Ah, controversy! Physics is of course not immune from it, and sometimes the participants in an argument can let anger get the better of them.
An example of this began last week, when the following video clip appeared, featuring Professor Brian Cox explaining to a lay audience the Pauli exclusion principle:
For reasons that I will try and elaborate on in this post, this short video was, to say the least, eyebrow-raising to me. Tom over at Swans on Tea picked up on the same video, and wrote a critique of it with the not quite political title, “Brian Cox is Full of **it“, in which he explained his initial critique of the video based on his own knowledge. I piped in with a comment,
Well put. I just saw this clip the other day and it was an eyebrow-raiser, to say the least. I thought I’d mull over the broader implications a bit before writing my own post on the subject, but you’ve addressed it well.
A more technical way to put it, if I were to try, is that the Pauli principle applies to the *entire* quantum state of the wavefunction, not just the energy, as Cox seems to imply. This is why we can, to first approximation, have two electrons in the same energy level in an atom: they can have different “up/down” spin states. Since the position of the particle is part of the wavefunction as well, electrons whose spatial wavefunctions are widely separated are also different.
Well, apparently being criticized was a bit upsetting for Professor Cox, because he fired off the following angry comment to both myself and Tom:
“Since the position of the particle is part of the wavefunction as well, electrons whose spatial wavefunctions are widely separated are also different.” What on earth does this mean? What does a wave packet look like for a particle of definite momentum? Come on, this is first year undergraduate stuff.
I’m glad that you, Tom, don’t need to know about the fundamentals of quantum theory in order to maintain atomic clocks, otherwise we’d have problems with our global timekeeping!
So, he basically insults both Tom and I in the course of several paragraphs, without addressing the comments at all, really. It gets worse. In addition to me later being referred to as “sensitive” by the obviously sensitive Dr. Cox (cough cough projection cough), he doubles down on his anger by referring on Twitter to the lot of those criticizing him (including Professor Sean Carroll of Cosmic Variance) as “armchair physicists”.
Well, there have been a number of responses to Cox’s angry rant, including a response on the physics from Sean Carroll and a further elaboration by Tom on his own case at Swans on Tea. I felt that I should respond myself, at the very least because I’ve been accused of not understanding “undergraduate physics” myself, but also because the “everything is connected” lecture in my opinion represents a really dangerous path for a physicist to go down.
We’ll take a look at this from two points of view; first, I’d like to comment on the style of Cox’s response to criticism, and then on the more important substance of the discussion.
First, on the style. When your response to criticism from research physicists is that they don’t understand undergraduate physics and that they are “armchair physicists”, you’ve basically admitted that you’ve lost the argument*. Though scientists certainly get into petty spats far too often, typically sparked by research disagreements, it is not considered a good thing. It is especially bad form for someone who is representing the field in a very public way to whine and name call: it is a very poor showing of what science is supposed to be all about.
Okay, let’s get to the substance! In order to get into the meat of the issue, I should say a few words about the quantum theory, since I don’t discuss it very often on this blog. Dr. Francis talks a bit about one of the issues — entanglement — over at Galileo’s Pendulum, also in reaction to this “controversy”.
Up through the late 19th century, “classical” physics served very well in describing the universe. As researchers started to investigate the behavior of matter on a smaller scale, they began to encounter phenomena that couldn’t be explained by the existing laws, such as the structure of the atom (more on this in my old post here).
Many of these issues were spectacularly resolved by the hypothesis that subatomic particles such as the electron and proton are not in fact point-like objects but possess wave-like properties. This idea was introduced by the French physicist Louis de Broglie in his 1924 doctoral thesis, and it naturally explained such phenomena as the discrete energy levels that electrons in atoms possess. The wave properties of matter can be demonstrated dramatically by using electrons in a Young’s double slit experiment; the electrons exiting the pair of slits produce a wave-like interference pattern of bright and dark bands, just like light.
But this explanation raised a natural and difficult question: what, exactly, is the nature of this electron wave? An example of the difficulties is provided by the electron double slit experiment. Individual electrons passing through the slits don’t produce waves; they arrive at a discrete and seemingly random points on the detector, like a particle. However, if many, many electrons are sent through the same experiment, one finds that the collection of them form the interference pattern. This was shown quite spectacularly in 1976 by an Italian research group**:
How do we explain that individual electrons act like particles but many electrons act like waves? The conventional interpretation is known as the Copenhagen interpretation, and was developed in the mid-1920s. In short: the wavefunction of the electron represents the probability of the electron being “measured” with certain properties. When a property of the electron is measured, such as its position, this wavefunction “collapses” to one of the possible outcomes contained within it. In the double slit experiment, for instance, a single electron (or, more accurately, its wavefunction) passes through both slits and has a high probability of being detected at one of the “bright” spots of the interference pattern and a low probability of being detected at one of the “dark” spots. It only takes on a definite position in space when we actually try and measure it.
