A retraction

Dear friends,

I think I may have made a terrible mistake, when I read a textbook on general relativity, which taught me that spacetime is real and can be distorted. From this I deduced that the group of local spacetime symmetries necessary for all of physics is SL(4,R). All the preprints posted on this blog are based on this assumption. They are all wrong.

The group of local spacetime symmetries in general relativity is indeed SL(4,R). But that does not mean the group of local spacetime symmetries in quantum gravity is necessarily SL(4,R). Now that I am actually beginning to understand how to quantise gravity, I realise that the correct group is SL(2,H). Both are real forms of SU(4), so the differences are relatively minor, but important.

Vector-spinor-tensor spacetime

In the previous post I described how to build a quantum spacetime out of the representations of Sym(6) over the field of two elements, 0 and 1. This group has four irreducible representations, 1, V, S and T=VxS, which I call vector, spinor and tensor. The vectors and spinors have 4 bits each, and the tensor has 16. The antisymmetric square of S has structure 1.V.1, and behaves like the permutation representation on 6 letters.

Now the odd permutations are non-physical: they correspond to things like reversing the direction of time, or ignoring the difference between the proton and the neutron, or between the electron and the neutrino. So we need to get rid of the odd permutations, and restrict to the alternating group Alt(6) of order 360. This makes no essential difference to V or S, but it converts T into an 8-dimensional representation over the field of order 4. This field is the quantum equivalent of the complex numbers, and its 3 non-zero elements form the group of order 3, with elements 1, v, w, say. Thus v.v=w, v.w=1 and w.w=v. Finally, 1+v+w=0, so that all arithmetical operations are now defined.

The important thing to know about modular representation theory is how to decompose the group algebra as a direct sum of indecomposables. In the case of Alt(6), the tensor representation never glues to anything else, and we have 8 copies of T in the group algebra. The vector representation glues into a tower V.1.S.1.V.1.S.1.V, which is of shape V.(VxV).V, where VxV=1.S.1.V.1.S.1 as we already saw. There are four copies of this tower in the group algebra, as well as four copies of S.(SxS).S. Finally there is a twin tower, with 1 on top and bottom, and in between the direct sum of two towers V.1.S.1.V.1.S and S.1.V.1.S.1.V. That’s really all you need to know about the 2-modular representation theory of Alt(6). Let’s just check the dimensions add up correctly: 8×16 + 8×24 + 40 = 360.

Last time I talked about VxV, which explains how vectors glue to vectors, and SxS, which explains how spinors glue to spinors. The former describes the structure of spacetime, and is therefore a quantum version of general relativity. The latter describes the forces between elementary particles, so is a version of quantum mechanics. I didn’t tell you about VxS=T, which explains how spinors glue to vectors. This is the story of how quarks (and electrons) glue to spacetime itself. It is rather like the strong force, but it isn’t really a force. That is, the quarks are not glued to each other, but are glued to spacetime. The fact that quarks are not glued to each other is what is known as asymptotic freedom.

Well, you can see that T is an 8-dimensional representation, so it has something in common with the 8-dimensional group SU(3) used in the standard model for the strong force. The quantum analogue of SU(3) is the finite group SU(3,2), of order 216. The 8-dimensional representation is the adjoint representation, so is a representation of PSU(3,2), which has order 72. There is a subgroup of order 36 that lies inside Alt(6), splitting the 6 letters into two sets of 3. The rest of PSU(3,2) is the supersymmetry that swaps V and S. If you do the calculations you will find that all this works out correctly, and the representation T is indeed the adjoint representation of SU(3,2), compatible with the action of Alt(6).

So there is my quantum theory of everything. There are three fundamental “forces” described by the structure of the representations V.1.S.1.V.1.S.1.V,  S.1.V.1.S.1.V.1.S and T. There is no mixing between these three “forces”, but they do not map precisely to the forces of nature as described in the standard model. The reason for this is really that the standard model does not distinguish between V and S in the same way that I do. The vectors and spinors in the standard model are mixtures of V and S, with a complex structure that depends on the observer. Hence the three fundamental forces are mixed together in the standard model. 

In principle, V.1.S.1.V.1.S.1.V describes how vectors relate to vectors, so describes the structure of spacetime, and therefore describes a theory of gravity that quantises general relativity. Similarly, S.1.V.1.S.1.V.1.S describes how spinors relate to spinors, and therefore describes the forces of quantum mechanics, that is the electro-weak forces. Finally T describes how spinors relate to vectors, and therefore describes the strong force.

It remains to embed these quanta into macroscopic spacetime, in order to explain the mixtures of V and S that occur in the standard model, and hence derive all the mixing parameters from the peculiar properties of an observer on the Earth. This takes us to a whole new level of representation theory, that relates complex representations to finite ones, so deserves a whole new post. By the way, I can’t decide whether to call this theory VeSpiT or SpiVeT. What do you think? I think I prefer VeSpiT, as it suggests a sting in the tail.

