C, P and T symmetries

The C, P and T symmetries of classical electromagnetism are quite clear: C negates the charge, P negates one or all three directions in space, and T reverses the direction of time. It is also clear that none of these symmetries can be realised in practice: C converts electrons, of which there are zillions everywhere, into positrons, which scarcely ever exist in the real world; P converts the real world into the looking-glass world, which as Lewis Carroll reminds us, is a figment of our imagination; and T causes time to go backwards, which, as we know, does not happen.

On the other hand, the combinations of any two of C, P and T are symmetries of classical physics, including electromagnetism: CP means that if you reverse a current, you reverse the poles of the electromagnet that the current creates. CT and PT both mean essentially the same thing, from a slightly different philosophical point of view, but with the same mathematical result. Therefore the combination of all three, CPT, is *not* a symmetry of classical physics. Please take careful note of this, I will test you on it later: CPT is NOT a symmetry of classical physics.

Now let us turn our attention to quantum mechanics. In quantum mechanics, there is a theorem called the CPT theorem, which says that CPT *is* a symmetry of quantum mechanics. Ergo, quantum physics is inconsistent with classical physics. Ergo, it is not possible to derive classical physics as a limiting case of quantum physics. Ergo, the measurement problem has no solution. Ergo, the search for a theory of everything is a pointless waste of time. Ergo, why are we wasting so much money on this problem?

I prefer to argue from a realist position, not from a mathematicist position, and take as an axiom the obvious fact that quantum mechanics *is* consistent with classical physics. If this axiom is false, then the universe could not exist. The universe does exist, ergo this axiom is correct. Ergo, the CPT theorem is false. Ergo, at least one of its assumptions is false. Now let us ask, which one of the CPT theorem’s hidden assumptions is false?

Well, I don’t want to get too technical, but the proof of the CPT theorem involves an “analytic continuation” from a Lorentzian spacetime (required for classical electromagnetism) to a Euclidean spacetime (required for quantum theory). It therefore requires spacetime to be a *complex* 4-space, not a real 4-space. But spacetime, in actual hard physical reality, is real, not complex. This is a clear, and obviously false, hidden assumption.

So, if any physicist cares to argue this with me, I will prove that if the CPT theorem holds, then the universe does not exist. Or, in contrapositive form, I will prove that if the universe exists, then the CPT theorem is false. It then depends whether the physicist is a theorist or an experimentalist: if the former, they will dogmatically assert that the CPT theorem is a correct statement about the universe, and will therefore be forced to deny their own existence and that of the entire universe; if the latter, they will (dogmatically?!) assert that they and the universe do exist, and will then set about designing an experiment to test the CPT theorem to destruction.

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Left-handedness in physics

The previous post has convinced me there is a need for a more penetrating analysis of the uses of left-handedness and right-handedness in physics. If you ask a particle physicist, they will tell you that there are two distinct sorts of handedness, which they call chirality and helicity. But if you ask them to explain the difference, they will be unable to do so. It is therefore necessary for you to explain it to them. Here is how you do it.

First take a circle. You can go round it either clockwise or anti-clockwise. This distinction is called helicity, because it distinguishes between the helix of a right-handed screw (that goes into the wall when you turn it clockwise) from that of a left-handed screw (that is only ever used in a few specialist applications, such as violin chin-rest attachments and a certain type of corkscrew). If you look at a right-handed screw in a mirror, you always see a left-handed screw. If you look at a clock in the mirror, the hands go anti-clockwise. (Well, that’s true for normal clocks, but there are a few anti-clocks in the world as well – I used to have one myself, until the burden of contrariness became too much for it, and the hands started to object to being pushed the wrong way round the face.)

The equation of a circle is x^2 + y^2 = 1. Now change one sign, to get x^2 – y^2 = 1. This is a hyperbola, and it comes in two pieces, disconnected from each other. This distinction is called chirality. If you put a mirror along the y axis, it reflects the right-handed part of the hyperbola into the left-handed part. If you put a mirror along the x axis, it does not. To be sure of changing the chirality, you must use two mirrors at right angles to each other.

