How to derive Born's rule for arbitrary observables from Bohmian mechanics?

[A. Neumaier]

An observable is defined by a measurement procedure

Only approximately, as measurements are always afflicted with errors, and the ''definitions'' must be changed from time to time to better match the theory.

The true definitions of basic observables like position, momentum, angumar momentum and energy is given by theory, to which any measurement ''definition'' must be calibrated to deserve the designation as measurement of something. Already the modern definiton of a second refers to nontrivial theory to be even understood!

Section 3.3 of your paper [...] (18) is surely not an angular momentum!

It is the macroscopic observable that describes perceptible outcomes associated with a measurement of angular momentum.

No. By definition, it is the macroscopic observable associated with a measurement of the operator defined by (18), whatever the right hand side works out to be. To claim that it is a measurement of angular momentum you'd need to show that (18) equals a component of the angular momentum operator!

 

[A. Neumaier]

Suppose that (17) is not true. Then what else the right-hand side of (17) could be?

If Eq. (3) in my paper can be wrong, then what else the right-hand side of Eq. (3) could be?

It is not my responsibility to give an interpretation or modification of equations whose claimed validity or interpretation is found wanting.

@A. Neumaier to understand quantum theory of measurements. Hence I would suggest you to redirect your curiosity in this more general direction. For instance, you can start with https://arxiv.org/abs/1406.5535 with emphasis on Sec. 6.1. The author is a world renowned quantum experimentalist who also has a good understanding of the theory.

Steinberg's 2014 paper ''Quantum Measurements: a modern view for quantum optics experimentalists'' that you cited is alright.

But where does it justify your formula (3) or (17) as being valid for the interaction responsible for the measurement of an observable corresponding to an arbitrary operator given? Section 6.1 to which you refer does not contain any formula resembling it!

 

[Demystifier]

To claim that it is a measurement of angular momentum you'd need to show that (18) equals a component of the angular momentum operator!

No, to say that A measures K means that A is well correlated with K. In the ideal case, when (17) reduces to (11), A is perfectly correlated with K due to (10). In your case, K is the angular momentum operator.

 

[Demystifier]

It is not my responsibility to give an interpretation or modification of equations whose claimed validity or interpretation is found wanting.

It's not a matter of responsibility, it's a matter of desire to understand. You are not my referee, you are someone who wants to understand something and I am someone who is willing to help you. If you are claiming that the right-hand side might be wrong, then you have to be able to say what else the right-hand side might be. Otherwise, I don't know why exactly do you think that it might be wrong. If you tell me what else it might be, I will be able to tell you why it cannot. It's for your own good, it's not my intention to prove you are wrong, I just want to help you understand the meaning of those equations.

 

[Demystifier]

Steinberg's 2014 paper ''Quantum Measurements: a modern view for quantum optics experimentalists'' that you cited is alright.

But where does it justify your formula (3) or (17) as being valid for the interaction responsible for the measurement of an observable corresponding to an arbitrary operator given? Section 6.1 to which you refer does not contain any formula resembling it!

The paper has Eq. (41) which is essentially the same as my (9). So if you accept my (9), then I guess you should accept everything which follows from my (9), including the derivation of the Born rule in my (16). If you consider a special case of his (41), namely $c_1=1$, $c_2=c_3=\cdots =0$, then you obtain my (3) (with $k'=k$).

My (17) is a generalization of my (11), he does not consider such a generalization explicitly. But it is implicit in his Sec. 3, for if he decided to describe generalized measurements in terms of wave functions of the apparatus, he would arrive at something like my (17).

 

[vanhees71]

Only approximately, as measurements are always afflicted with errors, and the ''definitions'' must be changed from time to time to better match the theory.

Sure, that's why the SI units still have finite uncertainties in their practical realization. Again, we discuss physics here, not mathematical abstractions. Any measurement is only complete with a thorough estimate for the "statistical" and "systematical" errors!