A tale of two cities

Today I want to talk about the differences between the representation theory of Lie groups over complex numbers, and the representation theory of finite groups over bits. Over complex numbers, representations (of finite or simple groups) always split as a direct sum of irreducible representations, so that one only ever needs to consider irreducible representations. Over bits, this is no longer true, so that the irreducible constituents of a representation do not have to sit side-by-side, but can be stacked on top of each other in arbitrarily complicated arrangements.

To illustrate the difference, compare SU(2) acting on a pair of complex numbers, with Sym(2) acting on a pair of bits. In the former case, there is an infinite series of irreducible representations, of dimensions 1, 2, 3, 4, …, where 1 is the trivial representation (no action), 2 is the spinor action, and 3 is the vector action, that is as SO(3). The important fact for physics is that 2×2=3+1. If, for example, this SU(2) represents the weak interaction, then the 2 represents the doublets of fermions (electron/neutrino, up/down quark, proton/neutron, etc.), while the 3 represents the Z, W^+ and W^- bosons. The challenge for physics is then to explain the “symmetry-breaking” between these three bosons, as well as the photon in the 1 representation.

Now consider instead the action of Sym(2) permuting two bits. Then the spinors are 00, 11, 10 and 01, and the symmetry breaking as 1+1+2 is obvious. So we can take 00 to represent the photon, 11 to represent the Z boson, and 10 and 01 to represent the W bosons, and Sym(2) representing charge conjugation. This representation is not irreducible, because there is an irreducible sub-representation on the even bit strings 00 and 11. But if the representation were a direct sum of irreducibles, these would both be trivial, and the whole representation would be trivial. So let us write this 2-dimensional representation as 1.1, to denote that it has a trivial sub-representation, with another trivial representation glued on top.

Physically, the representation on the bottom has mass, and the representation on the top has electric charge. This is an absolutely crucial property of charge in physics: it can never exist on its own, it always has to exist glued on top of mass.

Now let’s move up a gear, and consider the action of Sym(4) on 4 bits. we still have an “on switch” 1111 at the bottom of the representation, and we still have an “odd part” at the top of the representation. In the middle, we have a triplet of “even bits modulo complementation”, that is 0011/1100 and 1010/0101 and 1001/0110. The odd part similarly consists of four pairs of a single bit and the complementary three bits, thus 1000/0111, 0100/1011, 0010/1101 and 0001/1110. Now this representation can be used for many different things in quantum physics, depending on what four things one wants to study. For example, one could use 1111 for the Higgs boson, or some other place-holder for mass, and use the 3 even pairs for the Z and W bosons, now endowed with an extra “spin” bit to reflect the fact that they have spin 1, not spin 0. Then on top we could have four fermions, again each with a spin bit. Symmetry-breaking of the weak interaction then requires us to break the symmetry down to Sym(2) x Sym(2), so that we have two weak doublets of fermions, for example electron/neutrino and proton/neutron. (Incidentally, I leave it as an exercise for the reader to show that the sum of the masses of the even part (Z, W^+ and W^-), minus the masses of the odd part, is equal to twice the mass of the Higgs boson.)

The fun starts with 6 bits permuted by Sym(6). This group has only four irreducible representations on bits. One is the trivial 1-dimensional representation. There are two distinct representations on 4 bits, that I want to call the vector representation V and the spinor representation S. The fourth irreducible representation is the tensor product of V and S. For applications to physics, one wants the tensor product of S with itself. The structure of this tensor product happens to be a single tower with 7 floors, arranged as 1.V.1.S.1.V.1. This is the finite analogue of the Clifford Algebra used in the standard model, which has a structure like 1+V+A+V+1, where A is the anti-symmetric square of V. Indeed, in the finite case the anti-symmetric square of V is 1.S.1, so the matching of the irreducible parts of the Clifford Algebra works perfectly. But the finite case has a much more interesting structure, and permits the description of much more interesting quantum physics than is possible in the continuous case.

Since the anti-symmetric square of S is 1.V.1, we can obtain the vectors from the spinors in much the same way as is done in the standard model. But the vectors come with a scalar on the bottom (mass, remember), and a scalar on the top (charge). The vector cannot exist on its own without the mass, and the charge cannot exist without the vector and the mass. This is a good deal closer to how the real world behaves than is the standard model, in which mass and charge are optional extras.

The most astonishing thing about Sym(6), however, is its outer automorphism, that swaps V with S. Nothing like this exists for Lie groups, so that this “supersymmetry” between fermions and bosons cannot be expressed in the language of Lie groups. Indeed, as is well-known, attempts to model this supersymmetry with Lie groups led to predictions of all sorts of new and exotic particles, which experiment has not found. So it is pretty clear that Lie group supersymmetry does not describe the real universe. But I hope to convince you that Sym(6) supersymmetry does actually describe the real universe.