When we move up to higher-dimensional spaces, such as 4-dimensional spacetime, the details are more complicated, but the basic distinction is still the same. If you use an odd number of mirrors, it is a helicity, and if you use an even number it is a chirality. This is an absolutely clear-cut mathematical distinction about which there can be no argument whatsoever. Or can there? Physicists will argue the hind leg off a donkey (I know, I’ve seen them do it) and will argue black is white (I know, I’ve seen them do it). And since you can appear to change the chirality with only one mirror, if you put it in the right place, they will confidently assert that the mathematical distinction is not relevant to physics. But it is. Do anti-clocks tell negative time? No. Do anti-particles have negative mass? No. These facts are important.

Physicists always use complex numbers, so they can multiply by a square root of -1 any time they feel like it. Hence they can convert a circle into a hyperbola, and hence convert a helicity into a chirality. Never mind that that is cheating, or that it destroys the very thing they are trying to describe. They do it anyway. So that when Distler and Garibaldi try to define a chirality by complexifying the Lorentz group, it is not at all clear that they are not actually defining a helicity. Check your mirrors: how many have they used? One. They will argue till they are blue in the face that they have defined a chirality (I know, I have seen one of them do it), but no amount of argument will convince me that one is an even number.

Now let’s describe the handedness of the real universe we actually live in, instead of some fictitious universe described by some invented physical theory. First look at spacetime, with a unit distance defined by t^2 – x^2 – y^2 – z^2 = 1. This equation defines a 4-dimensional hyperboloid in two pieces. It therefore has a chirality. You need an even number of mirrors, that physicists called T (time reversal) and P (parity). The latter is either one mirror in space, that negates one coordinate, or three mirrors, that negate all three coordinates, but it really doesn’t matter. It only matters that T and P individually are odd numbers of mirrors, so describe helicities, but the two together have an even number, so describe a chirality. The TP-chirality is the obvious fact that you can’t go backwards in time, and you can’t convert a left-handed screw into a right-handed screw.

The existence of anti-particles with opposite charge to normal particles introduces a third mirror, that physicists call C (charge conjugation). Or does it introduce two new mirrors? Is charge conjugation a question of helicity, like the anti-clocks, or a question of chirality? I believe it is a chirality, that is a distinction between particles on one branch of the hyperbola, and anti-particles on the other. It is surely too fundamental a distinction to be a helicity. If so, then the basic CPT symmetry of quantum physics is also a chirality. It is this fundamental chirality of physics that models must be able to reproduce, and explain. (It is not enough just to implement the PT chirality, that is chirality of spacetime.)

The reason why all theories of quantum mechanics from Dirac (1928) onwards fail to do this is because by complexifying the Dirac algebra they have converted the fundamental chirality of the Lorentz group SO(3,1) into the helicity of SO(4), and they have converted the fundamental chirality of the hyperbola into the helicity of a circle. In a sense, therefore, Distler and Garibaldi are not to blame – they are only copying what everyone else has done for nearly a century – but still, they should know better. If I were to characterise this problem in one sentence, I would say that one left hand (chirality) doesn’t know what the other left hand (helicity) is doing.

But wait a minute – is the theory actually correct? Is the so-called `chirality’ of the weak interaction in fact a `helicity’ as the standard model actually implements it? Is it really, experimentally, a property of an odd number of mirrors, or an even number? Think about it, analyse the Wu experiment of 1957 for yourself – what does it say? It links the chirality of spin to the parity mirror. It is an odd number of mirrors. It is a violation of P-symmetry (one mirror or three). It is a helicity, not a chirality. Mathematically, this is also obvious, because the distinction between SU(2)_L and SU(2)_R in SO(4) is quaternion conjugation, which requires three mirrors. And because it is a helicity, not a chirality, you can understand it with a bunch of screws. You need three screws – they are called the Sun, the Earth and the Moon. Screw number one: which way does the Earth go round the Sun? Screw number two: which way does the Earth rotate on its axis? Screw number three: which way does the Moon go round the Earth? There is your helicity of the weak interaction. 

The problem of chirality in E8

I’ve heard it said many times that the problem with grand unified theories, and in particular with E8 theories, is getting the chirality of the weak force in there. This was a problem with Lisi’s model back in 2007 (which he claims to have dealt with, though I haven’t looked into it), and is a problem with the model presented in the recent Manogue/Dray/Wilson paper, although my co-authors insist that the ability to deal with chirality is one of the strong points of this model.

