I Many measurements are not covered by Born's rule

[dextercioby]

I don't. The balance shows me miligrams or even micrograms by properly transforming a gravitational effect (force in case of negligable space-time curvature effects) into mass via the gravitational acceleration.

Edit: this is an over simplified version. The working principles behind an electronic pharmaceutical balance showing the mass of a brick to be 1200,456 grams involve a very intricate way to mass calculation in terms of piezoelectric effects, microcurrents and quartz microcrystal physics and geometry.

 

[Paul Colby]

Born's rule neither predicts the possible values nor the frequencies of the results of measuring the total energy of any atom or molecule, though this is one of the most basic observables of quantum mechanics.

Is the total energy of an atom measurable? If so how is it done? I'm familiar with detecting photons either emitted or interacting with various atomic states but I know of no direct measurement of an atoms total energy. Perhaps this is more a theoretical construct than a direct observable? Usually the statement, "I have an hydrogen atom in state x" is not the result of direct measurement of the atoms energy but rather deduced from some other measurement. I worked with negative polarized ion sources in my formative years. Neutral hydrogen atoms were excited in an RF discharge. Then the proper spin ones were selected by focusing with a 6-pole magnet. Even though we "know" the atomic state (and by theory it's energy) it's only through indirect means that this is known. I know for a fact the Born rule was used 6 ways to Tuesday in the development of that polarized ion source.

 

[Paul Colby]

Actually, I would go one step further and point out that the energy of an atomic state has a natural line width due to vacuum polarization and such. So, even atoms don't have perfectly well defined energy levels.

[Edit] I see this is what you are pointing out. The Born rule does predict the average state energy. Yep, still don't know what you're on about.

 

[A. Neumaier]

the statement, "I have an hydrogen atom in state x" is not the result of direct measurement of the atoms energy

Of course. It is always the result of a preparation of the atom in the given state (though a judicious arrangement of sources and filters). That''s how the inputs to quantum experiments are created. Born's rule doesn't enter. To find out the state if it is not known (e.g., to find out whether the given state is an eigenstate of H or a thermal state) is impossible unless one has a large number of identically prepared atoms. Here Born's rule enters, but applied to very simple observables different from the total energy - typically binary observables of yes-no type, in case of photons also of quadrature measurements.

The Born rule does predict the average state energy.

The Born rule, taken literally, predicts that each measurement of the total energy produces exactly one energy level, at the exact value given by an eigenvalue of the Hamiltonian (normalized so that the ground state has zero energy). For a large ensemble of identically prepared atoms one can deduce from Born's rule a prediction of the average value of the total energy obtained from all these measurement values. This average value equals the expectation value computed from the theory. But because there is no way to perform the first part one cannot perform the second part which needs the first part as input. Thus Born's rule does not even predict the average total energy!

Yep, still don't know what you're on about.

An average interpretation makes not even sense if the individual measurements are fictitious of which it should be an average.

Thus if one starts with Born's rule one needs substantial handwaving to arrive for any observable $A$ at an interpretation of $Tr\rho A$ as an expectation value!

 

[A. Neumaier]

I know for a fact the Born rule was used 6 ways to Tuesday in the development of that polarized ion source.

Yes, Born's rule is often used, but quite often its use is not justified by its formulation, because most often it is used without direct reference to measurement, although the latter figures explicitly in its definition. And its conventional formulation is faulty because (unlike with any other physical rule) the assumptions are never stated under which it is valid, namely the conditions I gave in my post #28.

Thus everything surrounding Born's rule and its application contains a large dose of hand-waving!

This doesn't matter for experimental practice. But it is the cause of the century-long and wide-spread dissatisfaction with the foundations of quantum mechanics.

 

[A. Neumaier]

the energy of an atomic state has a natural line width due to vacuum polarization and such. So, even atoms don't have perfectly well defined energy levels.

My criticism still applies, though now to the more general version of Born's rule for measuring observables with a continuous spectrum.

On the other hand, in statistical mechanics it is commonly assumed that the (for a macroscopic object very densely spaced) spectrum of H is discrete, since otherwise the partition sum makes no longer sense. Thus I based my arguments on this widely used idealization.