This supersymmetry allows us to construct the spinors from the vectors in the same way that we constructed the vectors from the spinors. That is, the anti-symmetric square of V has the structure 1.S.1. This is extraordinary. Nothing like this exists in the standard model. It means that in the foundations of quantum mechanics it is not necessary to dream up an abstract concept of spin that is superimposed on spacetime. The concept of spin emerges from the quantum nature of spacetime itself. Or vice versa, however you want to look at it. Moreover, the “spin” representation, like the “vector” representation, is not really 4-dimensional, but 6-dimensional. It has an extra on/off bit at the bottom, and an extra even/odd bit at the top. These extra bits allow the inclusion of the concepts of left- and right-handed spin, but as a single integrated concept, not as two completely divorced types of spin.

Both 1.V.1 and 1.S.1 are described by permutation representations of Sym(6). One of them can be taken as the usual representation on 6 letters A,B,C,D,E,F, and then the other one is the image under the outer automorphism. That makes for some difficult notation, but one can also switch to a different set of six letters U,V,W,X,Y,Z for example, and match them up later. There is loads of stuff written on this, that I don’t want to go into just now.

Let’s just study the details of one of these representations at a time. First there is the overall on/off switch 111111. Then there are 15 pairs and their complements 110000/001111 etc. If you add together two pairs that overlap, you get another pair (remembering that 1+1=0), but if you add together two pairs that don’t overlap, you get the complement of a pair. That is why you cannot separate the pairs from their complements. The odd bit strings and their complements are of two types, 6 of type 1/5 such as 100000/011111, and 10 of type 3/3 such as 111000/000111. (Another exercise for the reader: choose a physical interpretation of this model to show that the total mass of three generations of electrons and neutrinos, plus three protons, is equal to the mass of five neutrons.)

To get the standard model 1+4+1 from this 1.4.1, we need to break the symmetry in some way. In particular, we need to fix one of the odd pairs. In fact, the standard model fixes one odd pair of each type, so breaking the symmetry simultaneously to 1+5 and 3+3, in other words to 1+2+3. Well, I suppose this is the symmetry-breaking on the bosons (vectors). The corresponding symmetry-breaking on the other 1.4.1 representation is then 1x2x3, which therefore gives us the three generations of fermions. Even this is not enough symmetry-breaking to get the standard model, because the 2 still has to be broken to 1+1 in order to make 1+1+1+3 in the vector representation, which is then subjected to Lorentz transformations to make 1+1+4. By this time, I am afraid, so much violence has been done to the mathematics that it no longer makes any sense at all. By all means continue to calculate with the standard model as before. Just don’t try to pretend that all that hocus-pocus has anything to do with how things really are.

There are various grand unified theories that attempt to model fundamental particles by doing less symmetry-breaking. The Georgi-Glashow model, for example unifies 1+2+3 into 1+(2+3), while the Pati-Salam model unifies 2+1+3 into 2+(1+3). But they do this with Lie groups, SU(5) in the former case, and SU(2) x SU(2) x SU(4) in the latter. There are also models that attempt the full unification of (1+2+3) with SU(6). But, as I have shown, a Lie group model can never reproduce the structure of the actual quantum world, in which representations are stacked on top of each other.

To see how these unification ought to work, we need to look at both the vector and the spinor representations, not just one of them. First look at Georgi-Glashow 1+5. The group reduces from Sym(6) to Sym(5), and the actions on one copy of 6+10+15 are (1+5)+(10)+(5+10), while the actions on the other are then (6)+(10)+(15), i.e without any splitting. A Lie group model can only look at one of these, and Georgi-Glashow uses (1+5)+(10)+(5+10). They then throw away the scalar (which contains the mass), and identify the odd and even parts as real and imaginary parts of a complex number, thereby throwing away the charge also. They are left with a complex 5-space and a complex 10-space, and assume that the latter is the anti-symmetric cube of the former (which it is, in fact).

The Pati-Salam model, on the other hand, reduces the symmetry to Sym(2) x Sym(4), so that one copy of 6+10+15 breaks as (2+4) + (4+6) + (1+6+8), while the other breaks as (6) + (4+6) + (3+12).

A third possibility is to split as 3+3, in which case one can also include the symmetry that swaps the two 3s. In this scenario we get (6) + (1+9) + (6+9). I do not know if there is a grand unified theory that does this, but general relativity certainly does. So this is the one to study if you want to quantise GR.

Well, there is so much more to say about six bits, but I think this is enough for one day. I have shown you how 6 bits are enough to encode all the fundamental forces of nature separately. Now we need to unify them. See you tomorrow for the quantum theory of everything.

Three bits

I know I said I was going to talk about six bits today, but six bits are HARD, so I wanted to start with something a bit easier. Three bits are what you need to understand the strong force. There are three “colours” each of which can be switched into two states. Hence the total number of states is 2^3=8. Very clear and easy, I think. Which is why I find it so difficult to understand that the standard model computes with 3^2 instead of 2^3. It is then necessary to use 3^2-1 rather than the full 3^2 in order to build a model that is compatible with experiment. Well, when I say compatible with experiment, it is actually quite hard to do experiments with the strong force, because it involves smashing protons into each other at ridiculously high speeds. So any model at all of the strong force is quite hard to falsify.

