A Evaluate this paper on the derivation of the Born rule

[vanhees71]

But theoretical physics does not need to be circular; one can have a good theory with a noncircular interpretation in terms of experiments.

While one is still learning about the phenomena in a new theory, circularity is unavoidable. But once things are known for some time (and quantum physics is known in this sense for a very long time) the theory becomes the foundation and physical equipment and experiments are tested for quality by how well they match the theory. Even the definitions of units have been adapted repeatedly to better match theory!But this gives you energy differences, not energy levels. This does not even closely resemble Born's rule!
Moreover, it is a highly nontrivial problem in spectroscopy to deduce from a collection of measured spectral lines the energy levels! And it cannot be done for large molecules over an extended energy range, let alone for a brick of iron.No. It depends also on selection rules and how much they are violated in a particular case. It is quite complicated.I mentioned everything necessary. To approximately measure the two quadratures of photons in a beam one passes them through a symmetric beam splitter and then measured the resulting superposition of photons in the two beams by a homodyne detection on each beam. Details are for example in a nice little book by Ulf Leonhardt, Measuring the quantum state of light. This is used in quantum tomography; the link contains context and how the homodyne detection enters.

If you test quantum theory, it's not given as the foundation but checked by observations. Physics is always circular in this sense, and a "test" means a "consistency check" between the theory used to construct your apparatus and the true outcome of the measurement in comparison what's really measured.

Concerning the hydrogen atom, in this sense you've never measured the energy levels but only differences by using spectroscopy, and the prediction of the seen spectrum, including the selection rules are, of course, based on Born's rule: You calculate transition-matrix elements and take their modulus squared! I didn't say that to get the spectrum of the gas is simple, but it's finally based on these very foundations of QT.

How a very similar problem is treated in heavy-ion physics, you can read here:

http://arxiv.org/abs/0901.3289

Concerning homodyne detection, what's measured according to the Wikipedia article (which is full of inaccuracies by the way, don't need to go into here) are intensities, as described in my example from Scully's textbook.

 

[A. Neumaier]

Well, I'm still lacking understanding the physics content of the thermal interpretation. Basically what you say is that what's measured are "expectation values", but I'm not allowed to define them via the usual probability interpretation (Born's rule, or rather Born's Postulate if you wish). So how do I understand the meaning of your "expectation values"? And what's "thermal" here? Are you taking always the expectation values with equilibrium distribution functions (equilibrium statistical operators)? I'm using the standard terminology here for lack of a better language. How do you call the "Statistical Operator", if you deny the statistical/probabilistic meaning?

What you call the statistical operator I call the density operator and denote it by $\rho$. Thermal does not mean thermal equilibrium, but refers to the fact that the thermal interpretation is borrowed from looking at how the results of statistical mechanics are interpreted in the applications to thermodynamics. The mathematical meaning of the expectation values (the q-expectation values in the terminology of Allahverdyan, Balian and Nieuwenhuizen) is defined by $\langle A\rangle :=$ trace $\rho A$, which is part of the shut-up and calculate stuff in statistical mechanics. This expression can be defined without any reference to an interpretation. The experimental meaning of this expression (like of anything in shut up and calculate) depends on the interpretation applied. In the thermal interpretation, the meaning is well-defined primarily for macroscopic variables (those considered in statistical equilibrium or nonequilibrium thermodynamics) where it gives the measured value of the macroscopic variable $A$ to an accuracy that grows with the system size like $O(N^{-1/2})$. This is sufficient for the interpretation of experiments since actual measurements are always taken of macroscopic objects (pointers, currents, spots, etc.). The meaning of a microscopic measurement is whatever the microscopic dynamics allows one to conclude about the correlations between the state of the microscopic system and the resulting state of the macroscopic variable actually read in the measurement. Thus it depends on how the measurement devices couple to the microscopic system. And of course this is as it has to be since a measurement device can function properly only if the coupling establishes the necessary correlations for a macroscopic event to be taken as a measurement of a microscopic observable.

 

[vanhees71]

Of course you can build a mathematical theory based on a system of postulates (axiomatic approach). As a physical theory it's empty. What I call "interpretation of a theory" is the particularly physical part of the theory, namely how to apply the formalism of the mathematical theory to observations with real measurement apparati in the real world.

You can call $\hat{\rho}$ "density operator", but it's NOT referring to the observable "density" of a many-body system. Take non-relativistic many-body theory in 2nd quantization (non-relativistic QFT) of scalar particles (Schrödinger particles sotosay). Then the particle-density operator is
$$\hat{n}(t,\vec{x})=\hat{\psi}^{\dagger}(t,\vec{x}) \psi(t,\vec{x}).$$
That's, at least, the usual language in many-body physics. Its expectation value is
$$n(t,\vec{x}) = \mathrm{Tr} (\hat{\rho} \hat{n}(t,\vec{x}).$$
Now it's clear what's meant by "density": It's a (local) observable. If you don't like to call $\hat{\rho}$ "statistical operator" (as is the standard name in modern textbooks, and everybody understands it who as successfully listened to the QM 1 lecture), I'd rather call it "state operator".