 

[Prathyush]

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 HH, does not apply. A measurment of HH (if itisi 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!

Point is the following, if you create the atom(say the ground state). If you probe it with particle x, we can precisely calculate the probability amplitudes of the given process. Consider situations where the helium atom remains in the ground state for instance. Situations where helium atom remains in an excited state is somewhat subtle to analyze(atleast for me.)

an amplitude such as |x,He> -> |y,He>, where x is some initial state of the photon. Y could be a multi particle state treated suitable using an appropriate second quantized framework.

This amplitude given by $<y,He|e^{-iHt}|x,He>$ has a sharp and precise meaning. And that is what is measured in a lab experimentally. The probability is $|<y,He|e^{-iHt}|x,He>|^2$ this quantity has a precise meaning and can only be interpreted using the Born's rule. How we measure the test particles is irrelevant for these considerations.

 

[A. Neumaier]

This amplitude ... has a sharp and precise meaning. And that is what is measured in a lab experimentally. ... and can only be interpreted using the Born's rule.

Yes, but it measures a state transition and not the total energy. I have no problem accepting Born's rule as the appropriate tool for interpreting scattering processes.

But it is already nontrivial to say which observable (in the sense of the conventional formulation of Born's rule) corresponds to the transition measured. In any case, a different observable figures for different transitions. And there are many more possible transitions than there are energy levels, hence one cannot even figuratively take one for the other.

 

[Prathyush]

Yes, but it measures a state transition and not the total energy.

That is not particularly important. Any experiment you construct can be analyzed using the framework of quantum mechanics using the Born's rule, which in general assigns probabilities to processes. I am trying to understand what it means to amplify microscopic information into macroscopic observables. The point is the following, the stage for final amplification can be separated from interaction of the probe particle with the system.

 

[ftr]

Arnold, what do you think of this experiment

https://physics.aps.org/featured-article-pdf/10.1103/PhysRevLett.110.213001

 

[vanhees71]

You repeat again and again this example about the energy of an atom and just don't realize that you cannot measure it in the way you imply. Of course you cannot measure things that are not observable in principle. You can measure the energy differences by spectroscopy or infer them from scattering experiments, including the corresponding transition probabilities (relative intensities of spectral lines) or cross section, but you cannot measure absolute energy levels (neither within QT nor classical mechanics).

 

[A. Neumaier]

Arnold, what do you think of this experiment

https://physics.aps.org/featured-article-pdf/10.1103/PhysRevLett.110.213001

If you open another thread, we can discuss it; it is an interesting experiment. But starting tonight I am on travel for a few days and cannot reply before next Monday or Tuesday. So please create then a new thread, and mention me (i.e., place @A.Neumaier in the text so that I am notified).

 

[A. Neumaier]

You repeat again and again this example about the energy of an atom and just don't realize that you cannot measure it in the way you imply.

You mean, in the way Born's rule says! I imply nothing else than what Born's rule says about measuring observables. And there is no doubt that the total energy is one of the most important observables in quantum mechanics.

 

[A. Neumaier]

Of course you cannot measure things that are not observable in principle.

So you want to claim that total energy (normalized as commonly done to ground state energy zero) is unobservable in principle?

Then why is $H$ called an observable?

And why can it be observed in the bulk, which is done routinely in thermodynamics?

How big must a quantum system be before its total energy becomes observable in principle?

 

[Paul Colby]

How big must a quantum system be before its total energy becomes observable in principle?

This depends almost entirely on how accurately the observable is to be determined. The more accurate, the larger the system or the larger the sample set. Even in the case where theory states the answer is certainty, one single measurement would not suffice to determine the answer experimentally.

 

[A. Neumaier]

This depends almost entirely on how accurately the observable is to be determined.

Born's rule says (in its textbook version) that measuring any observable gives as results an exact eigenvalue.

Thus Born's rule is only an approximate rule?

The more accurate, the larger the system or the larger the sample set.

The sample set has size 1 in all my arguments. Single measurements give a property of the single system.