But the problem of understanding why nature insists on using 3^2-1 rather than the full 3^2 is one of the unsolved problems of the standard model, known as the problem of colour confinement. It isn’t a problem of understanding nature. It is a problem of understanding human nature. Why on Earth is it so difficult for people to understand the difference between 3^2 and 2^3?

The law of small numbers

The law of small numbers is attributed to Richard Guy, who recently passed away at the age of 103. It is a somewhat jocular, or whimsical, law with a serious point. There is only a small number of small numbers. Small numbers occur everywhere. If you see the same small number appearing in different contexts, you should not be surprised. It is, in other words, a warning against numerology. You may not think that 103 is a particularly small number. But just be careful what you do with that number. If you start making a list of other people who died at the age of 103, and try to find connections between them, don’t complain to me (or Richard Guy) if you get accused of numerology. Still less if you try to make connections with people who live in a house numbered 103. Or if you try to read some significance into the fact that 103 is a prime number.

So I am acutely aware of the danger that awaits me when I embark on the process of trying to explain why I think the entire universe is based on the number 6. Clearly I have abstracted the structure of the universe down to an absurd level of abstraction, where no meaning remains. But the number 6 is not just a number, it has some structure. It has a structure as 1+2+3, and a structure as 1x2x3, both of which play fundamental roles in the standard model of particle physics. It also has a structure as 4×3/2×1, which has an even deeper significance. Either way, the number 6 is not the fundamental number, but can be derived from either 4 or 3 by various simple mathematical processes. What matters, then, is that space is 3-dimensional, or that spacetime is 4-dimensional. Even then, you can’t help noticing that 4=2+2=2×2=2^2, so that the number 2 is all we really need.

So why are there three dimensions of space? Because (2×2+2)/2 = 3, of course!

Why is there one dimension of time? Because (2×2-2)/2=1, of course!

So spacetime has dimension 4 = 3+1 = 2×2. Of course, I am not really saying that these are just numbers. They are dimensions of representations of fundamental symmetry groups, and it is the representation theory that matters, not the numbers.

The number 6 arises both as (4×4-4)/2 and as (3×3+3)/2. It is the fact that these are two different occurrences of the number 6 that is the reason behind the complexities of electro-weak unification. The version (4×4-4)/2 describes the electromagnetic field, and the version (3×3+3)/2 describes the weak interaction. Electro-weak mixing arises from the belief that these overlap in (3×3-3)/2. The representation theory, however, shows that they do not overlap in reality. The weak (3×3+3)/2 is part of (4×4+4)/2=10, and has nothing to do with electromagnetism. Electro-weak mixing arose from a mistaken impression that two occurrences of the number 6 were really the same, when the representation theory makes it abundantly clear that they are not.

So I take to heart my own warning about the law of small numbers. One must never rely on the numbers on their own. One must always find the structural reasons for the coincidence of two numbers. Only if the structural reasons agree, can one consider seriously the possibility that the coincidence has a deeper meaning.

One shilling

When I was a child, we called coins “bits”. Each bit of information is therefore one coin of information, that is the information “heads” or “tails”. A threepenny bit was more interesting, because it had 12 sides, like the pound coin today. In terms of bits, it behaved rather like a baryon, because you could go to a bank and change it for three pennies (or quarks, as they are called nowadays, because no-one knows what an old penny is any more!). The lowest energy state for three pennies is two heads up and one heads down, closely followed by the two-down, one-up state. But if you want to observe beta decay, you have to change the pennies into two ha’pennies, which are either left- and right-handed. (In case you didn’t know, the monarch’s head switches from left-facing to right-facing and back again with each new monarch.) Then you have to do something to the left-handed ha’pennies, which involves breaking them each into two farthings. So at the end of the day, if you want to understand how threepenny bits work, you have to break them down into 12 farthings. That at least is how the standard model of money worked, in the UK, in the early 1960s. And perhaps the 12 sides of the threepenny bit represented the 12 farthings.

Of course, the 12 sides of the threepenny bit were not observable: if you toss the coin, it always comes down heads or tails, and you never see it land on one of the 12 sides. So it is very difficult to observe the individual quarks.

In any case, for threepence you can buy one generation of quarks. For an extra three-farthings, you can add in the leptons (a ha’penny for the electron, a farthing for the neutrino – farthings are almost impossible to detect in the real world – they go right through the holes in your pocket and reappear on the other side of the Earth). Three generations will therefore cost you elevenpence-farthing, which leaves you exactly three farthings change out of shilling, which which to buy three dimensions of space to put your fermions into. So for a shilling you can buy the blueprint for the standard model of particle physics. In the usual rules of economics, all ha’pennies are exactly the same, so all electrons are exactly the same. Well, you get the picture.

