Quantum Properties

Should we understand quantum systems to have definite properties? In quantum interpretations values are usually taken to be the eigenvalues directly revealed in experiments and quantum systems generally have no definite eigenvalues. However, Sunny Auyang argues that this does not mean that they don’t have definite properties. The conclusion that they don’t arises from a restricted sense of what counts as a property. The conceptual structure of quantum mechanics is much richer and an expanded notion of properties facilitates an understanding of quantum properties that are more descriptive and structurally sophisticated.

One of the philosophical problems prompted by quantum mechanics is the nature of quantum properties and whether quantum systems can even be said to have properties. This is an issue addressed by Sunny Auyang in her book How is Quantum Field Theory Possible? And I will be following her treatment of the subject here.

One of the major contributors to the development of quantum mechanics, physicist Neils Bohr, whose grave I happened to visit when I was in Copenhagen, said: “Atomic systems should not even be thought of as possessing definite properties in the absence of a specific experimental setup designed to measure these properties.” Why is that? A lot of this hinges on what counts as a property, which is a matter of convention. For the kinds of things Bohr had in mind he was certainly right. But Auyang argues that it’s useful retain the notion and instead locate quantum properties in different kinds of things, in a way Bohr very easily could have agreed with.

Why are the kinds of things Bohr had in mind not good candidates as definite quantum properties? The upshot, before getting into the more technical description, is that in quantum systems properties like position don’t seem to have definite values prior to observation. As an example, in chemistry the electrons bound in atoms and molecules are understood to occupy orbitals, which are regions of space with probability densities. Rather than saying that a bound electron is at some position we say it has some probability to be at some position. If we think of a definite property as being something like position you can see why Bohr would say an atomic system doesn’t have definite properties in the absence of some experiment to measure it. Atomic and molecular orbitals don’t give us a definite property like position.

Auyang takes these kinds of failed candidates for definite properties to be what in quantum mechanics are called eigenvalues. And this will require some background. But to give an idea of where we’re going, Auyang wants to say that if we insist that properties are what are represented by eigenvalues then it is true that quantum systems do not have properties. However, she is going to argue that quantum systems do have properties, they are just not their eigenvalues; we have to look elsewhere to for such properties.

In quantum mechanics the characteristics of a quantum system are summarized by a quantum state. This is represented by a state vector or wave function, usually with the letter φ. A vector is a quantity that has both magnitude and direction. Vectors can be represented by arrows on a graph. So in a two dimensional graph the arrow would go from the center origin out into what is called the vector space. In two dimensions you could express the vector in terms of the horizontal and vertical axes; and the vector space would just be the plane these sweep out or span. It’s common to represent this in two, maybe three dimensions, but it’s actually not limited to that number; a vector space can have any number of dimensions. Whatever number of dimensions it has it will have a corresponding number of axes, which are more technically referred to as basis vectors. Quantum mechanics makes use of a special kind of vector space called a Hilbert space. This is also the state space of a quantum system. So recall that the description of the quantum system is its state, and this is represented by a vector. The state space then covers all permissible states that this quantum system can have.

Let’s limit this to two dimensions for the sake of visualization. And we can refer here to the featured image for this episode, which is a figure from Auyang’s book. We have a vector |φ> in a Hilbert Space with the basis, vectors {|α1>, |α2>}. So for this Hilbert Space |α1> and |α2> are basis vectors that serve as a coordinate system for this vector space. This is the system but it’s not what we interact with. For us to get at this system in some way we need to run experiments. And this also has a mathematical representation. What we get out of the system are observables like energy, position, and momentum, to name a few. Mathematically observables are associated with operators. An operator is a kind of linear transformation. Basically an operator transforms the state vector in some way. As a transformation, an operator usually maps one state into another state. But for certain states an operator will only result in the same state multiplied by some scaling factor. So let’s take some operator, upper case A, and have it operate on state |φ>. The result is a factor, lower case α multiplied by the original state |φ>. We can write this as:

A|φ> = α|φ>

In this kind of equation the vector |φ> is called an eigenvector and the factor α is called an eigenvalue. The prefix eigen- is adopted from the German word eigen for “proper”, “characteristic”, “own”, in reference to the fact that the original state or eigenvector is the same on both sides of the equation. In quantum mechanics this eigenvector is also called an eigenstate.

