a lot of photons to get a clear spectral line.

But a Born rule measurement is by the usual textbook definition a single measurement, which in the situation you describe would be a single photon impact. After averaging over many cases, you are measuring no longer eigenvalues but expectations, and Born's rule does not apply to such a measurement, whereas the thermal interpretation now applies.

the cross section is a typical probabilistic quantity as defined by the formalism of QFT.

Yes, but even a cross section is not a Born measurement - there is no Hermitian operator that is measured, but traces of particles (collision products), from which one determines their nature and deflection angle. It is very far from a Born rule measurement, which according to Wikipedia states (in this case reliably):

if an observable corresponding to a Hermitian operator A with discrete spectrum is measured [...] then

• the measured result will be one of the eigenvalues of A and
• the probability of measuring a given eigenvalue will equal [...]

To which operator do you apply this rule when you measure a cross section?? I never saw a Hermitian operator with the meaning of ''cross section''.

Note that I do not deny that QFT predicts cross sections correctly, but this is done using shut-up-and-calculate, and not using the Born rule! The latter is only used to motivate the unspoken folklore behind shut-up-and-calculate. The thermal interpretation spells out what is actually used.

The point is that very few measurements qualify as Born measurements; most real measurements are only very indirectly related to eigenvalues of Hermitian operators, and involve lots of other measurements and computations to get what is then referred to as a measurement. And the measurement of the mass of the Z-boson is even further away from Born's rule than a cross section.

How to evaluate averages in the QT formalism for given (pure or mixed) states is derived from Born's rule. There's no additional assumption going into that.

Born's rule is only a claim about measurements, with an implicit claim that all measurements fall under this rule. (It is precisely the latter claim that is conspicuously wrong, and that I am addressing in this thread.) The derivation of the rule for evaluating averages involves an element of handwaving since one has to imagine that one is performing unperformed measurements that are subsequently averaged. Worse, field expectations involve operators with a continuous spectrum, and for the derivation of the expectation rules one has to pretend that each of these unperformed experiments is done to infinite precision - since otherwise there would be extra terms coming from the measurement errors.

But unperformed experiments have no results, hence unperformed measurements have no results. Thus Born's rule doesn't apply to them, and one cannot average them. Except in a handwaving argument...

 

[A. Neumaier]

where is a contradiction here to the standard minimal interpretation of QT?

There is no contradiction, but the standard minimal interpretation is not really minimal since to lead to the quantum mechanical practice, it has to amend Born's rule by unspoken ad hoc rules for unperformed measurements over which to average, to get the rules that are really applied.

If you study your own work form the point of view of where you use foundations, you'll find that you almost never apply Born's rule!
But you make a lot of use of what I formulated as basic principle in my thermal interpretation!
In a sense, my thermal interpretation is far more minimal than Ballentine's minimal interpretation since unlike his, mine does not require anything about the measurement process, which is a complex subject of statistical mechanics, hence should not be built into the foundations!

 

[vanhees71]

I don't understand this. I apply Born's rule all the time when I calculate S-matrix elements or in-medium spectral functions or whatever from QFT. I also don't know, what you mean by "unperformed measurements". In the usual accelerator experiments, e.g., you count a lot of particles in a lot of scattering events binning them in energy and momentum. So you measure the kind of the particles and their energy and momentum, and the experimentalists make statistics, systematic error estimates, and so on which then can be compared to the outcome of the theoretial calculations we do using Q(F)T. These calculations are all standard calculations, using Born's rule. What else should they be?

Born's rule is just saying that, given a state $\hat{\rho}$ (statistical operator) the probability to find the result $a$ when measuring an observable $A$ is
$$P(a|\rho)=\sum_{\beta} \langle a,\beta|\hat{\rho}|a,\beta \rangle,$$
where $\beta$ labels probable degeneracies of the eigenvector of the associated operator $\hat{A}$ to the eigenvalue/spectral value $a$, and from this fundamental rule any other (probabilistic) outcomes encoded in S-matrix elements, expectation values, standard deviations, etc. etc. is derived.

 

[A. Neumaier]

I don't understand this. I apply Born's rule all the time when I calculate S-matrix elements or in-medium spectral functions or whatever from QFT.
Born's rule is just saying that, given a state $\hat{\rho}$ (statistical operator) the probability to find the result $a$ when measuring an observable $A$ is
$$P(a|\rho)=\sum_{\beta} \langle a,\beta|\hat{\rho}|a,\beta \rangle,$$
where $\beta$ labels probable degeneracies of the eigenvector of the associated operator $\hat{A}$ to the eigenvalue/spectral value $a$, and from this fundamental rule any other (probabilistic) outcomes encoded in S-matrix elements, expectation values, standard deviations, etc. etc. is derived.

In the usage in your first paragraph, Born's rule is just a metaphor for the quantum formalism, and not the rule you state in your second paragraph. You apply in the first paragraph not Born's rule (as defined in the second paragraph) but things that you regard as being its consequences. But it is a long way from the formula you called the Born rule to the interpretation of the S-matrix in QFT as giving observable transition rates. I had asked you

To which operator do you apply this rule when you measure a cross section?? I never saw a Hermitian operator with the meaning of ''cross section''.

and didn't get an answer. Please take the time to figure out to which measurements - in the long chain of events leading to obtaining an experimental scattering cross section, i.e., to have measured the cross section - the Born rule in the form you wrote it, in terms of operators and their eigenvalues, really applies!

