Evaluate this paper on the derivation of the Born rule

[RockyMarciano]

The reason is simply that the same definition does not imply the same results if the dynamical rules to which the definition applies are different.

So I guess you mean by this that the quantum dynamical rules answer my questions, how?

Moreover, even classically, measurements are often not predictable over a significant time scale. Classical Brownian motion (a dust particle in a fluid) is intrinsically undetermined, classically, since the initial state cannot be known to the accuracy required.

Yes, that's why I wrote "in principle", i.e. if that initial state was known it would be predictable, this is not the case in the quantum theory.

 

[A. Neumaier]

that the quantum dynamical rules answer my questions, how?

The relation between the pointer position of a macroscopic apparatus measuring the position of a particle, say, that can be inferred from the dynamics is never completely deterministic, not even in the classical case where the dynamics of the combined system is fully deterministic. See https://www.physicsforums.com/posts/5668841/

Similarly, the quantum rules predict that a reading of a quantum measurement tells nothing deterministic about the state of a single spin, unless the measurement is set up to measure the spin in exactly the same direction as the spin is prepared.

 

[RockyMarciano]

The relation between the pointer position of a macroscopic apparatus measuring the position of a particle, say, that can be inferred from the dynamics is never completely deterministic, not even in the classical case where the dynamics of the combined system is fully deterministic.

I already addressed this. In the classical case is not completely deterministic due to a practical imposibility to know the initial conditions completely, once more this is not the case in quantum mechanics, the question was how the dynamics justify this indeterminacy in the absence of the classical initial condition excuse, and also the different probabilities if not recurring to a postulate like the Born rule(however mysterious or vague it might be). I'm afraid just invoking the explanatory power of statistical mechanics for classical measurements is not enough.

 

[A. Neumaier]

. In the classical case is not completely deterministic due to a practical impossibility to know the initial conditions completely

Even when the initial state is fixed and the dynamics is deterministic, the information in the position of the particle at time t=0 is at no later time completely transmitted to the pointer of the detector. Thus the pointer cannot accurately reproduce the particle position.

How the Born rule appears in the quantum case is addressed by the papers we have been discussing here since post #1.

 

[Prathyush]

Is the clarification given in posts #85, #87, and #102 sufficient for you? Or what else needs to be clarified?

It is now clear enough for me to evaluate.

 

[RockyMarciano]

Even when the initial state is fixed and the dynamics is deterministic, the information in the position of the particle at time t=0 is at no later time completely transmitted to the pointer of the detector. Thus the pointer cannot accurately reproduce the particle position.

I'm asking you to explain why you claim this, it is not claimed nor explained in the classical theory(the classical pointer measures operationally by congruence between the measured object position and the measuring instrument or detector pointer, with all the uncertainty in the measure atributable to the initial state uncertainty) nor in statisitical mechanics that I'm aware of.

 

[RockyMarciano]

See https://www.physicsforums.com/posts/5668841/

Ok, I see there that you don't have an answer to my question, since you are asking it yourself. But then the only advantage I see for the statistical approach is that it makes explicit that the quantum formalism of the Born rule doesn't solve it either.

 

[A. Neumaier]

I see there that you don't have an answer to my question, since you are asking it yourself.

I had answered the question you posed. It is quite obvious that there is no natural way to make the pointer position give the exact value of the particle position, so it will be always somewhat inaccurate, which answers your question even without having the details in terms of a particular model.

it is not claimed nor explained in the classical theory(the classical pointer measures operationally by congruence between the measured object position and the measuring instrument or detector pointer,

There is no congruence on the microscopic level, as point particles do not form straight borders.

The question in the link given is different, since it asks for details how the pointer position gets its value, given the interaction.

 

[RockyMarciano]

I had answered the question you posed. It is quite obvious that there is no natural way to make the pointer position give the exact value of the particle position, so it will be always somewhat inaccurate, which answers your question even without having the details in terms of a particular model.

No, you hadn't. You conveniently left out of the quote the part where I say that classically that inaccuracy is attributed to the initial conditions lack of accurate knowledge. Now in the quantum case that is not an acceptable reason(the inherent HUP from noncommuting observables is, somewhat operationally encoded in the Born rule), but you seem not to be willing to recur to it in your interpretation using statistical mechanics reasoning.
So rather than stating the obvious about the fact that there will always be inaccuracy again, I'll ask you for the last time to explain the origin of the uncertainty according to your interpretation.

There is no congruence on the microscopic level, as point particles do not form straight borders..

straight borders??how is congruence achieved in geometry? Do points form straight borders?

The question in the link given is different

You should know, you linked it as an answer to my question.