Now, getting back to quantum properties, I mentioned before that Auyang takes the kind of definite properties that quantum systems are understood not to have prior to observation to be eigenvalues. Eigenstates are certainly observed and corresponding eigenvalues measured in experiments. But the issue is of properties of the quantum system itself. Any given eigenvalue has only a certain probability of being measured, among the probabilities of other eigenvalues. So any single eigenvalue can’t be said to be characteristic of the whole quantum system.

Let’s go back to the two-dimensional Hilbert space with state vector |φ> and basis vectors |α1> and |α2>. The key feature of basis vectors is that every vector in the vector space can be written as a linear combination of those vectors. That’s how they act as a coordinate system. So if we take our vector |φ> we can break it down into two orthogonal (right angle) components, in this case the horizontal and vertical components, and then the values for the coefficients for those components will be some factor, ci, of the basis vectors. So for vector |φ> the components will be c11> and c22>. In the more generalized form with an unspecified number of dimensions we can say that the vector |φ> is equal to the sum of cii> for all i.

|φ> = ∑cii>

The complex numbers ci are amplitudes, or probability amplitudes, though we should note that it’s actually the square of the absolute value of ci that is a probability. Specifically, the quantity |ci|2 is the probability that the eigenvalue ai is observed in a measurement of the operator A on the state vector |φ>. This is known as the Born rule. Another way of describing this summation equation is to say that the state of the system is a linear combination, or superposition, of all the eigenstates that compose it and that these eigenstates are “weighted” by their respective probability amplitudes. Eigenstates with higher probability amplitudes are more likely to be observed. And this touches again on the idea that observations of certain eigenstates are probabilistic and that’s the reason that the eigenvalues for these eigenstates are not considered definite properties. Because, they’re not definite; they’re probabilistic.

If we apply operator A to state |φ> we have a new vector A|φ>. In our Hilbert space this new vector’s components are expressible in terms of the coordinates, or basis vectors. If the basis vectors are eigenvectors of A then these components are expressible in terms of the probability amplitude ci. We could say that the application of this operator A to vector |φ> extracts ci and multiplies it by the eigenvalue ai. And this is good because remember eigenvalues are what we actually observe in experiments. So now we can express the state of the systems in terms of things we can observe.

This transformed vector A|φ> is equal to the sum of products of eigenvalue ai, amplitude ci, and eigenvector |αi>, for all i.

A|φ> = ∑aicii>

Now we’re ready to get into what Auyang considers what we can properly consider properties of quantum systems. For some observable A and its operator, the sequence of complex numbers {aici} can be called an A-amplitude and is, using the eigenvalues, expressed in terms of the probability amplitude ci. And this is where Auyang locates the properties of quantum systems. She interprets the probability amplitude ci or the A-amplitude as the definite property or the value of a certain quantum system in a certain state for the property type A. And she makes the point that we shouldn’t try to imagine what the amplitudes and A-amplitudes describe because they are nothing like classical feature; “they are literally unimaginable”. But they are calculable. And that’s their crucial, property-type feature.

We might ask why we should locate definite properties in something that we can’t imagine. Classical properties like classical energy, position, and momentum are more easily envisioned, so these prospective, unimaginable quantum properties might seem unsatisfying. But this touches on Auyang’s general Kantian perspective on the sciences, which is that our understanding of scientific concepts relies on a complex underlying conceptual structure. And in this case that underlying conceptual structure includes things like vectors, Hilbert spaces, bases, eigenvectors, eigenvalues, and amplitudes. If that structure is required to comprehend the system it’s not unreasonable that the system’s definite properties would be expressed in terms of that structure.

With that mathematical overview let’s look at the concept of properties more closely and at our expectations of them. And here I’d like to just quote an extended passage directly from Auyang’s book because this is actually my favorite passage:

“In quantum interpretations, the ‘values’ are usually taken to be eigenvalues or spectral values, which can be directly revealed in experiments, although the revelation may involve some distortion so that the veracity postulate does not hold. It is beyond a reasonable doubt that quantum systems generally have no definite eigenvalues. However, this does not imply that they have no definite properties. The conclusion that they have none arises from the fallacious restriction of properties to classical properties, of which eigenvalues are instances. Sure, quantum systems have no classical properties. But why can’t they have quantum properties? Is it more reasonable to think that quantum mechanics is necessary because the world has properties that are not classical?”