For me just to rename established names to something else, reminds me of the funny dialogue between a student and Mephisto, where Mephisto tries to explain the advantages and disadvantages of different subjects to study:

Schüler:

Doch ein Begriff muß bei dem Worte sein.

Mephistopheles:

Schon gut! Nur muß man sich nicht allzu ängstlich quälen
Denn eben wo Begriffe fehlen,
Da stellt ein Wort zur rechten Zeit sich ein.
Mit Worten läßt sich trefflich streiten,
Mit Worten ein System bereiten,
An Worte läßt sich trefflich glauben,
Von einem Wort läßt sich kein Jota rauben.

(I can't adequately translate this)

This refers to "theology"; however it seems to apply to the "quantum theology" of interpretations as well...

 

[A. Neumaier]

For me just to rename established names to something else

You seem to be familiar only with the terminology used in your particular field of application.

I am using terminology from established textbooks. (i) Reichl, in her modern course in statistical physics, calls $\rho$ the density operator. So do (ii) Walls and Milburn, (iii) Peng and Li, (iv) Paul, (v) Meystre and Sargent III and (vi) Scully and Zubairyin their books on quantum optics, and (vii) Messiah in vol. 1 of his books on quantum mechanics. In the quantum optics book by (viii) Klimov and Chumakov, and in (ix) Thirring's volume 4 of his course in mathematical physics, $\rho$ is called the density matrix (though it is an operator). So does (x) Oettinger in his book Beyond equilibrium thermodynamics.

Though $\rho$ is not an observable related to a density in space, it indeed deserves to be called a density operator since it is the quantum analogue of Boltzmann's phase space density, which is also not a density in the sense you are using it.

 

[vanhees71]

I only know the textbook by Zubairy and Scully, Quantum Optics, and they may call the Stat. Op. density operator, but in the usual statistical meaning. To call it density operator a relic of the misinterpretation of $|\psi|^2$ as density by Schrödinger. To call it "density matrix" is also a relic from times, where one preferred to write everything in some representation, where all operators become "matrices".

The observational facts finally lead to the probabilistic interpretation of this quantity (Born), and that's why it is better to call it statistical operator. A density is an observable, namely some quantity per volume (element). My example was the particle-number density.

 

[A. Neumaier]

Of course you can build a mathematical theory based on a system of postulates (axiomatic approach). As a physical theory it's empty.

It is as empty or nonempty as any shut up and calculate approach, as it only does computations. Interpretation enters, as always, only by relating the formal stuff to reality.

how to apply the formalism of the mathematical theory to observations with real measurement apparati in the real world.

I gave precise rules for interpretation (i.e., how to relate certain formulas to reality) in the thermal interpretation. The part of the interpretation common with any interpretation is given here. The part where I differ from tradition is that I do not assume anything about probabilities, and replace it by the uncertainty principle mentioned in posts #85 and #102 of the present thread. Instead of assuming it, the probability interpretation (where it applies) and Born's rule (where it applies) are derived in Chapters 8.4 and 10.3-5 of my online book.

 

[A. Neumaier]

I only know the textbook by Zubairy and Scully, Quantum Optics, and they may call the Stat. Op. density operator, but in the usual statistical meaning.

I think you also know Messiah's textbook, as you had referred to it in the past, and he uses the same terminology. The quantum optics book by Gerry and Knight also uses density operators; I had forgotten to mention it. Thus essentially all quantum optics people use it! Moreover, almost everything done in the textbooks (except the discussion of actual experiments) is shut up and calculate and doesn't depend on how you interpret the density operator.

To call it density operator a relic of the misinterpretation of $|\psi|^2$ as density by Schrödinger. To call it "density matrix" is also a relic from times, where one preferred to write everything in some representation, where all operators become "matrices".

It is not a misrepresentation since it is the quantum analogue of Boltzmann's phase space density. Densities need not refer to space only!

The observational facts finally lead to the probabilistic interpretation of this quantity (Born)

The interpretation matters only when you compare results with experiments. But the observational facts are compatible with a number of interpretations.

Since Born's rule is a consequence of the thermal interpretation whenever Born's rule applies to actual measurements, the observational facts are fully compatible with the thermal interpretation.