Now I know what the answer is to these problems of chirality in E8, but I don’t know what the question is, and I can’t get anyone to explain it in terms that give enough detail to match the answer to the question. The answer is that E8 contains a product of eight copies of SU(2), which can be split into a product of four copies of SO(4) in a multitude of ways, but not all of them are equivalent. Depending on how you do it, some copies of SO(4) are linked to others via which pairs of SU(2)’s can be swapped while preserving the decomposition into SO(4)’s.

To be more precise the number of ways of splitting 8 things into 4 pairs is 28 x 15 x 6 / 4 x 3 x 2 = 105, and there is a group of order 1344 acting on these. The chirality comes from the fact that the permutation action on the 8 things contains only even permutations. So if you swap two copies of SU(2), then you have to swap another two to compensate. But you do not have a complete choice of which other two to swap. That’s where the fun starts, and that’s why people who can ask the question cannot answer it, and why people who can answer the question cannot ask it. That’s why I am still sitting here with the answer, waiting for someone to be able to ask the question clearly enough.

The reason I am writing about this now is that I have found a way of asking the question that makes sense to me. It may not make sense to the people who want to know the answer, but that is their problem, not mine. The question is how to relate the gauge group SU(2)_L and its right-handed partner SU(2)_R, with the left-handed and right-handed spin groups Spin(3)_L and Spin(3)_R. I write it like this to make clear the physical distinction between the mathematically identical groups SU(2) and Spin(3). Experiment makes clear that there is a relationship. What is it?

This only uses four of the eight copies of SU(2), so we can ignore the strong force and electromagnetism, and answer the question in SO(4,4). BUT we must not extend to O(4,4), because that interferes with the other half of E_8, and negates the charge on the electron. Therefore we only have 12 of the 24 permutations of the four copies of SU(2). That is what physicists call chirality. In other words, there is a relationship between the weak force on the one hand and the splitting of the spinor into left-handed and right-handed parts on the other.

In fact the other half of the algebra is not split into copies of SU(2), but into U(1)+SU(3), for electromagnetism and the strong force, so we don’t need to worry about that. Hence E8 *implies* the chirality of the weak force. If I were a physicist, I would deduce that E8 must be the correct model of physics, but since I am a mathematician I will just say it suggests that it may be premature to say that “There is no E8 theory of everything”. Well, I’ve shown that E8 contains the complete standard model of particle physics, including chirality built in as standard. If you want mass, the three generations and gravity, then you won’t want to miss the next exciting episode, in which I reveal that there is in fact an E8 theory of everything. Ironic, really, considering I have spent the past several years believing there isn’t.

Two updated preprints

First some ancient history: Isaac Newton Institute preprint 19011, written two years ago, has been updated to take account of a referee’s comments, and is available at https://robwilson1.files.wordpress.com/2022/02/remarks3revws.pdf, since the arXiv refused to post it. As you know, my ideas have moved on significantly since then, so it is a bit of a compromise, and refers to later papers for the newer ideas, rather than being completely re-written.

Next the saga of “Quaternionic reflections on (non)locality”, which is still “on hold” at the arXiv after four weeks, and I can only assume will stay that way indefinitely. The 26-page version that I submitted to a journal yesterday is at https://robwilson1.files.wordpress.com/2022/02/nonlocal7lms.pdf, and does a reasonable job (I think) of superseding a great deal (though not all) of my earlier work. It certainly isn’t perfect, but I’m hoping it’s good enough that editors and referees will take it seriously, and help me to improve it, rather than reject without the option to resubmit.

What it does, or tries to do, is tie together a lot of separate ideas that have come and gone over many years into a single coherent story, that starts with the experimental properties of polarised light, and keeps going in a single logical argument until it reaches a quantum theory of gravity. You can quibble with some details, of course, but the overall architecture and engineering is sound. This bridge will not fall down.

Perishing lack of algebraic technique

When I was young, I learnt the violin from a teacher who was a great fan of the playing of Ruggiero Ricci, and not of the most famous violinist of the time, Yehudi Menuhin. I remember one of his comments on Menuhin, though I don’t remember the context: “Perishing lack of bow technique”. No doubt he would have used a stronger word if there wasn’t a child present. Now it seems to me that a great many famous physicists suffer in a similar way from a “perishing lack of algebraic technique”. Of course, this is a minority view, and most people are quite happy to accept the playing of Yehudi Menuhin without making a critical appraisal of his bow technique.

