Nor is isospin, but then that’s less well-known.
Most of what people say about quantum physics focusses on things like entanglement, acausality and uncertainty, with a kind of mystical bent, but there’s also something else which most people ignore which is equally weird, and on top of that is something else again which is as weird too, if not even weirder. These two things are spin and its oddly- but appropriately-named sibling isospin.
It’s been said that if you think you understand quantum mechanics, it means you don’t. This may or may not be true and there are different opinions about what it actually means, but I would say this is also true of spin and isospin. I’ll deal with spin first.
If you hold a spinning gyroscope, you can feel the difficulty of shifting it from the direction its pointing. If it’s one of those small toy ones, it won’t wrench you off your feet but its rotation will be shifted into your body if you’re standing. In a swivel chair, a sufficiently large and massive rotating object will rotate the chair if you try to move the object into a different angle. This principle is useful, and is for example employed with rifle bullets, spacecraft and compasses to stabilise them. Whereas magnetic compasses are useless near the poles, gyrocompasses can float around as they move and will therefore continue to point north if they’ve been set up that way in the first place. A spacecraft will tumble unpredictably in space unless it’s stabilised in some way, and one way of doing so is to make it spin as it launches, which keeps it pointing in the same direction. This particular spin is often counteracted by ejecting something spinning in the opposite direction to ensure the spacecraft instruments or devices stay facing the requisite direction later on.
These are all illustrations of angular momentum. Momentum in general is the tendency for an object to keep moving in the same way unless something stops it, that is, unless another force acts upon it. This is true of masses moving in straight lines, and of spinning masses. They will continue to spin in the same way around the same axis unless something acts on them to shift them or slow them down, and when this happens that momentum has to go somewhere as spin rather than in a straight line. This is called angular momentum.
We tend to think of atoms as consisting of nuclei surrounded by electrons in orbit around them: that is, rotating. Ferromagnetism happens when the atoms in a material are all lined up spinning in the same direction, and only applies to very few materials, notably iron but also cobalt and nickel. If you think of atoms as gyroscopes, which they are, what you’re doing when you magnetise something is shifting the axes of rotation of a load of gyroscopes, and that angular momentum shift has to go somewhere. And it does. If you suspend a piece of unmagnetised iron in space in zero gravity conditions, or more accessibly hang it from a thread, then apply a magnetic field to it, this will to some extent magnetise the block and shift the atoms, and it will start spinning. This is known as the Einstein-De Haas Effect. Yes, that Einstein.
This change in angular momentum can be measured quite easily because the mass of the iron is known and its rotation can be timed and observed. However, even if you take into account all the angular momentum involved in the shift, it doesn’t account for all of the spin. This is because the electrons themselves are tiny magnets pointing in a particular direction, and the magnetic field aligns not only the atoms but also the electrons. Now here’s the crucial question. How can an electron point in a particular direction? The answer is that it has an axis of rotation, and this accounts for the discrepancy in the rate of spin the lump of iron has. This difference in angular momentum just taking the orbitals into account and the actual difference allows the spin of the electron to be found.
And this is where things get really weird.
If you calculate the spin of an electron, either assuming the smallest probable size of the particle or the much more likely scenario of thinking of an electron as a point in space, there is an imponderable problem. If the electron has a size at all, in order to generate the amount of angular momentum it has, it would have to be moving faster than light. If, on the other hand, an electron is a point, it’s featureless except for location, so how can it be spinning at all? A point in space doesn’t have a direction or an axis of rotation in the conventional sense, so – huh?
This is not some abstract thing happening due to the vagaries of scientific theory either. A lump of iron really does start spinning if magnetised, and taking into account all of the rotational movement of the electrons in their orbitals shifting is not enough to account for the exact rate of rotation. In the end, then, there seems to be only one possibility: spin is a fundamental property of matter. From our usual perspective, it definitely looks as if there are just objects which are not spinning which we can rotate or might start or stop rotating, or speed up and slow down, and so on, as if it’s just another thing going on in the world, but that isn’t actually what’s happening. On a tiny scale, spin is an intrinsic property of matter like electrical charge or its absence. Moreover, it’s quantised. In the same way as there are jumps between values of something on a small scale rather than an infinitely smooth transition, so for example electrical charge is either neutral, equivalent to an electron, the reverse of an electron, and if a quark either -⅓ or + ⅔ or the opposite for their antiquarks, which add up to an equivalent charge to the electron when they form a nucleon, which is just as well because otherwise atomic matter couldn’t exist. This will become relevant.