Since the thermal interpretation derives Born's rule from a much simpler uncertainty principle (which unlike Born needs no knowledge of the - quite nontrivial - spectral theorem) one can infer from the derivation the domain of applicability of Born's rule, while putting it into the foundations makes the latter very fuzzy (since the undefined notion of measurement enters in a completely unspecified way) and doesn't exhibit the many situations where Born's rule is not applicable. The most notable counterexample is the measurement of the total energy of a system, which cannot be done in a way matching any of the conventional formulations of Born's rule.

 

[Prathyush]

Although you read this terminology very often, it's misleading. What you measure are observables, not states. The probability to find an outcome a when measuring the observable A if the system is prepared in a pure state represented by a normalized vector $|\psi \rangle$ is given by
$$P(a)=\sum_{\beta} |\langle a,\beta|\psi \rangle|^2,$$
where $|a,\beta \rangle$ denote a complete set of eigenvectors of $\hat{A}$ (the self-adjoint operator that represents the observable A).

It correct to say that we measure observables. In your view point, it is not relevant to talk about what happens to a system after observation, without an experimental context. I think there is a good reason to think that we should include the fact that the new state of the system is infact an eigenvector of the observable that we measure.

This may not be true in all experiments. However to me an ideal measurement of a given observable has this property. In the literature it is called Von Neumann measurement or something similar.

For instance if we measure a particle to be at position to be x at time t, for all future measurements the state of the particle must be taken to be |x,t>. I don't at this moment have a full argument to justify what I am saying. I will consider what you are saying, think about it and in a separate post evaluate this.

 

[vanhees71]

I think you also know Messiah's textbook, as you had referred to it in the past, and he uses the same terminology. The quantum optics book by Gerry and Knight also uses density operators; I had forgotten to mention it. Thus essentially all quantum optics people use it! Moreover, almost everything done in the textbooks (except the discussion of actual experiments) is shut up and calculate and doesn't depend on how you interpret the density operator.It is not a misrepresentation since it is the quantum analogue of Boltzmann's phase space density. Densities need not refer to space only!
The interpretation matters only when you compare results with experiments. But the observational facts are compatible with a number of interpretations.

Since Born's rule is a consequence of the thermal interpretation whenever Born's rule applies to actual measurements, the observational facts are fully compatible with the thermal interpretation.

Since the thermal interpretation derives Born's rule from a much simpler uncertainty principle (which unlike Born needs no knowledge of the - quite nontrivial - spectral theorem) one can infer from the derivation the domain of applicability of Born's rule, while putting it into the foundations makes the latter very fuzzy (since the undefined notion of measurement enters in a completely unspecified way) and doesn't exhibit the many situations where Born's rule is not applicable. The most notable counterexample is the measurement of the total energy of a system, which cannot be done in a way matching any of the conventional formulations of Born's rule.

Ok, you may call $\hat{\rho}$ a "density operator", because that is done in many textbooks. I've no problems with it, although I find it highly misleading. I don't know, how to make sense of an uncertainty principle, if I'm forbidden to use probability theory.

Except in GR absolute values of energies are not observable. That's nothing specific to QT but holds also for classical mechanics and electrodynamics. Take as the most simple example the hydrogen atom in non-relativistic approximation a la Schrödinger as taught in QM 1. It's our choice to write $V=-e^2/(4 \pi r)$, using the convention that the potential goes to 0 at $r \rightarrow \infty$. That's convention, you can add any constant you like to it without changing any observable predictions about the atom. What's measurable through spectroscopy are the energy differences, and the corresponding intensities (including the selection rules you quote) follow by the application of Born's rule.

 

[RockyMarciano]

I gave precise rules for interpretation (i.e., how to relate certain formulas to reality) in the thermal interpretation. The part of the interpretation common with any interpretation is given here. The part where I differ from tradition is that I do not assume anything about probabilities, and replace it by the uncertainty principle mentioned in posts #85 and #102 of the present thread. Instead of assuming it, the probability interpretation (where it applies) and Born's rule (where it applies) are derived in Chapters 8.4 and 10.3-5 of my online book.

Is your interpretation related to the phase space formulation of QM(deformation quantization)?, they have in common the classical statistics approach. is perhaps the necessary deformation assumed in your macroscopic measurement uncertainty?

 

[A. Neumaier]

What's measurable through spectroscopy are the energy differences, and the corresponding intensities (including the selection rules you quote) follow by the application of Born's rule.

No. Borns rule would assert that you observe Ei-E0 with a probabiility pi given by Boltzmann factors, whereas one in fact observes all Ei-Ek with i,k determined by selection rules and intensities given by a formula different from Born's. Thus this measurement is definitely not covered by Born's rule, and the latter does not justify the partition sum.