If your violin playing is larded with plenty of continuous (geometrical) vibrato (of the wave-function) and portato (of the mixing angles), the listener may be carried away with the beauty of your playing, but be unable to distinguish the elementary particles (notes) of the music, or distinguish between up bows and down bows (spins). You may be an expert at prestidigitation of the left hand, but if you haven’t got the right hand (algebraic) bow techniques of flying spiccato and the rest, you’ll never be able to play like Paganini, and you’ll never be able to distinguish all the discrete quantum states.

A discreet spiccato that distinguishes between up and down bows when crossing strings (from left to right or right to left) is a sine qua non of violin playing. Those violinists who think that the difference between up and down bows is a mere matter of convention (and I know quite a few violinists who do seem to think this!) do not understand the violin. Those string theorists who think the strings vibrate of their own accord are in complete denial of the fact that without the algebraic technique of the bow, there would be no music of the violin (or of the spheres). They should take a few lessons in bowing from the algebraic masters of the art.

The Free Will Theorem

The Free Will Theorem (FWT) is sometimes stated in the form “If humans have free will, then so do individual elementary particles”. Of course, the real statement is much more technical than that, and it was proved in various different versions by John Conway and Simon Kochen around 2006-2009. The proof is mathematically rigorous, so if we regard the conclusion as absurd, then we have to look for the hidden assumptions, and try to argue that they are not reasonable.

In fact, the theorem is not really about free will, it is about measurement of spin. And it is really an extension of Bell’s Theorem, and has similar implications for the nature of quantum reality. Bell’s theorem implies that no amount of “local hidden variables” can explain what actually happens in quantum mechanical experiments. The free will theorem implies that the direction of spin is itself a “local hidden variable” that cannot explain what actually happens in the experiments.

The way the proof works is by carefully choosing a set of 33 directions in space, and evaluating the spin measurement operator in those 33 directions, assuming that the operator is measuring a spin that has a “hidden” direction that can vary continuously in space. This leads to a contradiction, which implies that the spin “direction” can only take a finite number of values, strictly less than 33.

Now let’s think about this. What does it mean? It means that the very concept of “direction” is a non-local concept. In order to define a direction you need at least two particles, and in order to measure a direction you need far more than this. A good example to illustrate what is going is an iron magnet. The individual atoms of iron have a “spin”, and when the spins are aligned, the iron becomes magnetised. For this to work, we need a good concept of “direction” for an atom of iron. This isn’t a problem, because in the standard model an iron atom consists of 194 elementary particles all interacting with each other, so that although each individual particle has a very weak sense of direction, the combination of all 194 can have a very strong sense of direction.

Hence on the scale of atoms one can distinguish so many different directions in space that space is to a good approximation continuous on this scale. But as we get to smaller and smaller scales, the number of distinguishable directions decreases to the point where we have to treat these directions as discrete. That does not mean that space itself is necessarily discrete: the roads can be continuous, but they only go in a limited number of directions. The free will theorem implies the number of such directions is less than 33. In my model, the number is 31.

Left-handed and right-handed spin

Probably the biggest difficulty I have had in understanding the standard model of particle physics is trying to understand what the theorists mean when they talk about left-handed and right-handed spin. The textbooks do not explain it, and the theorists either cannot or will not explain it, and the mathematics behind it simply does not make sense. I am left with the sense that the distinction as currently made has neither physical nor mathematical meaning, and yet it is clear that there is something very important going on. Understanding what left-handed and right-handed spin are, and why they are as different as chalk and cheese, seems to me to be key to the understanding of mass, the quantisation of gravity, and the unification of the four fundamental forces.

Which is why I am intrigued by some bizarre properties of the action of the binary icosahedral group on the Dirac spinor. It is not at all what I expected, and it is nothing like any of the things I have written about in several papers and many posts on this blog. But suddenly the Dirac spinor splits into left-handed and right-handed parts that really are as different as chalk and cheese. The secret is in the symmetry-breaking. How exactly do we need to break the symmetry of the icosahedron/dodecahedron to get the standard model?

I think I have generally assumed that the symmetry needs to be broken to tetrahedral symmetry, which essentially means cubic symmetry. But the experimental evidence (which in this case is dismissed as meaningless coincidences by most physicists) told me years ago that it is actually broken to a dihedral symmetry. I really don’t know why I have ignored this evidence for so long. The basic fact of physical life is that space is not the same in all directions. The up-down directions are different from the sideways directions. The same seems to be true at the elementary particle level as well.