I also don't know, what you mean by "unperformed measurements". In the usual accelerator experiments, e.g., you count a lot of particles in a lot of scattering events binning them in energy and momentum. So you measure the kind of the particles and their energy and momentum

But the derivation of the interpretation of the meaning of the scattering amplitudes from the Born rule does not involve any of these measurements of particle tracks; instead it involves results of a lot of other, unperformed measurements.

 

[vanhees71]

Again, you measure cross sections by performing a lot of collision experiments identifying particles and binning them in momentum bins. So you basically measure the particle type (PID) and its momentum and make statistics over these events. This is what's measured according to Born's rule. What else should it be?

Of course, one example of how you perform such a measurement is to look at particle tracks in a magnetic field using a cloud chamber. It's well understood since 1928 how these tracks come about and that they can interpreted in terms of the momentum (extracted from the curvature of these tracks) in terms of QT, where also Born's rule (or just the standard QT formalism as a whole, if you wish, is applied). I don't see, where any scattering experiment is at odds with standard QT in minimal interpretation. If it were, we already had adapted QT to something else, but that's simply not necessary since the standard formalism works well so far.

 

[A. Neumaier]

Again, you measure cross sections by performing a lot of collision experiments

... and the final measurement result (the cross section) is not the eigenvalue of a Hermitian operator measured but a complicated function of other measurements. Thus matching theory to measured cross sections is not covered by Born's rule, which only tells how to match theory to measured Hermitian operator.
Thus Born's rule may apply to measuring the momentum of a particle (which has an associated operator), but it does not apply for measuring a cross section (which apparently has none).
Note that Born's rule says nothing about how to measure anything but only how the results of the measurement are related to the predictions given the state. I am asking for the justification for the latter when measuring the cross section, while you are answering about the former.

I don't see, where any scattering experiment is at odds with standard QT in minimal interpretation.

I never claimed any discrepancy. I just claim that though much lip service is paid to the Born rule, it is rarely applied. Even momentum measurements don't follow strictly Born's rule, as one never gets an exact value for the momentum but only an approximation, with an error determined by the experimental procedure and not by the state of the measured particle.

But my main criticism is about the derivation of the meaning of the S-matrix elements, which involves measurements never performed. The measurements that you describe, those of the momentum of particles, don't go into the derivation of the meaning of the S-matrix elements as describing transition rates - they only go into the experimental procedure to compare the meaning with. To derive the meaning one must instead use imagined Born-type measurements that are never performed!

 

[vanhees71]

Ok, then we agree to disagree.

 

[A. Neumaier]

I apply Born's rule all the time when I calculate S-matrix elements or in-medium spectral functions or whatever from QFT.

When you calculate something you are in shut-up-and-calculate mode and do not apply any interpretational axiom, in particular you don't apply Born's rule.

You apply the latter at best when you interpret your S-matrix element as a probability amplitude. But even this is not a straightforward application of Born's rule in any of its standard formulations (given, e,g, by wikipedia). The amplitude is neither a probability of finding a particle at a given position (Born's original definition of the rule) nor does its generalization by von Neumann that a measured value is one of the eigenvalues of the associate operator, apply.

Therefore the probabilistic interpretation of quantum mechanics is much more general than Born's rule. The latter just illustrates its application to certain very simple situations that require no theory and hence are suitable for an introductory textbook motivation of the quantum formalism.

 

[vanhees71]

When you calculate something you are in shut-up-and-calculate mode and do not apply any interpretational axiom, in particular you don't apply Born's rule.

You apply the latter at best when you interpret your S-matrix element as a probability amplitude. But even this is not a straightforward application of Born's rule in any of its standard formulations (given, e,g, by wikipedia). The amplitude is neither a probability of finding a particle at a given position (Born's original definition of the rule) nor does its generalization by von Neumann that a measured value is one of the eigenvalues of the associate operator, apply.

Therefore the probabilistic interpretation of quantum mechanics is much more general than Born's rule. The latter just illustrates its application to certain very simple situations that require no threorty and hence are suitable for an introductory textbook motivation of the quantum formalism.

Ok, we agree to disagree.

 

[tom.stoer]

Let me ask a question to see if I understand your ideas correctly:

Hendrik is claiming that the analysis of every measurement is based on Born's rule (or some mathematical generalization), whereas Arnold is claiming that many measurements are not related to Borns rule, at least not directly. Therefore he asks for the self-adjoint operators A to calculate expection values like

$\omega_{if} = \text{tr}\, \rho A$

$\frac{d\sigma}{d\Omega} = \text{tr}\, \rho A$

$M_{Z^0} = \text{tr}\, \rho A$

Right?

 

[vanhees71]

I never claimed that any measurement is described by Born's rule. What I, however, claim is that the only so far success interpretation of the state in QT is Born's probabilistic interpretation, i.e., Born's rule.

I don't know, what $\omega_{if}$ is. So I can't comment about this.