 

[vanhees71]

No, the Born rule exactly gives, what you describe. There are transition matrix elements as in my Insights article of the photoelectric effect in front of the Boltzmann (Bose-Einstein to be precise) factor for the emission rates. In the approximation presented there this implies the usual dipole-approximation superselection rules ($\Delta J \in \{-1,0,1 \}$ no transition for $J=0 \rightarrow 0$). The general thermal-field theory formula for photons from a dilute medium at rest (i.e., transparent for photons) is the McLerran-Toimela formula
$$E \frac{\mathrm N_{\gamma}}{\mathrm{d}^4 x \mathrm{d^3 \vec{q}}} \propto \mathrm{Im} \Pi_{\text{em}}^{(\text{ret})}(E,\vec{q}) f_{\text{B}}(E),$$
where $\Pi_{\text{em}}^{(\text{ret})}$ is the em. current-current-correlation function at the photon on-shell point, $E=|\vec{q}|$ (sum over two polarizations), and $f_{\text{B}}(E)=1/[\exp(E/T)-1]$. The corresponding spectral function, i.e., its imaginary part, takes care of all selection rules!

I don't understand, why you all of a sudden claim standard QT is invalid. I thought you only want to give another interpretation, whose logicI don't understand yet, I must admit, because I don't see, why denying that probabilities are at work by just not using the word but using the entire formalism based on the probability interpretation of the quantum state, should lead to any new insights about the nature of QT. I like the math of your great textbooks, but I don't see the merit for the physical interpretation compared to any standard treatment of QT based on the probabilistic interpretation, which so far is common to all interpretations. The reasons are wellknown. In the history of QT, the interpretation of the state as densities (you use this word obviously with some more reason than just using it as an old-fashioned misnomer from the old days of QT) has been given up very shortly after the formulation in three equivalent terms of wave mechanics (Schrödinger), matrix mechanics (Born, Heisenberg, Jordan), and "transformation theory" (Dirac). The probability interpretation, which is the only one compatible with the observational facts so far, is due to Born's famous footnote in his also famous paper on scattering theory and earned him a late Nobel prize finally in the 50ies.

 

[A. Neumaier]

I don't understand, why you all of a sudden claim standard QT is invalid.

I didn't claim that at all. I am just claiming that you frequently misuse the designation ''Born's rule'' for a lot of stuff that does not at all resemble Born's rule in its conventional formulation (upon which everyone but you agrees).

Born's rule says that if the spectrum of an observable $A$ has $k$ distinct eigenvalues, there are exactly $k$ distinct possible values of the measurement, and not that one measures up to $k(k-1)/2$ eigenvalue differences, as in the case of an observation of an optical spectrum (when the dipole approximation is no longer valid). Thus the experimental facts are in direct opposition with the claims of Born's rule stated everywhere.

why [...] using the entire formalism based on the probability interpretation of the quantum state, should lead to any new insights about the nature of QT.

Because the pure formalism itself (i.e., shut up and calculate alone) is silent about the interpretation, and anyone (such as the authors of the papers discussed in this thread, or myself) who wants to derive the probability interpretation (and thus explain why shut up and calculate is so successful in practice) is not allowed to assume it from the start.

Born's Nobel price worthy achievement cannot be the last words about the foundations; if they were, discussions about the interpretation of quantum mechanics would have subsided long, long ago.

 

[vanhees71]

Again, you cannot measure absolute energies but only energy differences. Also, you have introduced into the debate, how to measure the energy levels of an atom, and it's done since at least the 19th century by spectroscopy. In QT the measured frequences of the emitted light are the differences of the discrete energy levels. The (relative) intensity of the spectral lines, including the selection rules are given by Born's rule. That's all what I was saying, and that's what you find in any introductory textbook about atomic physics and usually also in QM 1 textbooks.

 

[A. Neumaier]

Again, you cannot measure absolute energies but only energy differences.

Sure, but this just means that you agree that for the measurement of the observable ''energy'', there is a deviation from Born's rule, which says that one measures eigenvalues. Your argument confirms the correctness of my assertion that energy measurements flatly contradict the claims of Born's rule about the possible values of a measured observable.

how to measure the energy levels of an atom, and it's done since at least the 19th century by spectroscopy. In QT the measured frequences of the emitted light are the differences of the discrete energy levels. The (relative) intensity of the spectral lines, including the selection rules are given by Born's rule. That's all what I was saying, and that's what you find in any introductory textbook about atomic physics and usually also in QM 1 textbooks.

You had suggested this in post #97 as the way to measure the Hamiltonian $H$, a key observable in quantum mechanics. I only note that it flatly contradicts the claims made by Born's rule concerning the measurement of the observable $H$.

Instead the optical measurement results conform to shut-up -and-calculate results about absorption lines, which make accurate predictions since these formulas are very different from what Born's rule claims about measuring $H$.