This interpretation is amazingly successful; coupled with the mathematics developed for the quantum theory (the Schrödinger equation, and so forth) it can reproduce and explain the behavior of most atomic and subatomic systems. However, the wave hypothesis raises many more deep questions! What, exactly is a “measurement”? How does a wavefunction “collapse” on measurement? If all particles are waves, why don’t we see their wave-like (or quantum) properties in our daily lives? Are the properties of a particle truly undetermined before measurement, or are they well-defined but somehow “hidden” from view?
This latter question formed the basis of a famous counterargument to the quantum theory called the Einstein-Podolsky-Rosen paradox, published in 1935. The paradox may be formulated in a number of ways; what follows is a simple model from optics. By the use of a nonlinear optical material, a photon (light particle) of a given energy can be divided into two photons, each with half the energy of the original, propagating in different directions, by the process of spontaneous parametric down conversion.
There is an important additional property of these half-energy photons, however; due to the physics of their creation, they have orthogonal polarizations. That is, if the electric field of one photon is oscillating horizontally the other must be oscillating vertically, and vice-versa. However — and this is the important part — nothing distinguishes between the two photons on creation, and nothing chooses the polarization of one or another. Just like the position of the electron in Young’s double slit experiment is genuinely undetermined until we measure it, the polarization of the photons is undetermined until we make a measurement. Nevertheless, there is a connection between the two photons: we don’t know which one has which polarization, but we know for certain that the polarizations are perpendicular. If we were to look at the photon polarization head-on, we might see something of the form shown below:
The photons are said to be entangled; though their specific behavior is undetermined, the physics of their creation still forced a relationship between the two.
Here’s where E, P & R felt there was a paradox: suppose we point our photons to opposite ends of the galaxy. If undisturbed, they remain in this entangled state and can in principle travel arbitrarily far away from one another. Now suppose we measure the polarization of one of the photons, and find the result is vertical; we’ve collapsed the wavefunction, and we now know with certainty that the other photon, at the other end of the galaxy, must be horizontally polarized. By measuring the polarization of one photon, we’ve automatically determined the state of the other one; apparently this wavefunction collapse must happen instantaneously, faster than even the speed of light!
This idea of entanglement and its “spooky action at a distance” was intended to demonstrate the ridiculousness of the Copenhagen interpretation of the quantum theory, but in fact it has been verified in countless laboratory experiments. Furthermore, E, P & R’s counter-explanation — that the polarizations of the photons are well-defined on creation, just “hidden” — has been demonstrated to not be true (though intriguing loopholes remain). It has also been shown that entanglement is consistent with Einstein’s special relativity. Although the collapse of the wavefunction can occur instantaneously, it is not possible to transmit any information this way, due (in short) to the random nature of the process.
We’ll get to the relevance of entanglement in a moment; we still need one more piece of the puzzle before we can discuss the “everything is connected” video, namely Pauli’s exclusion principle. As we have noted, the introduction of the quantum theory answered many questions, but raised many more. Among other things, the quantum theory predicts that electrons exist only in particular special and discrete “orbits” around the nuclei of atoms. This idea was first introduced in the Bohr model of the atom, as illustrated below:
An electron in a hydrogen atom can only exist in certain discrete stable orbits, labeled in this picture by the index n. Light is emitted from an atom when it drops from a higher energy (outer) orbit to a lower (inner) orbit. The existence and nature of these discrete orbits is explained by the wave properties of matter: electrons form a “cloud” around the nucleus, rather than orbiting in a well-defined manner.
But the wave nature of matter also raises a new problem: electrons are now somewhat “squishy”! In larger atoms with multiple electrons orbiting the nucleus, it was readily found that only a finite number of electrons can fill each orbital position/energy level. One is naturally led to wonder why all the electrons don’t just fill the lowest energy state of the atom, the “n=1″ state; because the electrons are wavelike and “squishy”, there doesn’t seem to be anything prohibiting this.
This was one problem that Wolfgang Pauli (1900-1958) concerned himself with. The answer he developed became known as the Pauli exclusion principle: no two identical fermions can occupy the same quantum state. “Fermions” include electrons, protons and neutrons: the constituent parts of ordinary matter. Under the Pauli principle, electrons cannot all pile into the ground state of an atom. Because electrons possess intrinsic angular momentum (“spin”) which can either be “up” or “down”, and this is part of the electron’s quantum state, two electrons can fit in the ground state with the same energy but with different spins.