You cannot define observables by pure math. Neither can you define observables without theory. It's a mutual "entangled" depency. E.g., to define the observable "position" you already need at least a model for "space". Galilei and Newton made a good guess with there absolute-time-absolute-space model, nowadays called Galilei space-time. This worked for quite a while (round about 300 years). Then with progress of measurement technique together with ingenious theoretical insight from Faraday (worked out mathematically by Maxwell) concerning electromagnetic phenomena the Galilei-space-time model turned out to be insufficient, and it was substituted with a more comprehensive relativistic space-time model (Minkwoski space in special and Einstein's pseudo-Riemannian manifold in general relativity theory). That's a paradigmatic example, how physics works as a mutual intertwined empirical and theoretical effort. A mathematical system of axioms is empty as a physical theory. E.g., our debates about POVMs show this: It's easy to formulate mathematically, but as long as it lacks concrete applications to real-world experimental setups it's useless for a physicist!

 

[A. Neumaier]

Steinberg's 2014 paper ''Quantum Measurements: a modern view for quantum optics experimentalists'' that you cited is alright.

But where does it justify your formula (3) or (17)

The paper has Eq. (41) which is essentially the same as my (9).

Indeed. But I was asking for a justification!

Equation (41) is not derived in the paper but postulated, motivated by an argument that cannot be made to work in general. And no use is made anywhere in the paper, showing that it is irrelevant for the experimentalist. Thus he doesn't need to bother about how correct it is.

@A. Neumaier by now it's quite clear that your true problem is [...] to understand quantum theory of measurements.

Rather, this seems to be your problem, since you take unchecked quotations from other work as the truth about the quantum theory of measurements!

A thorough discussion of the quantum theory of measurement is in Wigner's classic 'Interpretation of quantum mechanics', as printed in the reprint collection 'Quantum theory and measurement' by Wheeler and Zurek. On p.281, Wigner states equations (34), (35), which are more or less your (3) and (9), stating also the nondemolition qualification under which it is valid:

if the object did not change its state as a result of the measurement

This is an extremely strong restriction, as Wigner himself says on p.284:

the quantum-mechanical description of the measurement, embodied in (34) and (35), is a highly idealized description

In the subsequent discussion it appears that the nondemolition qualification is also necessary:

Only quantities which commute with all additively conserved quantities are precisely measurable

 

[Elias1960]

In an effective theory with cutoff there is no relativistic covariance, hence no relativistic symmetry, hence preferred frames are in principle observable.

Of course, the symmetry appears only in the large distance limit. Do you think this is somehow problematic? What we actually observe are, in comparison with the Planck length, very large distances. So, there should be no problem with this.

 

[A. Neumaier]

Please point to the page with the proof for the Born rule for general operators; I didn't see it there.

Part II sec. 2 Quantum theory of measurements p.180

Of course, this is a physical paper, thus, with physical requirements for proofs instead of mathematical.

Thanks. But p. 180 has no proof at all, and the outline at the top of the next page makes the (in general unwarranted, see post #42 above) assumption that one can neglect both the system Hamiltonian and the detector Hamiltonian and that one may therefore only consider the interaction term.

 

[A. Neumaier]

Of course, the symmetry appears only in the large distance limit. Do you think this is somehow problematic? What we actually observe are, in comparison with the Planck length, very large distances. So, there should be no problem with this.

Well, we are reaching experimentally smaller and smaller distances. Thus preferred frame effects should at some point become observable. When depends on the actual model you propose for an effective QED. None exists in the literature, there are only toy theories that significantly deviate from QED even in the large-distance limit.

 

[Demystifier]

In the subsequent discussion it appears that the nondemolition qualification is also necessary:

My analysis does not assume non-demolition because I allow that $|k'\rangle$ is not necessarily the same as $|k\rangle$.

 

[Demystifier]

But I was asking for a justification!

I have tried to explain you the justification in several ways, but you were not satisfied. You objected that it doesn't work for angular momentum, but I have not understood your objection. Can you try to rephrase your argument, why doesn't it work for angular momentum?

 

[A. Neumaier]

My analysis does not assume non-demolition because I allow that $|k'\rangle$ is not necessarily the same as $|k\rangle$.

But you assume that an eigenstate always changes into another (or the same) eigenstate, which is essentially as severe a restriction. Probably you cannot even give an example of a Hamiltonian where your formula results but k' does not equal k!

 

[vanhees71]

Indeed. But I was asking for a justification!

Equation (41) is not derived in the paper but postulated, motivated by an argument that cannot be made to work in general. And no use is made anywhere in the paper, showing that it is irrelevant for the experimentalist. Thus he doesn't need to bother about how correct it is.