$\mathrm{d}\sigma/\mathrm{d} \Omega$ is a (differential) cross section and derived from $S$-matrix elements, which directly uses Born's rule. You prepare very often the state of two asymptotic free particles with quite definite momenta (sometimes with somewhat definite polarization, if applicable and of interest) and measure with which frequency you find a certain final asymptotic free state of (the same or other) particles. Then you need $|S_{fi}|^2$ in accordance with Born's rule.

If $M_{Z_0}$ is the mass of the $Z_0$ then it's of course not an observable. It's defined as the real part of the corresponding pole of the Green's function in the corresponding channel of, e.g., lepton-antilepton scattering, and that's how it was finally discovered by Rubbia et al at CERN in the 1980ies.

 

[PeterDonis]

I never claimed that any measurement is described by Born's rule

This doesn't make sense to me. Born's rule says that if you make a measurement, the possible results are the eigenvalues of the measurement operator, and the probability of getting a particular eigenvalue is the squared modulus of the amplitude of the corresponding eigenvector in the expansion of the state in the appropriate basis. So how can measurements not be described by Born's rule?

 

[vanhees71]

Ok, for me Born's rule is the 2nd part only. As Arnold pointed out there are more general than ideal measurements, described in terms of socalled positive operator-valued measures (POVMs). This describes the imprecise measurement of one or several observables. Of course, it's just applying standard QT and thus Born's rule in the one or the other way to define them as a description of a physical measurement procedure in the lab.

 

[PeterDonis]

for me Born's rule is the 2nd part only

That doesn't make sense to me either, since the first part is a prerequisite for the second part. If your measurement result might not be an eigenvalue at all, then it makes no sense to apply the second part to predict the probabilities, since those probabilities add up to 1 and therefore if the second part is correct it is impossible for the result to not be an eigenvalue.

 

[A. Neumaier]

it's just applying standard QT and thus Born's rule

Thus you (unlike anyone else) call Born's rule everything that is based on standard QT, i.e., all the shut-up-and-calculate stuff. This makes the reference to Born's rule obsolete as as you effectively say everything is just applying standard QT.

 

[vanhees71]

One last time. Born's rule is the probability interpretation of the state, and it's indeed standard QT. The probability to find by measuring precisely the observable $A$, represented by the self-adjoing operator $\hat{A}$ with eigenspaces spanned by a (generalized) CONS $|a,\beta \rangle$
$$P(a)=\sum_{\beta} \langle a,\beta|\hat{\rho}|a,\beta \rangle,$$
implying the rule to evaluate expectation values
$$\langle A \rangle =\mathrm{Tr} (\hat{\rho} \hat{A}).$$
The rest of standard QT are the kinematical and dynamical rules associating the corresponding operators to observables and states.

 

[A. Neumaier]

Born's rule is the probability interpretation of the state

In your notation, Born's rule says that when you measure $A$ in the state $\rho$ you get the value $a$ with probability $\langle P(a)\rangle$, nothing else.

If you can deduce something from this rule it is a consequence of Born's rule but not Born's rule itself. Conflating a statement and its consequences by referring to it by the same name is utterly misleading!

In the present case, what you can conclude from Born's rule is that if you measure the observable $A$ in a large number $N$ of independently prepared realizations of the same experiment and then calculate the sample expectation value $\bar a =N^{-1}\sum_{k=1}^N a_k$ of the individual measurements $a_k$, it approaches the ensemble expectation value $\langle A\rangle$ with an accuracy of $O(N^{-1/2})$ times the magnitude of a typical value. The sample expectation value generally does not equal the ensemble expectation value.

Thus one cannot deduce from Born's rule the shut-up-and-calculate formula $\langle A\rangle=Tr(\rho A)$ but you need the latter as input to be able to formulate an exact asymptotic prediction of a large number of measurements. What you call a derivation is at best a heuristic motivation for the adoption of this formula into the shut-up-and-calculate toolbox.

Born's rule is silent about the value of the measured mass of a single brick of iron. Here $N=1$ in the above formula. The values can take any of an astronomically large number of values, and the Born probability of measuring any of these is extremely tiny. Since only a single measurement is made, the above derivation based on the law of large numbers does not apply.

But the rules of my thermal interpretation of quantum mechanics (i..e., the conventional interpretation in statistical thermodynamics) apply without any mumbling about eigenvalues or eigenvectors or probabilistic interpretation of the state. And in each single macroscopic measurement, they agree to high accuracy with experiment!

 

[vanhees71]

In probability theory given the probabilities as you said above the expectation value by definition is
$$\langle A \rangle = \sum_a P(a) a = \sum_{a,\beta} \langle a,\beta |\hat{\rho}| a,\beta \rangle a = \sum_{a,\beta} \langle a,\beta |\hat{\rho} \hat{A}|a,\beta \rangle =\mathrm{Tr} (\hat{\rho} \hat{A}).$$
In a real experiment, as you write, I make a large number of equal preparations and should get, according to the law of large numbers in the limit of an infinitely large ensemble $\overline{a}=\langle A \rangle$ (provided indeed that $P(a)$ are the probabilities to find a value $a$ when measuring $A$). I don't understand what you want to say when claiming the expectation value doesn't equal itself. Are you redefining the basic definitions of equality here?