Keep in mind that the Pauli principle applies to the complete state of an electron; this potentially includes its energy, its momentum, its spin, and its position in space. Any property of a pair of electrons that can be used to distinguish them counts against the exclusion principle.
Now we’ve hopefully got enough information to understand what Cox is trying to say in the video linked above. Let’s dissect it one step at a time:
For example, in this diamond, there are 3 million billion billion carbon atoms, so this is a diamond-sized box of carbon atoms. And here’s the thing, the pauli exclusion principle still applies, so all the energy levels in all the 3 million billion billion atoms have to be slightly different in order to ensure that none of the electrons sit in precisely the same energy level; Pauli’s principle holds fast.
This is a well-known and accepted property of matter. The electrons in a piece of bulk material are all “squashed together”, just like the multiple electrons in a complex atom are all squashed together. In an individual atom, the electrons must stack up into the different quantum states (different energies, different spins) that are permitted by the electron/nucleus interaction. In a bulk piece of crystal, a similar argument applies: there are a large number of permissible quantum states allowed, in which electrons are “spread out” over the size of the crystal; Pauli’s principle indicates that each electron must be in a different state, and they end up filling a “band” of energies.
But it doesn’t stop with the diamond, see, you can think that the whole universe is a vast box of atoms, that countless numbers of energy levels all filled by countless numbers of electrons.
Here’s where things start to go off the rails for me, and it seems like a dirty trick is being pulled. In a crystal, there are a large number of strongly-interacting electrons packed together, and it is natural — and demonstrable — that the wavefunctions of the electrons spread out over the entire bulk of the crystal, with the sides of the crystal forming a natural boundary. But jumping to the cosmological scale, we don’t “see” electrons whose wavefunctions stretch over the extent of the universe — our experiments show electrons localized to relatively small regions. Even if we treat the universe as a big box — and it’s unclear that this is even a reasonable argument to make — the behavior of electrons in the “universe box” is really, fundamentally different from the behavior of electrons in a “crystal box”. I think that Sean Carroll over at Cosmic Variance was saying something very similar when he notes, “but in the real universe there are vastly more unoccupied states than occupied ones.” That is: in a crystal, the electrons are “fighting” to find an unoccupied energy level to occupy, like a quantum-mechanical game of musical chairs. Over the entire extent of the universe, however, there are plenty of open energy levels — much like finding chairs at a Mitt Romney event in Michigan.
So here’s the amazing thing: the exclusion principle still applies, so none of the electrons in the universe can sit in precisely the same energy level.
Now we’re really getting into trouble. In a crystal, where the electrons are all essentially “smeared out” over the volume, the energy levels must necessarily split. But electrons in the universe don’t seem to be smeared out in the same way. It would seem to me that electrons separated widely in space — around different hydrogen atoms on opposite ends of the universe, for instance — would be perfectly well distinguished by their relative positions, and not need to have energy level splits. More on this in a moment.
But that must mean something very odd. See, let me take this diamond, and let me just heat it up a bit between my hands. Just gently warming it up, and put a bit of energy into it, so I’m shifting the electrons around. Some of the electrons are jumping into different energy levels. But this shift of the electron configuration inside the diamond has consequences, because the sum total of all the electrons in the universe must respect Pauli. Therefore, every electron around every atom in the universe must be shifted as I heat the diamond up to make sure that none of them end up in the same energy level. When I heat this diamond up all the electrons across the universe instantly but imperceptibly change their energy levels. So everything is connected to everything else.
Now the explanation has actually made the leap into being simply wrong! We have noted that, with tangled quantum mechanical particles, it is possible to instantaneously modify the wavefunction of one of the entangled pair by manipulating (measuring) the properties of the other. But, as we noted, nothing physical can be transmitted at this faster-than-light wavefunction collapse. Cox specifically says in this lecture that heating of electrons in his piece of diamond instantly changes the energy levels, i.e. the energy , of the electrons across the universe! A change in energy is a physical change of a particle, and this is specifically forbidden by the laws of physics as we know them.
Another thought came to me as I was reading this, and I found that it was already stated by Tom over at Swans on Tea. If all the electrons in the universe necessarily have different energies, then they are always in different quantum states — the Pauli exclusion principle would become irrelevant! It would seem to imply that we could pile an arbitrary number of electrons in the ground state of a hydrogen atom, although they would have slight indistinguishable energies. Obviously, we don’t see this. There may be a problem with this argument, as well, but it illustrates (as Tom says) that broad-reaching statements about atomic energy levels end up having potentially more implications than one would at first think.