 

[vanhees71]

Here you measure the frequency and intensity of spectral lines, not $H$ of the atom directly. All these spectral properties follow quantum mechanically via the formalism. In the semiclassical approximation (which is sufficient for absorption and induced emission in this case) it's given in time-dependent perturbation theory as explained in my Insights article, and this makes use of the probabilistic interpretation of states, i.e., Born's rule. How else should I, in your opinion, describe this measurement quantum theoretically? How else do you want to describe it? Where is a problem in the standard formulation of QT and where is the need for other terminology than the standard probabilistic one used since 1926? Atomic physics and spectra were among the very first applications of old QT (Bohr-Sommerfeld model) and lead to a clear disprove of this too classical pictures with ad-hoc "quantum rules". In the following it was among the very first applications of new QT and turned out a great success, including the explanation of fine and hyperfinestructure (later also with full QED)!

 

[A. Neumaier]

Here you measure the frequency and intensity of spectral lines, not H of the atom directly.

Well, it was you who called it a measurement of the energy in the first place.

Since you now say it isn't a measurement of the energy, does it mean that Born's rule cannot be applied to the measurement of the operator H (shifted such that the ground state has energy zero, so that all energy levels are uniquely defined and have a physical meaning)?

But if Born's rule cannot be applied to energy, your justification (in post #46) for explaining expectations in the canonical ensemble by means of Born's rule has completely evaporated. Indeed, this was the whole reason why I had asked (in post #94) about the measurement of energy.

 

[vanhees71]

Again, you cannot measure the absolute value of $H$; neither in classical nor quantum theory. The only place, where absolute values of energy densities (more precisely the absolute value of the energy-momentum-stress tensor of matter fields) are observable is GR, and there it's an unsolved problem to understand the observabled value of the cosmological constant.

 

[A. Neumaier]

Again, you cannot measure the absolute value of $H$; neither in classical nor quantum theory. The only place, where absolute values of energy densities (more precisely the absolute value of the energy-momentum-stress tensor of matter fields) are observable is GR, and there it's an unsolved problem to understand the observable value of the cosmological constant.

Would you please care to read what I wrote? I did not ask to measure the absolute value of $H$. I assumed that energies are shifted such that the ground state has energy zero, so that all energy levels $E_k$ are uniquely defined and have a physical meaning. This holds for any physical system, and one need not invoke general relativity to discuss its merits or problems.

These energy levels go into the rules for evaluating expectations in any canonical ensemble. To derive the canonical ensemble from Born's interpretation the very least that is needed is to show that a measurement of $H$ produces the value $E_k$ with probability $Z^{-1}e^{-\beta E_k}$. When I asked for a measurement of energy you first referred to a measurement of spectral information, but later you retracted your choice and said the latter does not measure energies but frequency and intensity of spectral lines.

Since there is no possibility to measure the energy according to Born's rule, Born's rule is obviously not applicable to the situation. Indeed, energy is hardly ever measured in applications of the canonical ensemble.

Thus the ''derivation'' of the canonical ensemble from Born's rule is spurious.

In order to uphold the derivation you need to give up the assertion that Born's rule refers to measurement. But then it completely loses its contact to experiment and hence its interpretational value.

 

[vanhees71]

How to assign probabilities is not within QT. If you know that the atom is in thermal equilibrium with a heat-bath, which you implicitly assume when you want to derive the canonical-ensemble interpretation. One way to argue is to use the Shannon-Jaynes maximum-entropy principle, which leads, when using the total energy as the one known variable, leads to
$$\hat{\rho}=\frac{1}{Z} \exp(-\beta \hat{H}).$$
The energy differences you can indeed measure by spectroscopy, and that's how it was done historically in the development of QT (it was an industry at the beginning of the 20th century with one important center at Sommerfeld's Munich institute), but we argue in circles here. The atomic spectra as energy differences were an empirical discovery of the 19th century. It's theoretical understanding helped a great deal to historically develop quantum theory. If there's one paradigmatic example for the measurement of quantum phenomena it's the energy levels of atoms!

 

[A. Neumaier]

One way to argue is to use the Shannon-Jaynes maximum-entropy principle, which leads, when using the total energy as the one known variable,

Well, yes, and it does not involve Born's rule, in contrast to what you had always claimed. Moreover, you need to know the expectation of the total energy. How do you know this? By a single macroscopic measurement, not by identically preparing many cases. Thus the thermal interpretation is assumed to even make sense of the maximum entropy principle - not Born's rule!

And if you use $H^2$ as the one known variable you get from max entropy a ridiculous density operator that does not match experiment. Thus the max entropy principle depends on what you believe is measured macroscopically. The correct result only comes out if you believe that the ensemble expectation of $H$ is measured - i.e. if you believe the thermal interpretation.