Born's rule says something definite about the possible results of each single measurement. How can it be that the result for a small system is inaccurate if only a discrete set of possibilities are allowed?

 

[vanhees71]

You mean, in the way Born's rule says! I imply nothing else than what Born's rule says about measuring observables. And there is no doubt that the total energy is one of the most important observables in quantum mechanics.

You do not even try to understand what I'm saying. Once more: The energy of a system is defined only up to an additive constant, and you can only measure energy differences, and that's also valid for an atom. We can in fact very accurately measure the energy levels of an atom by doing spectroscopy, and that's described, of course, by Born's rule. If that was not the case, nobody would ever have studied QT as discovered by Heisenberg, Born, Schrödinger, and Dirac in 1925/26! For the most simple atom, the hydrogen atom, it's among the most accurate measurements ever done, and the predictions of QT (in this case QED) are among the best confirmed in the history of physics.

 

[Paul Colby]

Born's rule says (in its textbook version) that measuring any observable gives as results an exact eigenvalue.

Thus Born's rule is only an approximate rule?

In cases where the observable spectrum is a continuum (like the energy transitions in a hydrogen atom interacting with the EM field) the Born rule states only one of a set of energies will result in a single measurement but not which one of the continuum will result. In addition it makes an exact (theoretical) statement about the frequency of observed values. Experimentally verifying this frequency or probability to infinite accuracy is for ever beyond ones capabilities which seems completely reasonable to me since exact measurement is never possible even in principle.

 

[vanhees71]

Also if the operator $\hat{A}$ representing the observable $A$ has a continuous spectrum (or if part of its spectrum is continuous), then Born's rule gives probability distributions (not probabilities) since the generalized eigenvectors for these spectral values are distributions not in the Hilbert space but in the dual of the nuclear space of the operator. E.g., for the position operator $\hat{x}$ the spectrum is entire $\mathbb{R}$, and for a normalized $|\psi \rangle$ representing the state $|\psi \rangle \langle \psi|$ the expression $P(x)=|\langle x|\psi \rangle|^2=\langle x|\hat{\rho} |x \rangle$ is the probability distribution, i.e., the probability to find the particle in an interval $[x-\epsilon,x+\epsilon]$ is given by $\int_{x-\epsilon}^{x+\epsilon} \mathrm{d} x' P(x')$. All this is no argument against Born's rule.

The more I follow Arnold's arguments the less I understand what his criticism against Born's rule is about :-(.

 

[Paul Colby]

The sample set has size 1 in all my arguments. Single measurements give a property of the single system.

Born's rule says something definite about the possible results of each single measurement. How can it be that the result for a small system is inaccurate if only a discrete set of possibilities are allowed?

This is only half of the Born rule which is why it's never applied this way in practice or theory. Case in point, the Stanford Magnetic Monopole. A single measurement is insufficient to establish a result. You should know this.

 

[A. Neumaier]

The energy of a system is defined only up to an additive constant, and you can only measure energy differences

The total energy of a system when normalized (as I always emphasized) such that the ground state energy $E_0$ is zero is fully defined, and each of its eigenvalues $\lambda_i$ is an energy level with a physical meaning; $\lambda_i=E_i-E_0$ is an energy difference, and hence measurable as you just state. Nevertheless its measurement has no resemblance at all to what Born's rule claims for the measurment of $H$.

You do not even try to understand what I'm saying.

I can say the same of you.

This is only half of the Born rule ...

If half the rule is already proved deficient in some case, the full rule with both parts is deficient in this case, too. Other cases don't change this.

... which is why it's never applied this way in practice or theory.

If it is never used in theory or practice, why is it part of the rule? A good rule never contains completely useless parts!

 

[A. Neumaier]

In cases where the observable spectrum is a continuum ...

But we are talking about energy levels, i.e., the discrete part of the spectrum.

... the Born rule states only one of a set of energies will result in a single measurement but not which one

The same is claimed in Born's rule for the discrete part. And it is claimed to hold exactly. If it were to hold holds only approximately, any of the many theoretical conclusions traditionally drawn from it would be slightly erroneous, too.

exact measurement is never possible even in principle.