“The no-property fallacy also stems from overlooking the fact that the conceptual structure of quantum mechanics is much richer than that of classical mechanics. In classical mechanics, the properties of a system are represented by the numerical values of functions, which assign real numbers to various states of the system. In quantum mechanics, functions are replaced by operators, which are structurally richer. A function is like a fish with only one swaying tail, its numerical value; an operator is like an octopus with many legs. Quantum mechanics employs the octopus with good reason, and we miss something important if we look only at the one leg that reminds us of the fishy tail. Quantum systems generally do not have definite eigenvalues, but they have other definite values. The stipulation that the values must be directly revealable in measurements confuses the empirical and physical meanings of properties.”

“I argue that we cannot give up the notion of objective properties. If we did, the quantum world would become a phantom and the application of quantum mechanics to practical situations sorcery. Are there predicates such that we can definitely say of a quantum system, it is such and so? Yes, the wavefunction is one. The wavefunction of a system is a definite predicate for it in the position representation. It is not the unique predicate; a predicate in the momentum representation does equally well. Quantum properties are none other than what the wavefunctions and predicates in other representations describe.”

And recall here that a wave function is another way of referring to the state of a quantum system. I think of this was moving things up a level. Or down a level depending on how you want to think of it. Regardless, at one level we have the eigenvalues that pop out with the application of an operator on a state vector. These are not definite properties of the system as a whole. In other words, the definite properties of the quantum system do not reside at this level. Rather they reside at the level prior to this, on which these outcomes depend. In the case of an atomically bound electron we could say that it is the orbital, the probability distribution of the electron’s location, that is a property of the quantum system, rather than any particular position. These sorts of properties have a lot more too them. As Auyang says, they are “structurally richer”. They’re not just values. They are amplitudes, from which we derive probabilities for various values. And what Auyang is saying is that there’s no reason not consider that the definite property of the quantum system.

Still, it is different from out classical notion of properties. So what is it that is common to both classical and quantum properties? Auyang borrows a term from Alfred Landé, proposing that a characteristic has empirical ramification if it is observable or “kickable”:

“Something is kickable if it can be kicked and kicks back, or it can be somehow physically manipulated and the manipulation produces observable effects. Presumably the property is remote and obscure if we must resort to the indirect kickability criterion alone. Thus kickability can only work in a well-developed theory in which the property is clearly defined, for we must be able to say specifically what we are kicking and how it is supposed to kick back.”

In the case of quantum properties we are indeed in a situation where the property is “remote and obscure”. But we also have recourse to “a well-developed theory in which the property is clearly defined”. So that puts us in a good position. Because of this it doesn’t matter if properties are easily visualizable. “Quantum properties are not visualizable, but this will no longer prevent them from being physical”. The physical surpasses what we are able to visualize.

So there is a well-developed conceptual structure that connects observables to the definite properties of the quantum system prior to these observables. To review a little how this structure and cascade of connections works:

We start with the most immediate aspect: what we actually observe, which enter into the conceptual structure as eigenvalues. Eigenvalues of an observable can be regarded as labels of the eigenstates. Eigenstates serve as axes of a coordinate system in the state space. This is an important point, so I’ll repeat it again in another way. As Auyang puts it: “An observable coordinates the quantum world in a particular way with its eigenstates, and formally correlates the quantum coordinate axes to classical indicators, the eigenvalues. An observable introduces a representation of the quantum state space by coordinatizing it.” So we have observations to eigenvalues, to eigenstates, to axes in a state space.

The coordinate system in the state space enables us to determine definite amplitudes. The state space is a vector space and any particular state or quantum system in this state space is a vector in this space. We can break this vector down into components which are expressed in terms of the coordinate system or basis, i.e. the eigenstates. This is the coefficient ci, which is a probability amplitude. This is why we’re able to determine definite amplitudes using the coordinate system. A quantum system has no definite eigenvalues but it does have definite amplitudes. When it’s broken down into its basis components a quantum state is series of eigenstate expansion, multiple terms that are added up to define the vector. Each of these terms has an amplitude associated with an eigenstate that is also associated with some observable. Practically, an indicator in the form of an eigenvalue is somehow triggered in measurements and experiments. And the probability of observing any particular eigenvalue will be defined by its amplitude. Specifically, the quantity |ci|2 is the probability that the eigenvalue ai is observed in a measurement of A on the state |φ>. But it is the probability amplitude ci that is the definite property of the quantum system rather than any particular eigenvalue that happens to be observed. What’s more, this is an objective property of the quantum system even in the absence of any experiment. As Auyang puts it: “Unperformed experiments have no results, but this does not imply that the quantum system on which the experiment might be performed has no properties.” Now to show the more complete cascade of kickability: we have physical observations, to eigenvalues, to eigenstates, to axes in a state space, to a state vector, to vector components, to component coefficients, to probability amplitudes. And it’s the probability amplitudes that are the definite properties of the quantum system.