I can’t help feeling that the problem today may be related to the fact that the threepenny bit has been replaced by the pound, and you cannot go to a bank and get your pound divided into three. A decimal, or SO(10), model, just doesn’t cut it. Perhaps you try an SU(5) model, and divide your pound into five 20p pieces. Then you notice that these coins have 7 sides, and you try to understand the 7 sides by using G_2 (which in monetary circles is better known as G7, formerly G8) and octonions. Or perhaps you use SU(2) to divide your pound into two 50p pieces, and again you have the 7-sided problem.

There are people, not in the mainstream of modern physics, who are still convinced that 240 pence in the pound is the correct way to go. Now 240 is the number of roots in the E_8 lattice, and re-arranging the 240 pennies in a pound is something that obsesses certain people. There are other people who point out that general relativity is based on the 10-dimensional Ricci tensor, and that therefore dividing a pound into 10×10 pence is obviously the correct way to buy a physical theory. I have explained how 10×10 breaks up as 20+35+45, in which the 20p is the Riemann curvature tensor (don’t forget, the 7 sides of the 20p are curved!), and the symmetry of the 35 and the 45 is broken, because there are no 35p or 45p coins. So perhaps the incompatibility between quantum mechanics and general relativity derives from the incompatibility of the imperial and decimal monetary systems.

No wonder they say that money is the root cause of all the problems in the world.

Four bits

Today I want to tell you how to build a toy model of physics using only four bits. It is not a correct model, for many reasons that I will explain in due course. It is a finite version of the Georgi-Glashow SU(5) Grand Unified Theory, with the complex numbers replaced by a single bit. The group therefore reduces from the (real or) complex Lie group SU(5) to the finite analogue, SO(5,2). This finite group has some other names, including Sp(4,2), that is the symplectic group, acting on a spinor with 4 bits (as opposed to the vector, which has 5 bits), and Sym(6), that is the group of all permutations on six letters.

For the purposes of exposition I shall use the representation as Sym(6), which is much more familiar than the other two representations. The translations to spinors and vectors are straightforward for those who know how to do them, but just obscure the essential group-theoretical core of the argument that I want to describe. The 15 fermions in a single generation are represented by the 15 (unordered) pairs from the 6 letters. Symmetry-breaking to the standard model is represented by splitting the 6 letters into 1+2+3, so that the 15 pairs split as 1+2+3+3+6. Since it is easy, let us actually do this: A+BC+DEF gives one special pair BC, a doublet AB, AC, a triplet AD, AE, AF and another triple DE, DF, EF, and the remaining six BD, CD, BE, CE, BF, CF.

If the pairing of B and C represents the weak force, acting on left-handed spinors, then we get four colours of weak doublets: AB and AC for leptons, and BD, CD etc for the three colours of quarks. So we get 2+6 left-handed spinors, and 1+3+3 right-handed, exactly as in the Georgi-Glashow model. In other words, we have built all the essential combinatorics of this model, and therefore of the standard model for a single generation.

Now imagine that we bolt on two more bits to take care of the three generations of fermions. Then we get a supersymmetry between the three fermions (generation bit(s) set) and a boson (generation bits not set). Then we can try to split 1+2+3+3+6 into the standard model mediators. Well, the 8 gluons must be either 2+6 or 2+3+3, leaving either 1+3+3 or 1+6 for the electroweak mediators. The former choice looks better, to allow a splitting between electromagnetism and the weak force. However, I emphasise the neither is correct. Supersymmetry of this kind has been decisively ruled out by experiment, and a model of this kind does not describe the universe as it actually is.

In the next post I will try to describe how these problems are resolved by incorporating the three generations into the model from the outset, rather than tacking them on as an afterthought. For this purpose, one needs a 6 bit model, and a good deal more mathematical sophistication. But the group Sym(6) still lies at the heart of the model. So to prepare you for some of the excitement to come, I would like to mention the outer automorphism of Sym(6).

You see, Sym(6) has not one, but two representations as permutations of 15 objects. One is on the 15 pairs, as we have seen. The other is on the 15 trisections, that is splittings of the 6 letters as 2+2+2. To see that this is completely different, it is enough to show how the splitting A+BC+DEF behaves: first, there are six trisections that pair A with B or C, thus AB/CD/EF, AB/CE/DF, AB/CF/DE, AC/BD/EF, AC/BE/DF and AC/BF/DE; second, there are three trisections that pair B with C, thus BC/AD/EF, BC/AE/DF and BC/AF/DE; third, there are six trisections that pair all of A,B,C with D,E,F, thus AD/BE/CF, AD/BF/CE, AE/BD/CF, AE/BF/CE, AF/BD/CE and AF/BE/CD. So, for the chosen subgroup, this other representation splits as 3+6+6, and not as 1+2+3+3+6.

In order to understand the “supersymmetry” between fermions and bosons, one first has to decide whether to use the outer automorphism of Sym(6) or not. Then one has to make three generations of fermions as 3×15=45, but the bosons instead work out as 3+15=18. This gives all 45+18=63 non-zero settings of 6 bits. In the next post I will explain how this works. For those in the know, the translation between vectors and spinors (i.e. bosons and fermions) for Sym(6) does in fact use the outer automorphism. But it is still not possible to build a correct model like this. The relationship between vectors and spinors is a good deal more subtle than the standard model takes account of.