The symmetry group permutes five things. Geometrically, they are five cubes inscribed in a dodecahedron. Algebraically, you can split them as 1+4 by fixing one cube, or as 2+3. The way the standard model of particle physics works is by using the splitting 1+4 for quantum electrodynamics (electricity, magnetism, that sort of thing), and the splitting 2+3 for the weak force (radioactivity). It is the latter splitting that I am only now beginning to understand in physical terms.

In mathematical terms, the way this 2+3 splitting manifests itself in the Dirac spinor is something that I read about 40 years ago, in a paper by Arjeh Cohen that classified the finite quaternionic reflection groups. You see, the way the binary icosahedral group acts on the Dirac spinor is as a quaternionic reflection group. The reflections (by definition) fix one half of the Dirac spinor and act on the other half. These are the left-handed and right-handed Weyl spinors of the standard model. One half has a triplet symmetry, and the other half has a doublet symmetry. They are as different as chalk and cheese.

In the standard model, the doublet symmetry is implemented in the left-handed spinor, but the triplet symmetry is not implemented in the spinor at all. That is what the standard model is missing. Read my papers to find out more.

Where does the standard model of particle physics come from?

When anything is measured in quantum physics, the answer is always this or that, never something in between. Therefore the fundamental processes must be discrete, not continuous. This does not mean that everything has to be discrete. But something has to be discrete. In particular, spin has to be discrete, because whenever and however you measure the spin of an elementary particle, it is either up or down, not something in between.

Spin is modelled in quantum mechanics with the continuous group SU(2), so we need a finite (discrete) subgroup of SU(2). Now we need to put this finite group into 4-dimensional spacetime, with the Lorentz group SO(3,1) acting on spacetime. That means we need our finite group to have a 4-dimensional absolutely irreducible real representation. There is only one group that has both of these properties, namely the binary icosahedral group, of order 120.

Now we need to do the representation theory. The representations are of two types: one type describes matter, and the other describes forces. The “matter” half of the real group algebra is a sum of four algebras of quaternion matrices, of sizes 1×1, 1×1, 2×2 and 3×3, so that the total number of degrees of freedom is 4x(1+1+4+9)=60. The finite group determines a compact subgroup of each matrix algebra, by taking all linear combinations of group elements that have determinant 1. These compact groups are so-called symplectic groups, and fit together into Sp(1) x Sp(1) x Sp(2) x Sp(3).

Physicists don’t like quaternions, but they like complex numbers. So they ignore the quaternions, and replace them with complex numbers. Hence the symplectic groups become unitary, and they get U(1) x U(1) x U(2) x U(3). Each factor contains a subgroup U(1) of complex scalar matrices, which in the mathematics are four distinct groups of scalars. In physics, however, all these scalars are multiplied together to give a single scalar, so that they end up with the group S(U(2) x U(3)), which is the gauge group of the standard model of particle physics.

This process of restricting from quaternions to complex numbers breaks almost all the symmetry. Physicists think that “symmetry-breaking” is something physically real that happened in the Big Bang when the universe was created. It is not. It is something that happened in their brains when they threw away the quaternions and insisted on building a theory using complex numbers instead.

Hamiltonian physics was the greatest revolution in theoretical physics since Newton, and Hamilton spent half his life in search of the quaternions. When he found them, he knew that he finally understood how to do physics properly. He was right. Throwing away the quaternions was the biggest mistake in the entire history of theoretical physics.

The McKay correspondence

John McKay is perhaps best-known for his observation that 196883+1=196884. Of course, this is a caricature. What happened was that in his wide reading of different areas of mathematics he found the number 196884 as the first Fourier coefficient of the j-function, and the number 196883 as the first non-trivial representation degree of the Monster. And he said to himself, “if that’s a coincidence, then I’m a Dutchman”, or words to that effect. Now I know that McKay isn’t a Dutchman, having met him on a number of occasions, and Richard Borcherds (who I also know isn’t a Dutchman, as we were close contemporaries in Cambridge) proved that it isn’t a coincidence.

Some other coincidences that McKay has pointed out have not (yet) had such profound consequences. The phenomenon that goes by the name of the McKay correspondence is a relationship between the finite subgroups of SL(2,C) and the ADE Dynkin diagrams. It is well-known to a certain type of physicists, because SL(2,C) is absolutely fundamental to quantum mechanics, and finite subgroups offer a promise of discretisation, and therefore a deeper understanding of the fundamental nature of reality.