The question of whether or not quantum systems have definite properties is a philosophical question rather than a question of physics, to the extent that those can be separated. It’s not necessary to engage in the philosophy in order to engage in the physics. One can measure eigenvalues and calculate probability amplitudes without worrying about whether any of them count as properties. But it’s arguably part of the scientific experience to step back on occasion to reflect on the big picture. To ask things like, “What is the nature of the work that we’re doing here?”, “What does all this tell us about the nature of reality?”, “Or about the way we conceptualize scientific theories?” For me one of the most fascinating insights prompted by quantum mechanics is of the necessity of the elaborate conceptual structures that support our understanding of the theory. To put it in Kantian terms, these conceptual structures are “transcendental” in the sense that they constitute the conditions that are presupposed and necessary for us to be able to understand the theory in the first place. And to me that seems quite significant.

Literal and Metaphorical Truths

In religion some things are literal, some things are metaphorical, and some things are both. I share a little diagram I’ve found helpful for organizing my thoughts around different combinations of the literal and metaphorical in religion.

I wanted to share a little diagram I put together a few years ago in a conversation with some friends about religion. I’ve found it a helpful way to organize some of my ideas and I’d like to capture it here so I can refer to it in the future. One of the fault lines that runs between what we could call orthodox and unorthodox, or conservative and liberal religious belief runs between literal and metaphorical interpretation. It is orthodox and conservative to interpret scripture literally. And it’s unorthodox and liberal to interpret scripture metaphorically. Or so the thinking goes. It’s not for no reason at all that this idea occurs. There’s something too it. And rather than disagree with it altogether I would just like to add more to it, but still with the end result of proposing a more complex picture of the possibilities.

One way I like to think of this is as an array of possible positions along two axes. One axis is truth. The other is interpretation. Along the truth axis things can be literally true or false. Along the interpretation axis things can be interpreted literally or metaphorically. This produces four combinations, four quadrants. I’ll through each of these.

As a quick technical note, I’m going to be using ‘metaphor’ in a less precise and careful way than I probably should do, but I’m doing it anyway. ‘Allegory’ might be a better word for some of these things but the popular use of ‘metaphor’ is common enough that I won’t worry about it.

The first quadrant is for those things that are literally true and are interpreted literally. So we could say, sweeping some complexity under the rug, that an orthodox, conservative believer would have more things in this quadrant than other people would. The Garden of Eden, Adam and Eve, the global Flood, Noah’s Ark are all literal, historical truths and are to be interpreted literally. And I just mention those ones first since those are things that more unorthodox, liberal believers might not put into that quadrant. For me this quadrant includes things like the existence of God and the resurrection of Christ. I believe these are things that are literally true and, in the case of Christ’s resurrection, actually happened. Those are the two big doctrines of greatest theological significance. But I also include lots of other stuff that may not be quite as significant but does happen to be literally true in my opinion. So for example, a lot of historical and political stuff in the Bible, the names of the different kings in the kingdoms of Israel and Judah, their relations and conflicts with other nations, their conquest by other nations, the rule under the Neo-Babylon and Persian Achaemenid Empires. All that stuff is pretty much accurate. Those may not be as important theologically and it’s stuff like that where I’d say literal interpretation is actually not as interesting as other interpretations. More on that later.