How many bits?

How many bits of information does an elementary particle carry? Obviously this depends what you mean by “elementary particle”, and it depends how you model them. In the standard model, elementary particles carry wave functions with them, so have infinite amounts of information. This is obviously metaphysically absurd, however well the theory describes the real world. I prefer to follow Einstein in his profound metaphysical disbelief in the conventional properties of the wavefunction. Of course, metaphysics, by its very nature, cannot be justified. But I prefer to believe that an elementary particle can carry only a finite number of bits of information. The apparently large number of bits that it carries must then be attributed to the interaction of the particle with a macroscopic environment. There is no experiment that can disprove my metaphysical stance, and I therefore profoundly wish that physicists would stop telling me that my ideas contradict experiment. They do not, and cannot, even in principle, contradict experiment.

Indeed, there are numerous experiments that strongly suggest the standard interpretation is wrong. The most obvious is neutrino oscillation: these observations demonstrate pretty conclusively that the generation of a neutrino, as usually defined by the weak interaction, is not an intrinsic property of the neutrino, but depends on the environment (i.e. on the particle it interacts with). Now I believe there is an intrinsic property of a neutrino that underlies these observations, but that the weak interaction does not define this intrinsic property. That is why the standard model requires the PMNS matrix to describe the relationship between the intrinsic property and the weak interaction property.

Anyway, how many bits are needed to define something that can be recognised as an “elementary particle”? The standard model elementary fermions number 45 in total. This requires 6 bits. In addition, one probably needs one bit for spin up/down, and another bit for particle/antiparticle, making 8 in all. This leaves 19 bit settings out of 64 for bosons, which should be enough for the standard model 13, plus 6 for contingencies. It is not completely obvious that all 8 bits are genuinely intrinsic to the particle, but this is a reasonable working hypothesis to be going on with.

Quite a number of attempted Grand Unified Theories can be considered in one way or another to be based on Lie groups that convert 8 bits into 8 copies of some real or complex vector space, and then tensor together two copies of this vector space to describe interactions. The Pati-Salam model, for example, uses a complex 8-space acted on by SU(2) x SU(2) x SU(4). This complex 8-space can be converted to a real 16-space, and an SO(16) model. Then SO(16) can be converted to Spin(16) and embedded in E_8, and so on and so forth. Attempts at E_8 models abound in string theory, and in more combinatorial models such as Garrett Lisi’s (in)famous “Exceptionally simple theory of everything”. All these models fall down essentially because they attribute vast numbers of continuous variables to the elementary particles themselves. If these continuous variables really exist, then protons decay and the universe falls apart.

But as long as the 8 bits are only used combinatorially, it is quite reasonable to use the E_8 root system as a system of elementary particles, as Lisi essentially does. The questions then are, firstly, is the combinatorics of quantum numbers correct, and secondly, how do we use the quantum numbers to derive the continuous variables needed for interaction probabilities and suchlike measurable properties? It is in considering this second question that conventional approaches fall down. I believe it is essential to recognise that these continuous variables are not intrinsic to the particle, but are properties of our macroscopic experiments.

First, however, we have to get the quantum numbers correct. It is here, in fact, that Distler and Garibaldi attack Lisi’s model, and claim that the combinatorics are not only incorrect, but cannot be corrected. But this claim rests on a number of unjustified assumptions about how the combinatorics represent reality. There are in fact many more possibilities than the ones they consider.

Dealing with 8 bits at once is horrendously complicated. There are 2^8=256 bit settings, and the interactions between them require a group of order something like 2^(8^2), which is a 20-digit number. We have to use as much symmetry-breaking as we possibly can to break this down to something manageable. One bit needs a group of order 2. Two bits need a group of order 24. Three bits, 1344, and four bits 322560. Two bits at a time is enough for me. So I’ll use two bits for 1+3 colours (leptons + quarks), and another two bits for 1+3 generations (bosons + fermions), and see how much information that gives me, before I get carried away with more bits.

If need be, I am not frightened of large finite groups. I am quite happy to work with the Monster, whose order is a 54-digit number. But if these large groups are not needed, I am even happier. There are many interesting groups that act on 2 bits, or 4 or 6, and I will tell you more about some of them another time.

Lie groups or finite groups?

In the previous post I mentioned Einstein’s unrealised dream of a discrete model of quantum mechanics. Such a discrete model must be based on the principle that an elementary particle (whatever that is) consists of a finite set of quantum numbers, and nothing else. All the continuous variables associated with a particle must then be considered as being properties of the relationship between the particle and its environment. “Environment” in this context can mean many things, from a single other particle that it interacts with, to a macroscopic measuring apparatus, to the whole Solar System or even larger environments.