 

[vanhees71]

The very formulation of the meaning of the statistical operator is based on Born's rule.

A argument for using $\langle H \rangle$ rather than $\langle H^2 \rangle$ (or any other non-linear function of $H$) might be that isolated systems should be uncorrelated, i.e., if I consider two non-interacting systems and look for equilibrium I should use additive conserved quantities in the entropy principle, e.g., energy. Then you have
$$\hat{H}=\hat{H}_1 + \hat{H}_2, \quad [\hat{H}_1,\hat{H}_2]=0,$$
and thus from the maximum-entropy principle
$$\hat{\rho}=\frac{1}{Z_1 Z_2} \exp(-\beta_1 \hat{H}_1) \exp(-\beta_2 \hat{H}_2).$$
If both systems are coupled to the same common "heat bath", of course, you have necessarily $\beta_1=\beta_2=1/T$, where $T$ is the temperature of the heat bath. This also leads to an additive entropy and additivity of all extensive thermodynamical quantities.

 

[A. Neumaier]

The very formulation of the meaning of the statistical operator is based on Born's rule.

Only in the vaguest sense, involving measurements never performed, and hence not subject to Born's rule.

But I see that your usage of the terms is so vague that it is impossible to discuss this with you. Effectively you are working in a shut up and calculate mode and invoke whatever interpretation appears to be needed to match predictions with experiment, but you use the catch word ''Born's rule'' (without actually using the rule) to justify what you do using hand-waving words - not logic, in terms of which nothing of this is justified.

With this hand-waving attitude there are no foundational problems at all, since they are all swept under the carpet of vagueness and imprecision in the usage of the language. On this level a fruitful discussion of foundations is impossible; we are just going in circles. The foundational problems appear only when each of the terms used gets a fixed meaning and arguments are based on that meaning only. Then the presence of problems that you don't see becomes obvious to anyone who cares.

 

[vanhees71]

It's you who doesn't define clearly what you mean with your "expectation values", if I'm not allowed to think in terms of probability theory, not me! The Born rule is very clear, and it has nothing to do with "shutup and calculate". It's one of the basic postulates (in my opinion indispensible) to relate the formalism of the theory to what's measured in the lab, and I use it in the usual textbook way to describe observations.

 

[A. Neumaier]

It's you who doesn't define clearly what you mean with your "expectation values",

I don't understand your criticism. A mathematical definition is the most precise definition one can give of anything. The interpretation depends on the application, and applied to macroscopic observable, it is very clearly defined that it means the actual value within its intrinsic uncertainty. This is enough to deduce the probabilistic interpretation in cases where it applies (sufficiently many independent replications of an otherwise very uncertain measurement).

The Born rule is very clear, and it has nothing to do with "shut up and calculate".

I agree. But the way you invoke the Born rule as being applied whenever the word probability or expectation appears is has nothing to do with the Born rule as given in the usual treatment, but is only camouflaged "shut up and calculate".

In particular, as discussed above, measuring energies in the lab is never done according to the description of a measurement according to Born's rule as given in the usual textbooks. But we are going again in circles...

 

[vanhees71]

I don't understand your criticism. A mathematical definition is the most precise definition one can give of anything. The interpretation depends on the application, and applied to macroscopic observable, it is very clearly defined that it means the actual value within its intrinsic uncertainty. This is enough to deduce the probabilistic interpretation in cases where it applies (sufficiently many independent replications of an otherwise very uncertain measurement).


I agree. But the way you invoke the Born rule as being applied whenever the word probability or expectation appears is has nothing to do with the Born rule as given in the usual treatment, but is only camouflaged "shut up and calculate".

In particular, as discussed above, measuring energies in the lab is never done according to the description of a measurement according to Born's rule as given in the usual textbooks. But we are going again in circles...

Maybe, I've not found it in your large book, but I haven't seen a clear definition of the meaning of expectation values, because you explicitly deny the usual probabilistic meaning. Now you say you want to derive it. Is it so difficult to give a clear definition of what your expectation value means, if not to be read in the usual probabilistic sense?