Spin has been described as “classically non-describable two-valuedness”, as it’s indescribable in the sense that it can’t actually be properly understood but must exist. Subatomic particles don’t literally spin in the same sense as a wheel or planet, but behave as if they do. All subatomic particles have a spin of either a whole number or a multiple of ½ other than a whole number. The former are bosons, the latter fermions. Fermions are “stuff” and bosons forces, so for example quarks and electrons are fermions and gluons and photons are bosons. Non-integral spin particles have a peculiar property which doesn’t seem to make sense, which is that to reverse their spin they need to be turned through not one but two full circles. How is this possible? Well, imagine a Möbius strip, which is a joined ribbon with an odd number of twists in it, usually simplified to just one. Following the edge around with your finger pointing to the right will reach the point where it points to the left after 360°. In order to get back to pointing the finger to the right, a further 360°of the strip have to be traversed. It’s easier to do this with a strip of paper or ribbon than try to imagine it, for me anyway. This is a good model for how half-integer spin particles work and how it’s possible for them to have to be turned right round twice before they’re back to their initial state. Incidentally, there’s a short story called ‘A Subway Named Möbius’ where a complicated underground train system has one more tunnel added to it which causes a train to disappear, and it doesn’t come back again until the tunnel gets blocked off again. I’m not by any means saying that’s anything more than a fanciful story, but if a topological analogy of that kind can be made regarding such a fundamental feature of physical reality, albeit on a quantum level, it does make me wonder what’s possible. For instance, it’s possible to imagine that space as a whole is “twisted” in all three dimensions, such that any journey round the Universe ends up with one finding one’s home planet is mirrored, or rather seems to be because one has in fact reversed, because the topology of three-dimensional space could in theory be analogous to a Möbius strip. A Möbius strip with an even number of twists is effectively not one at all.
This property of fermions, for complicated reasons I don’t understand, means that no two fermions can occupy the same energy state. This is not the case with bosons. For instance, a laser consists of innumerable photons in the same energy state because they’re bosons, but it effectively means that light cannot form structured matter. It can do things like form caustics and be focussed to a point, and the like, but fermions can build themselves into atoms. Nuclei have to consist of nucleons in different energy states, although they are less obviously in shells than electrons, but neutron stars also have to be in this state – every single neutron in a neutron star is in its own distinct energy state. The electrons in an atom organise themselves much more clearly into different levels, in the form of the shells which enable the periodic table to exist, with heavier elements having more shells and different properties. Without that, there would be no chemistry and no materials as we know them. The fact that I don’t understand this is a source of discomfort to me which I feel very driven to remedy, but right now that’s how things are. It also makes me wonder about Bose-Einstein condensates. These are an unusual state of matter which happens when a low-density gas consisting of bosons is cooled to almost absolute zero and the atoms become overlapping waves and ultimately a single, collective wave comprising all the atoms because they’re larger than the distances between them. Although atoms are made of fermions, each atom as a whole can be a boson if the total number of nucleons and protons is even, so this means that the possibility of attaining this state depends on isotope number as well as what kind of element the gas is, in a similar way to how helium-4 becomes a superfluid at a higher temperature than helium-3. If they were fermions, this would be impossible because they wouldn’t be able to occupy the same energy state.
For us to exist, spin must also, and there also have to be integral and non-integral varieties. It’s a sine qua non of our reality. The Multiverse presumably means there are other universes where there are, for example, only fermions or bosons, or perhaps universes where there is no spin, but these are all very boring places. A universe with just bosons would have no structured matter but instead consist of rays of energy, and one with just fermions would have no structured matter either but simply electrons.