A typical case is polarisation of a photon. Experiments on polarisation typically measure linearly polarised photons. Linear polarisation involves a continuous variable, namely the direction of polarisation, and therefore requires a Lie group U(1) for its description. Circular polarisation is discrete, as the photon has helicity either +1 or -1. In the standard model, linear polarisation is obtained from circular polarisation by “quantum superposition”. This is a well-defined mathematical procedure that works, but what it means physically is still not clear.

One thing that is clear, however, is that Bell’s Theorem prevents the direction of linear polarisation from being an intrinsic “local hidden variable” associated to the photon. I prefer to think of it therefore as a non-local variable, which means a variable associated with the macroscopic environment, or more accurately, with the relationship of the photon to its environment. How this works precisely, I cannot tell you, unfortunately. But experimental confirmation of Bell’s Theorem makes it clear that such non-local hidden variables must exist.

So the general principle must be that elementary particles themselves are described by finite groups, or specific representations of these finite groups, and their interactions are described by embedding these finite groups into Lie groups. Let us start at the beginning, and ask what finite analogue(s) of SU(2) we want. There is really only a choice of three, that is the binary tetrahedral, octahedral and icosahedral groups. The last of these has 5-fold symmetries, which are not obvious in the standard model, so this is probably ruled out. (Although it is possible that the three colours and three anti-colours actually hide a 5-fold symmetry obtained from the colour confinement condition.) The binary octahedral group contains the binary tetrahedral group as a normal subgroup of index 2, so we can probably consider these two possibilities together.

Now, combining two copies of SU(2) in the usual way to make SO(4), we can combine two copies of the binary octahedral group to make a group of order 1152. This contains the ordinary octahedral group (the rotation group of a cube or octahedron) as a finite analogue of SO(3). In the natural 4-dimensional representation, there are just four quantum numbers, but these can be combined in pairs to make 16 “particles”. For our finite analogue of SO(4), this representation breaks up as 1+3+3+9, while for the finite SU(2) it breaks as 1+1+1+1+3+3+3+3, and for finite SO(3) it breaks as 1+1+2+3+3+3+3. In particular, notice that the differences between SU(2) and SO(3) here are very subtle. It is therefore very easy to mistake one for the other, and thereby obtain an incorrect model.

For the Lie groups, SU(2) again breaks the representation as 1+1+1+1+3+3+3+3, while SO(3) breaks it as 1+1+5+3+3+3. In the standard model, weak SU(2) is used in the former incarnation. I believe it may be more productive to use the latter incarnation. One reason for this is that SO(3) is naturally bosonic, while SU(2) is naturally fermionic. Another is the occurrence of the 2-dimensional representation, which is required for modelling weak doublets, but does not occur in the SU(2) case. A third is that the mixing of 2+3=5 is an obvious place where the mixing of weak doublets with strong (or generation) triplets could conceivably take place.

The interpretation of these various representations depends on the context: are we modelling interactions, or measurements? Let us start with interactions. In this context, each particle has its own internal finite symmetry group, but when they meet these two finite groups are embedded in a Lie group, which for electro-weak interactions can probably be taken to be SO(4), but which in general must surely be SL(4,R). This Lie group is required for calculating the probabilities of the various possible outcomes of the interaction. Strong interactions probably require only SL(3,R), which acts as either 1+3+3+3’+6 or 1+1+3+3’+8, depending on whether we take two copies of the representation 1+3, or one copy of 1+3 and one of the dual 1+3′. How exactly one interprets these representations is not entirely clear, but there is no doubt that these numbers match up well with the numbers of particles of various types in the standard model.

Moving on to considering measurements, we embed the particle’s internal symmetry group and the observer’s macroscopic Euclidean spacetime symmetry group separately into SL(4,R), and measure one against the other. What we measure is invariant under SO(3), and therefore splits as 1+1+5+3+3+3. The 3-dimensional representations look like angular momentum representations, for three different types of fermions (electron, up and down quarks). The two scalars perhaps measure mass and charge, leaving 5 for colours, with colour confinement.

If we now want to attach the mass and charge to the angular momentum, we can take the anti-symmetric square of 1+3+3+3, where the scalar is a combination of mass and charge, since charge cannot occur without mass. For SL(4,R), this is an irreducible 45-dimensional representation, while for SO(3) it breaks up as three 1s, nine 3s and three 5s. So it looks like three generations of 1+5+3+3+3, which also appeared in the bosonic (rank 2) tensors and therefore gives us an unexpected “supersymmetry” between bosons and fermions. But it labels the 45 standard model fermions in quite a different way from the way the standard model does it. The reason for this is that the standard model insists that continuous variables like mass are intrinsic to the particle, whereas I insist that, on the contrary, a particle can only store a finite number of bits of information.

I may not have got all the details of the interpretations correct, but the general principles are sound, and the group theory and representation theory are sound.