Ok, let's define one last time, what's understood as Born's rule. It's the probabilistic interpretation of the meaning of "quantum state" no more no less:

A quantum state is represented by a positive semi-definite self-adjoint operator $\hat{\rho}$ with $\mathrm{Tr} \hat{\rho}=1.$

An observable $A$ is represented by a self-adjoint operator $\hat{A}$. The possible outcome of measurements of $A$ are the eigenvalues of the operator $\hat{A}$. Let $|a,\beta \rangle$ denote a complete set of orthonormalized eigenvectors of eigenvalue $a$, then the probality to measure the value $a$, if the system is prepared in the state described by $\hat{\rho}$ is
$$P(a)=\sum_{\beta} \langle a,\beta|\hat{\rho}|a,\beta \rangle.$$

It's a simple corrollary of this postulate that expectation values are (basis-independently!) given by
$$\langle A \rangle = \mathrm{Tr}(\hat{\rho} \hat{A}).$$
Now, even if you deny the probabilistic meaning given that the expectation value (which for me implies a probablistic meaning, because where else than in probability theory does the notion of an "expectation value" make sense?), you can reconstruct the probabilities from that rule, because you can define the projection operator to the eigenspace of $\hat{A}$ for eigenvalue $a$,
$$\hat{P}(a)=\sum_{\beta} |a,\beta \rangle \langle a,\beta|,$$
as an observable, and then you have of course
$$P(a) = \langle \hat{P}(a) \rangle=\mathrm{Tr}[\hat{\rho} \hat{P}(a)].$$
It's also clear, how to generalize all this for spectral values of $\hat{A}$ in the continuum. Then the sums become integrals as usual. I'm aware that for a mathematically rigorous treatment it's not that easy, but we discuss the physics here rather than the mathematical rigorous foundation of (non-relativistic) QT.

So my question is, how do you in your interpretation make sense of $\hat{\rho}$ if not this usual probabilistic one? What, then, is the meaning of expectation value defined by the trace? Why should such a complication be necessary for an interpretation superior to the standard one?

Note that, according to the book by Peres, also the generalized "incomplete measurement protocos" in terms of POVMs are derivable from the above summarized standard Born rule. So the standard Born rule is at least sufficient to include these more general modern notions of measurements.

 

[PeterDonis]

The possible outcome of measurements of $A$ are the eigenvalues of the operator $^\hat{A}$. Let $|a,\beta \rangle$ denote a complete set of orthonormalized eigenvectors of eigenvalue $a$, then the probability to measure the value $a$, if the system is prepared in the state described by $^\hat{\rho}$ is...

This I agree is Born's rule.

It's a simple corrollary of this postulate that expectation values are (basis-independently!) given by...

This is not. Probabilities are not expectation values, and Born's rule itself says nothing about expectation values.

Also, since you have already said that the possible outcome of measurements are eigenvalues, and the expectation value is not an eigenvalue (except in the special case that the state $\hat{\rho}$ happens to be an eigenstate of the operator $\hat{A}$), the expectation value clearly cannot be the outcome of a measurement, if we believe that Born's rule applies to all measurements--which you appear to be claiming. But @A. Neumaier has described measurements whose outcomes are not eigenvalues but expectation values. So it seems like Born's rule cannot apply to such measurements, which means statements about expectation values cannot be part of Born's rule.

 

[A. Neumaier]

So my question is, how do you in your interpretation make sense of ^ρρ^\hat{\rho} if not this usual probabilistic one? What, then, is the meaning of expectation value defined by the trace?

I had explained it multiple times:

Though traditionally called an ensemble expectation value, a more natural name - not suggesting a priori a probabilistic interpretation - for $\bar A=\langle A\rangle:=Tr \rho A$ would be the uncertain value of $A$. Quoting mostly from my web page, its physical meaning in general (without reference to probability or even measurement) is defined by the the following simple rule generalizing statistical intuition to situations where uncertainty is not required to be probabilistic:

Uncertainty principle: A Hermitian quantity $A$ whose uncertainty $\sigma_A:=\sqrt{\langle(A-\bar A)^2\rangle}$ is much less than $|\bar A|$ has the value $|\bar A|$ within an uncertainty of $\sigma_A$.

This is a very clear, practical principle. Physicists doing quantum mechanics (even those adhering to the shut-up-and-calculate mode of working) use this principle routinely and usually without further justification. The principle applies universally. No probabilistic interpretation is needed, so it applies also to single systems.

From this principle one can derive under appropriate conditions (see my online book) the following rule:

Measurement rule: Upon measuring a Hermitian operator $A$ in the state $\rho$, the measured result will be approximately $\bar A$, with an uncertainty at least of the order of $\sigma_A$. If the measurement can be sufficiently often repeated (on a system with the same or a sufficiently similar state $\rho$) then $\sigma_A$ will be a lower bound on the standard deviation of the measurement results.

Actually the above measurement rule should be considered as a definition of what it means to have a device measuring $A$. As such it creates the foundation of measurement theory. In order that a macroscopic quantum device qualifies for the description ''it measures $A$'' it must either be derivable from quantum mechanics, or checkable by experiment, that the property claimed in the above measurement rule is in fact valid. Thus there is no circularity in the foundations.