The second quadrant is for things that are literally false and are also interpreted literally. So a skeptic might put many of the same things into this quadrant that an orthodox, conservative believer would put in the first. So for a skeptic the Garden of Eden, Adam and Eve, the global Flood, Noah’s Ark are all to be understood literally and they are false. In a way the conservative and skeptic can be closer to each other in expectations than either is to the liberal. For my part there’s not much that I would put into this quadrant. And the things I would put here aren’t especially interesting. So for example, there are passages in Joshua and Judges that talk about mass destruction of Canaanite cities by the Israelites were archeology has found either or no evidence or contradictory evidence of those conquests. That’s historically interesting but not all that theologically important. Except I guess insofar as the violence in the mass violence in Bible might be exaggerated in some cases, which could impact our understanding of God and his expectations. So there’s a possible example. But there’s not a whole lot more that I’d put in that quadrant.

The third quadrant is for things that are literally false and interpreted metaphorically or allegorically. And this is where a lot of religious liberalism focuses or distinguishes itself. Scripture tells of Adam being created out of earth, Eve being made from a rib, or just in general that there were only two human individuals that gave rise to all of humanity. In the quadrant we can allow for the possibility that even if these things are not literally true they have metaphorical truth or metaphorical interpretation.

The story of Adam and Eve is one of the most fruitful and powerful stories in all of human history, not least for the variety of interpretations it can accommodate. Originally it may have been principally about the origin of human mortality, i.e. human death. Humans had to be mortal otherwise they would have rivaled God or the gods in power and dominion. But stories like the Adam and Eve story are not limited in their meanings, even by their original historical contexts. This story has also come to symbolize the fallen nature of humanity, these deeply-rooted but destructive instincts we have.  There’s a joke that Original Sin is the most empirically verified Christian doctrine. Certainly plenty of evidence, so that interpretation speaks to us. I think we can all relate to Paul when he said, “for what I want, that do I not; but what I hate, that I do” (Romans 7:15). I also think the story of Adam and Eve is a wonderful narrative device to think about growth, maturation, and all the struggles that come with that.

There are also things in this third quadrant that are pretty uncontroversially allegorical, like parables. We need not suppose that the parable of the Prodigal Son is a historical account of something that actually happened. Although we might say that something like it has probably happened many times. Myths and parables being the kind of things that aren’t true at one specific time and place but are rather true in many or all times and places. Most of Jesus’s parables are meant to be understood in this way. This becomes pretty evident in cases where people try to take him literally.

“And when his disciples were come to the other side, they had forgotten to take bread. Then Jesus said unto them, Take heed and beware of the leaven of the Pharisees and of the Sadducees. And they reasoned among themselves, saying, It is because we have taken no bread. Which when Jesus perceived, he said unto them, O ye of little faith, why reason ye among yourselves, because ye have brought no bread? …How is it that ye do not understand that I spake it not to you concerning bread, that ye should beware of the leaven of the Pharisees and of the Sadducees? Then understood they how that he bade them not beware of the leaven of bread, but of the doctrine of the Pharisees and of the Sadducees.” (Matthew 16:5-8,11-12)

Jesus is actually kind of getting after them hear for taking things literally when it really isn’t appropriate. Something to think about if you’re tempted to give someone a hard time for taking something metaphorically instead of literally.

There’s also the classic example with Nicodemus:

“Jesus answered and said unto him, Verily, verily, I say unto thee, Except a man be born again, he cannot see the kingdom of God. Nicodemus saith unto him, How can a man be born when he is old? can he enter the second time into his mother’s womb, and be born?” (John 3:3-4)

Yeah, so he’s taking things way to literally there, right? Jesus was so inclined to figurative speech that it was actually pretty unusual and noteworthy when he did speak directly. His disciples even remarked on it: “His disciples said unto him Lo now speakest thou plainly and speakest no allegory” (John 16:29).

Nevertheless parable was Jesus’s primary mode of teaching and he even marked the ability to understand things on this allegorical and metaphorical level as a distinguishing attribute in his disciples:

“Who hath ears to hear, let him hear. And the disciples came, and said unto him, Why speakest thou unto them in parables? He answered and said unto them, Because it is given unto you to know the mysteries of the kingdom of heaven, but to them it is not given. For whosoever hath, to him shall be given, and he shall have more abundance: but whosoever hath not, from him shall be taken away even that he hath. Therefore speak I to them in parables: because they seeing see not; and hearing they hear not, neither do they understand. And in them is fulfilled the prophecy of Esaias [i.e. Isaiah], which saith, By hearing ye shall hear, and shall not understand; and seeing ye shall see, and shall not perceive: For this people’s heart is waxed gross, and their ears are dull of hearing, and their eyes they have closed; lest at any time they should see with their eyes and hear with their ears, and should understand with their heart, and should be converted, and I should heal them. (Isaiah 6:10) But blessed are your eyes, for they see: and your ears, for they hear.” (Matthew 13:9-16)

It seems to have been important to Jesus that his potential disciples be made to think in a less literal, more metaphorical way, as a kind of productive trial; like this was especially conducive and even essential to the process of becoming a disciple.