Also we discuss QT and its foundations, and I stick to the standard formulation in (rigged Hilbert space) with operators representing states and observables within the statistical interpretation of the meaning of the state. You claim to need a much more vague "thermodynamical interpretation", and you still don't give a clea meaning of what "expectation value" means, at the same time forbidding to use the standard formulation in terms of probability theory.

The mass of a piece of iron is of course a parameter of that many-body system that can be measured quite easily by weighing it. Where should there be a measurement problem at all?

 

[A. Neumaier]

given the probabilities as you said above the expectation value

Your formula defines abstract ensemble expectation values (for an infinite ensemble of measurements). Sample expectation values (from measurements actually taken) are something quite different. Any course on statistics worthy its salt distinguishes between the two, as their properties are different. Just as the probability of casting a six is usually different from the relative frequency of sixes in an actual sequence of die throws. The exact connection between the two is that the (theoretical) probability is the ensemble expectation of the (measurable) relative frequency.

Sample expectation values are only approximations to ensemble expectation values, valid with an accuracy that scales only like $O(N^{-1/2})$ with the number $N$ of independent samples taken, hence say nothing at all for small $N$. Born's rule makes statements only about sample expectation values, while the ensemble expectation value (which does not involve any measurement issue) is a purely mathematical construct.

In particular, sample expectation values say nothing for sample size $N=1$, but the latter is the case in all macroscopic measurements.

The mass of a piece of iron is of course a parameter of that many-body system that can be measured quite easily by weighing it. Where should there be a measurement problem at all?

In quantum field theory it is not a parameter but one of the observables given by a Hermitian observable $M=mN$, where $N$ is an operator defined as integral over some quadratic expression in the fields. Its measurement therefore should be subject to Born's rule if the latter were the universal rule for measuring observables.

 

[vanhees71]

It's well known, that the expectation values of samples are "only" accurate to $\mathcal{O}(1/\sqrt{N})$, but why should this invalidate the standard probability interpretation of the state? Usually experimentalists try to "collect sufficient statistsics" to get a result at a "sufficient confidence level" (in HEP a discovery needs at least $5 \sigma$).

In the Standard Model the masses of the elementary particles are input parameters. There's no theory that predicts their values. Perhaps I don't understand what you mean. So what's $N$ concretely (for, e.g., a spin-0 particle to keep it simple). Of course you can measure the mass of particles in scattering experiments or by other means. That can be very difficult as the example of neutrinos show, but that's not a principle argument against the standard probabilistic interpretation of the quantum state either.

 

[A. Neumaier]

It's well known, that the expectation values of samples are "only" accurate to $\mathcal{O}(1/\sqrt{N})$, but why should this invalidate the standard probability interpretation of the state? Usually experimentalists try to "collect sufficient statistsics" to get a result at a "sufficient confidence level" (in HEP a discovery needs at least $5 \sigma$).

In the Standard Model the masses of the elementary particles are input parameters. There's no theory that predicts their values. Perhaps I don't understand what you mean.

I was considering a brick of iron in nonrelativistic quantum mechanics. Here $m$ is a constant (that can be taken from a book) but $N$ is a 1-particle operator whose value depends on the size of the brick. Measurement gives a single value ($N_m=1$) for the mass $M=mN$, no statistics is available, and no sample expectation value can be taken. Where should the probability interpretation come in?

 

[vanhees71]

If I'm not allowed to measure the mass, I can't measure it. I would weigh it, but without statistics, it's impossible to make a valuable measurement analysis ;-)).

 

[A. Neumaier]

If I'm not allowed to measure the mass, I can't measure it. I would weigh it, but without statistics, it's impossible to make a valuable measurement analysis ;-)).

Engineers measuring the weight of a brick measure once, or at most very few times. No statistics is needed since the value is anyway quite accurate.

 

[cosmik debris]

Engineers measuring the weight of a brick measure once, or at most very few times. No statistics is needed since the value is anyway quite accurate.

Doesn't this rely on many years of statistical measurements to calibrate the instrument to an acceptable accuracy?

Cheers

 

[A. Neumaier]

Doesn't this rely on many years of statistical measurements to calibrate the instrument to an acceptable accuracy?

The time and effort to develop a good measuring instrument does not matter for the fact that afterwards, the value of a macroscopic observable of a new quantum system (the brick of iron) is determined by a single measurement.

Note that also in Born's rule, the probability of measurement results is independent of the history of the measuring instrument. So why should you want to invoke it suddenly in a different situation?

 

[A. Neumaier]

Neither do you find precisely the value $\pm \hbar/2$ claimed to be measured by Born's rule. For in spite of many thousands of measurements of Stern-Gerlach type, Planck's constant $\hbar$ is still known only to an accuracy of 9 decimal digits.

Thus Born's rule is a fiction even in this standard textbook example!

The uncertainty of $\hbar$ is not fundamental but a technical problem, which will be solved next year or so by fixing its value, using either a Watt balance or a silicon ball. Then $\hbar$ will be exact as is the value of $c$ already since 1983. All this has absolutely nothing to do with any interpretation issues about QT!

So Born's rule was not valid in the past, and its validity depends on the choice of units?? This would be the only instance in physics where something depends in an essential way on units...