Moreover, Born's famous rule turns out to be derivable, too, (see my online book) but under special circumstances only, namely those where the Born rule is indeed valid in practice. (Though usually invoked as universally valid, Born's rule has severe limitations. It neither applies to position measurements nor to photodetection, nor to measurement of energies, just to mention the most conspicuous misfits.)

 

[vanhees71]

This I agree is Born's rule.
This is not. Probabilities are not expectation values, and Born's rule itself says nothing about expectation values.

Also, since you have already said that the possible outcome of measurements are eigenvalues, and the expectation value is not an eigenvalue (except in the special case that the state $\hat{rho}$ happens to be an eigenstate of the operator $\hat{A}$), the expectation value clearly cannot be the outcome of a measurement, if we believe that Born's rule applies to all measurements--which you appear to be claiming. But @A. Neumaier has described measurements whose outcomes are not eigenvalues but expectation values. So it seems like Born's rule cannot apply to such measurements, which means statements about expectation values cannot be part of Born's rule.

Ok, if it is not accepted here that probabilities can be defined as expectation values too, I try to forget this for a moment. It's not important for any argument. I hope we all agree that if the $P(a)$ for finding $a$ when measuring $A$ are given, the expectation value is
$\langle A \rangle=\sum_a a P(a)=\sum_{a,\beta} \langle a,\beta \hat{\rho} \hat{A} a,\beta \rangle=\mathrm{Tr}(\hat{\rho} \hat{A})$.
Of course, expectation values need not be eigenvalues of the corresponding operator. How do you come to that idea?

 

[vanhees71]

I had explained it multiple times:

Though traditionally called an ensemble expectation value, a more natural name for $\bar A=\langle A\rangle:=Tr \rho A$ (not suggesting a probabilistic interpretation a priori) would be the uncertain value. Quoting mostly from my web page, its physical meaning in general (without reference to probability or even measurement) is defined by the the following simple rule generalizing statistical intuition to situations where uncertainty is not required to be probabilistic:

Uncertainty principle: A Hermitian quantity $A$whose uncertainty $\sigma_A:=(A-\bar A)^2$ is much less than $|\bar A|$ has the value $|\bar A|$ within an uncertainty of $\sigma_A$.

This is a very clear and definite rule. Physicists doing quantum mechanics (even those adhering to the shut-up-and-calculate mode of working) use this rule routinely and usually without further justification. The rule applies universally. No probabilistic interpretation is needed, so it applies also to single systems.

This I don't understand. How are expectation values, including the standard deviation (there should be square root to meet the usual definition, i.e., $\sigma_A=\sqrt{\langle (A-\bar{A})^2}$), defined if not within probability theory? For me the notion of an expectation value is defined within some probability theory (e.g., the standard Kolomogorov axioms, which are for sure good enough for our discussion).

From this rule one can derive under appropriate conditions (see my online book) the following rule; the derivation is in my online book:

Measurement rule: Upon measuring a Hermitian operator $A$ in the state $\rho$, the measured result will be approximately $\bar A$, with an uncertainty at least of the order of $\sigma_A$. If the measurement can be sufficiently often repeated (on a system with the same or a sufficiently similar state $\rho$) then $\sigma_A$ will be a lower bound on the standard deviation of the measurement results.

I don't understand this, if I'm not allowed to think in terms of probability theory and the Law of Large Numbers, which is a one key result of probability theory. If I need to read an entire book for that, I'd like to know, which advantage it should have to redefine all the clear definitions used in the empirical sciences for centuries now!

Actually the above measurement rule should be considered as a definition of what it means to have a device measuring $A$. As such it creates the foundation of measurement theory. In order that a macroscopic quantum device qualifies for the description ''it measures $A$'' it must either be derivable from quantum mechanics, or checkable by experiment, that the property claimed in the above measurement rule is in fact valid. Thus there is no circularity in the foundations.

This I understand :-))). Of course, measurement apparati must be tested and calibrated to make sense. That's not a mathematical but an engineering task for experimentalists in the lab.

Moreover, Born's famous rule turns out to be derivable, too, (see my online book) but under special circumstances only, namely those where the Born rule is indeed valid in practice. (Though usually invoked as universally valid, Born's rule has severe limitations. It neither applies to position measurements nor to photodetection, nor to measurement of energies, just to mention the most conspicuous misfits.)

For me Born's rule very well applies to position measurements and photodetection. It's used in any book of quantum optics to describe photodetection within quantum theory. Why it shouldn't apply to position measurements, I also don't see (of course it cannot apply to photons, because you cannot even define a position observable in the usual sense). For massive particles, I don't see a problem to measure its position by simply putting a detector at a given place. Of course any such device has a finite resolution. To validate a given probability distribution for position, of course your device's resolution must be much better than the standard deviation of the probability distribution you want to measure, but I don't see a principle problem to measure position with arbitrary position.