All I want to say here is that under the McKay correspondence, the E_8 Dynkin diagram corresponds to the binary icosahedral group. Therefore, the McKay correspondence relates E_8 x E_8 heterotic string theory to the left and right multiplications of the binary icosahedral group on itself. Now I haven’t a snowball’s chance in hell of understanding E_8 x E_8 heterotic string theory, but I do understand the actions of the binary icosahedral group on itself quite well. So I thought, why not? McKay tells us we can understand E_8 x E_8 heterotic string theory by understanding the representation theory of the binary icosahedral group. What are we waiting for?!

Well, when I started working through it, I discovered some remarkable things. The complex regular representation of the binary icosahedral group embeds in E_8 in two distinct ways. But neither of these embeddings produces a model that is consistent with experiment. That isn’t the fault of the binary icosahedral group – it is the fault of E_8. The real group algebra of the binary icosahedral group produces a model that is both the left-regular and right-regular representation, and that, as far as I can tell so far, is consistent with experiment. But this model uses a mixture of the two embeddings, which has the unfortunate effect of losing the spinors. So E_8 only contains the bosonic half of the group algebra, so contains no elementary fermions, and therefore no matter. No matter, did you say? Yes, it does matter, very much!

There is a certain irony here, in that I only got interested in this problem because someone told me that Garrett Lisi had built a model of fundamental physics based on E_8. And now I have “proved” (based on a whole load of assumptions that you may not accept, of course) that there is no such model. This proof is much more subtle than the “proof” that Jacques Distler and Skip Garibaldi offered, that is a quite naive counting argument, and has some glaringly obvious unnecessary assumptions, most notably the assumption that both left-handed and right-handed Weyl spinors must carry a generation label. My proof relies instead on an analysis of how the gauge groups and gauge bosons actually act on the elementary fermions. While the numbers add up, the group actions don’t.

You may well say that I haven’t proved that E_8 x E_8 heterotic string theory is wrong, and that would be true. But I don’t need to, because it isn’t a theory, it doesn’t explain anything, and it doesn’t make predictions. I just need to build a theory that looks like E_8 x E_8 heterotic string theory under the McKay correspondence, and makes lots of testable predictions. Well, I’ve already done that for the E_6 and E_7 cases, so I don’t expect any major problems. Watch this space.

Another rejection

One of my papers that is on the arXiv, and that a real editor of a real (i.e. not predatory) journal asked me to submit to them, was rejected by a different journal that I (foolishly) decided to send it to, with a report (written by a real referee, who had spent some weeks on this) of three words: “lacks physical understanding”. I respectfully suggest that the referee “lacks mathematical understanding”.

It is of course perfectly normal for a referee to completely fail to understand what a paper is about. This has happened to me occasionally even when I have written papers in pure group theory. But it is almost inevitable whenever a paper is in any sense at all interdisciplinary. The paper talks about interactions between discipline A and discipline B, and is sent to a referee in discipline B, who doesn’t understand the part that belongs to discipline A, so ignores it and then (obviously) cannot understand the part that belongs to discipline B, so rejects it on the grounds that the (slightly surprising) implications for discipline B do not agree with the referee’s prejudices.

This is a huge problem for the progress of science in general. There is a huge prejudice against interdisciplinary research, which means that interdisciplinary scientists can’t publish their research, can’t get jobs, and whole areas of interdisciplinary research simply die. Funding agencies are desperate to support interdisciplinary research, and it is obviously crucial is so many areas, but they can’t do it, because the fundamental human imperative of survival undermines all their efforts. And the name of this imperative is subject prejudice.

If it was racial prejudice, or gender prejudice or any number of other prejudices, it would be illegal (in some countries). But because it is subject prejudice, it is not illegal, but is every bit as insidious. It is completely obvious to a mathematician that theoretical physics has got bogged down in some mathematics that doesn’t work. Or rather, it does work, up to a point, but it is inconsistent and therefore needs to be sorted out. It is completely obvious that physicists, left to themselves, have failed to sort out the problems that were obvious nearly a century ago. It is completely obvious that physicists need help from mathematicians. It is completely obvious that physicists reject help from mathematicians. It is completely obvious that this strategy will not work.