The last quadrant is the one I find most interesting and it sort of cuts across the more intuitive conservative-liberal interpretative divide. The last quadrant is for things that are literally true but that have metaphorical interpretation. Pretty much everything in this quadrant could also go in quadrant one: a literal interpretation is just as valid. But for my part I tend to the metaphorical interpretations more interesting.

I like a perspective on metaphor I picked up from Biblical scholar Marcus Borg. He said his students would often be disappointed about metaphorical interpretations, thinking of them as only metaphor. He tried to get people to change their perspective, to see metaphor as not second-best, what’s left over after the literal is stripped away, but as something added to, more than literal. Theology is poetry plus, not science minus.

My favorite example in this category is the death and resurrection of Jesus. I believe this is a literal truth. But it’s the metaphorical take on it that I dwell on in my religious practice. Metaphorical interpretations are strongly encouraged in the Bible. Paul many times spoke of Christ’s death and resurrection as something that the Lord’s disciples should act out in their own lives, dying and being born anew into a new life.

“Know ye not, that so many of us as were baptized into Jesus Christ were baptized into his death? Therefore we are buried with him by baptism into death: that like as Christ was raised up from the dead by the glory of the Father, even so we also should walk in newness of life. For if we have been planted together in the likeness of his death, we shall be also in the likeness of his resurrection: Knowing this, that our old man is crucified with him, that the body of sin might be destroyed, that henceforth we should not serve sin. For he that is dead is freed from sin. Now if we be dead with Christ, we believe that we shall also live with him: Knowing that Christ being raised from the dead dieth no more; death hath no more dominion over him. For in that he died, he died unto sin once: but in that he liveth, he liveth unto God. Likewise reckon ye also yourselves to be dead indeed unto sin, but alive unto God through Jesus Christ our Lord.” (Romans 6:3-11)

Knowing that Jesus literally died and rose again is an important part of Christianity. But there’s a more-ness there. Something more than literal to it. And we’re missing out if we don’t pay attention to it. In pondering the death and resurrection of Christ there’s a corresponding process of growth in us that involves letting old things die and new things flourish. “Except a corn of wheat fall into the ground and die it abideth alone but if it die it bringeth forth much fruit” (John 12:24). “Therefore if any man be in Christ he is a new creature old things are passed away behold all things are become new” (2 Corinthians 5:17).

I mentioned earlier that some of the political, historical facts of the Bible could have interpretations beyond the literal interpretations that could be more theologically interesting. For example, it’s pretty established and undisputed that the Kingdom of Judah was conquered by the Neo-Babylon Empire and then was under the control of the Persian Achaemenid Empire. Lots of corroborating and extra-Biblical evidence for all that. But the Biblical prophets make more out of this than just the bare historical facts. To them these events were replete with theological significance and the story of Judah’s fall, captivity, and redemption turns into the story of a people and eventually into the human story. No doubt about it, the conquest was a disaster of monumental scale that included the destruction of the temple. This too is the human story. It happened not just in this particular time and place but also anywhere and everywhere, all the time. And the message of redemption is similarly universally applicable.

“For a small moment have I forsaken thee; but with great mercies will I gather thee. In a little wrath I hid my face from thee for a moment; but with everlasting kindness will I have mercy on thee, saith the Lord thy Redeemer. For this is as the waters of Noah unto me: for as I have sworn that the waters of Noah should no more go over the earth; so have I sworn that I would not be wroth with thee, nor rebuke thee. For the mountains shall depart, and the hills be removed; but my kindness shall not depart from thee, neither shall the covenant of my peace be removed, saith the Lord that hath mercy on thee.” (Isaiah 54:7-10)

This is the kind of stuff that lies below the surface, literal level of things. And it’s just as important, sometimes more important than the literal matters of fact in religious doctrines. Learning to find and appreciate those deeper levels to things is a matter of, as Jesus said, having ears to ear.