 

[A. Neumaier]

You conveniently left out of the quote the part where I say that classically that inaccuracy is attributed to the initial conditions lack of accurate knowledge.

I had addressed this: My statement holds even when there is exact knowledge of the initial condition and exact deterministic dynamics. The problem is that the pointer is a complex system that cannot be made to exactly represent the position solely through the physical interactions.

how is congruence achieved in geometry? Do points form straight borders?

In geometry you have a small set of points, and congruence refers to matching these points by a rigid motion. But this is abstract mathematics.

In classical microscopic physics, there are no rigid objects, hence there is no way of performing a rigid motion. To measure the distance between two points one has to match the marked lines on a a macroscopic ruler (or a more sophisticated device) so that they approximate this distance, and this is never more exact than the width of the marked lines. Even if you do this under an electron microscope or another sophisticated device, you incur uncertainty, and it cannot be arbitrarily reduced, even when one would assume that the classical laws were valid down to arbitrarily small distances, since classical point atoms behave chaotically on small distances.

You should know, you linked it as an answer to my question.

I had referred to the last posting in this thread, where I had mentioned that

In fact, perfect information cannot be obtained. Whatever is obtained experimentally needs a justification why it deserves being called a particle position or momentum and how uncertain it is.

 

[RockyMarciano]

I had addressed this: My statement holds even when there is exact knowledge of the initial condition and exact deterministic dynamics. The problem is that the pointer is a complex system that cannot be made to exactly represent the position solely through the physical interactions.

Yes, ok, but I was trying to find out if you were claiming that the thermal interpretation gives an explanation to this complexity rather than just stating it. Now I know it doesn't, this is all I wanted to know.

In geometry you have a small set of points, and congruence refers to matching these points by a rigid motion. But this is abstract mathematics.

In classical microscopic physics, there are no rigid objects, hence there is no way of performing a rigid motion. To measure the distance between two points one has to match the marked lines on a a macroscopic ruler (or a more sophisticated device) so that they approximate this distance, and this is never more exact than the width of the marked lines. Even if you do this under an electron microscope or another sophisticated device, you incur uncertainty, and it cannot be arbitrarily reduced, even when one would assume that the classical laws were valid down to arbitrarily small distances, since classical point atoms behave chaotically on small distances.

I agree with this, again this complexity is not addressed by statisitcal mechanics in any way substantially better than by the Born rule, it is just a different rationalization of the difficulty that maybe makes more clear that this difficulty(that you address in the other thread linked) exists as much in the classic case as in the quantum but it is much more visible and problematic(infinite literature on the "measurement problem") in the latter that deals more specifically with the microscopic scale.

 

[A. Neumaier]

if you were claiming that the thermal interpretation gives an explanation to this complexity

The point of the thermal interpretation is to give better foundations to quantum mechanics than what Born's rule offers, more in line with what happens in actual measurements.

It was never intended to explain complexity. Complexity is a given and needs no explanation.

 

[RockyMarciano]

The point of the thermal interpretation is to give better foundations to quantum mechanics than what Born's rule offers, more in line with what happens in actual measurements.

That's a laudable intention.

It was never intended to explain complexity. Complexity is a given and needs no explanation.

This is highly debatable but it is off-topic here, maybe in the other thread.

 

[vanhees71]

I won't use the world collapse form now on, It has meanings that I don't intend. It is also very bad terminology. Let's use the following language from now on, We prepare a system in a state, described as $|\psi_{in}>$. The system was measured to be in a state described as $|\psi_{out}>$ with a probability $<\psi_{in}|\psi_{out}>^2$, When we use apparatus where we destroy the particle the appropriate clarification must be made. The wave function is our description of the system. What $|\psi_{in}>$ and $|\psi_{out}>$ are depend on the details of the experimental apparatus.

This must be non controversial to both of us.(Right?)

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 $A$).

Note that in general the time-evolution due to dynamics of state vectors and eigenvectors is different, depending on the chosen picture of time evolution. E.g., in the Schrödinger picture the states evolve according to the full Hamiltonian while the operators representing observables are time independent. In the Heisenberg picture it's the other way around, but you can arbitrarily split the time dependence between states and observable-operators. What's observable are the probabilities and related quantities like expectation values, and indeed the "wave function",
$$\psi(a,\beta)=\langle a,\beta|\psi \rangle$$
is picture independent. That only works if you properly formulate the probabilities for finding a certain result when measuring the observable, given the state $|\psi \rangle$ of the system!

 

[vanhees71]

The point of the thermal interpretation is to give better foundations to quantum mechanics than what Born's rule offers, more in line with what happens in actual measurements.

It was never intended to explain complexity. Complexity is a given and needs no explanation.

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?

 

[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!