But then Born's rule is valid only approximately, and cannot be fundamental.

The more I follow Arnold's arguments the less I understand what his criticism against Born's rule is about :-(.

You haven't even started to follow it. You react against things I never claimed and repeat things that have nothing to do with my arguments.

 

[dextercioby]

Many PF threads with debates in the QM section end up with "let's agree to disagree". This and the other current one will share no different fate.

 

[A. Neumaier]

Many PF threads with debates in the QM section end up with "let's agree to disagree". This and the other current one will share no different fate.

In my view, the point of such a discussion is not to agree but to express all the relevant facts and arguments so that the readers (not the combatants) can judge for themselves. Usually I contribute as long as new aspects come up, and a little longer just in case. And it is good to have sharp adversaries, since this forces one to put one's statement into the most clear and expressive form.

In the present case, I learned only through these discussions and my background studies caused by them what a can of worms Born's rule is if one takes if fully at face value and compares with real measurements. This made the discussion worthwhile for me, not whether vanhees71 or anyone else agrees or disagrees.

 

[dextercioby]

I am mostly interested in the mathematical formulation of physical theories, but reading through these two threads really made me learn something about the things I was in turn taught 13 years ago. That it is a very bold statement to claim that an axiom from a mathematical/logical construct/framework/model makes an unbreakable link to real life laboratory experiments. So the question remains? What is a quantum observable?

 

[Paul Colby]

That it is a very bold statement to claim that an axiom from a mathematical/logical construct/framework/model makes an unbreakable link to real life laboratory experiments.

If I understand your statement, why are you surprised? The situation where theory is devoid of a mandatory discussion of what's measurable is actually more shocking than the reverse. In classical mechanics one could get away with such abstraction. I find it very comforting that QM as a theory includes such an intimate connection with the world we interact with. In my opinion theory is just what we make up to fit phenomena we can measure and nothing more.

Also, my interpretation of the discussion in this thread is just a disagreement on what the rules "say" about nature. Some of it bordering on wrong or not even wrong.

 

[A. Neumaier]

a disagreement on what the rules "say" about nature.

Born's rule says nothing about nature (which doesn't know measurements), it is about measurements, which are not part of nature but of our scientific culture.

 

[Paul Colby]

Born's rule says nothing about nature (which doesn't know measurements), it is about measurements, which are not part of nature but of our scientific culture.

Yes, in my world view your statement is simply wrong but there is likely no way to make that point.

One parting shot. How can one claim that a "rule" that defines the probability of experimental observation doesn't "know" anything about said observation.

 

[A. Neumaier]

How can one claim that a "rule" that defines the probability of experimental observation doesn't "know" anything about said observation.

The knowing is in those using the rule and applying it very liberally to all sorts of situations without bothering about what the rule actually says, according to the traditional meaning of the terms used in expressing it.

I want to have foundations (and believe they can be clearly formulated) where the words match the meaning.

 

[Strilanc]

Sorry for quoting from way back at the start, but...

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.

Anytime you see measurement in the middle of some quantum mechanical model, you can add ancillae, replace that measurement with the relevant information leaking into the ancillae, and defer measuring the ancillae until the experiment finishes. That's the deferred measurement principle (...it's really more of a theorem). For example, instead of measuring then doing computations classically you can do the computations under superposition on a quantum computer before measuring.

The measurements you are describing in your examples are very complicated. We use computation to deal with those complications. In practice, for obvious pragmatic reasons, we do those computations classically and think about them classically. But, speaking theoretically, there's no fundamental obstacle to doing the computations under superposition on a quantum computer.

Doing all the statistics on a quantum computer, and only measuring at the very end, allows you to defer applying Born's rule until the final outcome has been computed. This fixes the theoretical problem you're pointing out.

Yes, you can apply Born's rule halfway through your experiment and then do some classical stuff outside of the regime that Born's rule is meant to apply to. But that doesn't mean it's impossible to translate what you are doing back into a regime that Born's rule does apply to. That translation may be ridiculously inconvenient, but it's still possible.