Rather, this seems to be your problem, since you take unchecked quotations form other work as the truth about the quantum theory of measurements!

A thorough discussion of the quantum theory of measurement is in Wigner's classic 'Interpretation of quantum mechanics', as printed in the reprint collection 'Quantum theory and measurement' by Wheeler and Zurek. On p.281, Wigner states equations (34), (35), which are more or less your (3) and (9), stating also the nondemolition qualification under which it is valid:

This is an extremely strong restriction, as Wigner himself says on p.284:

In the subsequent discussion it appears that the nondemolition qualification is also necessary:

Sure, there's not a mathematical justification for (41) in this paper, but it provides precisely what I'm still lacking in explaining the meaning of POVMs. Now it would be great if somebody could write a paper merging this paper by a practitioning experimenter, providing the physical meaning of the formalism in an intuitive way such that he can work with them as an experimentalist, with the very abstract definitions of mathematical physicists, i.e., something for a phenomenological theoretical physicist like me.

The so far best treatment of measurement theory for quantum optics I've seen is the book

J. Garrison, R. Chiao, Quantum optics, Oxford University
Press, New York (2008).
https://dx.doi.org/10.1093/acprof:oso/9780198508861.001.0001
though there's no discussion about the POVM formalism but "only" the usual treatment of photon-detection measurements based on the standard Born rule. Obviously that's sufficient for most of the standard phenomenology in quantum optics covered (in my opinion very well) in this book.

Is there something out there, which explains the POVM formalism on this level and with this notation? That would be really very helpful!

 

[vanhees71]

I have tried to explain you the justification in several ways, but you were not satisfied. You objected that it doesn't work for angular momentum, but I have not understood your objection. Can you try to rephrase your argument, why doesn't it work for angular momentum?

In this connection: Does anybody have a reference to a Stern-Gerlach measurement, where not the magnetic moment of a spin-1/2 angular momentum has been measured but some higher angular-momentum state, like an atomic SGE with atoms of larger total $\vec{j}$?

 

[Demystifier]

But you assume that an eigenstate always changes into another (or the same) eigenstate, which is essentially as severe a restriction.

No, I don't assume that.

Probably you cannot even give an example of a Hamiltonian where your formula results but k' does not equal k!

Of course I can. For example, when $|k\rangle$ is a photon momentum eigenstate, then $|k'\rangle=|0\rangle$, the photon vacuum. That's because the measurement of photon destroys the photon.

 

[A. Neumaier]

I have tried to explain you the justification in several ways, but you were not satisfied.

You only justified it by (i) calling it standard in measurement theory, (ii) referring to other papers which were not justifiying it either, and (iii) by a request that I should provide corrected right hand sides for your equations. Of course, neither is a satisfying justification.

You objected that it doesn't work for angular momentum, but I have not understood your objection. Can you try to rephrase your argument, why doesn't it work for angular momentum?

That it doesn't work for angular momentum was just a guess. I found a more detailed discussion of the angular momentum case in London and Bauer (also reprinted in Wheeler and Zurek, whose page numbers I am using). But precisely at the point where I'd have needed details the argument given there is incomplete: On p.251 they claim your equation (9) with $v_k(y)$ being eigenfunctions of the pointer, and refer to Section 12 for the argument. But there $v_k(y)$ denotes instead an arbitrary time-dependent function of $y$ (defined in the footnote * on p.254) satisfying (4). Thus there is a gap in their discussion of the measurement of angular momentum.

After having checked the literature myself, my most important objection is that the derivation of the formula (3) or (9) you assume is based on the far too strong assumption of nondemolition.

 

[vanhees71]

In the SGE the "pointer" is the particle's position, right? If you accept this, it's the most simple example for a measurement describable completely by quantum dynamics (sic!), i.e., the motion of a neutral particle with a magnetic moment through an inhomogeneous magnetic field!

 

[A. Neumaier]

No, I don't assume that.

So the $|k'\rangle$ are not the eigenstates of $K$ for the eigenvalue $k'$? This is not what the notation would have suggested.

But you still assume without argument that separable states remain separable, while the evolution of a separable state by a general interacting Hamiltonian destroys separability. Thus you need a justification for this separability. The references you gave give none.

 

[Demystifier]

my most important objection is that the derivation of the formula (3) or (9) you assume is based on the far too strong assumption of nondemolition.