A particle is supposed to have mass, charge and spin. Of the charge values, these can be either positive, negative or neutral, and of the integral and non-integral varieties mentioned above depending on whether quarks, leptons or both make them up. This addition also occurs with spin. Neutral particles clearly do exist, for instance neutrons, whose existence can be deduced fairly easily with precise enough measurements. Chlorine has two common stable isotopes, and if one does something like react salt with something else in distilled water or tries to make a saturated solution of pure sodium chloride in it, one is soon confronted with the fact that the ratio between the weights of the same amount of salt and other substances has to involve a fraction. This is because all normal chlorine is a mixture of the two types of atoms with different numbers of neutral particles, and these are neutrons. Mass, charge and spin all have to be conserved in nuclear and other processes, so for example if a potassium-40 atom emits a positron, one of its protons must become a neutron and it becomes argon-40, and unstable particles decay into various different “fragments”, but they must all add up to having the same charge and spin. Hence a negatively charged muon may become an electron with the same charge but since an electron is so much less massive than a muon, the spin, i.e. the angular momentum, still has to go somewhere, which is into a muon neutrino and an electron neutrino. Likewise, when a neutron leaves the safe confines of an atomic nucleus it only has about a quarter of an hour to “live”, and will decay into a proton and an electron, conserving charge, and also an electron neutrino. They have extremely low mass but observation of the 1987 CE supernova 168 000 light years away revealed that they do have some because of the timing of their arrival here compared to light. Supernovæ produce bursts of neutrinos because the protons in them collapse into a neutron star, converting themselves to neutrons in the process and emitting neutrinos. There are three types of neutrinos, associated with tauons, muons and electrons.
Neutrinos are a bit mind-boggling because they have no mass or charge but only spin,but they must exist because otherwise the accounts wouldn’t balance, as it were. However, there was a problem with solar neutrinos detected in the 1960s, when it turned out the Sun was producing fewer of them than current physics said it should. Until this was resolved, it was possible, though of course extremely unlikely, that for some reason the Sun had stopped working and that the light and heat we were getting from it was simply the last blast of a defunct star, so in a way it was quite worrying, but it’s okay now.
Before I get to the next bit, I want to mention a much older form of philosophy than nuclear physics. Back in the day, there were supposed to be four elements opposed to each other: earth, air, fire and water. Each had two qualities opposed to each other, namely dry and wet and cold and hot. Their atoms were also supposed to correspond to the five Platonic polyhedra, which is why there are five elements rather than four. All of this makes mathematical sense and you can imagine flipping the eight-pointed star round, turning it through 90° and so on – it’s symmetrical. It could even have predictive power in that if one of them was missing, its qualities could be determined, and it has correspondances in alchemy, psychology, astrology and humoral medicine, the last of which is actually useful in herbalism. However, it isn’t applicable to science as it’s usually practiced today, and someone claiming to use it, as I just did, might be seen as undermining their ethos. Nonetheless, the symmetry is real.
There’s also a symmetry in the physics of elementary particles, which allows one to anticipate where gaps may exist implying other particles yet to be discovered. Symmetries can be analysed by group theory in mathematics. One of the most obvious places where this crops up is with Rubik’s cubes, where certain turns may or may not be performed in a particular order to return the cube to its original state. With Rubik’s cubes there are also “orbits”. If you take one to pieces and put it back together arbitrarily, the chances are you will have placed it in another orbit in which there is no arrangement with all the faces the same colour. I think there are eight of these. Groups also apply to arithmetic, so it makes sense to introduce them with that familiar subject. A group is a set with some operations of a certain kind performable on it. It has an identity element, inverse elements and these are associative. For instance, addition and integers form a group because adding zero doesn’t change a number, adding a positive number can be undone by adding the same negative number and it doesn’t matter where you put the brackets: (2+1)+3 = 2+(1+3). Likewise with a Rubik’s cube, keeping it in the same position and turning the top row one twist to the right and then the right hand side one twist downward can be undone by turning the right hand side one twist upward and the top row one twist to the left, and there’s also an identity element in that if you leave the cube alone, it stays the same, which sounds a bit silly but these are just two examples of groups which can be easily understood. Group theory is relevant to crystallography and cryptography. Take this sentence, for example. ROT13 turns it into “Gnxr guvf fragrapr sbe rknzcyr”, and applying ROT13 to it again turns it back into “Take this sentence for example”.