As stated, Cox’s argument is really incorrect, and violates relativistic principles. One can argue, in his defense, that this is a consequence of trying to simplify things for a popular audience, and that he really meant something a little more subtle. However, on an undergraduate physics page he makes a similar argument, and linked to it in defense of his lecture.
Imagine two electrons bound inside two hydrogen atoms that are far apart. The Pauli exclusion principle says that the two electrons cannot be in the same quantum state because electrons are indistinguishable particles. But the exclusion principle doesn’t seem at all relevant when we discuss the electron in a hydrogen atom, i.e. we don’t usually worry about any other electrons in the Universe: it is as if the electrons are distinguishable. Our intuition says they behave as if they are distinguishable if they are bound in different atoms but as we shall see this is a slippery road to follow. The complete system of two protons and two electrons is made up of indistinguishable particles so it isn’t really clear what it means to talk about two different atoms. For example, imagine bringing the atoms closer together – at some point there aren’t two atoms anymore.
You might say that if the atoms are far apart, the two electrons are obviously in very different quantum states. But this is not as obvious as it looks. Imagine putting electron number 1 in atom number 1 and electron number 2 in atom number 2. Well after waiting a while it doesn’t anymore make sense to say that “electron number 1 is still in atom number 1″. It might be in atom number 2 now because the only way to truly confine particles is to make sure their wavefunction is always zero outside the region you want to confine them in and this is never attainable.
We can try and explain this more elaborate and detailed argument pictorially for two electrons. What Cox seems to be espousing here is essentially that two electrons naturally evolve into an entangled state after some period of time. How this would work: we start with two electrons (labeled “red” and “blue” for clarity, though they are in reality completely indistinguishable particles) surrounding two different hydrogen nuclei. Let us suppose they start spatially separated, as shown below:
Because the red electron’s wavefunction stretches into the domain of the blue atom, and the blue electron’s wavefunction stretches into the domain of the red atom, as time goes on it becomes increasingly likely that the red and blue electrons have switched places. The wavefunctions may evolve to something like this:
Eventually, after sufficient time has passed, each electron is equally likely to be in either atom, which we crudely sketch as:
That is, we expect the wavefunctions to be identical for the two electrons. But they can’t be identical, according to the Pauli principle! Therefore something else must shift in the wavefunctions to make them distinguishable — one ends up with a slightly higher energy, one ends up with a slightly lower energy.
What we’ve got here is what I imagine would be considered a form of entanglement: we know with certainty that there is only one electron around each nucleus, but the specific location of either is undetermined.
This idea isn’t particularly controversial: this is essentially what happens in crystals, as we have discussed, and what happens to the electrons in molecules or otherwise interacting atoms. But this is just a description for two atoms — can we make the same sort of arguments on a universal scale? Here I have two problems. The first is that there are too many questions that get raised when trying to extend this to this degree, among them:
- Time. How long does it take to get such an entanglement between two electrons? For two electrons next to one another, I imagine it would be nearly instantaneous, but for two electrons separated by light-years? I’m guessing the period of time is very, very large, which brings me to my next point…
- Stability. It is not easy to produce significantly entangled photons in the laboratory, and it is hard to maintain that entanglement. Keeping two particles entangled for long periods of time is experimentally nontrivial, due to external interactions with other particles: in essence, our quantum system is being continually “measured” by outside influences. Do widely separated electrons ever form an appreciable degree of entanglement? Completely unclear, and rather doubtful.
- Infinity. Part of the argument for this universal entanglement is built on the idea that the spatial wavefunctions of electrons are of infinite extent, i.e. they are spread-out throughout all of space. Indeed, stationary (definite energy) solutions of the Schrödinger equation are infinitely spread out, but I would use a lot of caution to make concrete conclusions from that observation. In optics, classical states of “definite energy” are monochromatic waves, which are used all the time to make optics calculations convenient. It follows from the mathematics that monochromatic waves are always of infinite extent, just like wavefunctions, but here’s the thing: nobody with any sense in optics assumes that this infinite extent is a physical behavior that one should derive concrete physical conclusions from. A monochromatic wave is just a convenient idealization of the real physics***.