Fine, but I explained how my (minor) generalization avoids this assumption without significantly affecting the further analysis. Again, if you think that nondemolition could have a more radical change of the right-hand side of (3), then you should be able to write down how do you imagine that it might look like.

 

[A. Neumaier]

Does anybody have a reference to a Stern-Gerlach measurement, where not the magnetic moment of a spin-1/2 angular momentum has been measured but some higher angular-momentum state, like an atomic SGE with atoms of larger total $\vec{j}$?

London and Bauer (see post #52) discuss an atom with arbitrary spin, though (as mentioned) the discussion contains a gap.

 

[vanhees71]

So the $|k'\rangle$ are not the eigenstates of $K$ for the eigenvalue $k'$? This is not what the notation would have suggested.

But you still assume without argument that separable states remain separable, while the evolution of a separable state by a general interacting Hamiltonian destroys separability. Thus you need a justification for this separability. The references you gave give none.

Indeed for a measurement you need precisely the opposite: A good measurement entangles the measured observable with the pointer state of the apparatus!

 

[Demystifier]

So the $|k'\rangle$ are not the eigenstates of $K$ for the eigenvalue $k'$? This is not what the notation would have suggested.

You are right about that, I should have choosen a different notation or make an additional comment in the paper. But the paper is now accepted for publication in that form, so I will not change it.

But you still assume without argument that separable states remain separable, while the evolution of a separable state by a general interacting Hamiltonian destroys separability. Thus you need a justification for this separability. The references you gave give none.

By separable, do you actually mean factorizable? If so, then you are of course right that general Hamiltonian destroys it. But we are not studying a general Hamiltonian. We are studying a special Hamiltonian, chosen such that it serves as a measurement of K. A Hamiltonian that radically destroys that property could not be interpretd as measurement of K.

 

[A. Neumaier]

Fine, but I explained how my (minor) generalization avoids this assumption without significantly affecting the further analysis.

You assume a generalization of a conclusion of an analysis that holds only under the assumption of nondemolition. This does not imply that your generalization holds without the assumption of nondemolition, because this is not the way logic works.

To prove your formula you need to repeat the standard argument without the nondemolition assumption and show that your generalized formula still follows.

if you think that nondemolition could have a more radical change of the right-hand side of (3), then you should be able to write down how do you imagine that it might look like.

Without assuming nondemolition, (3) would look like
$$|k\rangle|A_0\rangle \to \sum_{k'} |k'\rangle|A_{kk'}\rangle,$$
summed over a complete set of basis vectors $|k'\rangle$ (which might or might not be the original basis).

 

[vanhees71]

London and Bauer (see post #52) discuss an atom with arbitrary spin, though (as mentioned) the discussion contains a gap.

I look for an experimental (!) paper!

 

[Demystifier]

Without assuming nondemolition, (3) would look like
$$|k\rangle|A_0\rangle \to \sum_{k'} |k'\rangle|A_{kk'}\rangle,$$
summed over a complete set of basis vectors $|k'\rangle$ (which might or might not be the original basis).

Consider $|A_{kq}\rangle$ and $|A_{kp}\rangle$ for $q\neq p$. Are $|A_{kq}\rangle$ and $|A_{kp}\rangle$ macroscopically distinguishable? My further analysis will depend on your answer.

 

[A. Neumaier]

By separable, do you actually mean factorizable?

Yes; for a state, separable and factorizable are synonymous.

But we are not studying a general Hamiltonian. We are studying a special Hamiltonian, chosen such that it serves as a measurement of K. A Hamiltonian that radically destroys that property could not be interpreted as measurement of K.

Yes, but it must be shown that there are reasonable Hamiltonians that represent (i) a sensible system dynamics in isolation, (ii) a sensible detector dynamics in isolation, and (ii) sensible (physically realizable) interaction terms mimicking the measurement setting for an arbitrary system observable.
And it must be shown that these Hamiltonians actually produce the form (3) or (9) you want to assume for the subsequent analysis.

That this analysis is necessary but lacking in your paper is visible from the fact that you get the fully decohered result (15) by a very simple and general argument, while papers on decoherence must work quite hard to prove decoherence for given various simple idealized measurement settings.