Physics has various symmetries. For instance, there’s symmetry between matter and antimatter, and there are other symmetries such as the correspondence between leptons and quarks. Electrons, muons and tauons accord with up and down, charm and strangeness and top and bottom. The names of up/down and top/bottom are not accidental, although there were moves to name top and bottom truth and beauty instead.
Group theory can be used to classify different forms of symmetry. Spin falls into the SU(2) group. This is one of the Lie groups, which are groups which also behave like spaces. SU groups are “Special Unitary” groups, and I should point out here that I have never knowingly understood matrices and they were a significant hole in my mathematical knowledge at school, because I could never understand how to multiply them, so I’m just going to have to let this pass and say this is this thing, this is out there, and that’s it. I believe I can safely assume that anyone with at least a CSE in maths will get them and understand this better than I can because it’s just my personal blind spot. Having said that, I will kind of give it a go.
There are six flavours of quark: up, down, strangeness, charm, top and bottom. These can be arranged in a hexagon and can be swapped to some extent. A neutron is two down quarks and one up, and a proton two up and one down. The names seem to relate to these properties, because if up and down were swapped in an atomic nucleus it would swap the neutrons and protons. Mathematically this can be envisaged as being part of the SU(3) group, and this is the other area in which the word “spin” has been used: isospin. Isospin is another property of matter which has the same kind of symmetry as spin but is not spin. Then again, spin in the subatomic sense is really quite far from our intuitive understanding of rotating objects, so the fact that this is also called spin, relatively speaking, is not a big leap from the other kind of spin. It’s also why the words “top”, “bottom”, “up” and “down” are used. Just as an electron can be thought of as having an arrow pointing up which can be flipped through two turns to be an arrow pointing down, although it has no link with gravity which determines up and down in everyday parlance, so can some quarks be thought of as “up”, flippable conceptually to “down”, and “top”, flippable to “bottom”. If SU(3) is applied to hadrons (mesons or nucleons), they can be flipped to other hadrons with similar properties. Another application of group theory revealed a gap in the pattern which turned out to be the omega-minus particle consisting of three strange quarks, which was detected and confirmed that group theory could be fruitfully applied to isotopic spin.
Why is it called “isospin” or “isotopic spin”? Well, nuclei are isotopes of various kinds, so for example there’s helium-3, made of two protons and one neutron, as well as helium-4, consisting of two protons and two neutrons, and tritium, an isotope of hydrogen comprising two neutrons and one proton. If the nucleons in these were swapped, they would respectively be tritium, helium-4 and helium-3. This is a form of symmetry pertaining to isotopes, and it influences their stability because there are certain isotopes of elements which would be stable whether or not the neutrons in them were protons and vice versa, and these are particularly stable isotopes. Extending this into the transuranic realm of synthetic elements, it’s possible to predict which isotopes of heavy elements are likely to be the most stable.
It’s also a system of classification, because at one point in the mid-twentieth century a large number of hadrons were known, almost all of which seemed to have no prospect of being part of ordinary matter or having any special relevance to it, which was very puzzling. Another, more recent, puzzle is whether this is just a case of making pretty patterns, albeit useful ones, out of elementary particles or whether it reflects something profound about the nature of physical reality. Murray Gell-Mann, who thought this up, referred to it as the Eightfold Path à la Buddhism, and Fritjof Capra has written extensively on the idea of links between subatomic physics and Eastern spiritual concepts such as Daoism. Western philosophers tend to think of this as jejune and crass.
There is an issue regarding what appears to be the appropriation of quantum physics ideas by the New Age movement in such films as ‘What The Bleep Do We Know?’ and ‘The Secret’. On the other hand, there is also the question of whether this is an excessively proprietorial attitude on the part of some nuclear physicists and academics. But that’s a topic for another post.
A Box Of Chocolates 