A natural question to ask at this point: isn’t physics all about deriving general conclusions from simple physical laws? Why are you being more cautious with Pauli, and quantum mechanics, than you are with, say, gravity and electromagnetism? Part of the difference is, as we have noted above, that we simply do not understand the quantum theory well enough to derive boldly such universe-wide conclusions. An even more important difference, though, is that I can see the universal consequences of gravitation and electromagnetism experimentally, whereas it is not clear what consequences, if any, this “universal Pauli principle” provides. Which brings me to my final observation; returning to Cox’s lecture notes:
The initial wavefunction for one electron might be peaked in the region of one proton but after waiting for long enough the wavefunction will evolve to a wavefunction which is not localized at all. In short, the quantum state is completely specified by giving just the electron energies and then it is a puzzle why two electrons can have the same energy (we’re also ignoring things like electron spin here but again that is a detail which doesn’t affect the main line of the argument). A little thought and you may be able to convince yourself that the only way out of the problem is for there to be two energy levels whose energy difference is too small for us to have ever measured in an experiment.
Emphasis mine. HOLY FUCKING PHYSICS FAIL. Here Cox explicitly acknowledges that his “universal Pauli principle” consequences are something that not only cannot be measured today, but in principle can never be measured, by anyone.
THEN WHY THE HELL ARE WE TALKING ABOUT IT??!!
At its core, physics is all about experiment. Experimental tests of scientific hypothesis are what distinguish physics (and all science, really) from general philosophy and, worse, mysticism and pseudoscience. Consider the application of Cox’s conclusion to a few other situations:
- Astrology is the influence of the stars upon human beings via quantum mechanical influences whose energy difference is too small for us to have ever measured in an experiment.
- Homeopathy is the lingering effect of chemical forces on water via quantum mechanical changes to the water whose energy difference is too small for us to have ever measured in an experiment.
- The human soul exists materially in the quantum wavefunction of a human being, manifesting itself in changes whose energy difference is too small for us to have ever measured in an experiment.
In my eyes, there really is not much difference between various pseudoscientific shams being propagated in the world today and the logical argument of a “universal Pauli principle”. (When I mentioned this argument to a colleague, he said, “Ask him how many angels dance on the head of a pin.“) In a sense the whole discussion of this blog post has been a waste of time: my theoretical counterarguments may be reasonable or they may not; we can never draw any conclusion about the reality of this universal principle because it lies outside our ability to ever detect it.
I tend to be rather forgiving of using simple, arguably misleading, models to introduce physical principles. For instance, I’m a defender of the use of the Bohr model as a good tool to expose students to quantum ideas in a simple and historical way. My criterion, however, is this: a model or explanation must, as a whole guide students in the right direction towards the greater “truth”, such as it is in science. The “universal Pauli principle” fails this on two parts: it gives a false impression of the importance of completely unexperimental conclusions, and it opens the door to pseudoscientific nonsense. Nevertheless, Cox doubled down on his statements in a Wall Street Journal article, somehow arguing that his original argument is a necessary evil in a world where the public needs to be excited about science.
In a sense, though, we have ironically come full circle on Pauli. It was none other than Wolfgang Pauli who coined the phrase “not even wrong” to describe theories that cannot be falsified or cannot be used to make predictions about the natural world. It has been most recently used to describe string theory, with the argument that the predictions of string theory cannot be tested with any experimental apparatus that exists. However, string theory can at least in principle be tested, albeit not today, where it seems that the “universal Pauli principle” described by Cox has no measurable consequences, in principle, and is immune to any test imaginable. It serves no useful purpose in the world of physics, and as we have noted there are many objections to it actually working the way it is advertised to work.
I was recently thinking of the many advantages to the explosion of science communicators on the internet, and one that struck me is that we no longer have to rely on a single or a small number of “authority figures” to tell us what is right and wrong in the scientific world. This entire fiasco emphasizes how important this new abundance of voices will be in an ever more complex universe.
With no hypothesis to test, and no measurable consequences for science, I conclude my thoughts on the “universal Pauli principle”.
Requiescat in pace,
“omnia conjuncta est”¹
* If someone wants to get in a pointless pissing match of who is more of an “armchair physicist” based on CVs, though, I’m your huckleberry.
** P.G. Merli, G.F. Missiroli and G. Pozzi, “On the statistical aspect of electron interference phenomena,” Am. J. Phys. 44 (1976), 306-307.
*** Curiously, in arguing against the use of the spatial distribution of a quantum wavefunction in providing “distinguishability” of electrons, Cox uses a “momentum eigenstate” — a particle of perfectly specified momentum and infinitely uncertain position. This is pretty much the equivalent of a monochromatic plane wave in optics, which again nobody would use as a realistic example of how the world works.
¹Thanks to Twitter folks for suggesting the translation of the latter phrase. Alternate translation: “omnes continent”.
Postscript: A couple of friends (including @minutephysics) have pointed out that none of the discussions so far have included quantum field theory, which makes things even more complicated (non-conservation of particle number, for instance).