Thus arriving at decoherence is quite nontrivial - it is indeed the only real difficulty of decoherence theory. Not that it cannot be done in particular settings, but it is done with sophisticated machinery, not with the tools of the 1930s that you employ. I haven't seen a decoherence analysis for measuring arbitrary system operators. Should you know one, it might solve the problem, and I'd be very interested in a reference.

 

[A. Neumaier]

In the SGE the "pointer" is the particle's position, right? If you accept this, it's the most simple example for a measurement describable completely by quantum dynamics (sic!), i.e., the motion of a neutral particle with a magnetic moment through an inhomogeneous magnetic field!

It is the recorded particle position (i.e., with irreversible amplification of the result). Otherwise the whole experiment is unitary and one is stuck with superpositions and no measurement.

 

[A. Neumaier]

Consider $|A_{kq}\rangle$ and $|A_{kp}\rangle$ for $q\neq p$. Are $|A_{kq}\rangle$ and $|A_{kp}\rangle$ macroscopically distinguishable? My further analysis will depend on your answer.

I don"t know. I just know what to expect in general, and that it is nontrivial to prove that something more specific happens.

Thus to decide this is for you to analyse for a specific Hamiltonian modeling the desired measuring process. It would be part of the required justification.

 

[A. Neumaier]

I look for an experimental (!) paper!

Ah, this was not clear from your query. I don't know about Stern-Gerlach, perhaps
https://arxiv.org/abs/cond-mat/0401526?

But you should look at

• Franke‐Arnold, Allen & Padgett, Advances in optical angular momentum. Laser & Photonics Reviews, 2 (2008), 299-313.

The optical version produces nice pictures; I have seen live demonstrations!

 

[A. Neumaier]

I look for an experimental (!) paper!

I don't know about Stern-Gerlach, perhaps
https://arxiv.org/abs/cond-mat/0401526?

Actually, Stern-Gerlach experiments for higher spin started with

• Breit & Rabi, Measurement of nuclear spin, Phys. Review 38 (1931), 2082.

In 1944, Rabi received the Nobel price for related work. I also looked at

• Bellamy & Smith, IV. The nuclear spins and magnetic moments of 24Na, 42K, 86Rb, 131Cs and 134Cs. The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science 44 (1953) , 33-45.

It contains results for the fine structure of spin 1/2 nuclei; the 2 levels of the textbook treatment split due to relativistic corrections.

 

[vanhees71]

Great! Thanks!

 

[A. Neumaier]

We are studying a special Hamiltonian, chosen such that it serves as a measurement of K.

I haven't seen a decoherence analysis for measuring arbitrary system operators. Should you know one, it might solve the problem, and I'd be very interested in a reference.

Specifically, what is lacking is a valid argument that shows that, for some Hamiltonian with a sensible physical interpretation (i)-(iii) as stated in post #62, the pointer states have decohered, i.e., (15) is approximately true. If this were shown it would indeed follow by your argument that the Bohmian pointer positions reproduce Born's rule for the measurement of $K$.

 

[Demystifier]

Yes; for a state, separable and factorizable are synonymous.

Yes, but it must be shown that there are reasonable Hamiltonians that represent (i) a sensible system dynamics in isolation, (ii) a sensible detector dynamics in isolation, and (ii) sensible (physically realizable) interaction terms mimicking the measurement setting for an arbitrary system observable.
And it must be shown that these Hamiltonians actually produce the form (3) or (9) you want to assume for the subsequent analysis.

That this analysis is necessary but lacking in your paper is visible from the fact that you get the fully decohered result (15) by a very simple and general argument, while papers on decoherence must work quite hard to prove decoherence for given various simple idealized measurement settings.

Thus arriving at decoherence is quite nontrivial - it is indeed the only real difficulty of decoherence theory. Not that it cannot be done in particular settings, but it is done with sophisticated machinery, not with the tools of the 1930s that you employ. I haven't seen a decoherence analysis for measuring arbitrary system operators. Should you know one, it might solve the problem, and I'd be very interested in a reference.

OK, now we more or less agree. You are right that actually proving the existence of appropriate decoherence is nontrivial. I have assumed it, not proved it. But I hope you will agree that, from what is already known about decoherence (analytically solved toy models and numerically solved more complicated models), the assumption of existence of appropriate decoherence is a rather plausible and reasonable assumption. If it would turn out that a more rigorous analysis shows nonexistence of appropriate decoherence, it would be very surprising. So, strictly speaking, I have not rigorously proved my claim, but I have given a very plausible argument for that.