But there are problems with the experiment even when $\hbar$ is fixed: The measurement of angular momentum in a Stern-Gerlach experiment is a more complicated thing. One doesn't get an exact value $\pm\frac{\hbar}{2}$ even when $\hbar$ is fixed.

For in spite of what is claimed to be measured, what is really measured is something different -- namely the directed distance between the point where the beam meets the screen and the spot created by the particle on the screen (by suitable magnification). This is a macroscopic measurement of significant but limited accuracy since the spot needs to have a macroscopic extension to be measurable. From this raw measurement, a computation based on the known laws of physics and the not (or not yet) exactly known value of $\hbar$ is used to infer the value of the angular momentum a classical particle would have so that it produces the same spot. This results for the angular momentum in a value of approximately $\pm\frac{\hbar}{2}$ only, with a random sign; the accuracy obtainable is limited both by the limited accuracy of the distance measurement and (at present) the limited accuracy of the value of $\hbar$ used.

Thus for a realistic Stern-Gerlach measurement, Born's rule is only approximate, even when $\hbar$ is exactly known.

Only the idealized toy version for introductory courses on quantum mechanics satisfies Born's rule exactly since the two blobs at approximately the correct position and the assumed knowledge of exact 2-valuedness obtained from the quantum mechanical calculation count for demonstration purposes as exact enough. If the quantization result is not assumed and a true measurement of angular momentum is performed, one gets no exact numbers!

I think, it's non-sensical to discuss further. I'm out at this point, to prevent provoking more off-topic traffic.

But it is certainly on-topic here.

Note that I don't claim that Born's rule is always wrong. Like anything in physics, Born's rule has its range of validity but leads to problems when applied outside this range of validity. After a careful study of lots of alleged cases where the Born rule applies I conclude the following:

Born's rule in its standard version, i.e., upon measurement one obtains some eigenvalue, with a probability given by the usual formula, is valid precisely for measuring observables

1. with only discrete spectrum,
2. measured over and over again (to make sense of the probabilities), where
3. the difference of adjacent eigenvalues is significantly larger than the measurement resolution, and where
4. the measured value is adjusted to exactly match the spectrum, which must be known exactly prior to the measurement.

In particular, Born's rule does not apply in cases such as the total energy, one of the key observables in quantum mechanics, where the spectrum is often very narrowly spaced and the energy levels are usually only inaccurately known. Therefore Born's rule cannot be used to justify the canonical ensemble formalism of statistical mechanics; it can at best motivate it. Born's rule also doesn't apply to situations where typically only single measurements of an observable are made. Therefore Born's rule does not apply to typical macroscopic measurements, whose essentially deterministic predictions are derived from statistical mechanics.

The measurement of angular momentum is a case where the 4th assumption is clearly needed.

Consider a hypothetical student who performs a Stern-Gerlach experiment with a polarized silver beam (deflected always upwards) but (having been absent when the matter was discussed in the course) has no idea what the value of the angular momentum should be in this case. The student follows instead the standard rules for the measurement of angular momentum will have to measure as described in my post #160 in the other thread the distance of the single spot produced and then perform some calculations to get some value for the angular momentum. The inevitable inaccuracy of the distance will inevitably lead to an inaccuracy of his measurement result. Measuring multiple times will lead to slightly different distances (agreeing to within the measurement accuracy) and hence to slightly different values of the deduced angular momentum, against Born's rule taken literally, which requires that each time the exact value $\hbar/2$ is measured.

However, under the additional assumption 4., the student will know (from theory that must already be assumed to be correct to apply) that only two ideal measurement values are possible, namely $\pm\hbar/2$. The measurement accuracy suffices to exclude the minus sign. Thus the student silently sharpens the inaccurate measurement results to make them conform to the theory and reports a constant value $\hbar/2$ with that value of $\hbar$ that is the currently accepted norm. The student's supervisor, Hendrik van Hees, approves this, although compared to the exact Born rule (with the true value of $\hbar$), the measured value is still slightly in error, in the 10th decimal place. He counts this (today not checkable) discrepancy as only a technical problem, not as a limitation of the (in fact highly idealized) textbook description of Born's rule.

But in a few years, when the value of $\hbar$ is fixed exactly by a norm yet to be defined, the world of quantum interpretation is in order again.

 

[dextercioby]

Here's my take on the subject: Born's probabilistic interpretation is taken to make the necessary link between ideal measurements (these are assumed when one measures an observable with a discrete spectrum), their ability to be reproduced (this is the assumption of the "frequentist" interpretation of probabilities) and the mathematical formalism in terms of abstract (rigged) Hilbert spaces and their machinery.
Real - laboratory made - measurements can be tied to the mathematical formalism I believe in terms of relaxing the PVM formulation due to von Neumann to the POVM formulation. I believe that any sensible discussion which relates Born's rule (not the original one of 1926, for sure) to real life measurements should include the proper mathematical setting as well as proper experiments.
Inaccuracy in determining fundamental constants is not an issue. The inaccuracy of measuring "c" was never a subject in relativity, the change of 1983 was simply to offer a clearer definition of the meter.

 

[A. Neumaier]

their ability to be reproduced

No, nothing is reproducible in a stochastic setting, and indeed this is not necessary.

Instead, for making experimentally testable sense of probabilities one needs the ability to prepare the system repeatedly in an identical state. This is another severe limitation, as preparation and state calibration is in practice also of limited accuracy.

