# I Many measurements are not covered by Born's rule

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1. Nov 20, 2016

### A. Neumaier

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.

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

2. Nov 20, 2016

### vanhees71

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.

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.

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).

3. Nov 20, 2016

### A. Neumaier

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.
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.
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):
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.
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....

Last edited: Nov 21, 2016
4. Nov 21, 2016

### A. Neumaier

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!

5. Nov 21, 2016

### 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.

6. Nov 21, 2016

### A. Neumaier

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
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!
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.

7. Nov 21, 2016

### 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.

8. Nov 21, 2016

### A. Neumaier

... 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 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!

9. Nov 21, 2016

### vanhees71

Ok, then we agree to disagree.

10. May 30, 2017

### A. Neumaier

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.

Last edited: Jun 5, 2017
11. May 30, 2017

### vanhees71

Ok, we agree to disagree.

12. Jun 4, 2017

### 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?

13. Jun 4, 2017

### 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.

14. Jun 4, 2017

### Staff: Mentor

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?

15. Jun 4, 2017

### 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.

16. Jun 4, 2017

### Staff: Mentor

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.

17. Jun 5, 2017

### A. Neumaier

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.

18. Jun 5, 2017

### 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.

19. Jun 5, 2017

### A. Neumaier

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!

Last edited: Jun 5, 2017
20. Jun 5, 2017

### 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?

Last edited: Jun 5, 2017