 

[Demystifier]

I don"t know. I just know what to expect in general, and that it is nontrivial to prove that something more specific happens.

Thus to decide this is for you to analyse for a specific Hamiltonian modeling the desired measuring process. It would be part of the required justification.

Well, I am sure that they are not macro distinct for some Hamiltonians but macro distinct for other Hamiltonians. I have no intention to explicitly study the evolution by such Hamiltonians, but I can easily explain what are the physical consequences of each case.

First let me discuss the aspects which are common to both cases. Instead of my Eq. (3), more generally we have
$$|k\rangle|A_0\rangle \rightarrow \sum_q a_q |q\rangle |A_{kq}\rangle$$
where, due to unitarity,
$$\sum_q |a_q|^2=1$$
Hence, due to linearity, we have
$$\sum_k c_k |k\rangle|A_0\rangle \rightarrow \sum_k c_k \sum_q a_q |q\rangle |A_{kq}\rangle \equiv |\Psi\rangle$$

Now consider the case in which $|A_{kq}\rangle$ with the same $k$ but different $q$ are macro distinct. This means that one value of $k$ may result in more than one different measurement outcomes, so in this case the interaction cannot be interpreted as a measurement of $K$. Nevertheless, it is still some kind of measurement (because we do have some distinguishable measurement outcomes) . In fact, it is a generalized measurement discussed in Sec. 3.3 of my paper. To see this, let us introduce the notation
$$(k,q)\equiv l, \;\;\; c_ka_q \equiv \tilde{c}_l, \;\;\; |q\rangle\equiv|R_l\rangle$$
With this notation, the $|\Psi\rangle$ above can be written as
$$|\Psi\rangle = \sum_l \tilde{c}_l |R_l\rangle |A_l\rangle$$
which is nothing but Eq. (17) in my paper.

Now consider the case in which $|A_{kq}\rangle$ with the same $k$ but different $q$ are not macro distinct. We write $|\Psi\rangle$ as
$$|\Psi\rangle = \sum_k c_k |\Psi_k\rangle $$
where
$$|\Psi_k\rangle \equiv \sum_q a_q |q\rangle |A_{kq}\rangle$$
In the multi-position representation we have
$$\Psi(\vec{x},\vec{y})=\sum_k c_k\Psi_k(\vec{x},\vec{y})$$
where
$$\Psi_k(\vec{x},\vec{y})=\sum_q a_q \psi_q(\vec{y}) A_{kq}(\vec{x})$$
Using the Born rule in the multi-position space we have
$$\rho(\vec{x},\vec{y}) =|\Psi(\vec{x},\vec{y})|^2
\simeq \sum_k|c_k|^2 |\Psi_k(\vec{x},\vec{y})|^2$$
In the second equality we have assumed that $A_{kq}(\vec{x})$ are macro distinct for different $k$, which we must assume if we want to have a system that can be interpreted as a measurement of $K$. Hence we obtain
$$\rho^{\rm (appar)}(\vec{x})=\int d\vec{y} \rho(\vec{x},\vec{y})
\simeq \sum_k|c_k|^2 \sum_q |a_q|^2 |A_{kq}(\vec{x})|^2$$
where we have used the orthogonality of the $|q\rangle$ basis in the form
$$\int d\vec{y} \psi^*_{q'}(\vec{y}) \psi_q(\vec{y}) =\delta_{q'q}$$
Finally, by denoting with $\sigma_k$ the region in the $\vec{x}$-space in which all $A_{kq}(\vec{x})$ with the same $k$ are non-negligible, we have the probability
$$p_k^{\rm (appar)}=\int_{\sigma_k} d\vec{x} \rho(\vec{x})
\simeq|c_k|^2 \sum_q |a_q|^2 \int_{\sigma_k} d\vec{x} |A_{kq}(\vec{x})|^2
\simeq|c_k|^2 \sum_q |a_q|^2 = |c_k|^2$$
which is the derivation of the Born rule in the $k$-space from the Born rule in the multi-position space. This derivation is nothing but a straightforward generalization of the derivation in Sec. 3.2 of my paper. The point is that the derivation works even when my (3) is replaced by a more general relation as you suggested.