Thus Born's rule needs idealization both on the preparation and on the measurement side to be valid. Being dependent on these idealizations, Born's rule is only a theoretical instrument. To relate to reality, one still needs an additional step that says how real preparations and measurements are related to the ideal counterparts. It is here where statistical mechanics models and calibration techniques come in.

 

[vanhees71]

Of course, quantum theory is, as the name says, a theory, but with overwhelming success. The key point to relate the formalism to real-world experiments is Born's rule. To rename things or to bring up totally unrelated metrological problems doesn't change this. That not all measurements are ideal, is also not an issue. You can always coarse grain if necessary, and QT has the perfect formalism for it, namely the corresponding density operators. The total mass, the total energy and other "bulk quantities" obey the same rules as any other observables in QT, and classical variables of this kind behave classical, because the relevant accuracy of their measurement and observation is way coarser than the fundamental uncertainties due to QT.

The energy levels of a hydrogen atom are very precisely measureable (it's in fact among the best measured values ever). Nothing hints at an invalidity of standard QT. To the contrary it's a pretty convincing measurement for QED. What's not fully understood in this connection has to do with our lack of understanding of the proton's structure, but there's no hint that this has to do with any general fundamental structure of QT either.

Of course, as any theory, QT may one day be disproven by observation and may be what we consider today a mere normalization factor as $\hbar$ to relate our arbitrary SI units to the natural units may turn out not to be a "natural constant" in this sense. This, however, has nothing to do with Born's rule. Then maybe one finds a new even more accurate theory, where Born's rule is not necessary anymore or it's derived as an approximation, but nothing in today's uncertainty in some decimal place of $\hbar$ hints in this direction.

Also the description of coarse grained measurements formalized into the formalism using POVM is derived from Born's rule. There's no new quantum theory only because of this new kind of prescriptions of special kinds measurements.

I hope my point of view is now sufficiently clear, and we really don't need further empty debates about really settled foundations of QT concerning the meaning of Born's rule. I've just found the "Unwatch button on top of the page"

 

[A. Neumaier]

I hope my point of view is now sufficiently clear

It is very clear but - concerning Born's rule - simply wrong!

Nothing hints at an invalidity of standard QT.

You are arguing against a straw man. My critique was never directed against the validity of quantum mechanics (which I don't question at all), only against the present interpretations of quantum mechanics, and in this thread in particular against the universal validity of Born's rule, which is part of most interpretations.

The total mass, the total energy and other "bulk quantities" obey the same rules as any other observables in QT,

But not with respect to Born's rule. You have been unable to come up with a scheme for measuring the total energy with probabilities according to Born's rule. This has nothing to do with coarse-graining; total energy exists in any small or large system.

According to Born's rule, the recipe for obtaining the energy levels is very simple: Measure the total energy (shifted to zero ground state energy) at a fixed temperature and record all the values obtained in 100000 runs. Remove the duplicates to get the spectral lines, and count their multiplicity to get approximate probabilities and hence (up to a constant factor) the Boltzmann factors. This is what Born's rule says when specialized to the observable called total energy. But nothing is further from the truth!

The energy levels of a hydrogen atom are very precisely measurable (it's in fact among the best measured values ever).

The energy levels of a helium atom are very precisely measurable, too. But in contrast to what Born's rule claims, the measured energy levels are not the exact eigenvalues of $H$. This holds both since the measured values of the energy levels changed over time with improving spectroscopic techniques, and since the form of $H$ is not even exactly known. Also in contrast to what Born's rule claims, measuring the helium energy is not a random process whose result is one of the energy levels, with the probabilities given by Born's rule. Instead, it is a complicated process taking as input hundreds of spectra (which don't measure energy but something else) and sophisticated numerical algorithms that fit the whole set of energy levels to the wavelengths obtained from the spectral lines. As a result you get all energy levels of the helium atom at once - you cannot get them one by one, as Born's rule claims! Finally, having all the energy levels doesn't even tell you what the total energy of a particular helium atom is - as you needed zillions of them to do the computation. So if you ask about the total energy of the next helium atom you encounter, you are as clueless as before doing the computation. But you assert dogmatically (because you claim that Born's rule is universally valid) that its total energy is (or would be when measured) one of the approximately computed energy levels, with some associated probability. A completely untestable statement!

I've just found the "Unwatch button on top of the page"

You also need to find out how not to be informed when someone quotes you...

 

[vanhees71]

And you haven't come up with any clear description of how you think that the total mass or energy of a system is contradicting Born's rule. If QT wasn't able to desribe the spectra of small atoms like He, nobody would take QT very seriously.

 

[Prathyush]

And you haven't come up with any clear description of how you think that the total mass or energy of a system is contradicting Born's rule. If QT wasn't able to desribe the spectra of small atoms like He, nobody would take QT very seriously.

I have discussed coarse grained observables before in specific post #14 in the thread Born Rule and thermodynamics.

I emphasized that when we discuss coarse grained observables it is important to recognize the importance of the probe, in specific its lightness compared to other scales in the problem. With a suitable and well defined probe, it is in principle possible to study the interaction between the probe and the system under investigation, even for very large systems. We can always measure the probe particle suitably, without worrying how an experiment can be constructed to measure the probe particle. We can use this to infer information about the macroscopic system. From such a consideration one has to motivate the use of the formula $<A> = <tr \rho A>$ using a suitably constructed probe particle.