Quantum reality

Both Peter Woit and Sabine Hossenfelder have recently posted reviews of Jim Baggott’s book “Quantum reality”. Jim Baggott commented on some of the comments on Peter Woit’s blog, and made what seemed to me to be very sensible remarks. So I ordered his book and looked at his website. In particular, I saw that he recommends the book by Alastair Rae “Quantum physics: illusion or reality?” that I have just finished reading. This book was first published in 1986, but it seems to me has hardly dated at all.

The problems that Alastair Rae discusses are the same problems that are still around today. Although millions of words have been written abut them, there has been little or no tangible progress in the intervening three and a bit decades. In a sense, both quantum mechanics and continuum mechanics work perfectly well in their respective realms, and the problems arise mainly in the places where they impinge on each other, that is in the understanding of entanglement and measurement.

Most authors regard these as problems of interpretation of quantum mechanics, rather than as problems within quantum mechanics. I disagree. Continuum mechanics is, almost by definition, described by differential equations in which various quantities or “fields” are differentiated with respect to space and time. In quantum mechanics these fields are no longer differentiable, or even continuous, and therefore, as Einstein remarked, differential equations cannot accurately describe what is going on. But physicists are so used to using differential equations for everything, that they even use differential equations for quantum mechanics. Then they have problems interpreting the “quantum fields” that appear in these differential equations. Well, surprise, surprise.

The first sign that something is wrong with this approach is that in continuum mechanics all the differential operators commute with each other, whereas in quantum mechanics they anti-commute with each other. Dirac “solved” this problem by introducing the gamma matrices, as a way of forcing the differential operators to anti-commute. This is really a mathematical sleight-of-hand, rather than a physical solution to anything. What it does is drive a wedge between quantum mechanics, where the operators anti-commute, and continuum mechanics, where the operators commute, and prevents any sensible unification of quantum and continuum approaches.

In the general situation, operators neither commute nor anti-commute, which implies that they cannot be modelled by differential operators, but must be modelled by algebraic operators. Moreover, these algebraic operators generate not a Lie algebra (anti-commuting), nor a Jordan algebra (commuting), but a Clifford Algebra (both). The differential operators of continuum mechanics then arise from ignoring the anti-commuting parts of the algebra, and averaging over large numbers of instances of the commuting parts. But in fact this does not work for electromagnetism, in which the electromagnetic field is best expressed in terms of anti-symmetric rank 2 tensors. So the emergence of macroscopic electromagnetism out of quantum electrodynamics must be more subtle than this.

In continuum mechanics, one can build a great deal of theory from a single scalar field, say h. One can differentiate h with respect to space and time to get a scalar field dh/dt (really a partial derivative, of course) and a vector field grad(h). Differentiating again we get only a symmetric tensor, with 10 degrees of freedom, since the equations d(grad h)/dt=grad(dh/dt) and curl(grad h)=0 destroy the other 6 degrees of freedom. But we need these 6 degrees of freedom for the electromagnetic field, so we have to generalise from the scalar and vector fields dh/dt and grad(h) to an arbitrary scalar field phi and an arbitrary vector field A. Then it is no longer necessarily true that curl A = 0, and we can use B = curl A instead for the magnetic field. And it is no longer necessarily true that dA/dt = grad(phi), so we can use E = -dA/dt – grad(phi) for the electric field.

At this point we can write down Maxwell’s equations and get the standard theory of electromagnetism. But in doing so we have to accept that the scalar and vector fields phi and A are not continuous (because if they were, they would be zero), but quantised. Moreover, phi and A are only defined up to addition of dh/dt and grad(h), which form a copy of the dual of spacetime. Similarly, E and B are only really defined up to addition of the second derivatives of h. But it still appears at this stage to be possible to treat h as continuous, so that its second derivatives define the stress-energy tensor. Then we can build general relativity on top of this, while keeping phi and A quantised, so that electromagnetism is quantised with photons. A model of this type resembles the current state of fundamental physics, without a quantum gravity. Quantum gravity may therefore be a luxury we can do without.

However, quantisation of the weak interaction via the Z and W bosons implies that we have not finished with the quantisation yet. Indeed, we haven’t really started, because we haven’t quantised phi and A yet. Since phi and A, like E and B, depend on the observer, they cannot be quantised individually, but only together. It is not completely clear whether phi and A transform like time and space, or the dual thereof, and therefore it is not completely clear whether quantisation of the forces should take place in the rank 2 tensors or the rank 4 tensors. For the moment let us assume that E and B are antisymmetric rank 2 tensors, so that phi and A form a copy of the dual of spacetime. Since they are odd rank tensors, they must be quantised by fermions. My best guess is that they are massless, and are therefore rather like the neutrinos in the standard model. But there are other possibilities.

However the details may work out (for which see the various papers posted on this blog), the striking thing about this approach is that by removing the physically meaningless differentiation operators from the Dirac equation, we remove the need for the Dirac matrices, the wave-function and the spinor, and can express everything in terms of classical concepts of spacetime and 4-momentum, mass, charge and current, albeit quantised. Thus we remove all the troublesome concepts from the theory, and avoid the apparently intractable problems of interpretation.