 

[A. Neumaier]

And you haven't come up with any clear description of how you think that the total mass or energy of a system is contradicting Born's rule. If QT wasn't able to desribe the spectra of small atoms like He, nobody would take QT very seriously.

I decribed it clearly and even had boldfaced the contradictions with Born's rule, in case of the Helium atom. No coarse-graining is involved!

It does not contradict QT (so everybody is right to take it very seriously) but it very clearly contradicts Born's rule in the usual formulation.

Born's rule is not QT but only a very fallible part of it, with a very limited domain of applicability!

 

[Prathyush]

I decribed it clearly and even had boldfaced the contradictions with Born's rule, in case of the Helium atom. No coarse-graining is involved!

It does not contradict QT (so everybody is right to take it very seriously) but it very clearly contradicts Born's rule in the usual formulation.

Born's rule is not QT but only a very fallible part of it, with a very limited domain of applicability!

Why doesn't Born's rule with work with Helium atom? You predict the energy levels and you see them? You can precisely calculate transition matrices, and the results can be interpreted in an experiment using a probe particle. How we measure the probe particle can be postponed into a separate investigation. To interpret the results of such an investigation we use the Born's rule.

 

[Prathyush]

But in contrast to what Born's rule claims, the measured energy levels are not the exact eigenvalues of H.

Now why do you say this?

 

[ftr]

Arnold, Let's say we measure the position of electron in many hydrogen atoms in ground state, are you saying the expectation value will be only 3/2a0(a0 Bohr radius) and we cannot actually prove Born rule even in principle because it cannot be done in practice. So that we can only use the density matrix to calculate expectation value only, is that correct. If not, what are you saying for such case, i.e. what is the electron doing, is it playing hide and seek, dancing or what.

 

[Paul Colby]

Born's rule is silent about the value of the measured mass of a single brick of iron. Here N=1N=1N=1 in the above formula. The values can take any of an astronomically large number of values, and the Born probability of measuring any of these is extremely tiny. Since only a single measurement is made, the above derivation based on the law of large numbers does not apply.

It's unclear to me one may even assure the total number of iron atoms is fixed. The act of just looking at the brick may knock atoms on or off. I think the hair you're trying to split here is too fine for me to see so let me concede defeat and go back to my calculations.

 

[vanhees71]

I decribed it clearly and even had boldfaced the contradictions with Born's rule, in case of the Helium atom. No coarse-graining is involved!

It does not contradict QT (so everybody is right to take it very seriously) but it very clearly contradicts Born's rule in the usual formulation.

Born's rule is not QT but only a very fallible part of it, with a very limited domain of applicability!

Again, there is no contradiction between QT and the Helium spectrum nor is there any contradiction between observations and Born's rule, which is an integral part of QT as a physical theory. If such a contradiction were discovered, this would mean the most sensational result since the discovery of QT itself, and we'd be well aware of this. We really discuss in circles.

 

[A. Neumaier]

Why doesn't Born's rule with work with Helium atom? You predict the energy levels and you see them? You can precisely calculate transition matrices, and the results can be interpreted in an experiment using a probe particle. How we measure the probe particle can be postponed into a separate investigation. To interpret the results of such an investigation we use the Born's rule.

I only claimed that Born's rule doesn't work for the measurement of the total energy of a helium atom. For this it doesn't matter whether other things can be interpreted with the Born rule, but only that Born's rule, applied to the observable $H$, does not apply. A measurment of $H$ (if it is at all possible) never gives a single energy level (neither exactly as claimed by Born's rule nor even approximately as a relaxed version would perhaps claim), picked at random from the full list according to the probabilities stated in the rule. I'd like to see the experimental arrangement that would do this!

[''the measured energy levels are not the exact eigenvalues of H.''] Now why do you say this?

I had explained it already in the post cited by you. The exact energy levels are still (and will probably always be) unknown, in spite of the fact that the energy levels have been measured many times.

Lets say we measure the position of electron in many hydrogen atoms in ground state

This is an essentially imposiible thing to do. independent of that, this is not the problem addressed by me. I just point to examples of measurements where Born's rule clearly doesn't make sense. This is not meant to address all possible or impossible other applications of Born's rule.

It's unclear to me one may even assure the total number of iron atoms is fixed.

In the grand canonical ensemble the number of atoms need not be fixed. nevertheless, the total energy is a well-defined entity.

Born's rule, which is an integral part of QT as a physical theory.

No. it is part of the interpretation, and as such like any other interpretation not an integral part of QT.

 

[vanhees71]

Again, not knowing the exact Hamiltonian, has nothing to do with the foundations of QT we are discussing here. You made a very bold claim, namely that the very foundation of QT (and thus QT itself) is disproven by claiming that Born's rule is wrong. You should give a clear evidence for that bold statement, but all you do is to give examples for incomplete knowledge about sufficiently complicated systems.

Indeed we don't know exactly how to describe even a proton, which is a complicated bound state of "partonic constituents", where even this phrase is not completely understood. All this, however, has nothing to do with the fundamental structure of QT and thus doesn't disprove QT as a fundamental theory, including Born's rule.

 

[A. Neumaier]

You made a very bold claim, namely that the very foundation of QT (and thus QT itself) is disproven

You deliberately misread what I write. I never claimed this. Born's rule is needed only to interpret some measurements, approximately, and is therefore not an intrinsic part of quantum theory, which is a theory supposed to hold exactly (i.e., to arbitrary precision), not only approximately.

 

[ftr]

I just point to examples of measurements where Born's rule clearly doesn't make sense.

Are you saying in effect that the expectation value of the density matrix( wavefunction derivative) has a physical significance, is that correct.

 

[A. Neumaier]

Are you saying in effect that the expectation value of the density matrix( wavefunction derivative) has a physical significance, is that correct.

The expectation value and the associated uncertainty have indeed a physical significance, independent of measurement. They are the only values that can be consistently be assigned to an observable in a given state before a measurement is made. The expectation value gives a prediction for the observed value within this uncertainty, the best possible prediction without having actually performed the measurement.

 

[ftr]

The expectation value and the associated uncertainty have indeed a physical significance, independent of measurement. They are the only values that can be consistently be assigned to an observable in a given state before a measurement is made. The expectation value gives a prediction for the observed value within this uncertainty, the best possible prediction without having actually performed the measurement.

But we still have to assign probabilities to the spectrum to calculate expectation values, don't we.

Edit: in that case what does those assignments mean/imply.

 

[vanhees71]

I'm also unable to understand, what A. Neumaier means. On the one hand, he doesn't want to use probabilities, on the other hand, all he says is using probabilities, because he has not defined, what the words "prediction of the observed value" and "within this uncertainty" means.

In normal language, what's behind these words is probability theory, and if I know about a random variable only its expectation value and its standard deviation, I'd use the maximum-entropy method to associate a probability distribution with "least prejudice" in the sense of the Shannon entropy and end up with a Gaussian distribution.

To repeat it one zillionth time again: The state (defined as an equivalence class of preparation procedures) is described in the formalism by a positive semidefinite self-adjoint operatator $\hat{\rho}$ (the statistical operator) implies all probabilitis (or probability distributions) to measure a value of any observable definable on the system in question. In the formalism an observable (defined as an equivalence class of measurement procedures) is described by a self-adjoint operator $\hat{A}$. If then $|a,\beta \rangle$ is a complete set of orthonormalized (generalized) eigenvectors of $\hat{A}$, the probability (distribution) for measuring $A$ given the state described by $\hat{\rho}$ is
$$P(a)=\sum_{\beta} \langle a,\beta|\hat{\rho}|a,\beta \rangle,$$
where the sum has to be understood in the usual sense as a sum over the discrete part of $\beta$ and an integral over the continuous part of $\beta$.

 

[Paul Colby]

In the grand canonical ensemble the number of atoms need not be fixed. nevertheless, the total energy is a well-defined entity.

Well defined so what? The number of atoms present in the brick isn't constant from measurement to measurement in any real world measurement. To argue so is to apply an accuracy limit or averaging argument. Born rule looks safe to me.

 

[vanhees71]

The (non-relativistic) grand-canonical ensemble by definition applies to the situation that you consider a smaller subsystem of a large (also macroscopic) system where both energy and (a conserved!) particle number can be exchanged within the subsystem and the "rest". Only the average energy and particle number are fixed via their "conjugate" thermodynamic quantities $\beta=1/T$ and $\alpha=-\mu/T$. The corresponding statistical operator is
$$\hat{\rho}(\beta,\alpha) =\frac{1}{Z} \exp(-\beta \hat{H} -\alpha \hat{N}),$$
where
$$Z=\mathrm{Tr} \exp(-\beta \hat{H} -\alpha \hat{N}).$$

 

[A. Neumaier]

he has not defined, what the words "prediction of the observed value" and "within this uncertainty" means.

In general, this is an approximate notion independent of probability. The position of a car is always uncertain to within about 1 meter, a statement that can be verified without recourse to probability

The meaning of observed values and uncertainty are discussed in the NIST Reference on Constants, Units, and Uncertainty, which may be regarded as the de facto standard for representing uncertainty. This source explicitly distinguish between uncertainties ''which are evaluated by statistical methods'' and those ''which are evaluated by other means''. For the second class, it is recognized that the uncertainties are not statistical but should be treated ''like standard deviations''.

The number of atoms present in the brick isn't constant from measurement to measurement in any real world measurement.

So what? Measured positions of a moving car, or measured times or measured currents or measured temperatures or whatever else people measure are not constant either, and still people trust their single measurements. Except when the noise is so large that repetition is necessary. Only then statistics enters.

But we still have to assign probabilities to the spectrum to calculate expectation values, don't we.

Expectations are calculated from the defining formula $\langle A\rangle =Tr \rho A$, and probabilities nowhere enter.

all he says is using probabilities

I nowhere use them.

if I know about a random variable only its expectation value and its standard deviation

But quantum observables are not random variables, as you know very well! Random variables always commute!

(a conserved!) particle number

Only locally conserved. The total particle number in the ''rest'' may well be infinite, and hence globally meaningless. Only the particle density matters, and figures in the formulas.